The Cosmic Universe

This page considers some of the key issues in cosmology, the science that attempts to grasp the biggest picture of all: how did the universe start, where is it going, and was there anything before?

The Origin of the Universe

In 1970, Stephen Hawking and Roger Penrose showed that general relativity predicted that the universe would have started with a singularity, a single point with infinite density and infinite curvature of spacetime (see the Penrose-Hawking singularity theorems). This might sound like an elegant and compact solution to how the universe started. However, this poses a problem for physics as our current models break down at infinite densities (this is nicely explained here). For example, if we attempt to calculate the initial density of the universe:

In this case we have complete freedom to choose any initial conditions for the universe. So the situation can be summed-up as: anything can happen at a singularity. Again, this might appear to be an elegant, compact solution to how the universe started. But if the universe really did begin with a singularity that would render the situation beyond mathematical analysis (fortunately, we don't generally find infinities in physical reality, and the presence of infinities in a theory is usually taken as a sign that the theory is flawed). The result has been a succession of imaginative ideas by which the singularity at the start of the universe might be avoided.

In 1973, Edward Tryon proposed the theory that the universe (as a small point - a quantum particle) might have emerged out of nothing (ex nihilo creation) due to a quantum fluctuation. According to the Heisenberg uncertainty principle, there can be enough uncertainty about the energy at a point, , to allow the creation of a particle/antiparticle pair, but only for a very short time, :

In 1982, Alexander Vilenkin extended Tryon's proposal to consider the idea that the universe could have inflated from one of these "virtual particles" (see here). Andre Linde realised that a succession of further quantum fluctuations within the expanding universe would result in areas of secondary inflation. This process can continue indefinitely in a process known as eternal inflation. In an eternally inflating universe, "bubble" universes can bubble-up out of an eternally inflating space like bubbles in a bottle of champagne (see Andrei Linde's paper Inflation, Quantum Cosmology and the Anthropic Principle).

Inflation is now generally accepted as describing conditions in the earliest moments after the origin of the universe. But inflationary theories should not be considered as providing the "whole story" of the origin of the universe: inflationary theories do not seem to avoid the initial singularity, as Stephen Hawking point out on page 52 of his paper The Nature of Space and Time: "Inflation alone can not explain the present state of the universe. One can see this by taking any state for the universe now and running it back in time. Providing it contains enough matter, the singularity theorