cosmological argument model essay

A Level Philosophy & Religious Studies

The Cosmological argument

Cosmological arguments attempt to justify the conclusion that God exists as the required explanation of the existence of the universe.

A posteriori. Cosmological arguments are typically a posteriori arguments, which means they are based on experience. The cosmological argument is based on observation of everything in the universe having a cause, being in motion or being contingent and therefore requiring a creator. These observations form the premises of cosmological arguments. On the basis of those observations, an inference is then made to the nature of the origin of the universe being God.

Cosmological arguments from causation

Aquinas’ first and second ways.

Aquinas’ first two ways are developed from Aristotle’s theory of efficient causation, which is an attempt to explain the change we observe. Aristotle thought that change required a prime mover which sustains the motion and causation we experience.

Efficient causation involves sustaining causes, those which bring about their effect continuously, such that if they ceased to exist then their effect would also cease to exist. E.g., the gravity of the earth causes the moon to be in orbit, which in turn causes the sea tides to rise and fall on earth.

Aquinas’ 1 st way (motion)

P1. We observe that there are things in motion. P2. Motion is the actualization of a thing’s potential to be in motion. P3. A thing can only come to be in motion by being moved. P4. A mover must be something that is actual, i.e., in a state of actuality. P5. A thing cannot move itself. C1. So, all things in motion must have been moved by a mover, which was also moved by another mover. P6. There cannot be an infinite regress of movers, otherwise there would be no first mover and then no motion. C2. Therefore, there must be a first mover which must itself be unmoved (as it is pure actuality). That thing we call God.

Change requires going from potential to actual, which depends on something that is actual, which cannot depend merely on other potential things, so there must be something of pure actuality. A thing that is purely actual with no potential cannot change, it is an unmoved mover or uncaused causer.

Aquinas’ 2 nd way (atemporal causation)

P1. We observe efficient causation. P2. Nothing can cause itself. P3. There is a logical order to sustaining causes: the first cause, then intermediate causes, then an ultimate effect. P4. If A is the efficient cause of B, then if A doesn’t exist neither does B. C1. There must be a first sustaining cause, otherwise P1 would be false as there would be no further sustaining causes or effects. C2. As there is a first cause, there cannot be an infinite regress of causes. C3. The first cause must itself be uncaused. That thing we call God.

Aquinas’ first two ways treat the relationship between cause and effect as ontologically real but not temporal, although they are consistent with a temporal understanding of cause and effect. They point to the logical implications of there being sustaining causes. This is why especially Aquinas’ 2 nd way is called a cosmological argument from ‘atemporal causation’.

The first and second way attempt to show God must exist as the first mover or causer. The word ‘first’ in the concept of a first cause or first mover is not meant to indicate it being ‘first’ in time, but ontologically first in the sense that motion and causation are ontologically dependent on it.

The Kalam cosmological argument from temporal causation

The Kalam cosmological argument explicitly involves temporal causation, which is when a cause brings about its effect after it and the continued existence of the effect is independent of the existence of the cause.

The Kalam infers a beginning cause rather than a sustaining cause. The causal sequence being temporal, with God as the beginning cause, is a central feature of the argument.

One advantage of the Kalam argument is that it’s easier to explain how an atemporal God could create the world in one act compared to a sustained act of creation over time.

W. L. Craig brought this argument to prominence in the late 20 th century and named it ‘Kalam’ after the Islamic philosophy which first invented in the 11 th century.

P1. Everything that begins to exist has a cause of its existence. P2. The universe began to exist (an infinite regress is not possible). C1. So, the universe has a cause of its existence.

Further steps are required to show that the cause of the universe is God. Craig firstly argues that scientific explanation applies within the universe and therefore cannot apply to its actual creation. The cause of the universe must therefore have a personal explanation, i.e., intentionally created by an intelligent mind. This being must have the power to create a universe from nothing (ex nihilo). It must be outside time and space since it created time and space. As a timeless, eternal being, God did didn’t begin to exist so it’s then no contradiction in claiming that God doesn’t have a cause. These are qualities that God would have, so the cause of the universe is God.

A strength of the Kalam argument is W. L. Craig’s arguments for the impossibility of an ‘actual infinite’ meaning an infinite in reality. The problem is that sets with infinite members can be equal in size to their subsets. Craig uses the illustration of a library with an infinite number of books, half of which are red. Half of infinity is still infinity, so half of infinity is not actually smaller than infinity. This might make sense theoretically, but Craig claims problems arise when applying it to reality. It would mean the set of red books ‘X’ is apparently smaller than the set of all the books ‘Z’, and yet paradoxically is also equal in size, since both are infinite. Infinities therefore cannot actually exist, since then they could be both smaller than and the same size as other infinities.

Counter: An infinite series is actually possible.

G. Cantor argued that the mathematical properties of infinite sets/things are simply radically different to those that are finite, making Craig’s library or Hilbert’s Hotel not absurd. Craig takes it to be obviously absurd for a subset (red books) to be both smaller than and equal to its set (all the books). However, it’s only absurd for finite sets. For infinite sets it’s not absurd, it’s actually their defining characteristic. When we think of libraries or hotels, we have in mind our ideas about finite sets of things, but Cantor argued such intuitions are not applicable to infinite sets. Infinite sets simply have different mathematical properties, one of which is the possibility of a one-to-one relation between the number of members of infinite sets and those of their subsets.

Hume’s objection to the ‘causal principle’

The strength of cosmological arguments from causation is that they are based on the causal principle, which is that every event has a cause, or that every contingent being has a cause of its existence.

Ex nihilo nihil fit (nothing comes from nothing) goes back to the ancient Greeks like Parmenides who influenced Aristotle and later Aquinas. It’s not possible for an event to happen without a cause, otherwise something could come from nothing, which is absurd.

Arguing for the Kalam cosmological argument, William Lane Craig claims that the causal principle is:

“based on the metaphysical intuition that something cannot come out of nothing.” – W, L. Craig

A causal principle is one of the essential pillars of cosmological arguments, alongside arguments against the possibility of infinite regression and inferences to the divine nature of the ultimate origin of the universe.

Weakness: Hume’s objection to the causal principle.

Hume’s fork tells us that propositions (such as the causal principle) can either be analytic or synthetic.

Hume argues that the causal principle is not true by definition (analytic). There doesn’t appear to be anything incoherent in the idea of an event or thing existing without a cause. It is conceivable and not obviously self-contradictory. We can imagine something popping into existence without a cause. The idea of a four-sided triangle is obviously self-contradictory because the idea of triangle contradicts the idea of four-sides. Yet, the idea of an event doesn’t seem contradicted by the idea of no cause.

So, the causal principle can only be justified on a posteriori grounds, which makes it a synthetic truth. The problem is, claims based on experience cannot be known with certainty to be true in all cases. All we can justifiably claim is that every event we have observed has a cause. It does not follow from this that all events have a cause, since we have not experienced all events.

So, the universe could exist without a cause. The cosmological argument therefore fails because in attempting to argue for God’s existence as the required explanation of the universe, it assumes that the universe has a cause.

Lawrence Krauss & Alan Guth add evidence from modern theoretical physics which resonates with Hume’s point. The universe has zero total energy and therefore required no energy to be created So, it could have come from nothing.

Evaluation defending the cosmological argument:

The cosmological argument could be defended by arguing that the universality of the causal principle is justified through induction. We have experienced many causal interactions, all of which involved the constant conjunction of cause and effect. From this we can infer that all effects have a cause. It is possible that this may be false, that in some cases the causal principle may not hold. However, the evidence so far suggests that it always holds. Therefore, we are empirically justified in accepting the causal principle.

Evaluation criticizing the cosmological argument:

Furthermore, we cannot even be empirically justified in holding the causal principle. All the evidence that we have for the causal principle comes from our observation of change within the universe itself. We are not justified in assuming the relevance of that experience to the conditions under which the universe itself came to exist. Those conditions, whatever they are, could be radically different to anything we now observe within the universe. So, we cannot even justify the causal principle on an empirical basis.

Cosmological arguments from contingency

Aquinas’ 3rd way (contingency).

P1. We observe that there are contingent beings. P2. A series of contingent beings cannot regress infinitely into the past. C1. So, a series of contingent beings must be finite. P3. If this finite series was all that existed, then before it would be nothing. P4. If there was once nothing, there would be nothing now, which is absurd. C1. So, there must be more than this finite series of contingent beings, i.e., a necessary being. P5. There cannot be an infinite regress of necessary beings. C3. There must be a necessary being “having of itself its own necessity … That thing we call God.”

Leibniz’ principle of sufficient reason

Leibniz improves on Aquinas’ 3 rd way by removing unnecessary reasoning about nothing once existing.

Leibniz’ argument is a priori, it doesn’t require inference from experience. The downside of a posteriori arguments is that they are defeasible, meaning in principle future experiences could always prove them false. A priori arguments based on logic are stronger.

Leibniz bases his argument on the principle of sufficient reason, which does both the job of a causal principle and an argument against the infinite regress. It shows that there must be not just any causal explanation, but a causal explanation which provides an ultimately sufficient reason for everything that exists. This strengthens the argument by making it dependent on only one claim.

P1. For every true fact or assertion, there is a “sufficient reason why it is thus and not otherwise.” P2. There are two types of truth: truths of reasoning and truths of fact. P2a. Truths of reasoning are necessary, so their opposite is impossible. The sufficient reason for truths of reasoning can be discovered a priori. P2b. Truths of fact are contingent, so their opposite is possible. The sufficient reason for truths of fact cannot be discovered through other contingent truths, because they too require a sufficient explanation, and so on. C1. A sufficient reason for contingent facts must be found outside a series of contingent things. C2. The sufficient reason for contingent facts must be a necessary substance. C3. That necessary substance is God. C4. So, God exists.

Leibniz claims that the principle of sufficient reason (P1) can be known as a necessary truth. Even if we can’t know or even find out what the reason is, there must be one. ‘From nothing, nothing comes’ because nothing is not sufficient to create something. Only a necessary being is sufficient to explain the universe because otherwise there would be an infinite chain of contingent beings, but an infinite regress alone can have no sufficient explanation.

If things have always existed going back forever then nothing would have a sufficient reason for its existence. Everything’s reason for existence would consist something for which its reason for existence consists in something else. There would simply be an infinite deferring of the reason for existence and thus there would not actually be a reason for existence. So, a necessary being must have begun the chain of contingent beings and is the sufficient explanation of the universe.

The fallacy of composition

The strength of cosmological arguments from contingency is their seeking an ultimate explanation rather than only a first cause.

They focus more fundamentally on the nature of things. Beings have causal relations, but also an ontological status, i.e., contingency or necessity. Contingent beings & series must have an external explanation since it is their nature to depend on something else for their existence.

Weakness: Hume & Russell on the fallacy of composition.

This criticism explicitly attacks cosmological arguments from contingency, but if successful would also undermine arguments from causation. It is a fallacy to assume that what is true of a thing’s part(s) must also be true of the whole. Bertrand Russell illustrated this by pointing out that just because every human (parts) has a mother, that doesn’t mean the human race (whole) has a mother.

Hume uses the example of a finite set of contingent beings (20 particles). Just because the parts of a set/series of contingent beings have an explanation, that doesn’t mean the whole set/series has an explanation.

Experience shows that parts of the universe are contingent & have a cause/explanation. This doesn’t mean the universe itself as a whole must also be contingent (have a cause/explanation).

The only empirical way to believe a whole series has an explanation is to commit the fallacy of composition by assuming the whole is like the parts. So, a posteriori cosmological arguments commit the fallacy of composition by assuming that the universe has a cause when all we experience is that parts of the universe have a cause. That’s the problem with Aquinas’ 3 rd way.

Evaluation defending the cosmological argument

However, Copleston rightly points out that contingency arguments don’t actually appear to make an inference from parts to whole.

They argue that a series of contingent things must have an external cause. Copleston argues:

P1. A series is either caused or uncaused. P2. If a series is uncaused, the reason for its existence must be internal to it, making it necessary. P3. No amount of contingent things can be necessary, not even an infinite number of them, so a series of contingent things cannot be necessary. C1. So, a series of contingent things must have an external cause like a necessary being.

However, Hume’s empiricism drives his point further. Hume questions how we know a series must have a cause/explanation. In P2, Copleston assumes that an uncaused series must have an internal explanation. Hume doesn’t see how experience can justify thinking a series must have an explanation at all.

However, Leibniz’ a priori argument may avoid this problem as it’s derived from intuitively known a priori necessary truths, not experience. Leibniz is a rationalist and believes the whole has an explanation because of his a priori principle of sufficient reason. Nothing can be true, like a whole series existing, without a sufficient explanation for why it is thus and not otherwise.

Copleston vs Hume & Russell on the universe as a brute fact

Hume and Russell take the fallacy of composition objection further in a way that can respond to Leibniz’ a priori approach.

Hume turns to the example of an infinite series of contingent beings. He argues it is possible and doesn’t need an ultimate sufficient reason. Each being is explained by the being it depends on.

Of course, Leibniz wants to object that this leaves the whole chain itself without explanation, without reason for existence.

Hume, however, denies the validity of claiming that there is ‘whole’ chain. If experience only shows that contingent beings exist, we can question whether there actually is a ‘whole’.

“uniting of these parts into a whole, like the uniting of several distinct counties into one kingdom … is performed merely by an arbitrary act of the mind and has no influence on the nature of things.” – Hume

“I think the word “universe” is a handy word in some connections, but I don’t think it stands for anything that has a meaning … The whole concept of cause is one we derive from our observation of particular things; I see no reason whatsoever to suppose that the total has any cause whatsoever … it doesn’t need to be its own cause, what I’m saying is that the concept of cause is not applicable to the total.” – Russell.

We cannot say the ‘whole universe/series’ is even a valid concept, let alone that it is contingent. So, we can’t conclude there is a need for an explanation like a necessary being. Cosmological arguments from contingency fail. Note that this isn’t quite saying the universe could be necessary. A necessary being contains its own reason for existence, but Hume and Russell are proposing that the universe could simply have no explanation at all.

All we can justifiably say is that the universe/series is “just there, and that’s all” (Russell) . It is a brute fact, one which has no explanation, whether causal or otherwise. Russell points to evidence from Quantum mechanics:

“The physicists assure us that individual quantum transitions in atoms have no cause.” – Russell.

So, arguments from contingency baselessly assume that a series (as a whole) must have an explanation at all.

This argument also counters the arguments from causation. If we have no basis for thinking the universe has an explanation of any kind, that includes causal explanation.

Copleston firstly responds that only some interpretations of quantum mechanics propose uncaused events. Copleston further argues:

“I cannot see how science could be conducted on any other assumption than that of order an intelligibility in nature” – Copleston.

Copleston’s argument is successful because the ‘brute fact’ argument is self-defeating. Science and philosophy are about finding out why things are the way they are; their causes and explanations. If we say that there is no reason, then we undermine the purpose of science and philosophy itself.

Furthermore, Russell argues that while a scientist may look for causes, they do not assume that there is one to find. Even if quantum transitions are only uncaused on some interpretations of quantum mechanics, the mere idea of it still shows that physicists are able to conceive of events that have no cause. It is therefore at least logically possible for events in nature to have no cause or explanation. So, a contingent series doesn’t need an external cause. The principle of sufficient reason cannot be a necessary truth. The universe could be one of those natural events that has no cause or explanation. Science should accept that possibility, since science should be open to whatever could be true.

Copleston attempts to show that Russell is assuming that there isn’t a cause and Russell attempts to show that Copleston is assuming that there is. Ultimately, defenders of the cosmological argument are the ones making the positive claim about reality, so they have the burden of proof. It looks like for cosmological arguments from contingency, the conclusion that God exists as the explanation/cause of the universe cannot be reached without assuming that the universe has an explanation/cause. In that case, the mere possibility of the universe being a brute fact is enough to undermine the cosmological argument. It’s not irrational to look for a cause, even a cause like a God, but since there might not be a cause it is irrational to think that there must be one.

Issues around the possibility of a ‘necessary being’

A strength of cosmological arguments from contingency is that their conclusions achieve more than arguments from causation. They can establish God’s necessity, meaning inability to cease existing, which is a key element of Christian theology. Aquinas understands necessity to mean the inability to cease existing, which fits with the concept of omnipotence.

Weakness: Hume’s rejection of the possibility of a necessary being.

Hume’s fork:

A priori reasoning can only tell us about the relations between ideas , i.e. analytic knowledge (true by definition). E.g. “a bachelor is an unmarried man”.

A posteriori reasoning can only tell us about matters of fact , i.e. synthetic knowledge (true by the way the world is). E.g. “The sun will rise tomorrow”.

“there is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by any arguments a priori” – Hume.

A being whose existence is logically necessary is an absurdity to Hume. A thing’s existence is a matter of fact. All matters of fact can be conceived false and denied without contradiction. So, anything which could exist could also not exist. The concept of a being which ‘must’ exist, whose existence cannot be denied without contradiction, is therefore absurd and meaningless.

Hume’s justification for the fork is that truths of logic/definition are necessary, because they will be true no matter what happens regarding the factual state of the universe. E.g., 1+1 will always = 2, because there is no possible change to the universe which could make it false.

This shows there is a disconnect between logical (analytic) truths which are necessary (cannot be otherwise) and factual (synthetic) truths which can. The term “necessary existence” violates this disconnect. It is about what factually exists, so must be synthetic. Yet, it somehow also has the “cannot be otherwise” property that only belongs to logical analytic truths. We cannot know that a being’s existence is logically necessary, since a being’s existence cannot be established through logic.

“The words, therefore, necessary existence, have no meaning.” – Hume.

Any argument which attempts to conclude that God exists necessarily therefore fails, including the ontological argument and some cosmological arguments.

The masked man fallacy. Hume’s argument depends on conceivability entailing possibility. It is therefore susceptible to the masked man fallacy, which shows that we can conceive of the impossible. Imagine someone heard of a masked man robbing a bank. They can conceived that it is not their father. Yet, if it was their father, then it is impossible that it is not their father. Yet, that was what they conceived of. So, we can conceive of the impossible. When Hume argues that our ability to conceive of God not existing shows that it is possible for God to not exist and that God therefore cannot be necessary, he assumes that conceivability entails possibility. Conceiving of God’s non-existence could be conceiving of something impossible because God is necessary.

Hume’s argument is successful because it is epistemological, i.e., focused on what we can know. Hume isn’t really saying a necessary being is ‘impossible’, just that it is impossible for us to know that such a being exists. Aquinas, Leibniz and Craig say that it is a being with the property of the impossibility of non-existence, but Hume simply throws up his hands and says he cannot see how anyone can ascribe meaning to that statement. We might technically be able to understand the words, but we cannot conceive or understand how they could map onto reality. If we are to accept the term on that basis, Hume rightly goes on to say that we might as well say that, even though we can’t understand how, reality itself has this mysterious property of the impossibility of non-existence, so positing a God is unnecessary anyway.

Issues around the possibility of an infinite series

A crucial strength of all cosmological arguments are the justifications its proponents provide for the premise that an infinite series is impossible. This is essential, for if there is an infinite regress going back in time forever then all forms of the cosmological argument fail. God could not be concluded to exist as the origin of what exists if there was no origin. These arguments attempt to show that the idea of an infinite regress leads to paradox or absurdity and therefore must be false.

Aquinas argues an infinite regress is impossible. If there was an infinite regress, there would be an infinite amount of time before the present moment. That means to get to the present moment, an infinite amount of time must have passed. However, an infinite amount of time cannot pass. No matter how long you wait, even if you never stop waiting, the amount of time passed can never reach infinity. So, there cannot be an infinite amount of time before the present moment and therefore there cannot be an infinite regress. You cannot traverse an infinite through successive addition.

Weakness: The possibility of an infinite series.

The idea of an infinite regress may be difficult to comprehend, but there is no obvious logical contradiction in the idea. The concept of ‘time’ does not seem contradicted by the predicate of ‘infinitely regressing into the past’.

Critics of the infinite regress attempt to show that while there is no obvious contradiction, nonetheless paradoxes arise from the concept. This aims to show that it is illogical and therefore impossible.

However, Hume thinks we simply lack the evidence on which to properly judge whether it is possible or not. We can strengthen Hume’s point with modern science which is full of examples where physicists speculate about ways an infinite regress could exist without paradox.

Some physicists argue for the block universe, that the passing of time is an illusion. Others suggest the universe eternally cycles between expansion and collapse and that a new timeline begins each cycle. In that case, an infinite amount of time never passes (so Aquinas fails) and in fact a time line containing an actual infinite never existed (so Craig fails). Yet, an infinite series of cycles happened (just not on any particular time-line).

The meta point of these examples is that we know very little about what time and infinity are and how they work, so claims about the impossibility of an infinite series are unjustified. Philosophers who think they can draw conclusions about time through pure logic alone fail because our actual concept of time is so poorly scientifically understood and could bare very little relation to what time actually is. It seems Hume would say we should admit that we are not in a position to conclude either that there is or is not an infinite regress.

There is certainly much we don’t know about time and infinity. However, what we currently understand suggests Aquinas’ reasoning is right. The eternal bang/crunch model may be logically possible, but we have no reason to believe it. The evidence suggests the universe started with a big bang and there is no evidence it will collapse again. We are justified in believing what we currently have most reason to believe.

Hume’s stance is the right one. We simply lack the required evidence to have a reasonable belief one way or the other regarding whether time regresses infinitely or not. Ultimately it is a matter for scientists and mathematicians and since they are far from decided, we are not justified in claiming an infinite regress is possible or impossible. Cosmological arguments therefore fail as they have the burden of proof. They are making a positive claim about what exists on the assumption of an infinite regress’ impossibility.

Extra credit:

Modern science’s rejection of the cosmological argument.

A reasonably popular view in physics had been the steady state theory, which was that the universe had always existed and that there is a continuous creation of matter which expands. However in the mid 20 th century, the discovery of the microwave background radiation provided very strong evidence against steady state and for the big bang theory. Steady state theory leaves no room for a creator God because it claims the universe was not created but has always existed, whereas the big bang arguably does leave room for God because the big bang could simply be how God created the universe. So, the big bang theory was actually welcomed by many Christians for this reason.

Pope Pius XII declared in 1951 that the big bang theory does not conflict with the Catholic conception of creation.

The question then becomes whether science can explain why the big bang happened without God. One possibility is the oscilating universe theory, that what we call our universe is just one of the cycles of big bangs and big crunches that has been oscilating forever.

However, another theory with more evidence for it is Alan Guth’s inflation theory. Laurence Krauss , a physicist, explains Guth’s theory that the universe came from nothing because it actually is nothing. Gravity has negative energy, the total amount of which in the universe happens to exactly cancel out the positive energy of the matter in the universe, so the total energy of the universe is zero; it is nothing. Krauss claims this answers Leibniz’ question of why there is something rather than nothing. Quantum mechanics has existed eternally and causes quantum fluctuations which can create a zero-energy universe from nothing because such a universe requires zero energy to create.

W. L. Craig’s response to Krauss is that Krauss’ definition of nothing is faulty since it’s actually something. Nothing really means total absence of anything.

Defence of Krauss: Arguably the definition of nothing philosophers have traditionally used is in fact not what nothing actually is though. Science has proven the philosophical conception of nothing to be inaccurate.

Essay Plan: Cosmological Argument

October 30, 2012.

AS Religious Studies Revision: The Cosmological Argument

Ao1 material: i.e. ‘what goes in part a)’, a) explain the cosmological argument. (25).

The cosmological argument begins with the observation that the universe exists. It then asks the question ‘why is the universe here?’ or ‘why is there something rather than nothing?’

P1: Everything in the universe has a cause. P2: The universe itself must have a cause. P3: To avoid infinite regress of causes there must be an uncaused cause. C: This uncaused cause is God.

Type of argument

Inductive: inductive reasoning is where the premises support the conclusion, but they do not entail it. It is usually based upon information coming from the senses (the order and complexity we observe with our eyes). It is therefore not deductive, which is where the premises of an argument do entail the conclusion, i.e. the conclusion is necessary e.g. 1+1=2..

A posteriori: it is based upon experience: it comes ‘after the fact’ of order and complexity, it is not a priori which is based upon reasoning before experiencing. Synthetic: a proposition whose predicate concept is not contained in its subject concept. In other words, if I say ‘all triangles have three sides’; the fact that a triangle has three sides (predicate) is contained in the definition (subject). In the statement ‘there is design in the universe’ there is doubt because the predicate (design) is not contained in the subject (universe). We have to use our senses to verify the truth of this statement.

Scholars whose versions of the argument you must explain…(you need to do it in detail)

Thomas Aquinas: The first 3 of the 5 Ways.

Unmoved mover: everything that moves is moved by something else. There must be an initial cause of movement in the universe. There is an unmoved mover called God.

Uncaused cause: everything has a cause. There cannot be an infinite regress of causes. The first cause is God. Possibility and Necessity: everything in the universe exists contingently i.e. it could not exist. It is conceivable for everything in the universe to go out of existence. There must be something which cannot not exist (that exists necessarily). That something is God.

Aquinas argued that the world depends on God now for its existence.

The Kalam Argument:

This was suggested by Islamic scholars such as al-Ghazzali and argues that God is the originating cause of the universe. It goes: everything that has a beginning of existence must have a cause; the universe began to exist; the universe has a cause; the cause is God. Modern versions come from scholars such as William Lane-Craig.

Frederick Copleston: The Radio Debate

Copleston debated the existence of God with Bertrand Russell on the radio. Copleston provided a new version of Aquinas’ Five ways.

AO2: Critical evaluation i.e. ‘what do I put in part b)?’

b. The strengths fo the cosmological argument outweigh the weaknesses. Discuss (10)

Remember to read the question on the exam paper first before just regurgitating.

The strengths of the cosmological argument

The strengths of the cosmological argument are the strengths of inductive reasoning: inductive arguments begin with something that we can observe. Inductive reasoning begins with experience which may be universal (i.e. everyone has had it) or it may at least be testable. The argument does not rely upon fixed definitions that we must accept (unlike the Ontological Argument). The Cosmological argument fits in with the God of classical theism (omnipotent, omnibenevolent, omniscient). It makes sense to think that there is an initial cause to the universe: this fits with our experience of events within the universe. Most scientists would argue that the universe has a beginning, which fits in with the Cosmological argument.

The weaknesses of the Cosmological Argument.

The weaknesses of this argument are the weaknesses of inductive reasoning: the conclusion does not necessarily follow from the premises. Just because things in the universe have causes, doesn’t mean that the universe as a whole has a cause: we have no experience of universes being caused so cannot claim we know that they need a cause. (Bertrand Russell). The universe is just here and that is that: we do not need to ask why. (Bertrand Russell). There is no logical absurdity in claiming that things can come into existence without a cause. (Hume). Only analytic propositions (e.g. 1+1=2) can exist necessarily. (Kant). Perhaps the universe has always been here (Buddhism) and fluctuates in and out of existences (Big Crunch theory).

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Cosmological arguments.

Published online by Cambridge University Press: 25 August 2018

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cosmological argument

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An influential argument (or family of arguments) for the existence of God. Its premises are that all natural things are dependent for their existence on something else; the totality of dependent beings must then itself depend upon a nondependent, or necessarily existent, being, which is God. Like the argument to design, the cosmological argument was attacked by Hume and Kant. Its main problem is that it requires us to make sense of the notion of necessary existence. For if the answer to the question of why anything exists is that some other thing of a similar kind exists, the question merely arises again. So the ‘God’ that ends the question must exist necessarily; it must not be an entity of which the same kinds of question can be raised. The other problem with the argument is that it unfortunately affords no reason for attributing concern and care to the deity, nor for connecting the necessarily existent being it derives with human values and aspirations.

From: cosmological argument in The Oxford Dictionary of Philosophy »

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cosmological argument

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Stanford Encyclopedia of Philosophy - Cosmological Argument

cosmological argument , Form of argument used in natural theology to prove the existence of God . Thomas Aquinas , in his Summa theologiae , presented two versions of the cosmological argument: the first-cause argument and the argument from contingency . The first-cause argument begins with the fact that there is change in the world, and a change is always the effect of some cause or causes. Each cause is itself the effect of a further cause or set of causes; this chain moves in a series that either never ends or is completed by a first cause , which must be of a radically different nature in that it is not itself caused. Such a first cause is an important aspect, though not the entirety, of what Christianity means by God. The argument from contingency follows by another route a similar basic movement of thought from the nature of the world to its ultimate ground.

Descartes’ cosmological and ontological arguments

Traditional arguments for God’s existence include:

3. Design Argument (the universe shows evidence of design, a designer must exist).

The existence of God is crucial to Descartes because in the sustained argument of the Meditations, God is the bridge from the hyperbolic doubt of the Cogito back to knowledge of the empirical world and the abstract world of logic and mathematics.

Descartes does not set out his arguments in formal deductive terms (he antedates predicate logic and was no fan of syllogistic logic). He uses scholastic terminology. At times he seems to think that God’s existence is readily evident to any diligent, attentive meditator, and arguments are just heuristic devices to help the slower meditator to the almost self-evident truth that God’s existence is known by clear and distinct perception. For all these reasons, the meditator has to do some work to penetrate the arguments.

P1: I have the idea of a most perfect (infinite, eternal,omnipotent, benevolent) being (God).

P2 is less easy to grasp. Discussion is couched in technical, scholastic terms. Two types of reality (being) are distinguished regarding ideas. The existence of an idea (its formal reality) is distinguished from the content of the idea (its objective reality). “Objective” refers to the object contained in the idea, rather like the modern use of “subjective” – it refers to the tree (say) in the mind not the tree in the garden. The notion of degrees of reality is then introduced. Ideas all have the same degree of formal reality, all being states of mind, but they differ in degrees of objective reality – lowest in a “mode” (modification of a substance e.g. shape), intermediate in a finite substance, highest in an infinite substance.

P2: Whether expressed in scholastic or modern terms, P2 is simply an assertion. No evidence is given for it. To assume a finite mind needs an infinite mind to cause it begs the question as to God’s existence. As far as I can see simple things plus simple rules can lead to complex things e.g. laws of nature plus simple initial conditions has produced atoms, compounds, galaxies, life and minds, so that the Causal Principle is false.

Ontological Argument

Originally due to Anselm, declared invalid by Aquinas, the argument lapsed, and Descartes’ use of it surprised his contemporaries.

P2: this is fine if we mean that the conceived entity can be thought of AS IF it existed necessarily. It doesn’t mean that any such entity actually exists, or indeed could possibly exist.

Also, the traditional objection to the ontological argument applies, that we can prove the existence of anything e.g. I have a clear and distinct idea of a necessarily existing perfect pizza, holiday, partner etc.

2 thoughts on “ Descartes’ cosmological and ontological arguments ”

The Cosmological Argument Exploratory Essay

To find inspiration for your paper and overcome writer’s block
As a source of information (ensure proper referencing)
As a template for you assignment

The Cosmological argument is defined as the argument which proves that the universe was created by God. This argument clearly claims that the world is in existence because it was created by God. The first arguments are said to have been generated by Aristotle around three hundred years before Christ.

In these arguments, it is well explained that everything that is in existence has a cause. This means that if one was to decide to go back far enough, a cause would be determined. It is also said that the first initial cause was God. The cosmological argument was initially developed by philosophers of Muslim decent in the middle ages. On a day to day basis, it seems like everything comes in and out of existence, depending on their coming to existence and also the fact that they might have not existed at all.

Seeking an explanation to these kinds of things is not in any way improper. God is defined as the Supreme Being (Oddie para 2). Unlike other beings, God is not one to come into existence and also fade out of existence. God is termed as being an eternal being. God is not dependent on any other beings for existence thus he is termed as an Independent Being. God’s existence is not a matter of luck or chancing unlike every other being that are a matter of chance.

Hence God is a necessary Being. In addition, God can be said to be a necessary, independent, and eternal Being who the rest of the universe or cosmos depends on for their well being. God is regarded as a creator and sustainer of lives and the universe. Among the many arguments surrounding this topic is that there cannot be a true infinite. This is because even a small part of what is referred to as infinite is itself seen as an infinite since it has been derived from the entire infinite.

For example, an infinite collection of the blue and white marbles would be best suited to explain this theory. The number of blue marbles in this pack is directly proportional to the total number of all marbles in the pack, because both are infinite. The same holds for the number of marbles in the pack.

Thus, the number of blue marbles equals the number of white marbles which equals the sum of all blue and white marbles. It would be incongruous to hold to the claims of existence of a true infinite. Similarly, thinking that infinite happenings can occur prior to a given time is awkward.

The second argument talks about the formation of actual infinite. It states that it cannot be formed. It is fascinating that philosopher Craig also states that the cause of the universe is as a result of the Personal Creator. He goes on to state that “the only way to have an eternal cause but a temporal effect would seem to be if the cause is a personal agent who freely chooses to create an effect in time” (Craig p. 88)

The creation theory has been defined numerous times and there are a number of different stories telling on how the earth was formed. The evolution theory is somewhat believable, probably by pagans and evolutionists. Most communities have come up with theories in a bid to explain their existence.

This is most common in African cultures where they have learned to coin theories explaining their existence. Christians believe in the existence of a God who is considered to have willed the existence of all that there is. This God is not a physical being that is part of the universe. God’s mighty power rules over the entire universe.

The Bible clearly shows us that God is indeed the uncaused First Cause that created the universe by willing it into existence. Another cosmological argument that is consistent with the Bible is the Kalam Cosmological Argument. Aristotle’s theory was of great importance in the cosmological arguments about the existence of God.

He argued that everything that takes place is as a result of something else. For example, if one was to slip on a banana peel and fall, there would be the question of who left the peel there? Why was the peel left there? Where did it come from? This can be traced back as far as one would want. Aristotle thought that it would have been easier by reaching only the first cause which is causing but itself uncaused. This theory is now known as the prime Mover.

Creation is the only widely accepted theory on the existence of the universe. It may vary depending on one’s religion but it all leads to the same, one natural and Supreme Being. The Christians believe in God and his son Jesus Christ who came to earth in humanly form and ascended into heaven.

Their holy book is known as The Bible. The Muslims believe in Allah and Mohamed his Prophet, their holy book is known as The Koran. The Hindus believe in a number of gods but there is a supreme being they believe in and they refer to their holy book as the Bahagdvita. These are just the main religions in the world and a little of their beliefs. It is evident that God is a super natural Being.

Works Cited

Craig, William Lane. “The existence of God and the beginning of the universe.” Truth: A Journal of Modern Thought 3 (1991): 85-96.

Oddie Graham. The cosmological argument: Why is there something rather than nothing? 2011. Web.

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Cosmology and Theology

As long as humans have been trying to make sense of the universe, they have been proposing cosmological theories. Furthermore, the notion of a deity often plays a central role in these cosmological theories. According to most monotheistic religions, God is the sole creator and sustainer of the universe.

But the last one hundred years have seen a different sort of cosmology: a scientific cosmology. Without running afoul of the demarcation problem, the notable characteristics of scientific cosmology are that it uses the tools of mathematical physics (it is formalizable) and that it makes precise and testable predictions. What has this new scientific cosmology to do with traditional (often theistic) cosmologies? Has the new cosmology replaced the older cosmologies? Does the new cosmology inform or interpret the older cosmologies?

Our subsequent discussion will be restricted almost completely to the case of western monotheism—Judaism, Christianity, and Islam—and even more specifically to variants of Christianity. Even so, we note a wide range of diversity among Christian beliefs and Christian attitudes towards science in general, and towards scientific cosmology in particular. At one extreme, we find ultra-traditional versions of Christianity that emphasize literal interpretation of Scripture, and that often interpret theological doctrines in terms of ancient Greek philosophical categories (e.g., God as eternal, unchangeable, etc.). Even within this more traditional camp, we find differences in terms of amount of literalism, and amount of flexibility with regard to traditional theological doctrines. (e.g., there are ongoing debates among traditional theologians about the relation of God to time.) At another extreme, we find more recent incarnations of Christianity that draw heavily on ideas from German idealism and/or process philosophy. There are also subtle, but not negligible, differences in attitude among Protestant, Roman Catholic, and Orthodox Christian theists. Thus, we should not think of theism as one fixed set of doctrines that is simply consistent or inconsistent with scientific cosmology.

Furthermore, even though most theological interactions with cosmology have taken place within the Christian tradition, it has rarely—if ever—been the case that the defining feature of Christianity (viz. the unique role of Christ) has played any explicit role in these interactions.

1. Overview: Cosmology, theology and religion

2.1 whose theism is the big bang supposed to confirm, 2.2 should the theist look for confirmation from scientific cosmology, 2.3 which cosmological models support a doctrine of creation ex nihilo , 2.4 can we trust general relativity, 2.5 does the big bang provide evidence for atheism, 3. steady-state theories, 4. quantum and string cosmologies, 5.1 cyclic cosmologies, 5.2 the multiverse, 6. infinity and the universe, 7. physical eschatology, 8. conclusions: cosmology and god, other internet resources, related entries.

Christianity and other monotheistic religions (Islam and Judaism) assume a transcendent and sovereign God who created the universe and continually maintains its existence. The world only exists because of an ultimate and supernatural cause which is, as Newton said, “not blind and fortuitous, but very well skilled in Mechanicks and Geometry” (Cohen 1978, 282). Whether in a general philosophical sense or in a scientific sense, cosmology has always been part of theism, but it is only relatively recently that cosmology based on physics and astronomy has entered the discussion concerning the existence and role of God. A limited application of physics to the study of the universe can be found in the second half of the nineteenth century when the cosmological consequences of the law of entropy increase were eagerly discussed in relation to the Christian doctrines of a world with a beginning and end in time. However, physical cosmology is essentially a twentieth century science which emerged as a result of the discovery about 1930 that the universe is in a state of expansion that possibly started a finite time ago. Cosmology as a subdiscipline of physics differs in some respects from mathematical, philosophical and classical observational cosmology, but of course the different approaches are in constant interaction. In a modern sense, physical cosmology became established after the discovery of the cosmic microwave background in 1965 which quickly turned the hot big bang model into the standard model of the universe. Jim Peebles’ Physical Cosmology of 1971, possibly the first book with this title, may be taken as the beginning of modern physical cosmology.

Although physical cosmology based on general relativity theory and elementary particle physics is thus a modern science, many of the theologically relevant questions related to current cosmology are old. Did the universe come into existence a finite time ago? Will it come to an end? Why are the cosmic evolution and the laws of nature of just such a kind that they permit intelligent life to exist? These and other questions of obvious relevance to theism are currently being discussed in the light of the most recent cosmological theories and observations, but the questions themselves (and, indeed, many of the answers) were familiar to medieval philosophers and theologians. This is also the case with the question that is sometimes considered the ultimate one: Why is there a cosmos? There is no reason to expect that today’s advanced physical cosmology, or the even more advanced of tomorrow, will provide final answers that satisfy theists and atheists alike.

2. Creation and the big bang

Einstein’s general theory of relativity shows that the structure of spacetime is itself a dynamical variable, subject to causal influence by the material constituents of the universe. Indeed, Einstein immediately saw the potential to apply general relativity to large-scale cosmological questions. The first cosmological model of Einstein (1917) described a static universe, i.e. one whose spatial geometry is constant over time. However, Einstein’s static universe was empirically inadequate: it could not account for the redshift data gathered by Edwin Hubble and others in the 1920s. The redshift data indicated that distant stars are moving away from us, and moving faster in direct proportion to their distance. Thus, the data indicated an expanding universe.

In the 1920s and 1930s, a number of cosmological models of general relativity were proposed that predict the expansion of the universe. The most accurate account of the data is given by the family of Friedmann-Robertson-Walker (FRW) models. The key characteristic of these models is that space is homogeneous , and hence isotropic (i.e. looks the same in all directions). From the homogeneity assumption, it follows that the entire 4-dimensional spacetime divides neatly into a stack of 3-dimensional “spaces” each of which has constant curvature. The three possibilities for this curvature correspond to the three classical geometries: Euclidean (flat), spherical (positive), or hyperbolic (negative). In a given FRW spacetime, the geometry of space at one time is related to the geometry at any other time by means of a scale factor S ( t ). Indeed, pick a reference time T , such as 2021, pick two reference galaxies, and let d ( T ) be the distance between these galaxies at time T . Then the distance between the two galaxies at any other time t is given by d ( t )= S ( t ) d ( T ), where we set S ( T )=1. This number S ( t ) is called the scale factor, and its behavior encodes the dynamics of a FRW universe.

In those FRW spacetimes that can reasonably be thought to model our cosmos (e.g., those with massive objects), the time parameter t has an absolute lower bound t 0 . In particular, as t decreases towards t 0 , the scale parameter S ( t ) goes to zero. What happens when t reaches t 0 ? In short, these models cannot say what happens, because there are no points of spacetime with time coordinate t 0 . That is, t 0 is an ideal point that is never reached: the universe exists at all times after t 0 , but not before or at time t 0 . A spacetime model with this feature is called singular , and the ideal point that is never reached is called a singularity . In other words, the big bang is a singularity in a FRW spacetime.

The FRW spacetimes are extremely accurate descriptions of the large scale structure of our universe. Since these models describe a universe with a finite lifetime , it is reasonable to conclude that the universe has not always existed.

But many physicists and philosophers hesitate to draw this conclusion. In fact, the standard view in the 1950s and early 1960s was that the singularities of the FRW models were consequences of false idealizing assumptions, namely assumptions of perfect isotropy and homogeneity. But this escape route from singularities was definitely closed when Robert Geroch, Stephen Hawking, and Roger Penrose proved the “singularity theorems,” according to which almost all spacetimes are singular, and in particular, almost all cosmological models describe a finitely old universe.

A number of theists take the past-singular nature of cosmological models as confirmation of the claim that God created the universe ex nihilo . The list of advocates of this “big bang theology” includes Pope Pius XII, Francis Collins (director, US National Institutes of Health), and apologists William Lane Craig and Hugh Ross. And indeed, big bang cosmology does provide prima facie support for theism. After all, big bang cosmology says that the universe has a finite age, and (traditional) theism says that God created the universe out of nothing. Does big bang cosmology not confirm traditional theism? We give several reasons to be cautious about such claims.

Advocates of big bang theology are most interested in the claim that the universe is finitely old. Thus, the chain of inferential support should run as follows:

Big Bang Model → supports → Universe Finitely Old → supports → Theism

Before discussing the first supposed inferential relation, we note that not all theists are committed to the claim that the universe is finitely old. For example, Aquinas claims (in several places, including the Summa Theologica ) that Reason cannot demonstrate the finitude of the universe. But Aquinas also thinks that Reason can demonstrate the existence of God; therefore Aquinas does not think that the very concept of God as creator implies that the universe is finitely old. (Contrary to some contemporary theologians, Aquinas claims that a Christian theist should believe that the universe is finitely old. For Aquinas, the finite age of the universe is a revealed doctrine, like the divinity of Christ.)

Contemporary theologians Arthur Peacocke and Ian Barbour also claim that the doctrine of the “creation” of the universe is best interpreted as one of the universe’s timeless dependence on God, and that such dependence does not demand a temporal creation event. This is also the view of William Stoeger (2010), a Jesuit priest and cosmologist, who argues that scientific cosmology can purify theology but never be in conflict with what theology legitimately asserts. For the remainder of this article, we will not discuss further the question of whether theism requires, or strongly supports, the claim that the universe is finitely old. (Arguments for this claim are assessed in (Copan & Craig 2004).) For now we focus on versions of theism that are committed—in a perhaps naive way—to creation ex nihilo . Even on this understanding of theism, there are still reasons to exercise caution in seeing the big bang as confirming the prediction that God created the universe.

The big-bang theologian argues that the claim

“The universe is 13 billion years old”

provides evidential support for theism. But there are many theists for whom the discovery that the universe is 13 billion years old would actually serve as a disconfirmation of their theistic belief. For example, Bishop Ussher of Ireland (1581–1656) claimed to derive from the Bible that the universe was created in 4004 B.C.; and even in the twenty-first century, some Christian thinkers claim Biblical warrant for an age of the universe much less than 13 billion years (see Kelly 2000 and Byl 2001). For these thinkers, then, the big bang would disconfirm theism—or at least their version of theism, which is committed to the literal accuracy of the Biblical account of creation. More strongly, it seems that Christian theism is committed to a belief in a finitely old universe primarily on the basis of its commitment to the accuracy of Biblical accounts of creation. Thus, if a theist comes to believe the big-bang account of the universe’s origin, and hence to doubt the literal Biblical account, then she will lose one reason—and possibly her only reason—for believing that the universe is finitely old.

Of course, there are also theists who interpret Genesis metaphorically as implying that the universe was created, but not indicating a specific age for the universe. For these theists, finding that the universe is finitely old might confirm rather than undermine their belief.

According to traditional Christian theism, creation ex nihilo is miraculous—something which the laws of nature cannot explain. But then why should a theist expect to be able to derive creation ex nihilo from the laws of nature? Compare with other supposed miracles, e.g., within Christianity the claim that Jesus changed water into wine. Do Christian theists claim that chemistry should predict that water can transform into wine? Of course not: God is supposed to be able to transcend the laws of nature, and the laws of nature are defeasible when it comes to describing what actually did happen (since God could have intervened). But then couldn’t the best (most explanatory, most elegant) cosmological theory posit an infinite past, whereas in reality God created it some finite time in the past?

The puzzle we have just encountered trades on the special status of cosmology as a historical, yet law-based science, with only one actual model. While theists would certainly not expect the laws of chemistry to predict that water can transform into wine, they do believe that an accurate historical account would include reference to those miracles that did occur. So, is cosmology more like history, or more like chemistry? If God did create the universe, should a cosmological theory report (or predict or entail) such a fact? Or should cosmology only be required to provide laws for universes, laws which might in fact have been broken in our universe?

Suppose that the theist takes a harder line and says that theism requires (or favors) cosmological models with a finitely old universe. In this case, the time parameter in our cosmological models should never take values lower than some fixed number, which we can conveniently set to zero.

But the interval (0, t ) is topologically isomorphic to (−∞, t ), suggesting that duration of time (finite versus infinite) might lack intrinsic physical or theological significance. Such a point was made already by E.A. Milne in 1935, and then independently by Charles Misner in 1969. In particular, Misner replaces the time parameter t with the negative of its logarithm (i.e., −log t ) in order to assuage worries that a bounded time parameter makes no sense. According to Misner (1969, 1331), even in models that begin with singularities, “the Universe is meaningfully infinitely old because infinitely many things have happened since the beginning.” Interestingly, Misner’s move can hardly be motivated by a desire to obviate the need for a creator: Misner is a self-described Catholic Christian.

The possibility that the finite/infinite time distinction is conventional was noted by the Catholic philosopher of science Ernan McMullin, who concludes that the theological doctrine of creation ex nihilo must not be given a metric interpretation (McMullin 1981). Rather, claims McMullin, the ex nihilo doctrine should be interpreted order-theoretically: the time series has a first point. But this order-theoretic criterion will not help theism, at least not in regard to current cosmological models. On the one hand, FRW cosmological models fail the order-theoretic criterion: they have no first moment of time. On the other hand, an ideal first moment of time could be adjoined to any spacetime, even those that have a metrically infinite past (see Earman 1995). Thus, a simple order-theoretic criterion is a bad guide to whether cosmological models are consistent with the doctrine of creation ex nihilo .

A more adequate criterion of when a cosmological model is consonant with creation ex nihilo would require a detailed analysis of singular spacetimes (for extensive discussion of the latter topic, see Earman 1995). The best current account of when a spacetime is truly singular (as opposed to merely being described with inadequate coordinates) is when it contains inextendible geodesics of finite length. Intuitively, a geodesic is the path that would be followed by a clock in freefall. If a clock were travelling on a past-inextendible geodesic, then at some finite time in the past, the clock did not exist; more strongly, spacetime itself did not exist. Thus, the big-bang theologian would do best to claim that creation ex nihilo is confirmed precisely by those cosmological spacetimes that have inextendible geodesics. (Indeed, this criterion does hold for FRW models.) The main problem with such a proposal is that it ties a robust, intuitive theological doctrine down to an extremely precise technical feature of Lorentzian manifolds (as described by differential geometry). The risk then is that by doing so, one would add extraneous content to the theological doctrine: a future model might fail the technical criterion while still being consistent with the theological doctrine. Furthermore, many Christian theists claim that core theological doctrines are perspicuous—in particularly, not understood exclusively by an elite class of priests or scholars. But the notion of a Lorentzian manifold having incomplete geodesics can hardly be said to be accessible to the average layperson.

Finally, big-bang theology overreaches if it says that general relativity and the singularity theorems have settled once and for all that the universe had a beginning in time. In fact, relativistic cosmology predicts its own invalidity for times close to a dynamic singularity, such as the big bang. (For a dissenting opinion, see Misner 1969.) The reason that relativistic cosmology predicts its own own invalidity is that in the neighborhood of singularities, gravitational effects are intense, and quantum effects can be expected to play a predominant role. But general relativity does not incorporate quantum effects, and indeed it is untested in such regimes of intense gravitational force. Thus, there is little reason to believe that the singularity theorems make a valid prediction about the structure of a future successor theory of general relativity that includes quantum effects. We discuss this issue further in Section 4.

Most philosophers and physicists have thought that big bang cosmology is either neutral towards, or supportive of, traditional theism. Thus, atheists have usually taken a defensive stance, trying to defeat the arguments of the big-bang theologians. But a vocal minority—we might call them “big-bang atheologians”—have made the stronger claim that big bang cosmology disconfirms theism. The most visible proponents of this big-bang atheology are the philosophers Adolf Grünbaum and Quentin Smith. In the case of Smith, quantum cosmologies are taken to provide even stronger evidence against theism.

In putting forward their arguments, big-bang atheologians make a number of points that seem to have been overlooked by their theistic counterparts. One such point is that FRW cosmological models have no first state. Thus, a theist who invokes the big bang cannot say that there is a state of the universe, say Α, such that God created the universe in state Α. He or she will have to invoke a more sophisticated notion of God creating initial temporal intervals, or something like that. Are these more sophisticated developments still consonant with traditional theology?

Big-bang atheologians also argue that it makes no sense to accept both that there were no times before the big bang (since time itself comes into existence with the universe) and that the universe was caused. Of course, many theists claim that God causes the universe timelessly, and they would attempt to defend the coherence of such a notion in the face of these criticisms.

The case of quantum cosmology provides further complications for theology. Without going into excessive detail of specific proposals, a common theme of many quantum cosmologies is that they postulate a probability distribution over universes themselves. In other words, quantum cosmologies provide a measure of likelihood that certain sorts of universes exist. Some older quantum cosmologies (e.g., Hawking’s early proposals) still predict that the universe is finitely old. And yet, one might consider their further explanatory power—over and above classical general relativity—as undermining theistic explanations of the universe. In particular, quantum cosmology predicts with a high probability that a universe like ours would exist. Seen from this perspective, one might take quantum cosmologies to offer a competing, non-theistic explanation for the origin of the universe.

Theists have argued, in response, that quantum cosmologies do not provide unconditional probabilities for the existence of the universe. For example, Craig (1997), Deltete & Guy (1997), and Oppy (1997) argue that quantum cosmologies provide only conditional probabilities for the existence of some universe configurations given other universe configurations. Smith (1998) responds by defending the claim that the probabilities in quantum cosmology are unconditional. But neither side of this debate has attended to the special complications involved in interpreting quantum , rather than classical, probabilities. For example, Smith treats the universal wavefunction Ψ( h i j , f ) as providing a probability distribution over universe configurations ( h i j , f ). But we know, from elementary quantum mechanics, that it is literally inconsistent (i.e., leads to contradictions) to treat a wavefunction as giving probabilities in an absolute sense. (This contradiction is derived by the famous Kochen-Specker theorem.) We conclude that in order for the metaphysical significance of quantum cosmologies to be assessed, a more nuanced consideration of the interpretation of quantum mechanics will be required.

Aristotle’s cosmology belonged to the class of steady-state theories in so far that his universe was changeless and eternal. When Einstein in 1917 proposed the first relativistic model of the universe, he unwittingly pictured a universe which had qualitative features in common with Aristotle’s: it was finite in space, but infinite in time. The discovery of the expansion of the universe excluded the steady state from relativistic cosmology, but not from other forms of cosmology. Robert Millikan, Nobel laureate and famous physicist, was among those who in the 1930s favored an eternally recurrent universe with a continual creation of matter and energy to counter the increase of entropy. He thought that such an eternal and evolving universe revealed the creator’s continual activity and explicitly presented his cosmological view as support for the doctrines of Christianity in general and for the immanence of God in particular.

Contrary to the earlier ideas of a steady-state universe, the theory that Fred Hoyle, Hermann Bondi and Thomas Gold introduced in 1948 accepted that the universe is expanding. Conceptually the theory was founded on the “perfect cosmological principle,” that is, the postulate that the universe in its large-scale features is not only spatially but also temporally homogeneous. Although this classical steady-state theory was abandoned in the 1960s because of its inability to account for new discoveries (such as the cosmic microwave background and the redshifts of quasars), it remains an instructive case in the cosmology-theology discussion. Moreover, the theory is not quite dead yet, as some of its characteristic features survive in the quasi-steady-state cosmology (QSSC) still defended by Jayant Narlikar and a few other cosmologists. This model does not satisfy the perfect cosmological principle, but it assumes an indefinite cosmic time scale during which matter is continually created. In this respect it is an alternative to the big-bang theory and its supposed association with divine creation. In 1994, at a time when he was developing the QSSC model, Hoyle referred to big-bang cosmology as “a form of religious fundamentalism” (Hoyle 1994, 413). According to the classical steady-state theory, the universe has expanded for an infinity of time and will continue to do so for ever; yet the average density of matter remains constant because matter, or rather matter-energy, is continually being created out of nothing. (In later versions of the theory, matter creation was not ex nihilo .) Both features—the infinite time scale and the continuous creation of matter—were controversial and caused concern of a philosophical and also a theological nature.

It was widely assumed in the 1950s that the steady-state universe was contrary to theism or at least made God superfluous as a creator of the cosmos. After all, how can God have created a universe which has existed in an infinity of time? According to the astronomer, science popularizer and non-believer Carl Sagan, “this is one conceivable finding of science that could disprove a Creator—because an infinitely old universe would never have been created” (1997, 265). However, although the argument may seem to pose a real problem for theism, the theologians were well prepared—it had been discussed since the thirteenth century when Thomas Aquinas suggested that God could indeed have created an infinitely old universe. Moreover, theological responses to an infinitely old universe were far from new, for they had already been developed in relation to eternally cyclic models, either in the more speculative versions of the nineteenth century or the relativistic models that were proposed in the 1930s onwards.

According to the Thomistic doctrine of creatio continuans , God causes things to exist in the sense that their existence depends wholly on his power. If they were left to themselves they would turn into, or return into, nothingness. From this point of view creation is basically a metaphysical rather than a physical and temporal concept, and an eternal yet created universe is perfectly possible. Interestingly, the leading steady-state physicist William McCrea was also a devout Christian who argued that cosmology, in whatever form, must necessarily include the postulation of a divine creator. As theologians in the 1950s, both Protestants and Catholics, were quick to point out, Hoyle’s eternal universe was not particularly heretical, for it was still in need of a creator. Not only did they mobilize the old concept of continual divine creation, emphasizing that cosmic creation is primarily about the ontological dependence of the world on God, they also stressed that faith in God has little to do with physical cosmology in whatever of its versions. Erich Mascall, a priest and philosopher of religion, saw no reason why the steady-state model should cause worry among the faithful. As he said in 1956, “The whole question whether the world had a beginning or not is, in the last resort, profoundly unimportant for theology” (Mascall 1956, 155).

Views similar to Mascall’s have been held by many later theologians and Christian philosophers, but not by all. There is disagreement about how solidly based in the Bible the concept of atemporal continual creation is, and also about the significance of an absolute beginning of the world (for opposite views, see Copan and Craig 2004 and May 1994). The view that cosmology is essentially irrelevant to Christian belief has not gone uncontested. As Ernan McMullin has pointed out, Christian doctrines are more than metaphysics and codes for moral conduct; they are also cosmic claims that say something about the universe and what it contains of things. For this reason theologians need to pay attention to cosmology in particular and to science in general.

Some Christian scientists and philosophers have seen the continual creation of matter, as posited by the steady-state theory, as a manifestation of perpetual divine creation. Thus, the Catholic philosopher Philip Quinn (1993) adopted the old notion of creatio continuans to the case of steady-state cosmology. The argument is essentially that since ex nihilo creation of matter violates energy conservation, there must be an external creative cause that accounts for the violation, and this cause he identifies with perpetual divine creation. This kind of reasoning has been severely criticized by Adolf Grünbaum, who flatly dismisses the claim that underlies the idea of perpetual divine creation, namely that nothingness is the natural state of the universe. This claim has also been argued in detail by Richard Swinburne (1996), who finds it extraordinary that there exists anything at all and from the fact that something exists infers the existence of God. But according to Grünbaum there is no room for divine creation in either big-bang or steady-state cosmology. “Steady-state cosmology,” he concludes, “is indeed logically incompatible with [the] claim that divine creative intervention is causally necessary for the nonconservative popping into existence of new matter in the steady-state universe” (Grünbaum 1996, 529).

Whereas steady-state cosmology is at least problematical from the point of view of traditional theology, it goes well together with the ideas of process theology or philosophy, where God is seen as interacting creatively and incessantly with natural processes. In a general sense, Whitehead’s philosophy is more in harmony with the steady state than the big bang universe. The prominent British astronomer Bernard Lovell (1959), a devoted Christian inspired by process thinking, was in sympathy with the steady-state theory and saw no reason why it should be a threat to belief in a divine being. To him, the creation of matter was a sure sign of God’s activity.

As we mentioned previously, there are reasons to suspect the invalidity of classical general relativity in regions near a singularity—most importantly, for times very close to the big bang. In particular, when lengths are very small, and curvature and temperatures are very high, then—if the gravitational force behaves like all other known forces of nature—quantum effects will take over, and we should accordingly expect different outcomes. This observation is itself sufficient to destroy the aspirations of big-bang theology—unless there are good reasons to think that the finite-age prediction of relativistic cosmology will be preserved in a quantum gravity or in string theory. In this section, we briefly review the known data about singularities in theories that attempt to unify gravity and quantum mechanics. Our review supports two conclusions: (1) We do not know yet if the best model will predict a finitely old universe, but (2) there are good reasons to think that the big bang is not necessarily an absolute beginning.

There have been a number of proposed theories of quantum cosmology. Perhaps best known of these is the proposal of Stephen Hawking, which results in a universe with no boundary—motivating the famous question, “what place, then, for a creator?” The bearing of Hawking’s cosmology on theism has already been discussed extensively by Craig & Smith (1995), Deltete & Guy (1997), Craig (1997), and Smith (1998). But it would be ill advised to take Hawking’s theory as giving the final version of quantum cosmology. As noted by Drees (1990), Hawking’s approach is just one among several competing attempts to incorporate quantum effects into relativistic cosmology, and we are not compelled to accept its idiosyncratic metaphysical picture. More to the point, Hawking’s cosmological model is ad hoc in the sense that it does not flow from a more comprehensive unification of general relativity quantum theory. In this section we turn to two cosmological theories that do result from systematic and comprehensive unifications of general relativity and quantum theory: loop quantum cosmology and string cosmology.

Loop quantum cosmology (LQC) is an approach to cosmology within the framework of the loop quantum gravity (LQG) program (Rovelli 2004), which itself starts with the idea that unifying quantum theory and general relativity will require “quantizing” the gravitational field—and hence the structures of spacetime itself. Roughly speaking, to quantize a theory means that the quantities (e.g., position, momentum, scalar curvature, etc.) are replaced by “matrices,” or more generally with “operators on a Hilbert space.” This replacement can have profound physical consequences, most particularly the spectrum of a quantity (i.e., the numerical values it can possess) can become discretized where it was previously continuous, or bounded where it was previously unbounded, and quantities can be forced to obey a Heisenberg uncertainty principle.

For our purposes, the important question is what happens to those quantities (e.g., spatial curvature) that grow unboundedly large in classical FRW spacetimes as the time parameter t approaches the initial boundary time t 0 ? To answer this question requires going through intricate technicalities involving domains of definition of operators, etc.. To summarize, however, the most prominent proposal (championed primarily by Martin Bojowald and collaborators; see Bojowald 2009) results in a scale parameter S ( t ) that is bounded away from zero, entailing that curvature is bounded from above. More is true: the dynamical equations of LQC extend through the big bang, i.e., the universe existed before the big bang.

It would be premature to take loop quantum cosmology as having decisively overturned the big-bang, finite-age cosmological model. Nonetheless, there is a non-negligible probability that it will do so in the near future; and hence there is a non-negligible probability that the big bang is not the beginning of the universe, and a fortiori, not the creation event (even if there was one).

However, loop quantum gravity is not the most popular—in terms of sheer number of researchers—approach to unifying quantum theory and gravity. The title of most popular belongs to string theory, and so string theory’s perspective on the big bang event is of crucial interest for those wishing to assess the bearing of physical cosmology on traditional theological doctrines.

All indications from string cosmology point to the fact that the universe existed before the big bang. In particular, string theory claims that if we apply fundamental symmetry transformations to cosmological models of the recent universe, then we get a copy of the universe (with important quantities inverted) that might be called the “pre-big-bang universe.” In this scenario, the absolute big bang disappears and is replaced by a saddle point in the dynamical evolution of spacetime curvature: before this point, curvature is increasing, and after this point, it is decreasing.

According to Gasperini (2008), string cosmology’s prediction of a pre-big-bang universe results from a principled application of symmetry principles. Furthermore, string theory has a built in mechanism (namely a minimum string length) that seems to rule out singularities of infinite curvature or spatial length shrinking to zero. As was the case in LQC, the values of physical quantities in string theory are constrained by quantum mechanical laws; and so some quantities that grew beyond bounds in classical theory are well-behaved in quantized versions of that theory.

We currently lack the empirical data that would distinguish between competing models of quantum cosmology. But these models make different empirical predictions from each other, and they also make different predictions from classical relativistic cosmology. Hence, it is an empirical, rather than a metaphysical, question whether the big bang was the beginning of the universe.

However, just as the big-bang is not unambiguously friendly to theism, so quantum cosmology is not unambiguously hostile to theism. Indeed, Chris Isham (1993, 405) has suggested that quantum cosmology’s description of a boundary-free universe accords quite well with theism’s insistence that God sustains the universe at all times. (See also the discussion in Drees 1988, 1990, 1991.) Clearly, theism has shown some flexibility in integrating its doctrines with prevailing scientific worldviews—as evidenced, e.g., in the integration of Aristotelian cosmology, and in the exploitation of big-bang cosmology. Should we expect the situation to be any different with quantum cosmology?

5. Other non-standard cosmologies

Apart from the quantum-based cosmologies mentioned above, there are several other theories of the universe that differ from the generally accepted big bang theory and are, in this sense, “non-standard.” A few of these models have been discussed within a religious context. We shall limit ourselves to two groups of theories, cyclic cosmologies and multiverse theories.

Origen, a third-century Christian philosopher, speculated that God, before he created our universe, had busied himself with the creation of an endless series of earlier worlds. His idea of an eternal cyclic universe was however condemned by the church and has since then generally been regarded as associated with atheism and materialism. Indeed, from about 1850 to 1920 the classical cyclic or recurrent universe was popular among many atheistic thinkers who found it to be incompatible with Christian doctrines. Nonetheless, some theists have endorsed such a universe, e.g., Joseph Smith’s claim that “as one earth shall pass away, and the heavens thereof even so shall another come” (Mormon Book of Moses 1:38).

Although Einstein’s cosmological field equations do not justify a series of pulsating universes, many cosmologists have suggested ways in which a collapsing universe can reappear from a nonsingular state and thus give birth to a new universe, perhaps ad infinitum. It should be pointed out that Lemaître, the cosmologist-priest, did not suggest such a “Phoenix universe,” in spite of numerous claims to the opposite. In some cases atheism has been part of the motivation for proposing singularity-free models with an unlimited past and future. For example, the British physicist William Bonnor considered the new big bang theory “the opportunity Christian theology has been waiting for ever since science began to depose religion from the minds of rational men” (Bonnor 1964, 117). His own favored candidate, a universe oscillating smoothly and eternally between two states of finitely high density, avoided a divine miracle and theistic exploitation of cosmology, in which respect it was similar to the steady-state theory. Steven Weinberg noticed the similarity: “Some cosmologists are philosophically attracted to the oscillating model, especially because, like the steady-state model, it nicely avoids the problem of Genesis” (Weinberg 1977, 154).

Classical-relativistic cyclic models presupposed a closed universe disagreeing with current observations, for which reason they are no longer considered viable alternatives. But the twenty-first century has witnessed several new proposals, of which we shall mention only two: the “conformal cyclic cosmology” developed by Roger Penrose, and the “new cyclic cosmology” developed by Paul Steinhardt and Neil Turok.

In his theory of conformal cyclic cosmology, Penrose claims that as the big bang is approached, massive objects play a negligible role, so that the physics is governed by degrees of freedom that are invariant under rescaling of lengths and times. Such degrees of freedom are called “conformal invariants.” Thus, Penrose claims, we make a mistake to model the early universe by a Lorentzian manifold with a metric (as is done in classical general relativity). Rather, spacetime should be described by a conformal manifold , which is essentially a conformal equivalence class of general relativistic spacetimes. The “cyclic” part of Penrose’s cosmology comes from noticing that the future of one ever-expanding universe can be smoothly bridged to the past of another big-bang universe by means of such a conformal manifold. In this case, the big bang is not a true beginning, but only a sort of phase change from one “epoch” to another (Penrose 2010).

The new cyclic cosmology of Steinhardt and Turok develops ideas from string theory to describe a universe without inflation going through an endless sequence of cycles—in which case, the big bang is not the beginning of time (Steinhardt and Turok 2007). In this respect the model is similar to the steady-state model, and Steinhardt and Turok have indeed described it as a “remarkable reincarnation” of Hoyle’s old theory. Although the new cyclic model has attracted a fair amount of attention, it is not widely accepted. Nor is this the case with the pre-big-bang bouncing cosmology advanced by Gabriele Veneziano and Maurizio Gasperini on the basis of string theory. According to the pre-big-bang model the universe is not only eternal into the future, it is also eternal into the past, the two cosmic phases (contracting and expanding) being separated by a non-singular big bang.

Eternal bouncing models qualitatively similar to the pre-big-bang scenario have been proposed earlier, either on the basis of the relativistic field equations or on the idea of a plasma universe. According to the Swedish physicist Hannes Alfvén, a Nobel laureate of 1970 who developed the latter idea, the plasma universe was an alternative to the theistic big bang theory. Since none of the models considered in this section operate with an absolute beginning, they may seem to be problematic from a theistic point of view. However, the theist can always appeal to perpetual divine creation, just as in the case of the steady-state universe.

The modern idea of the multiverse is theologically more controversial. In its so-called landscape version, which since 2002 has been promoted and developed by Leonard Susskind and many other physicists, it is based on the apparent non-uniqueness of the equations of string theory. The solutions of the equations describe, in a sense, possible worlds with different physical parameters, interactions, types of particles, and even dimensionality; the multitude of solutions are then identified with really existing worlds which generally are causally separate from ours. As a mechanism for generating the huge number of universes, multiverse physicists make use of the eternal inflation scenario. Moreover, the multiverse is closely associated with anthropic reasoning: we find ourselves in our universe, with its particular physical laws and content of particles, not because other universes are impossible or improbable, but because our kind of life cannot exist in other universes. The theory of the multiverse has seductively great explanatory power (while it has almost no predictive power), which is a major reason why many physicists and cosmologists find it attractive. On the other hand, other physicists dismiss it as pseudoscience because it is practically untestable.

It is common among supporters of the multiverse to conceive it as an alternative to a divinely created world and ideas of natural theology. Because it represents our universe as a chance universe, special only by the fact that we live in it, the multiverse has been likened to another and more famous anti-design theory, neo-Darwinianism. Weinberg puts it as follows: “Just as Darwin and Wallace explained how the wonderful adaption of living forms could arise without supernatural intervention, so the string landscape may explain how the constants of nature that we observe can take values suitable for life without being fine-tuned by a benevolent creator” (Weinberg 2007, 39). At least to some theists, the multiverse stands in sharp contrast to Christian belief. As Richard Swinburne sees it, “To postulate a trillion trillion other universes, rather than one God in order to explain the orderliness of our universe, seems the height of irrationality” (Swinburne 1996, 68).

Swinburne joins several other philosophers and physicists in seeing physical cosmology as providing key data for a new design argument. The strength of such fine-tuning arguments continues to be a hotly contested issue among philosophers (see Collins 2009, Colyvan et al. 2005, McGrew et al. 2001, White 2000).

On the other hand, there is no one-to-one correspondence between the multiverse and belief in a divine creator. Several philosophers have argued that if theism is true, we should expect the actual world to be a multiverse: being a perfect being, God would create a multiverse rather than just a single universe (Kraay 2010). It is possible to answer affirmatively to the question, “does God love the multiverse?”, such as the physicist Don Page did at a symposium in 2008 (see Page 2008). Even if there are 10 500 universes (but not, perhaps, if there are an infinite number of them), they could have been providentially created by the almighty God with a purpose we cannot fathom. Why not? It has even been suggested (by Paul Davies) that multiverse explanations are reminiscent of divine explanations and unintentionally reintroduce a transcendent creator.

Mormon theology differs in several respects drastically from the theology of traditional Christianity. Not only is God personal and held to have been created by a prior god (who was again created by a prior god, etc.), according to the central doctrine of “eternal progression” human beings will ultimately become like God himself. There is an infinite number of beings and it takes an eternity for them to become gods. Standard big bang cosmology, based as it is on a universe of finite age, is incompatible with Mormonism, where existence has neither beginning nor end. Whereas traditional theologians have no problem with a universe created ex nihilo , and many subscribe to this doctrine, Mormons flatly reject it. In order to overcome the conflict with physical cosmology, some Mormon thinkers have turned to the multiverse. Realizing that the attempt to harmonize the Mormon dogma of eternal progression with modern cosmology is problematic, Kirk Hagen says: “For Mormonism, a compelling reason to consider a multiverse cosmology is to attempt a reconciliation of modern cosmological ideas and the central tenet of Mormon doctrine” (Hagen 2006, 28).

The anthropic principle, an integral part of multiverse cosmology, has similarly been discussed in theological contexts and, again similarly, without any consensus emerging from the many discussions. In its most common version, called the weak anthropic principle, it states that what we observe is selected by our existence in a universe with just such properties that allow us to exist. Swinburne and some other theists in favor of design arguments find the anthropic principle to be, at best, unnecessary and obfuscating. To them, the values of the cosmic parameters and constants of nature appear to be fine-tuned because they are fine-tuned, the designer being God. The atheist Richard Dawkins goes further, arguing that the anthropic principle is an alternative to the design hypothesis and provides strong evidence for a world without God. However, theists do not generally see anthropically based arguments as a problem for a divinely created world. William Lane Craig and John Polkinghorne are among those who hold that the anthropic principle is compatible with divine design and can even be seen as indirect support for theism. According to the South-African cosmologist (and Quaker) George Ellis, anthropic fine-tuning is the result of a purposeful design of the universe. He has suggested a “Christian anthropic principle” as the basis for an ultimate understanding of the universe that combines scientific and religious perspectives (Ellis 1993).

In relation to the design argument, as reinvigorated by the discussions of the anthropic principle, some physicists and philosophers have returned to an old objection to it, namely that it is not an argument for the Christian God; it is at best an argument for a cosmic architect in a deistic sense, or for that matter several such architects. On the other hand, theists have replied that even if this objection be true it does not constitute a proof that the God of theism does not exist. Although design arguments frequently occur in connection with the anthropic principle, it needs to be said that they were not part of the original anthropic program initiated by Brandon Carter in 1974.

Cosmological theory has gone through many phases and proposals over the past 100 years. Some have included universes with infinite pasts and these were mentioned previously. However, for the past forty years there has been a strong consensus on the modern big bang theory which has a finite past. But even if the universe is temporally finite in the past, it may well be spatially and materially infinite. If space is infinite and the cosmological principle is assumed to be valid, the universe will contain an infinite number of galaxies, stars, atoms and everything else. Such actual infinities not only cause philosophical and logical problems, they may also cause problems of a theological nature. In his thought experiment known as “Hilbert’s hotel” the famous mathematician David Hilbert argued that (countable) actual infinities are so bizarre that they cannot have anything to do with the real world we live in. In this thought experiment, Hilbert imagines that he arrives at a hotel with an infinite number of rooms, all of which are occupied. The desk clerk claims nonetheless that there is a vacancy. He then proceeds to ask each guest to move to the room with the next highest number in the sequence (see Oppy 2006, 8; Kragh 2014, Other Internet Resources).

Hilbert himself was uninterested in religion, but later philosophers and theologians have occasionally used his peculiar hotel apologetically, as an argument for the existence of God and also for the finitude of the universe.

The theological problems related to an infinitely large universe are not specifically related to modern physical cosmology but have been discussed since the early days of Christianity. On the other hand, they may be seen as even more relevant today, when the favored cosmological model has zero curvature, meaning that space is flat. Although a flat cosmic space does not necessarily imply an infinite universe, many cosmologists assume that the universe is indeed spatially infinite.

The theological implications of an infinite universe were discussed by the church fathers and, in greater detail, by Johannes Philoponus in the sixth century. Many of the arguments were of the same kind as those used in the attempts to prove the impossibility of a temporal infinity. At the time of the scientific revolution it was commonly assumed that physical space cannot be truly infinite, only indefinitely large. Infinity was seen as a divine attribute not to be found elsewhere; to claim that nature is infinite would be to endow it with divinity, a heretical view characteristic of pantheism. While the generally accepted view among theists was, and to some extent still is, that an infinite universe is philosophically absurd and theologically heretical, there was no consensus on the issue. In fact, several Christian thinkers, from Descartes in the seventeenth century through Kant in the eighteenth to Edward Milne in the twentieth, have argued that an infinite universe is in better agreement with God’s will and omnipotence than a finite one. According to Milne, “It requires a more powerful God to create an infinite universe than a finite universe; it requires a greater God to leave room for an infinity of opportunities for the play of evolution than to wind up a mechanism, once and for all” (Milne 1948, 233). The correlation between finitism and theism, and infinitism and atheism, should be seen as historically contingent rather than justified by either scientific or theological reasons.

During the early period of modern cosmology, relativistic models with zero or negative curvature were sometimes associated with materialism and atheism because they implied a universe of infinite size. Conversely, Einstein’s closed and finite universe was welcomed by theists. According to Ernest W. Barnes, the mathematically trained bishop of Birmingham, infinite space was “a scandal to human thought,” (Barnes 1931, 598). His argument was epistemic as well as theological: only if God’s universe is finite can we hope to understand the full range of his activity. Lemaître thought likewise that the universe had to be finite in order to be comprehensible. In agreement with his later warning against the “nightmare of infinite space” (Kragh 2004, 139), both of his two innovative cosmological models, the expanding model of 1927 and the big-bang model of 1931, were spatially closed. The steady-state model of the 1950s was not only unpopular among Christians because of its lack of a cosmic creation, but also because it implied a homogeneous universe of infinite extent. According to Stanley Jaki, a Benedictine priest and historian of science, the infinite universe is a scientific cover-up for atheism. Mormons do not agree, though, for they need a universe which is infinite in both time and space.

The present consensus model of a geometrically flat accelerating universe is usually taken to imply an infinite cosmos. The general attitude of cosmologists is to ignore the troublesome philosophical problems and speak of the infinite universe as just an indefinitely large one. They rarely reflect on the weird epistemic consequences of an actual infinity and even more rarely on the theological consequences. Ellis is an exception to the rule. He and his collaborators have argued forcefully against an infinite universe, suggesting that the flat space of the consensus model is probably an abstraction that does not hold physically (Ellis, Kirchner and Stoeger 2004). If the universe is really infinite and uniform it can be (and has been) argued that there will be an infinity of identical copies of all human beings and indeed of everything. Such a consequence, as discussed by Ellis, Max Tegmark, Alan Guth and others, clearly is theologically disturbing.

Even more disturbing, says Ellis, is that God may then not be able to keep track of and give attention to the infinite number of beings in the universe. Moreover, if there is a multitude of cosmic regions, each of which is inhabited with intelligent beings, one may need to contemplate a multitude of Christ-figures, incarnations and crucifixions. Ellis was not only willing to consider such a scenario, he also thought that it strengthened the case for a finite universe, for then “we would have to countenance only a finite number of civilizations needing redemption. Surely an infinite number of Christ-figures must be too much, no matter how one envisages God” (Ellis 1993, 394).

The cosmological field equations are time-symmetric and the fundamental laws of physics are assumedly valid at any time. Thus, modern cosmology is not only about the past of the universe, it also offers scenarios about its far future, including speculations about the fate of intelligent life. Given that the apocalyptic passages in the Bible speak of an end of the world and a possible new creation (e.g., 2 Peter 3:10–13), the cosmic future may seem to offer another point of contact between cosmology and theistic religion. But can there be a secular or scientific eschatology?

Scientifically based speculations about the state of the cosmos in the far future and the possibility of endless life were first discussed in the late nineteenth century in connection with the controversy over the heat death predicted by the second law of thermodynamics. Some of the German scientists involved in the controversy argued that life might persist even in the very high-entropic environment of the far future, and they explicitly referred to the eschatological aspects of cosmology. Characteristically, while the heat death scenario was welcomed by Christian authors, it was vehemently opposed by materialists and atheists who argued for an eternal universe with eternal life. As Eddington, a Quaker and an advocate of the inevitable heat death, later asked: “Since when has the teaching that ‘ heaven and earth shall pass away’ become ecclesiastically un-orthodox?” (Eddington 1935, 59).

Since the 1970s “physical eschatology” has emerged as a new subfield of astrophysics and cosmology, pioneered by Freeman Dyson, Jamal Islam and others (see the survey in Kragh 2011, 325–353). The field deals primarily with the state of the universe in the remote future as based on extrapolations of cosmological models and the assumption that the presently known laws of physics will remain indefinitely valid. The favored scenario is the open ever-expanding universe where extrapolations typically result in an ultimate future (at about 10 100 years from now!) in which the universe consists of nothing but an exceedingly thin electron-positron plasma immersed in a cold radiation of neutrinos and photons. Other studies presume a closed universe collapsing in a big crunch and others again investigate the nearer future of humankind, say a few million years from now. While many of these studies are not concerned with the final state of life, some are, and it is this latter group that constitutes physical eschatology proper. According to John Barrow and Frank Tipler, the research field is, “the study of the survival and the behavior of life in the far future” (Barrow & Tipler 1986, 658).

Physical eschatologists usually ignore the religious associations of their studies or deny that they exist. Tipler is a controversial exception, however. Not only does he argue that some kind of life can continue forever in a closed universe, he also claims that it is the very collapse of the universe that permits eternal life. When the final eternity has been reached at what he calls the “omega point,” life becomes omniscient and the temporal becomes atemporal. According to Tipler, the final singularity is God and “theology is nothing but physical cosmology based on the assumption that life as a whole is immortal” (Tipler 1995, 17). In his book The Physics of Christianity (Tipler 2007), he continues his idiosyncratic exploration of modern cosmotheology according to which theology is merely a branch of physics. Tipler’s views are undoubtedly extreme, but (and perhaps for this reason) they have caused much discussion among theologians.

The term physical eschatology indicates a connection to biblical eschatology, but it is far from clear that the two are related in any meaningful sense. The message of the Bible is not so much the end of the physical universe as it is about the imminent return of Christ, the transformation of humans from flesh to spirit, and the final kingdom of God. It is about the ultimate destiny and goal of humans, not of self-reproducing robots. As Jefferson Davis (1999) notes, the ultimate hope which is crucial in theology cannot be supplied by the laws of physics. The scenario of a closed universe, such as argued by Tipler, may appear to be more compatible with the biblical view than the case of the ever-expanding universe, but even in the former case it is hard to establish a meaningful connection. While the end of the world does not conflict with the Bible, the claims of immortality of intelligent life forms (not necessarily humans) do. The Bible says that God alone is immortal and that all his created beings are doomed to extinction unless God decides otherwise.

Several theologians have expressed concern about the cosmologists’ scenarios of the end of the universe and stressed that there is a world of difference between these scenarios and proper eschatology. According to Wolfhart Pannenberg the Christian affirmation of an imminent end of the world is scarcely reconcilable with the cosmological extrapolations of the state of the universe zillions of years ahead. Karl Peters probably speaks for the majority of theologians when he writes: “If the expanding universe is indeed open, expanding forever, then how can one speak of God recreating the universe? If the universe is closed, then it is likely to end in a ‘big crunch’ of mammoth black-hole proportions. Again, it is difficult to see how a new creation can take place” (Schwarz 2000, 180). According to Peters, the physical end of the universe would in effect imply the non-existence of God as understood in the Christian tradition. Whereas Pannenberg, Peters, Arthur Peacocke and others tend to think that physical and Christian eschatology are either contradictory or incommensurable, Craig has taken a more reconcilable view. According to him, the cosmologists’ versions of secular eschatology furnish grounds for taking seriously the hypothesis of a transcendent creative and omnipotent agent. This agent may not be the classical God, but more likely God in a panentheistic version.

Finally, Robert Russell (2008) argues that the potential conflict might be resolved by appealing to God’s omnipotence and freedom to perform miracles: the future of the universe would have been what science predicts had God not decided to act at Easter and bring about, and will continue to bring about, the new creation. This view is allegedly not in conflict with science, but only with the philosophical assumption that the events predicted by science must happen, and this assumption Russell sees no reason to accept.

The question, “why does the universe exist?” admits of answers from traditional religions as well as from contemporary cosmological theories. However, according to Bede Rundle (2004), neither of these answers are needed, for philosophical analysis is sufficient to prove the existence of a physical universe. While some claim that the scientific answer has superseded all theological answers, others claim that the scientific answer reinforces the claim that God created the universe. Indeed, the story of the interaction between scientific cosmology and theology is by no means a simple tale of a better theory replacing an inferior; nor a simple tale of the convergence of diverse sources of knowledge. A naive or ideological reading of twentieth century cosmology might count big bang cosmology as providing new support for theism, and alternatives such as steady-state cosmology as atheistic backlashes. (And of course, the work of apologists such as W.L. Craig lends credence to this sort of picture.) But such a view misses many nuances, both in the historical record, as well as in the logical structure of these issues. From a historical point of view, there has been little correlation between religious views of scientific cosmologists and their proposed cosmological models. From an epistemological point of view, there are numerous obstacles to claiming that the big bang confirms the hypothesis that God exists. And from a metaphysical point of view, God’s hand is not manifest even in big bang models: these models have no first state for God to create, and these models have no time for God to exist in before the big bang.

By pointing out some of the subtleties in the relationship between scientific cosmology and theology, we do not intend to claim that the two are nonoverlapping magisteria (to borrow a phrase from Stephen Jay Gould). To the contrary, contemporary cosmology is fascinating precisely because it has such intricate logical relations with traditional metaphysical and theological issues.

A good source for modern cosmology and its philosophical and religious contexts is Hetherington 1993. The historical interaction between cosmology and religion in the twentieth century is dealt with in Kragh 2004, and, from a different perspective, in Worthing 1996. An interesting dialogue about the (a)theistic implications of physical cosmology can be found in Craig & Smith 1995. A popular account of loop quantum cosmology is given in Bojowald 2009; more technical accounts are given in Ashtekar 2009 and Wüthrich 2006. Popular accounts of string cosmology are given in Gasperini 2008 and Veneziano 2004, 2009. For Steinhardt and Turok’s cyclic cosmology, see Steinhardt & Turok 2007. For Penrose’s conformal cyclic cosmology, see Penrose 2010.

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How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

Kragh, H., 2014. “ The True (?) Story of Hilbert’s Infinite Hotel ,” manuscript at arXiv.org.
infidels.org , has several open-access articles on related topics.
counterbalance.org , has bibliographies and discussions of these topics.
Partial bibliography on theism and physical cosmology , by Hans Halvorson.

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The Kalam Cosmological Argument

This article is the text of Dr. Craig's 2015 lecture at the University of Birmingham, where he did his doctoral studies which led to the revival of the kalam cosmological argument in our day.

As a boy I wondered at the existence of the universe. I wondered where it came from. Did it have a beginning? I remember lying in bed at night trying to think of a beginningless universe. Every event would be preceded by another event, back and back into the past, with no stopping point—or, more accurately, no starting point! An infinite past, with no beginning! My mind reeled at the prospect. It just seemed inconceivable to me. There must have been a beginning at some point, I thought, in order for everything to get started.

Little did I suspect that for centuries—millennia, really—men had grappled with the idea of an infinite past and the question of whether there was a beginning of the universe. Ancient Greek philosophers believed that matter was necessary and uncreated and therefore eternal. God may be responsible for introducing order into the cosmos, but He did not create the universe itself.

This Greek view was in contrast to even more ancient Jewish thought about the subject. Hebrew writers held that the universe has not always existed but was created by God at some point in the past. As the first verse of the Hebrew holy scriptures states: “In the beginning God created the heavens and the earth” (Genesis 1:1).

Eventually these two competing traditions began to interact. There arose within Western philosophy an ongoing debate that lasted for well over a thousand years about whether or not the universe had a beginning. This debate played itself out among Jews and Muslims as well as Christians, both Catholic and Protestant. It finally sputtered to something of an inconclusive end in the thought of the great eighteenth century German philosopher Immanuel Kant. He held, ironically, that there are rationally compelling arguments for both sides, thereby exposing the bankruptcy of reason itself!

I first became aware of this debate only after graduating from university. Wanting to come to terms with this question, I decided upon completion of my Master’s degree work in philosophy to find someone who would be willing to supervise a doctoral thesis on this question. The person who stood out above all others was Prof. John Hick at the Universty of Birmingham. We did come to Birmingham, and I did write on the cosmological argument under Prof. Hick’s direction, and eventually three books flowed out of that doctoral thesis. I was able to explore the historical roots of the argument, as well as deepen and advance the analysis of the argument. I also discovered quite amazing connections to contemporary astronomy and cosmology.

Because of its historic roots in medieval Islamic theology, I christened the argument “the kalam cosmological argument” (“ kalam ” is the Arabic word for medieval theology). Today this argument, largely forgotten since the time of Kant, is once again back at center stage. The Cambridge Companion to Atheism (2007) reports, “A count of the articles in the philosophy journals shows that more articles have been published about . . . the Kalam argument than have been published about any other . . . contemporary formulation of an argument for God’s existence. . . . theists and atheists alike ‘cannot leave [the] Kalam argument alone’” (p. 183).

What is the argument which has stirred such interest? Let’s allow one of the greatest medieval protagonists in this debate to speak for himself. Al-Ghazali was a twelfth century Muslim theologian from Persia, or modern day Iran. He was concerned that Muslim philosophers of his day were being influenced by ancient Greek philosophy to deny God’s creation of the universe. After thoroughly studying the teachings of these philosophers, Ghazali wrote a withering critique of their views entitled The Incoherence of the Philosophers. In this fascinating book, he argues that the idea of a beginningless universe is absurd. The universe must have a beginning, and since nothing begins to exist without a cause, there must be a transcendent Creator of the universe.

Ghazali formulates his argument very simply: “Every being which begins has a cause for its beginning; now the world is a being which begins; therefore, it possesses a cause for its beginning.” [1]

Ghazali’s reasoning involves three simple steps:

1. Whatever begins to exist has a cause of its beginning.

2. The universe began to exist.

3. Therefore, the universe has a cause of its beginning.

Let’s look at each step of this argument.

Notice that Ghazali does not need a premise so strong as (1) in order for his argument to succeed. The first premise can be more modestly stated.

1'. If the universe began to exist, then the universe has a cause of its beginning.

This more modest version of the first premise will enable us to avoid distractions about whether subatomic particles which are the result of quantum decay processes come into being without a cause. This alleged exception to (1) is irrelevant to (1'). For the universe comprises all contiguous spacetime reality. Therefore, for the whole universe to come into being without a cause is to come into being from nothing, which is absurd. In quantum decay events, the particles do not come into being from nothing. As Christopher Isham, Britain’s premier quantum cosmologist, cautions,

Care is needed when using the word ‘creation’ in a physical context. One familiar example is the creation of elementary particles in an accelerator. However, what occurs in this situation is the conversion of one type of matter into another, with the total amount of energy being preserved in the process. [2]

Thus, this alleged exception to (1) is not an exception to (1').

Let me give three reasons in support of premise (1'):

1. Something cannot come from nothing. To claim that something can come into being from nothing is worse than magic. When a magician pulls a rabbit out of a hat, at least you’ve got the magician, not to mention the hat! But if you deny premise (1'), you’ve got to think that the whole universe just appeared at some point in the past for no reason whatsoever. But nobody sincerely believes that things, say, a horse or an Eskimo village, can just pop into being without a cause.

2. If something can come into being from nothing, then it becomes inexplicable why just anything or everything doesn’t come into being from nothing. Think about it: why don’t bicycles and Beethoven and root beer just pop into being from nothing? Why is it only universes that can come into being from nothing? What makes nothingness so discriminatory? There can’t be anything about nothingness that favors universes, for nothingness doesn’t have any properties. Nor can anything constrain nothingness, for there isn’t anything to be constrained!

3. Common experience and scientific evidence confirm the truth of premise 1'. The science of cosmogeny is based on the assumption that there are causal conditions for the origin of the universe. So it’s hard to understand how anyone committed to modern science could deny that (1') is more plausibly true than false.

So I think that the first premise of the kalam cosmological argument is surely true.

The more controversial premise in the argument is premise 2, that the universe began to exist. This is by no means obvious. Let’s examine both philosophical arguments and scientific evidence in support of premise 2.

First Philosophical Argument

Ghazali argued that if the universe never began to exist, then there has been an infinite number of past events prior to today. But, he argued, an infinite number of things cannot exist. Ghazali recognized that a potentially infinite number of things could exist, but he denied that an actually infinite number of things could exist.

When we say that something is potentially infinite, infinity serves merely as an ideal limit which is never reached. For example, you could divide any finite distance in half, and then into fourths, and then into eighths, and then into sixteenths, and so on to infinity. The number of divisions is potentially infinite, in the sense that you could go on dividing endlessly. But you would never arrive at an “infinitieth” division. You would never have an actually infinite number of parts or divisions.

Now Ghazali has no problem with the existence of merely potential infinites, for these are just ideal limits. But he argued that if an actually infinite number of things could exist, then various absurdities would result. If we’re to avoid these absurdities, then we must deny that an actually infinite number of things exist. That implies that the number of past events cannot be actually infinite. Therefore, the universe cannot be beginningless; rather the universe began to exist.

It’s very frequently alleged that this kind of argument has been invalidated by developments in modern mathematics. In modem set theory the use of actually infinite sets is commonplace. For example, the set of the natural numbers {0, 1, 2, . . .} has an actually infinite number of members in it. The number of members in this set is not merely potentially infinite, according to modern set theory; rather the number of members is actually infinite. Many people have inferred that these developments undermine Ghazali’s argument.

But is that really the case? Modern set theory shows that if you adopt certain axioms and rules, then you can talk about actually infinite collections in a consistent way, without contradicting yourself. All this accomplishes is showing how to set up a certain universe of discourse for talking consistently about actual infinites. But it does absolutely nothing to show that such mathematical entities really exist or that an actually infinite number of things can really exist. If Ghazali is right, then this universe of discourse may be regarded as just a fictional realm, like the world of Sherlock Holmes, or something that exists only in your mind.

The way in which Ghazali brings out the real impossibility of an actually infinite number of things is by imagining what it would be like if such a collection could exist and then drawing out the absurd consequences. Let me share one of my favorite illustrations called “Hilbert’s Hotel,” the brainchild of the great German mathematician David Hilbert.

Hilbert first invites us to imagine an ordinary hotel with a finite number of rooms. Suppose, furthermore, that all the rooms are full. If a new guest shows up at the desk asking for a room, the manager says, “Sorry, all the rooms are full,” and that’s the end of the story.

But now, says Hilbert, let’s imagine a hotel with an infinite number of rooms, and let’s suppose once again that all the rooms are full. This fact must be clearly appreciated. There is not a single vacancy throughout the entire infinite hotel; every room already has a flesh-and-blood person in it. Now suppose a new guest shows up at the front desk, asking for a room. “No problem,” says the manager. He moves the person who was staying in room #1 into room #2, the person who was staying in room #2 into room #3, the person who was staying in room #3 into room #4, and so on to infinity. As a result of these room changes, room #1 now becomes vacant, and the new guest gratefully checks in. But before he arrived, all the rooms were already full!

It gets worse! Let’s now suppose, Hilbert says, that an infinity of new guests shows up at the front desk, asking for rooms. “No problem, no problem!” says the manager. He moves the person who was staying in room #1 into room #2, the person who was staying in room #2 into room #4, the person who was staying in room #3 into room #6, each time moving the person into the room number twice his own. Since any number multiplied by two is an even number, all the guests wind up in even-numbered rooms. As a result, all the odd-numbered rooms become vacant, and the infinity of new guests is easily accommodated. In fact, the manager could do this an infinite number of times and always accommodate infinitely more guests. And yet, before they arrived, all the rooms were already full!

As a student once remarked to me, Hilbert’s Hotel, if it could exist, would have to have a sign posted outside: “No Vacancy (Guests Welcome).” Can such a hotel exist in reality?

Hilbert’s Hotel is absurd. Since nothing hangs on the illustration’s involving a hotel, the argument can be generalized to show that the existence of an actually infinite number of things is absurd.

Sometimes people react to Hilbert’s Hotel by saying that these absurdities result because the concept of infinity is beyond us and we can’t understand it. But this reaction is mistaken and naïve. As I said, infinite set theory is a highly developed and well-understood branch of modern mathematics. The absurdities result because we do understand the nature of the actual infinite. Hilbert was a smart guy, and he knew well how to illustrate the bizarre consequences of the existence of an actually infinite number of things.

Really, the only thing the critic can do at this point is to just bite the bullet and say that a Hilbert’s Hotel is not absurd. Sometimes critics will try to justify this move by saying that if an actual infinite could exist, then such situations are exactly what we should expect. But this response is inadequate. Hilbert would, of course, agree that if an actual infinite could exist, the situation with his imaginary hotel is what we would expect. Otherwise, it wouldn’t be a good illustration! But the question is whether such a hotel is really possible.

So I think Ghazali’s first argument is a good one. It shows that the number of past events must be finite. Therefore, the universe must have had a beginning. We can summarize Ghazali’s argument as follows:

1. An actual infinite cannot exist.

2. An infinite temporal regress of events is an actual infinite.

3. Therefore an infinite temporal regress of events cannot exist.

Second Philosophical Argument

Ghazali has a second, independent argument for the beginning of the universe. The series of past events, Ghazali observes, has been formed by adding one event after another. The series of past events is like a sequence of dominoes falling one after another until the last domino, today, is reached. But, he argues, no series which is formed by adding one member after another can be actually infinite. For you cannot pass through an infinite number of elements one at a time.

This is easy to see in the case of trying to count to infinity. No matter how high you count, there is always an infinity of numbers left to count.

But if you can’t count to infinity, how could you count down from infinity? This would be like someone’s claiming to have counted down all the negative numbers, ending at zero: . . ., -3, -2, -1, 0. This seems crazy. For before he could count 0, he would have to count -1, and before he could count -1, he would have to count -2, and so on, back to infinity. Before any number could be counted an infinity of numbers will have to have been counted first. You just get driven back and back into the past, so that no number could ever be counted.

But then the final domino could never fall if an infinite number of dominoes had to fall first. So today could never be reached. But obviously here we are! This shows that the series of past events must be finite and have a beginning.

Ghazali sought to heighten the impossibility of forming an infinite past by giving illustrations of the absurdities that would result if it could be done. For example, suppose that for every one orbit that Saturn completes around the sun Jupiter completes two. The longer they orbit, the further Saturn falls behind. If they continue to orbit forever, they will approach a limit at which Saturn is infinitely far behind Jupiter. Of course, they will never actually arrive at this limit.

But now turn the story around: suppose Jupiter and Saturn have been orbiting the sun from eternity past. Which will have completed the most orbits? The answer is that the number of their orbits is exactly the same: infinity! (We can’t slip out of this argument by saying that infinity is not a number. In modern mathematics it is a number, the number of elements in the set {0, 1, 2, 3, . . . }.) But that seems absurd, for the longer they orbit, the greater the disparity grows. So how does the number of orbits magically become equal by making them orbit from eternity past?

Another illustration: suppose we meet someone who claims to have been counting down from eternity past and is now finishing: . . . -3, -2, -1, 0! Whew! Why, we may ask, is he just finishing his countdown today? Why didn’t he finish yesterday or the day before? After all, by then an infinite amount of time had already elapsed. So if the man were counting at a rate of one number per second, he’s already had an infinite number of seconds to finish his countdown. He should already be done! In fact, at any point in the past, he has already had infinite time and so should already have finished. But then at no point in the past can we find the man finishing his countdown, which contradicts the hypothesis that he has been counting from eternity.

Alexander Pruss and Robert Koons have recently defended an engaging contemporary version of Ghazali’s argument called the Grim Reaper Paradox. There are infinitely many Grim Reapers (whom we may identify as gods, so as to forestall any physical objections). You are alive at midnight. Grim Reaper 1 will strike you dead at 1:00 a.m. if you are still alive at that time. Grim Reaper 2 will strike you dead at 12:30 a.m. if you are still alive then. Grim Reaper 3 will strike you dead at 12:15 a.m., and so on. Such a situation seems clearly conceivable—given the possibility of an actually infinite number of things—but leads to an impossibility: you cannot survive past midnight, and yet you cannot be killed by any Grim Reaper at any time. Pruss and Koons show how to re-formulate the paradox so that the Grim Reapers are spread out over infinite time rather than over a single hour, for example, by having each Grim Reaper swing his scythe on January 1 of each past year if you have managed to live that long.

These illustrations only strengthen Ghazali’s claim that no series which is formed by adding one member after another can be actually infinite. Since the series of past events has been formed by adding one event after another, it can’t be actually infinite. It must have had a beginning. So we have a second good argument for premise 2, that the universe began to exist. We can summarize this argument as follows:

1. A collection formed by successive addition cannot be an actual infinite.

2. The temporal series of events is a collection formed by successive addition.

3. Therefore, the temporal series of events cannot be an actual infinite.

First Scientific Confirmation

One of the most astonishing developments of modern astronomy, which Ghazali would never have anticipated, is that we now have strong scientific evidence for the beginning of the universe. The first scientific confirmation of the universe’s beginning comes from the expansion of the universe.

All throughout history men have assumed that the universe as a whole was unchanging. Of course, things in the universe were moving about and changing, but the universe itself was just there, so to speak. This was also Albert Einstein’s assumption when he first began to apply his new theory of gravity, called the General Theory of Relativity, to the universe in 1917.

But Einstein found there was something terribly amiss. His equations described a universe which was either blowing up like a balloon or else collapsing in upon itself. During the 1920s the Russian mathematician Alexander Friedman and the Belgian astronomer Georges LeMaître decided to take Einstein’s equations at face value, and as a result they came up independently with models of an expanding universe. In 1929 the American astronomer Edwin Hubble, through tireless observations at Mt. Wilson Observatory, made a startling discovery which verified Friedman and LeMaître’s theory. He found that the light from distant galaxies appeared to be redder than expected. This “red shift” in the light was most plausibly due to the stretching of the light waves as the galaxies are moving away from us. Wherever Hubble trained his telescope in the night sky, he observed this same red-shift in the light from the galaxies. It appeared that we are at the center of a cosmic explosion, and all of the other galaxies are flying away from us at fantastic speeds!

Now according to the Friedman-LeMaître model, we are not really at the center of the universe. Rather an observer in any galaxy will look out and see the other galaxies moving away from him. This is because, according to the theory, it is really space itself which is expanding. The galaxies are actually at rest in space, but they recede from one another as space itself expands.

The Friedman-LeMaître model eventually came to be known as the Big Bang theory. But that name can be misleading. Thinking of the expansion of the universe as a sort of explosion could mislead us into thinking that the galaxies are moving out into a pre-existing, empty space from a central point. That would be a complete misunderstanding of the model. The theory is much more radical than that.

As you trace the expansion of the universe back in time, everything gets closer and closer together. Eventually the distance between any two points in space becomes zero. You can’t get any closer than that! So at that point you’ve reached the boundary of space and time. Space and time cannot be extended any further back than that. It is literally the beginning of space and time.

To get a picture of this we can portray our three-dimensional space as a two-dimensional disk which shrinks as you go back in time (Fig. 2).

Fig. 2. Geometrical representation of space-time. The two-dimensional disc represents our three-dimensional space. The vertical dimension represents time. As one goes back in time, space shrinks until the distance between any two points is zero. Space-time thus has the geometry of a cone. The point of the cone is the boundary of space and time.

Eventually, the distance between any two points in space becomes zero. So space-time can be represented geometrically as a cone. What’s significant about this is that while a cone can be extended indefinitely in one direction, it has a boundary point in the other direction. Because this direction represents time and the boundary point lies in the past, the model implies that past time is finite and had a beginning.

Because space-time is the arena in which all matter and energy exist, the beginning of space-time is also the beginning of all matter and energy. It’s the beginning of the universe.

Notice that there’s simply nothing prior to the initial boundary of space-time. Let’s not be misled by words. When cosmologists say, “There is nothing prior to the initial boundary,” they do not mean that there is some state of affairs prior to it, and that is a state of nothingness. That would be to treat nothing as though it were something! Rather they mean that at the boundary point, it is false that “There is something prior to this point.”

The standard Big Bang model thus predicts an absolute beginning of the universe. If this model is correct, then we have amazing scientific confirmation of the second premise of the kalam cosmological argument.

So is the model correct, or, more importantly, is it correct in predicting a beginning of the universe? Despite its empirical confimation, the standard Big Bang model will need to be modified in various ways. The model is based, as we’ve seen, on Einstein’s General Theory of Relativity. But Einstein’s theory breaks down when space is shrunk down to sub-atomic proportions. We’ll need to introduce sub-atomic physics at that point, and no one is sure how this is to be done. Moreover, the expansion of the universe is probably not constant, as in the standard model. It’s probably accelerating and may have had a brief moment of super-rapid expansion in the past.

But none of these adjustments need affect the fundamental prediction of the absolute beginning of the universe. Indeed, physicists have proposed scores of alternative models over the decades since Friedman and LeMaître’s work, and those that do not have an absolute beginning have been repeatedly shown to be unworkable. Put more positively, the only viable non-standard models have been those that involve an absolute beginning to the universe. That beginning may or may not involve a beginning point. But on theories (such as Stephen Hawking’s “no boundary” proposal) that do not have a point-like beginning, the past is still finite, not infinite. The universe has not existed forever according to such theories but came into existence, even if it didn’t do so at a sharply defined point.

In a sense, the history of twentieth century cosmology can be seen as a series of one failed attempt after another to avoid the absolute beginning predicted by the standard Big Bang model. That prediction has now stood for nearly 100 years, during a period of enormous advances in observational astronomy and creative theoretical work in astrophysics.

Meanwhile, a series of remarkable singularity theorems has increasingly tightened the loop around empirically tenable models by showing that under more and more generalized conditions, a beginning is inevitable. In 2003 Arvind Borde, Alan Guth, and Alexander Vilenkin were able to show that any universe which is, on average, in a state of cosmic expansion throughout each history cannot be infinite in the past but must have a beginning. That goes for multiverse scenarios, too. In 2012 Vilenkin showed that models which do not meet this one condition still fail for other reasons to avert the beginning of the universe. Vilenkin concluded, “None of these scenarios can actually be past eternal.” [3] “All the evidence we have says that the universe had a beginning.” [4]

The Borde-Guth-Vilenkin theorem proves that classical space-time, under a single, very general condition, cannot be extended to past infinity but must reach a boundary at some time in the finite past. Now either there was something on the other side of that boundary or not. If not, then that boundary just is the beginning of the universe. If there was something on the other side, then it will be a region described by the yet-to-be discovered theory of quantum gravity. In that case, Vilenkin says, it will be the beginning of the universe. Either way, the universe began to exist.

Of course, scientific results are always provisional. We can fully expect that new theories will be proposed, attempting to avoid the universe’s beginning. Such proposals are to be welcomed and tested. Nevertheless, it’s pretty clear which way the evidence points. Today the proponent of Ghazali’s cosmological argument stands comfortably within the scientific mainstream in holding that the universe began to exist.

Second Scientific Argument

As if this weren’t enough, there is actually a second scientific confirmation of the beginning of the universe, this one from the Second Law of Thermodynamics. According to the Second Law, unless energy is being fed into a system, that system will become increasingly disorderly.

Now already in the nineteenth century scientists realized that the Second Law implied a grim prediction for the future of the universe. Given enough time, all the energy in the universe will spread itself out evenly throughout the universe. The universe will become a featureless soup in which no life is possible. Once the universe reaches such a state, no significant further change is possible. It is a state of equilibrium . Scientists called this the “heat death” of the universe.

But this unwelcome prediction raised a further puzzle: if, given enough time, the universe will inevitably stagnate in a state of heat death, then why, if it has existed forever, is it not now in a state of heat death? If in a finite amount of time, the universe will reach equilibrium, then, given infinite past time, it should by now already be in state of equilibrium. But it’s not. We’re in a state of dis equilibrium, where energy is still available to be used and the universe has an orderly structure.

The nineteenth century German physicist Ludwig Boltzmann proposed a daring solution to this problem. Boltzmann suggested that perhaps the universe is, in fact, in a state of overall equilibrium. Nevertheless, by chance alone, there will arise more orderly pockets of disequilibrium here and there. Boltzmann refers to these isolated regions of disequilibrium as “worlds.” Our universe just happens to be one of these worlds. Eventually, in accord with the Second Law, it will revert to the overall state of equilibrium.

Contemporary physicists have universally rejected Boltzmann’s daring Many Worlds Hypothesis as an explanation of the observed disequilibrium of the universe. Its fatal flaw is that if our world is just a chance fluctuation from a state of overall equilibrium, then we ought to be observing a much smaller patch of order. Why? Because a small fluctuation from equilibrium is vastly more probable than the huge, sustained fluctuation necessary to create the universe we see, and yet a small fluctuation would be sufficient for our existence. For example, a fluctuation that formed a world no bigger than our solar system would be enough for us to be alive and would be incomprehensibly more likely to occur than a fluctuation that formed the whole universe we see!

In fact, Boltzmann’s hypothesis, if consistently carried out, would lead to a strange sort of illusionism: in all probability we really do inhabit a smaller world, and the stars and the planets we observe are just illusions, mere images on the heavens. For that sort of world is much more probable than a universe which has, in defiance of the Second Law of Thermodynamics, moved away from equilibrium for billions of years to form the universe we observe.

The discovery of the expansion of the universe in the 1920s modified the sort of heat death predicted on the basis of the Second Law, but it didn’t alter the fundamental question. Recent discoveries indicate that the cosmic expansion is actually speeding up. Because the volume of space is increasing so rapidly, the universe actually becomes farther and farther from an equilibrium state in which matter and energy are evenly distributed. But the acceleration of the universe’s expansion only hastens its demise. For now the different regions of the universe become increasingly isolated from one another in space, and each marooned region becomes dark, cold, dilute, and dead. So again, why isn’t our region in such a state if the universe has already existed for infinite time?

The obvious implication of all this is that the question is based on a false assumption, namely, that the universe has existed for infinite time. Today most physicists would say that the matter and energy were simply put into the universe as an initial condition, and the universe has been following the path plotted by the Second Law ever since its beginning a finite time ago.

Of course, attempts have been made to avoid the beginning of the universe predicted on the basis of the Second Law of Thermodynamics. But none of them has been successful. Skeptics might hold out hope that quantum gravity will serve to avert the implications of the Second Law of Thermodynamics. But in 2013, the cosmologist Aron Wall of the University of California was able to formulate a new singularity theorem which seems to close the door on that possibility. Wall shows that, given the validity of the generalized Second Law of Thermodynamics in quantum gravity, the universe must have begun to exist, unless one postulates a reversal of the arrow of time (time runs backwards!) at some point in the past, which, he rightly observes, involves a thermodynamic beginning in time which “would seem to raise the same sorts of philosophical questions that any other sort of beginning in time would.” [5] Wall reports that his results require the validity of only certain basic concepts, so that “it is reasonable to believe that the results will hold in a complete theory of quantum gravity.”

So once again the scientific evidence confirms the truth of the second premise of Ghazali’s cosmological argument.

On the basis, therefore, of both philosophical and scientific evidence, we have good grounds for believing that the universe began to exist. It therefore follows that the universe has a cause of its beginning.

What properties must this cause of the universe possess? This cause must be itself uncaused because we’ve seen that an infinite series of causes is impossible. It is therefore the Uncaused First Cause. It must transcend space and time, since it created space and time. Therefore, it must be immaterial and non-physical. It must be unimaginably powerful, since it created all matter and energy.

Finally, Ghazali argued that this Uncaused First Cause must also be a personal being. It’s the only way to explain how an eternal cause can produce an effect with a beginning like the universe.

Here’s the problem: If a cause is sufficient to produce its effect, then if the cause is there, the effect must be there, too. For example, the cause of water’s freezing is the temperature’s being below 0 degrees Celsius. If the temperature has been below 0 degrees from eternity, then any water around would be frozen from eternity. It would be impossible for the water to begin to freeze just a finite time ago. Now the cause of the universe is permanently there, since it is timeless. So why isn’t the universe permanently there as well? Why did the universe come into being only 14 billion years ago? Why isn’t it as permanent as its cause?

Ghazali maintained that the answer to this problem is that the First Cause must be a personal being endowed with freedom of the will. His creating the universe is a free act which is independent of any prior determining conditions. So his act of creating can be something spontaneous and new. Freedom of the will enables one to get an effect with a beginning from a permanent, timeless cause. Thus, we are brought not merely to a transcendent cause of the universe but to its Personal Creator.

This is admittedly hard for us to imagine. But one way to think about it is to envision God existing alone without the universe as changeless and timeless. His free act of creation is a temporal event simultaneous with the universe’s coming into being. Therefore, God enters into time when He creates the universe. God is thus timeless without the universe and in time with the universe.

Ghazali’s cosmological argument thus gives us powerful grounds for believing in the existence of a beginningless, uncaused, timeless, spaceless, changeless, immaterial, enormously powerful, Personal Creator of the universe.

Al-Gha-zalı-, Kitab al-Iqtisad fi’l-I’tiqad , cited in S. de Beaurecueil, “Gazzali et S. Thomas d’Aquin: Essai sur la preuve de l’exitence de Dieu proposée dans l’Iqtisad et sa comparaison avec les ‘voies’ Thomiste,” Bulletin de l’Institut Francais d’Archaeologie Orientale 46 (1947): 203.

Christopher Isham, “Creation of the Universe as a Quantum Process,” p. 378.

Audrey Mithani and Alexander Vilenkin, “Did the universe have a beginning?” arXiv:1204.4658v1 [hep-th] 20 Apr 2012, p. 5. For an accessible video, see see http://www.youtube.com/watch?v=NXCQelhKJ7A (accessed February 23, 2014), where Vilenkin concludes, “there are no models at this time that provide a satisfactory model for a universe without a beginning.”

A.Vilenkin, cited in “Why physicists can't avoid a creation event,” by Lisa Grossman, New Scientist (January 11, 2012).

Aron C. Wall, “The Generalized Second Law implies a Quantum Singularity Theorem,” arXiv: 1010.5513v3 [gr-qc] 24 (Jan 2013), p. 38.

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AQA A-Level Philosophy - Cosmological Arguments

Subject: Philosophy and ethics

Age range: 16+

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12 December 2022

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AQA A-Level Philosphy 25 mark essay, plan, mark scheme and 5-mark questions Essay titled ‘Do cosmological aruguments prove that God exists?’ and is marked as a band 5 essay (21-25 marks) This resource also includes a complete set of 5 mark questions and answers for Cosmological Arguments relating to the existence of ‘God’ This was written for the 2020 exams, be aware of any changes to the curriculum or marking criteria, these should form an overview of the topic but be sure to complete your own research with more relevant materials where neccessary.

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Testing the consistency of early and late cosmological parameters with BAO and CMB data

Liu, Guanlin

The recent local measurements of the Hubble constant H0, indicate a significant discrepancy of over 5σ compared to the value inferred from Planck observations of the cosmic microwave background (CMB). In this paper, we try to test the standard cosmological model ΛCDM by testing the consistency of early and late cosmological parameters in the same observed data. In practice, we simultaneously derive the early and late parameters using baryon acoustic oscillation (BAO) measurements, which provide both low and high-redshift information. CMB data are also included in the analysis as a "distance prior", which tracks the same BAO feature and resolve parameter degeneracy. By using the parameter ωm=Ωmh2, we introduce ratio(ωm), defined as the ratio of ωm which are constrained from high and low-redshift measurements respectively, to quantify the consistency between early and late parameters. We obtained a value of ratio(ωm)=1.0069±0.0070, indicating there is no tension between early parameters and late parameters in the framework of ΛCDM model. As a result, our test does not expose the defects of the ΛCDM model. In addition, we forecast the future BAO measurements of ratio(ωm), using several galaxy redshift surveys and 21 cm intensity mapping surveys, and find that these measurements can significantly improve the precision of cosmological parameters.

Astrophysics - Cosmology and Nongalactic Astrophysics

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Issue		687, July 2024


Article Number		A57
Number of page(s)		14
Section		Extragalactic astronomy
DOI
Published online		28 June 2024

1. Introduction

2. models of bbhs with circumbinary blr systems, 4. discussion, 5. conclusion, acknowledgments.

Appendix A:
List of tables
List of figures

Variations of light curves and broad emission lines for periodic QSOs from co-rotating supermassive binary black holes in elliptical orbits

Junqiang Ge 1 ,2 , Youjun Lu 2 ,1 , Changshuo Yan 1 ,2 and Jifeng Liu 1 ,2 ,3 ,4 ,5

1 National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Beijing 100101, PR China e-mail: [email protected], [email protected] 2 School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, PR China 3 New Cornerstone Science Laboratory, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, PR China 4 Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, PR China 5 WHU-NAOC Joint Center for Astronomy, Wuhan University, Wuhan, PR China

Received: 12 October 2023 Accepted: 8 April 2024

Context. Periodic quasi-stellar objects (QSOs) are considered as candidates of supermassive binary black hole (BBH) systems in galactic centers. The periodicity of their light curves can be interpreted as being due to the Doppler boosting caused by the rotation of the two black holes (BHs). Further confirmation of these candidates may require different lines of observational evidence.

Aims. Assuming the Doopler boosting scenario, in this paper we investigate the (coherent) variations of broad emission lines (BELs) and continuum light curves for active BBH systems surrounded by a circumbinary broad-line region (cBLR) and focus on their dependence on the eccentric orbital configuration.

Methods. We calculated the variation of continuum light according to the motion of BBHs on elliptical orbits, with simplified orbital orientation for demonstration, the Doppler enhanced or weakened photoionization of each BLR cloud by the central BBH sources and its variation by assuming a shifted Γ-distribution of BLR clouds for a simple BLR geometry, and finally obtain the coherent variation of the continuum and the BELs.

Results. We find that the amplitude and variation pattern of the continuum light curves and the evolution of the BEL profiles both sensitively depend on the eccentric orbital configuration of BBH systems, especially when the eccentricity is high. If only the secondary BH is active, the variation amplitudes of continuum light curves and BELs both increase with increasing BBH inclination angles and orbital eccentricities, but decrease with increasing BBH mass ratio. If both BHs are active, the asymmetry in the ionization of BLR clouds at different areas caused by the Doppler boosting effect of the secondary BH is weakened due to that of the primary BH at the opposite direction, which leads to systematically smaller variation amplitudes of both continuum light curves and BELs compared with those in the cases where only secondary the BH is activated.

Conclusions. The coherent variations of the BEL profiles with the continuum light for those periodic QSOs provide an important way to confirm the existence of BBHs in their center. Future joint analysis of the light curves and multi-epoch observed BEL profiles for periodic QSOs may lead to the identification of a number of BBH systems.

Key words: black hole physics / line: profiles / galaxies: active / quasars: emission lines / quasars: supermassive black holes

This article is published in open access under the Subscribe to Open model . Subscribe to A&A to support open access publication.

Supermassive binary black hole (BBH) systems are natural products of the hierarchical galaxy mergers in the Λ cold dark matter (ΛCDM) cosmological frame (e.g., Begelman et al. 1980 ; Yu 2002 ) since almost every massive galaxy hosts a supermassive black hole (BH) at its center (e.g., Kormendy & Richstone 1995 ; Magorrian et al. 1998 ; Kormendy & Ho 2013 ). Recently, Pulsar Timing Array (PTA) observations reported evidence of the stochastic nanohertz gravitational-wave (GW) background with detected Hellings–Downs correlation ( Hellings & Downs 1983 ) at the ∼2 − 4 σ confidence level (e.g., Agazie et al. 2023a ; Antoniadis et al. 2023 ; Reardon et al. 2023 ; Xu et al. 2023 ), which is compatible with the predicted GWs produced by the cosmic BBHs (e.g., Agazie et al. 2023b ; Chen et al. 2023 ; Ellis et al. 2024 ). Individual subparsec BBHs with periods of about a year are also the main targets of PTA experiments (e.g., Burke-Spolaor et al. 2019 ) and are expected to be detected soon (e.g., Chen et al. 2023 ). Electromagnetic observational evidence of such BBHs have also been searched for over more than a few decades. A large number of candidates were selected according to various observational signatures, but definitive evidence for subparsec BBHs is still elusive (see a recent review by D’Orazio & Charisi 2023 ).

Current BBH candidates at subparsec separations are mainly found by indirect signatures, such as double-peaked and/or asymmetric broad emission lines (BELs) (e.g., Eracleous & Halpern 1994 ; Gaskell 1996 ; Tsalmantza et al. 2011 ; Eracleous et al. 2012 ; Popović 2012 ; Decarli et al. 2013 ; Ju et al. 2013 ; Shen et al. 2013 ; Liu et al. 2014 ; Guo et al. 2019 ), periodic optical/UV light curves (e.g., D’Orazio et al. 2015 ; Graham et al. 2015 ; Charisi et al. 2016 ; Liu et al. 2019 ; Chen et al. 2024 ), or other methods (e.g., Yan et al. 2015 ; Zheng et al. 2016 ). However, each type of signature could have alternative model interpretations other than the BBH model, which means that only taking one of the above signatures makes it hard to confirm the existence of a BBH system. For example, the Keplerian rotation of a disk can also produce double-peaked BELs as observed (e.g., Chen & Halpern 1989 ; Eracleous et al. 1997 ). The periodic light curves can also be produced by damped random walk of the quasi-stellar object (QSO) flux variation among a large QSO sample (e.g., Vaughan et al. 2016 ; Liu et al. 2019 ). Direct resolving and identifying of these BBH candidates may require a spatial resolution of several microarcsec or smaller, which is not achievable by current facilities.

Current numbers of BBH candidates selected from various methods are on the order of 10 2 , which are mainly obtained by the SDSS QSO spectroscopic survey ( Tsalmantza et al. 2011 ; Eracleous et al. 2012 ; Liu et al. 2014 ), the Catalina Real-time Transient Survey (CRTS), Panoramic Survey Telescope and Rapid Response System (Pan-STARRS), Palomar Transient Factory (PTF), and Zwicky Transient Facility (ZTF) photometric surveys ( Graham et al. 2015 ; Charisi et al. 2016 ; Liu et al. 2019 ; Chen et al. 2024 ). Confirmation of these BBH candidates may require further observations and detailed theoretical modeling, in which observations include continuous photometric monitoring to verify the periodicity (e.g., Liu et al. 2018 ) and spectroscopic observations to reveal the response of BELs (e.g., Li et al. 2016 ), and detailed theoretical modeling is necessary to establish combined binary black hole and broad-line region (BBH+BLR) systems to study how the BELs vary with the co-rotating BBHs (e.g., Ji et al. 2021a , b ; Songsheng & Wang 2023 ). The combination of photometric–spectroscopic monitoring and theoretical modeling may improve the interpretation to these BBH candidates at subparsec separations (e.g., D’Orazio et al. 2015 ; Song et al. 2020 , 2021 ).

When applying theoretical models to interpret observed light curves or BEL profiles of QSOs, a simple assumption was made frequently that only the secondary BH is active and the two BHs are co-rotating on a circular orbit (e.g., D’Orazio et al. 2015 ; Song et al. 2020 , 2021 ; Ji et al. 2021a , b ; Songsheng & Wang 2023 ). However, both observations and simulations indicate that BBH systems may have high orbital eccentricities (e.g., Valtonen et al. 2021 ; Jiang et al. 2022 ; Lai & Muñoz 2023 ). Hydrodynamic simulations also suggest that the accretion mode of BBHs depends on several parameters, including the mass ratio, eccentricity, disk thickness, and kinematic viscosity of the accretion disk(s) (e.g., Miranda et al. 2017 ; Duffell et al. 2020 ; D’Orazio & Duffell 2021 ; Dittmann & Ryan 2022 ; Siwek et al. 2023a , b ). The accretion rate of the secondary BH (with mass M 2 ) tends to be substantially higher than that of the primary BH (with mass M 1 ) when q M = M 2 / M 1 is substantially less than 1, and it becomes equal to that of the primary BH when q M ∼ 1 (e.g., Farris et al. 2014 ; Kelley et al. 2019 ; Duffell et al. 2020 ; Dittmann & Ryan 2023 ).

The ongoing and planned photometric and spectroscopic surveys will find a large number of QSOs with periodic light curves and will obtain their multi-epoch spectra. These surveys include the Vera Rubin Observatory’s Legacy Survey of Space and Time (Rubin, Ivezić et al. 2019 ), WFST ( Wang et al. 2023 ), ZTF ( Bellm et al. 2019 ), DESI ( DESI Collaboration 2016 ), the Nancy Grace Roman Space Telescope ( Spergel et al. 2015 ), SDSS-V ( Kollmeier et al. 2017 ), and the Sitian project ( Liu et al. 2021 ). Combining the observed periodic light curves and the variation of the BEL profiles together could provide coherent evidence for the existence of BBHs in these QSOs. Therefore, it is of great importance to explore the dependence of the variation pattern of the periodic light curves and the BEL profiles on the orbital parameters of BBH systems.

In this paper we focus on investigating the variation pattern of periodic light curves and BELs of active BBH systems with different accretion modes and orbital eccentricities associated with a circumbinary BLR (cBLR). This paper is organized as follows. In Sect. 2 we describe the basic procedures for establishing the BBH model system. In Sect. 3 we show the main results of the light curve and BEL variations of hypothesized BBH systems with different accretion modes and eccentricities. In Sect. 4 we discuss the uncertainties and applications of the current model. The conclusions are given in Sect. 5 .

Sketch of the configuration of a BBH+cBLR system, with the BBHs and BLR clouds rotating in the same direction. For the BBHs co-rotating in elliptical orbits, the longitude of ascending node is set to be Ω = 0° for simplicity, i.e., the sky plane (yellow) and the BBH orbital plane (gray) share the same -axis in the barycentric coordinate system. The argument of periapsis is denoted as . The observer’s line of sight is set in the − plane with an inclination angle of to the BBH orbital plane. The BLR has an opening angle of and its middle plane is the same as the BBH orbital plane.

The detailed model parameters are listed in Table 1 . The mass ratios of the two BHs q M are set to be either 0.1 or 0.5, which corresponds to a remnant by a minor or major merger, respectively. According to numerical simulations, the difference of mass accretion rates Ṁ 2 and Ṁ 1 can be simply represented as a function of q M : Ṁ 2 / Ṁ 1 = 1/(0.1 + 0.9) q M ) ( Farris et al. 2014 ; Duffell et al. 2020 ). When q M = 0.1, the value of Ṁ 2 is approximately five times higher than Ṁ 1 . When q M = 0.5, Ṁ 2 is approximately two times higher than Ṁ 1 . In this work we simply assume two kinds of accretion modes for BBH systems. In the first, only the secondary BH is active (i.e., q L = 1 : 0), which corresponds to those cases that the secondary BH dominates the flux variations in the Doppler boosting scenario. In the second mode both BHs are active with q L = 1 : 1, which corresponds to those cases with a mass ratio of q M ≲ 1 where the accretion rates of the two BHs are about the same. Considering that the real observations are more complex than simulations with a simplified parameter input, the analyses of the current case study includes the setup of ( q M , q L ) = (0.1, 1 : 0) and (0.5, 1 : 1), as suggested from simulations, and the cases of ( q M , q L ) = (0.1, 1 : 1) and (0.5, 1 : 0) for comparison.

Model parameters for BBH systems co-rotating in elliptical orbits surrounded by a circumbinary BLR.

We consider that the BELs (e.g., H α and H β ) are photoionized by UV photons emitted from the accretion disk. For the BBH+cBLR system, the BBHs and BEL profiles are correlated only if the accretion disk of the active BH could radiate UV photons. Here we simply assumed that the photoionization of BLR clouds by the UV photons emitted from the accretion disk of an active BH is effective only if the radius of the Roche lobe ( R crit ) is larger than the typical radius of UV continuum emission ( r UV ) (see Appendix A for a detailed analysis). Only with R crit ≳ r UV can we study the photoionization of BLR clouds by the central BHs. As shown in Fig. A.1 , e ≲ 0.6 is a reasonable criterion for studying the Doppler boosting effect of BBH+cBLR systems with M •• ∼ 10 9 M ⊙ and T orb ∼ 2 yr. In the case study we set e = 0 and 0.5 to compare the effects of circular and elliptical orbits of BBHs to the observed continuum and BEL flux variations.

We derived the semimajor axis a BBH of BBH systems with given BBH total mass and orbital period as well as the orbital radii of the two BHs, the azimuthal and radial velocity components V ϕ and V r according to the elliptical motion of the two BHs (see Appendix B ). According to the kinematic motion of the two BHs and the accretion mode settings, we can calculate the Doppler boosted continuum light curves and also the Doppler enhanced or weakened photoionization to BLR clouds. In the simplified BLR model (Appendix C ), by assuming the BLR size of the BBH system R BLR follows the same empirical relationship with optical luminosity as the single BH case, we find that R BLR is roughly eight times larger than the separation of BBHs for those candidates selected by periodic light curves. This suggests that the BBH candidates have circumbinary BLRs. By setting a shifted Γ-distribution (e.g., Pancoast et al. 2014 ) of BLR geometry, we were able to calculate the response of BLR clouds to the continuum variation of the central ionizing source (Appendix D ) by considering the following factors: 1) position variation of the two BH components, 2) time-dependent Doppler boosting or weakening effect of the ionizing flux to each BLR cloud, 3) the line emission from each BLR cloud with reverberation response to the central ionizing source, 4) the gravitational redshift of the photons emitted from each BLR cloud, and 5) the Doppler shifts of the photons emitting from each BLR cloud to the observer. By considering all these factors, we finally derived the coherent variation of BEL profiles with the continuum.

In the Doppler boosting scenario, the amplitudes of enhanced or weakened continuum and BEL fluxes depend on many parameters, such as those listed in Table 1 . For an elliptical orbit (as shown in Fig. 1 ), by assuming the longitude of ascending node Ω = 0° for simplification, we sample the argument of periapsis ω ranging from 0° to 270° linearly with a step of 90° to show the dependence on the variation of the observed continuum light curves on it. We took six snapshots (i.e., ϕ = 0, π /3, 2 π /3, π , 4 π /3, and 5 π /3) to monitor the phase evolution of BEL profiles modulated by the orbital motion of BBHs. The observer was set to have a viewing angle of either i obs = 30° (close to face-on) or 60° (close to edge-on). The half opening angle of the BLR was set as 45° (flattened disk geometry; GRAVITY Collaboration 2018 ), with all the BLR clouds rotating in circular orbits. For the rotating direction of the two BHs and BLR clouds, all of them are assumed to be rotating counterclockwise.

3.1. Variation of continuum light curves from BBHs in elliptical orbits

For a BBH system co-rotating in an elliptical orbit surrounded by a circumbinary BLR, the continuum light curve received by the observer depends not only on M •• , T orb , q M , and q L , but also on e and ω , compared to the case of circular orbit.

3.1.1. q L = 1 : 0

In the case of q L = 1 : 0, Fig. 2 compares the continuum light curves obtained for cases with circular ( e = 0, black dashed line) and elliptical orbits ( e = 0.5 and ω = 0°–270°, colored from blue to red) with mass ratio q M = 0.1 (left panel) and 0.5 (right panel). When only the secondary BH is active, its rotational velocity increases with decreasing q M , which consequently leads to smaller variation amplitudes of the light curves for the system with q M = 0.5 than those with q M = 0.1.

Comparison of continuum light curves emitted by BBH systems when only the secondary BH is active with mass ratio = 0.1 (left panel) and 0.5 (right panel) at = 60°. The black dashed line shows the light curve from the system with circular orbit ( = 0), the other four lines colored from blue to red show those resulted from elliptical orbits ( = 0.5), with four different position angles of their major axes varying from 0° to 270°. The horizontal dotted line shows the light curve without the Doppler Boosting effect.

For a counterclockwise-rotating BBH+cBLR system, the evolution trend of the observed continuum light curve is mainly determined by the increase of decrease in the projected velocities to the observer. As shown in Fig. 1 , for the cases of ω = 0° and 180°, the tangential lines at the pericenter are parallel to the observer’s plane ( Y − Z plane for the current setup), which result in symmetric light curves at the first half and second half of each period, similar to that of the circular orbit but with different Doppler enhanced or weakened amplitudes and time durations. In addition to the counterclockwise rotation of the secondary BH, one can observe a continuum light curve with the strongest Doppler enhanced and the smallest Doppler weakened amplitudes at ω = 0°, and a light curve with the smallest Doppler enhanced and the largest Doppler weakened amplitudes at ω = 180°. For other cases (i.e., ω ≠ 0° or 180°) the observed continuum light curves are asymmetric. The shape of a continuum light curve is determined by the projected velocities of the secondary BH at the pericenter. When 0° < ω < 180°, the light curves show a variation pattern of a slow increase and rapid decrease (e.g., ω = 90° in Fig. 2 ). When 180° < ω < 360°, the light curves show a variation pattern of a rapid increase and slow decrease (e.g., ω = 270° in Fig. 2 ).

In the case of q M = 0.5 (right panel of Fig. 2 ), the rotational velocities at each phase are systematically lower than that of q M = 0.1 (left panel of Fig. 2 ), the corresponding Doppler enhanced or weakened amplitudes are hence also systematically smaller, but keeping the variation patterns the same as that of q M = 0.1.

3.1.2. q L = 1 : 1

When both the BHs are active, the amplitude of the continuum light curves depends on the luminosity ratio of the two BHs. The secondary BH has a higher rotational velocity than the primary BH, and hence contribute more to the Doppler enhanced or weakened amplitudes. As shown in the left column of Fig. 3 , for q L = 1 : 1 and q M = 0.1, the emitted continuum light curve by the secondary BH is the same as that shown in Fig. 2 , but with the assumed intrinsic flux decreased from 1 to 0.5. The amplitude of continuum light curve emitted by the primary BH is approximately ten times weaker than the secondary, and the shape of the light curve is opposite to that of the secondary BH due to the momentum conservation. With the mass ratio q M increasing to 0.5 (right column of Fig. 3 ), the Doppler boosted amplitude of the secondary BH is systematically smaller than that of q M = 0.1. On the other hand, the rotational velocity of the primary BH increases, which causes a higher amplitude of the light curve than the case of q M = 0.1. The increasing mass ratio q M indicates a decreasing (increasing) amplitude of light curves by the secondary (primary) BH, which means that at any time, the enhanced (weakened) flux modulated by the secondary BH always corresponds to the weakened (enhanced) flux by the primary BH. This means that the observed continuum light curves have smaller amplitudes than the case of q L = 1 : 0. With increasing q M , resolving the periodicity of these light curves hence requires a higher quality of photometric observations.

Comparison of continuum light curves emitted by BBH systems when both the BHs are active, with = 1 : 1, = 60°, = 0.1 (left column) and 0.5 (right column). The top panel shows the observed continuum light curve contributed by Doppler Boosting effect from the two co-rotating black holes. The Doppler boosted or weakened flux modulated by the secondary and the primary BHs are shown in the middle and bottom panels, respectively. The lines in different shapes and colors are the same as in Fig. .

With the interpretation on the behaviors of the light curve variations from BBH systems, we then investigate how the response of the circumbinary BLR varies for different cases.

3.2. The BEL profile variations for different BBH systems in circular orbits

To clarify how the elliptical orbits of BBHs can affect the response of BLR clouds, we first study the variation pattern of profile shapes and amplitudes with different mass and luminosity ratios of BBHs co-rotating in circular orbit. Figure 4 shows the profile variations in one period at six phases ( ϕ = 0, π /3, 2 π /3, π , 4 π /3, and 5 π /3) for the four models: ( q M , q L ) = (0.1, 1:0), (0.5, 1:0), (0.1, 1:1), and (0.5, 1:1). For each model, we show the flux variations relative to the mean profile obtained from the six phases in each corresponding bottom panel. In the case of BBHs co-rotating in the same direction with the BLR clouds, given an observer with i obs > 0, the blueshifted clouds (moving toward the observer) contribute higher flux variations than the redshifted BLR clouds, due to the time-delay effect, and are characterized by the relative flux variations in each bottom panel of Fig. 4 .

BEL profile variations in a single orbital period for BBHs co-rotating in circular orbits with = 60°. The two left columns show BBH systems when only the secondary BH is active, i.e., = 1 : 0, but with = 0.1 (left column) and 0.5 (middle-left column). The two right columns show profile variations when both the two BBHs are active with = 1 : 1, in the case of = 0.1 (middle-right column) and 0.5 (right column). In each column the top panel shows BEL profiles of the six phases (0 to 5 /3) in one period color-coded from blue to red (see legend), and the bottom panel presents the relative flux variation of BEL profiles compared to the mean profile obtained from that of the six phases.

When only the secondary BH is active ( q L = 1 : 0) the amplitude of the light curve for q M = 0.5 is slightly smaller than that for q M = 0.1 (Fig. 2 ). The BEL profile variations for q M = 0.1 (left column of Fig. 4 ) and 0.5 (middle-left column of Fig. 4 ) are quite similar and the flux variation caused by the Doppler boosting effect for the model of q M = 0.5 is also slightly smaller than that of q M = 0.1.

The amplitudes of the BEL profile variations decrease significantly with increasing q M for the case of q L = 1 : 1. As shown in the two right columns of Fig. 4 , the amplitude of flux variations for q M = 0.5 is substantially smaller than that for q M = 0.1. This dramatic decrease is not due to the Doppler boosting effect by the secondary BH, but is dominated by the ionizing photons emitted from the accretion disk of the primary BH. Although the profile variation caused by the secondary BH is slightly different between q M = 0.1 and 0.5, the difference of rotating velocities and hence the Doppler boosting effect of their primary BHs for the two models increases dramatically from the cases with q M = 0.1 to those with q M = 0.5 (see the bottom panels of Fig. 3 ).

3.3. BEL profile variations for different BBH systems in elliptical orbits: dependence on ω and ϕ

When the two BHs are co-rotating in elliptical orbits, the two orbital parameters e and ω would make more complex profile variations compared to those in circular orbits. For a BBH+cBLR system with fixed M •• and T orb , the velocity difference between the apocenter and pericenter increases with increasing eccentricity. Given a viewing angle i obs , the variation patterns of profile shapes and fluxes of observed BELs are mainly dominated by e and ω . Different values of ω (i.e., different projected velocities to the observer) and a different area of BLR clouds enhanced by the central BH, would make the observed BEL profiles have varied shapes at a certain phase. Instead, increasing e (i.e., rapidly increasing velocities at the pericenter; Fig. A.1 ) would increase the difference in profile shapes among different phases.

Figures 5 and 6 show the resulting BEL profiles from the Doppler boosting hypothesis of four BBH+BLR systems viewed at i obs = 60° and 30°, respectively. In these two figures the columns from left to right show the cases with ω = 0°, 90°, 180°, and 270°. In each panel phase 0 means that the counterclockwise-rotated secondary BH is located at the pericenter, and phase π corresponds to that rotated to the apocenter.

Comparison of continuum light curves emitted by BBH systems when only the secondary BH is active (top two rows) and both the two BH are active (bottom two rows) for = 0.5 and = 60°. In each row, the top panels show the profiles at six phases (0 to 5 /3) in one period, and the relative flux variations are shown in the corresponding bottom panels as that in Fig. . Columns from left to right show the cases with = 0° ,90° ,180°, and 270°, which reflects the profile variations caused by different eccentricity vectors of the elliptical orbits observed at a fixed viewing angle.

Legends are the same as those for Fig. , except that = 30°.

In Fig. 5 , when the BBH+BLR system is viewed close to edge-on ( i obs = 60°), double-peaked profiles appear for different ω and phases. As analyzed in Sects. 3.1 and 3.2 , the system with ( q M , q L ) = (0.1, 1:0) has the highest light curve variation amplitude with respect to the other three systems and the greatest BEL profile variations. The amplitudes of BEL profile variations in one period for elliptic BBH systems are different from those circular ones (Fig. 4 ). In the case of e = 0.5, the BBHs at phases from ϕ = π /2 to 3 π /2 that cross the pericenter have shorter rotating timescale and higher velocities than those from 3 π /2 to π /2. For the case when all the BLR clouds are rotating in circular orbits, given the fixed inclination angle of an observer, the BEL profile at a certain phase would not vary. However, for BBHs in elliptical orbits the varying ω can cause profile variation at a certain phase.

With the argument of periapsis ω varying from 0° to 360° (left to right columns of Fig. 5 ), BEL profiles have large variations at a given phase as the positions of each of the two BH components relative to the observer at that phase are different for cases with different ω . Co-adding with the time-delay effect, the envelope of BEL profile variations in each panel are hence different for those cases with different ω . For example, at ϕ = 2 π /3 (green line) the double-peaked profile has similar fluxes at the blue and red peaks at ω = 0°. When ω changes to 90°, the blue peak has higher flux than the red peak. At ω = 180°, the total flux of the profile increases, and the red and blue peak fluxes become similar again. While for ω = 270°, the flux of the red peak is significantly higher than that of the blue peak. Therefore, in the elliptic BBH system, two periodic signatures appear. The first is a periodic profile variation in one period (e.g., ϕ = 0 to 5 π /3 shown in each panel), which is dominated by rotation of the central BHs. The second signature is the periodic profile variation in a certain phase (e.g., ω = 0° to ∼270° at ϕ = 2 π /3), which is caused by ω . In the case when the BBH systems have no orbital precession or the timescale of the precession is hugely longer than the rotating timescale, it is only possible to see the periodic BEL profile variation in one period.

When observing the BBH+cBLR system at a viewing angle close to face-on (e.g., i obs = 30° in Fig. 6 ), the BEL profiles only present Gaussian or asymmetric shapes, and the amplitudes of flux variation decrease significantly for all four model systems compared to that observed for i obs = 60°. However, the relative flux variation trends are still similar to that shown in Fig. 5 .

3.4. Varied amplitudes of BELs for different continuum light curves and orbital eccentricities of BBHs

When only the secondary BH is active, as shown in the two left panels of Fig. 7 , at fixed e , higher q M means lower velocity of the secondary BH, and hence lower A BEL and A conti . For the case of M •• = 10 9 M ⊙ , when the eccentricity e increases from 0 to 0.6, the rotation velocity at the pericenter for the secondary BH increases, which corresponds to both increased A conti and A BEL . For A BEL , its increasing trend slows down with increasing e . The variation of A BEL is caused by two competing factors: (1) the increasing rotation velocity caused by increasing e at phases ϕ = 3 π /2 to π /2 (at the pericenter ϕ = 0) could contribute stronger Doppler enhancement to the ionization of BLR clouds; (2) the decreasing time duration at phases ϕ = 3 π /2 to π /2 would decrease the fraction of BLR clouds with stronger Doppler enhancement.

Variation of BEL amplitudes ( ) with different continuum amplitudes ( ) and orbital eccentricities ( ) of BBHs with = 10 at = 0°. The two left columns show as a function of (left panel) and (middle-left panel) for the case when only the secondary BH is active ( = 1 : 0), and the two right columns show vs. (middle-right panel) and vs. (right panel) for the case when both the BHs are active with = 1 : 1. In each panel, the points connected by lines of the same color show results of a fixed mass ratio, and the four different mass ratios = 0.1, 0.3, 0.5, and 0.7 are colored in blue, green, violet, and red, respectively. Each group of four points with different but the same are linked by gray lines.

For the case when both the BHs are active with q L = 1 : 1 (two right panels of Fig. 7 ), the response of BLR clouds to the central ionization source is affected by both the secondary and primary BHs. The ionization behavior of the secondary BH is the same as that of q L = 1 : 0, while the behavior of the primary BHs is opposite to that of the secondary BH. The differences between flux amplitudes and variation trends shown in the two left and two right panels are caused by the activity of the primary BH. The intensity ratio of the two beams depends on the luminosity ratio q L and on the mass ratio q M . The higher luminosity fraction taken by the primary BH or higher mass ratio of the two BHs indicate enhanced ionization of BLR clouds by the primary BH, which appear in the Doppler weakened regions that are modulated by the secondary BH. Therefore, the total fluxes of BELs in the case of q L = 1 : 1 increase and A BEL decreases compared to the case of q L = 1 : 0. This explains why A BEL of q L = 1 : 1 is systematically smaller than that of q L = 1 : 0.

Although the activity of the primary BH decreases both the observed A conti and A BEL (the left and middle-right panels of Fig. 7 ), the BEL profiles still have an increasing A BEL with increasing e (the right panel of Fig. 7 ) since the response of BLRs to the central ionizing sources is still dominated by the secondary BH.

For q L varying from 1 : 0 to 0 : 1, the increasing luminosity fraction of the primary BH corresponds to decreasing A conti and A BEL when the secondary BH dominates the Doppler boosting effect. Then the variation trends and phases of the continuum light curve and BEL profiles become inverse when the primary BH dominates the Doppler Boosting effect.

When changing the viewing angle i obs from 0° (face-on) to 90° (edge-on), a similar variation pattern of A conti and A BEL with e appear at different i obs . With increasing e , the value of A conti and A BEL increase. On the other hand, the values of A conti and A BEL also increase with increasing i obs for both q L = 1 : 0 and 1 : 1 cases (Fig. 8 ), which means that the periodic variations of continuum and BEL fluxes are more detectable with larger inclination angles.

Variation of continuum ( ) and BEL amplitudes ( ) with different inclination angles for BBHs with = 10 , = 0°, and = 0.0–0.6. The two left panels show the case when only the secondary BH is active ( = 1 : 0), and the two right panels show the case when both the BHs are active with = 1 : 1. In each panel, the points colored from dark blue to red represent the orbital eccentricity of BBHs = 0.0–0.6, respectively.

In this paper, as a case study, we focus on the continuum and BEL profile variations of the BBH+cBLR system by assuming Ω = 0° (i.e., cosΩ = 1, for simplification). For those observed periodic QSOs with arbitrary Ω ≠ 0°, if all other parameters are fixed, increasing Ω means decreased amplitudes of both A conti and A BEL . However, their coherent variation behaviors remain the same as those analyzed above.

We investigated the coherent variations of light curves and BEL profiles for co-rotating BBH systems with different orbital eccentricities, by assuming either that only the secondary is active or that both the BHs are active with equal luminosity. However, the observed cases should be more complex. In the following we discuss the complexity in the BBH+cBLR modeling and the observational strategy on how to identify BBH systems.

4.1. Different BLR geometries

To explain the observed BEL profiles that vary from single-peaked to asymmetric or double-peaked shapes, the inclination angle could be changed from being viewed face-on to edge-on, and the BLR may also be changed from elliptical (e.g., Shen & Loeb 2010 ) to disk-like (e.g., Eracleous & Halpern 1994 ) geometries. For the observed line asymmetry, the disk-like BLRs with clouds in elliptical orbits are preferred for modeling (e.g., Kovačević et al. 2020 ). Some complex BLR structures have also been investigated, for example BLRs in thin and thick disks with inflow or outflow substructures (e.g., Wang et al. 2018 ) or with two flattened disks perpendicular to each other (e.g., Ji et al. 2021b ). If considering co-planar BBH+cBLR systems with the two BHs co-rotating in circular orbit, the BLRs varying from spherical to thin disk indicate an increasingly sensitive response of BLR clouds to the central source (e.g., Ji et al. 2021a ).

4.2. Orbital decay of BBHs

The orbital evolution of two BHs at different semimajor axes is controlled by different decay mechanisms, for example from disk-dominated viscous evolution at large separations to the secondary-dominated viscous evolution stage (the distance of BBHs at the pericenter D peri ≲ 10 4 R Sch for q M ∼ 0.1), then the gravitational wave-dominated (GW-dominated) evolution stage ( D peri ≲ 500 R Sch ), and finally the gas-disk decoupled stage at D peri ≲ 100 R Sch , where R Sch is the Schwarzschild radius corresponding to the total mass of BBHs (e.g., Haiman et al. 2009 ).

For BBHs dominated by the GW-driven orbital decay mechanism, the GW decay timescale is

where F ( e ) is the enhancement factor caused by orbital eccentricity e :

From Eq. ( 1 ) we can derive that at M •• = 10 9 and T orb = 2 yr the GW-driven orbital decay timescales are ∼1.2 × 10 5 and ∼4.5 × 10 4 yr for q M = 0.1 and 0.5 at e = 0.5, respectively. The orbital decay does not affect the periodicity of BBHs in this work significantly.

4.3. An optimized way of identifying BBH candidates

Currently, there are two efficient ways of searching for BBH candidates: by selecting periodic light curves (e.g., D’Orazio et al. 2015 ; Graham et al. 2015 ; Charisi et al. 2016 ; Liu et al. 2019 ; Chen et al. 2024 ) and based on asymmetric or double-peaked BEL profiles (e.g., Eracleous & Halpern 1994 ; Gaskell 1996 ; Tsalmantza et al. 2011 ; Eracleous et al. 2012 ; Popović 2012 ; Decarli et al. 2013 ; Ju et al. 2013 ; Shen et al. 2013 ; Liu et al. 2014 ; Guo et al. 2019 ). However, both the dynamical modeling of BBHs (e.g., D’Orazio et al. 2015 ; Jiang et al. 2022 ) and BEL profile modeling (e.g., Shen & Loeb 2010 ; Nguyen et al. 2019 , 2020 ) are difficult to distinguish from the single-BH case, due to a set of model degeneracies (see reviews by Wang & Li 2020 ; D’Orazio & Charisi 2023 ). Limited by photometric and spectroscopic observations, current efforts on the selection and identification of BBH candidates have mainly focused on the continuum light curves or BEL profiles separately. In Ji et al. (2021a , b) , we analyze the detailed response of BLR clouds and BEL profiles to the central BBHs that are co-rotating in circular orbits by focusing on the cases where the BBH orbital plane is co-planar ( Ji et al. 2021a ) or misaligned ( Ji et al. 2021b ) with the middle plane of the circumbinary BLR. In this paper we investigate the coherent variation of BBH continuum light curves and BEL profiles for BBHs orbiting at different eccentricities with simplified orbital orientation, which indicates that the joint analyses of periodic light curves and multi-epoch observed BELs could lead to successful identification of BBH candidates.

As analyzed in this work, the amplitudes of periodic continuum light curves and BELs are correlated with each other (see Figs. 7 and 8 ). Because of the momentum conservation, the active primary BH could cause non-sinosoidal continuum light curves and decreased A conti (Figs. 2 and 3 ). The BEL profiles at different phases in one period not only have significant differences in shapes and fluxes (Figs. 4 – 6 ) caused by orbital eccentricity and argument of periapsis of BBHs at certain ( q M , q L ), but also have a decrease in flux variation amplitudes with increasing mass ratio and luminosity contribution by the primary BH (Figs. 7 and 8 ). When taking into account the inclination angle, the amplitudes of continuum light curves and BEL profiles both increase with increasing inclination angles. This means that modeling the continuum light curves and multi-epoch observed BEL profiles together would improve the probability of identifying BBH candidates, and the BBH scenario can be confirmed if a clear coherent variation of the BELs with the continuum can be found.

On the other hand, the ongoing (e.g., WFST, Wang et al. 2023 ) and future photometric surveys (e.g., the Rubin and Sitian project, Ivezić et al. 2019 ; Liu et al. 2021 ) would find more BBH+cBLR systems occupied at much wider parameter space compared with those currently obtained by CRTS, Pan-STARRS, PTF, and ZTF projects (e.g., Graham et al. 2015 ; Charisi et al. 2016 ; Liu et al. 2019 ; Chen et al. 2024 ). There will be more spectroscopic data obtained from DESI ( DESI Collaboration 2016 ), the Nancy Grace Roman Space Telescope ( Spergel et al. 2015 ), and SDSS-V ( Kollmeier et al. 2017 ) surveys. This makes the simultaneous modeling of periodic light curves and BEL profiles being possible for many BBH candidates.

In this paper we investigated the response of the line emission from BLR clouds to the continuum emission from central BBH systems co-rotating in elliptical orbits by focusing on the evolution trends of both observed light curves and BEL profiles. For QSOs with the central BBHs surrounded by a circumbinary BLR (BBH+cBLR) system, we investigated two cases of BH activity, one where only the secondary BH is active ( q L = 1 : 0), and the second where two BHs are active with equal luminosity ( q L = 1 : 1), with simplified orbital orientation (i.e., Ω = 0°).

For BBHs co-rotating in elliptical instead of circular orbits, the Doppler boosting effect caused by their orbital eccentricity and argument of periapsis could affect both the shape of continuum light curve, for example sharply (slowly) increasing but slowly (sharply) decreasing trends in a period, and two kinds of periodicities in BEL profile variations in the Doppler boosting hypothesis. On the other hand, BBHs in different active behaviors can result in quite different amplitudes of light curves and BEL profiles. At q L = 1 : 0 the variation amplitudes of continuum light curves and BELs both increase with increasing eccentricity, but decrease with increasing mass ratio of BBHs. Instead, for q L = 1 : 1, the activity of the primary BH can cause systematically decreased amplitudes of continuum light curves and BELs, due to its enhanced ionization to cBLR clouds in the Doppler weakened area that are modulated by the secondary BH. Currently, we only consider BHs that accrete with stable accretion rate; in the future, we will also explore how BBH systems vary with different fluctuations in accretion rates, such as the damped random walk variation.

The confirmation of BBH candidates becomes more complex when considering the elliptically co-rotated orbits and accretion activities of BBHs rather than those cases with simply assumed circular orbits or single BH activity. With the theoretical understanding on BBH systems at varied mass ratios, luminosity ratios, orbital eccentricities, and BH accretion activities, the incoming huge amount of photometric observations (e.g., by Rubin, Sitian, and WFST) and multi-spectroscopic observations (e.g., by DESI, Roman space telescope, and SDSS-V) would search for BBH candidates with more complete parameter spaces than mentioned in this work, based on which we could constrain the merger rate of massive black hole binaries.

We would like to thank the anonymous referee for the suggestions that helped us to improve this paper. This work is supported by the National Key program for Science and Technology Research and Development (grant Nos. 2020YFC2201400 and 2022YFC2205201), the Beijing Municipal Natural Science Foundation (No. 1242032), the National Natural Science Foundation of China (NSFC) (grant Nos. 11903046, 12273050, 11991052, 11988101, and 11933004), and the Strategic Priority Program of the Chinese Academy of Sciences (Grant No. XDB23040100). JG acknowledges support from the Youth Innovation Promotion Association of the Chinese Academy of Sciences (No. 2022056) and the science research grants from the China Manned Space Project, and JFL acknowledges support from the New Cornerstone Science Foundation through the New Cornerstone Investigator Program and the XPLORER PRIZE.

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Appendix A: Stability of the accretion disks for BBH systems

For the active BH in the BBH system, we simply assume that the BH accretion disk is steady inside the Roche lobe. When studying the photoionization process of UV photons that radiated from the accretion disk to BLR clouds, the minimum radius of the accretion disk should make sure the quantity of radiated UV photons is enough for ionizing BLR clouds and result in matchable continuum and BEL fluxes, as observed from those single BH cases; in other words, the radius of Roche lobe R crit is equal to or larger than the half light radius of the accretion disk at UV band ( r UV ). Only with R crit ≳ r UV could the BEL profiles be observed and the dynamics of the BBH system encoded. For BBHs systems ( Eggleton 1983 ), the radius of the Roche lobe for the BH could be appoximated within 1% of its value with the formula

Radii of Roche lobes of BBHs and rotating velocity of the secondary BH at the orbital pericenter as a function of orbital eccentricity of the two BHs. The left and middle panels show the radii of Roche lobes ( and ) in units of the Schwarzschild radii for the primary ( ) and secondary ( ) BHs, respectively. The typical radius of the UV emission ( ) in the accretion disk for a = 10 is shown as a horizontal dashed line for comparison. The right panel presents the rotating velocity of the secondary BH at the pericenter ( ), which is the maximum velocity in each period. In each panel are shown the results of = 10 (blue), 10 (green), and 10 (red) with = 2 yr and = 0.5. Considering that the UV emission might disappear when ≲ , the current model configuration only focuses on those systems with ≳ , i.e., ≲ 0.6 for = 10 as a case study.

Appendix B: Dynamics of the BBH system

Assuming a BBH system with an orbital period of T orb = 2 yr, which is roughly the median intrinsic period of BBH candidates in the sample of Charisi et al. (2016) , the semimajor axis of the two black holes with total mass M •• = 10 9 M ⊙ is

In the case of elliptical orbits (e.g., Valtonen et al. 2008 ; Charisi et al. 2022 ), the velocity of the secondary BH depends on its location at different position angle. Given the secondary BH rotating in elliptical orbit with eccentricity e and mass ratio q M in the X-Y plane (Figure 1 ), the radius of the secondary BH is

and the velocity components V ϕ and V r of the secondary BH are

where G is the acceleration of gravity and ϕ is the true anomaly of the secondary BH. The position and velocity of the primary BH can then be derived by momentum conservation.

Appendix C: The simplified BLR model

For a BBH system with a cBLR (BBH+cBLR), we simply assume its BLR size has the same relation with the luminosity of the central ionization source as that of single-BH systems ( Kaspi et al. 2000 , 2007 ; McLure & Jarvis 2002 ; Peterson et al. 2004 ), although it could be different. The BLR size of the BBH system can then be derived by following the empirical relationship between the BLR size and optical luminosity (e.g., Lu et al. 2016 ). For convenience of model construction, we convert the optical luminosity to the Eddington ratio ( λ Edd ) and M •• by assuming λ Edd = 0.1 for the BBHs:

The typical BLR size is approximately eight times larger than the separation of the BBHs, significantly far away from the central two BHs, and can be taken as a cBLR affected little by the central BBH dynamics. Therefore, in this work, we consider the configuration of BBHs surrounded by a cBLR, for which we take a shifted Γ-distribution (see details in Pancoast et al. 2014 ) as the radial emissivity distribution of BLR clouds:

Here R S is the Schwarzschild radius, R BLR is the mean BLR radius, F = R min / R BLR is the fractional inner BLR radius ( R min ), β is the shape parameter, and g = p ( x |1/ β 2 , 1) is drawn randomly from a Gamma distribution.

As to the angular distribution of the BLR clouds, for a BLR with an half opening angle θ o (Figure 1 ), we assume

where θ represents the angle between the orbital angular momentum of a cloud and the normal to the BLR middle plane, and U is a random number ranging from 0 to 1 and is set to describe the randomly distributed clouds, but constrained with θ ≤ θ o (see Pancoast et al. 2014 , for details).

Appendix D: Response of BLR clouds to the central source

For the response of BLR clouds to the central BBH systems co-rotating in elliptical orbits, the equations are similar to that with BBHs co-rotating in circular orbits as introduced in our previous work ( Ji et al. 2021a ), except for differential rotating time in each orbital phase and the variation of projected major axis of the elliptical orbit.

For the continuum emission dominated by the active secondary black hole, the time in the observer’s frame ( t obs ) is determined by the intrinsic time ( t in ) in the BBH mass center rest frame (hereafter BBH frame) and the cosmological time dilation:

The above equations are the same for the BBH systems co-rotating both in circular and elliptical orbits.

For the modeling of the BBH+cBLR system under the Doppler boosting scenario, we assume the accretion rates are constant (e.g., D’Orazio et al. 2015 ; Duffell et al. 2020 ), whenever only the secondary BH is active or when both BHs are active. Five factors are taken into account to derive the finally observed BEL profiles (see details in Ji et al. 2021a , b ):

the position variation of the two BH components;

the time-dependent Doppler boosting effect on the ionizing flux emitted from the primary or secondary BH accretion disks and recieved by the BLR clouds;

the reverberation process of the BLR clouds to the central source(s);

the gravitational redshift of the emission from BLR clouds ( Tremaine et al. 2014 );

the Doppler blue- and redshifts of photons from the BLR clouds to the observer.

All Figures

	Legends are the same as those for Fig. , except that = 30°.

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The Cosmological argument

Cosmological arguments from causation

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Aquinas’ 2 nd way (atemporal causation)

The Kalam cosmological argument from temporal causation

Hume’s objection to the ‘causal principle’

Cosmological arguments from contingency

Leibniz’ principle of sufficient reason

The fallacy of composition

Copleston vs Hume & Russell on the universe as a brute fact

Issues around the possibility of a ‘necessary being’

Issues around the possibility of an infinite series

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Testing the consistency of early and late cosmological parameters with BAO and CMB data

1. Introduction

Variations of light curves and broad emission lines for periodic QSOs from co-rotating supermassive binary black holes in elliptical orbits

3.1. Variation of continuum light curves from BBHs in elliptical orbits

3.1.1. q L = 1 : 0

3.1.2. q L = 1 : 1

3.2. The BEL profile variations for different BBH systems in circular orbits

3.3. BEL profile variations for different BBH systems in elliptical orbits: dependence on ω and ϕ

3.4. Varied amplitudes of BELs for different continuum light curves and orbital eccentricities of BBHs

4.1. Different BLR geometries

4.2. Orbital decay of BBHs

4.3. An optimized way of identifying BBH candidates

Appendix A: Stability of the accretion disks for BBH systems

Appendix B: Dynamics of the BBH system

Appendix C: The simplified BLR model

Appendix D: Response of BLR clouds to the central source

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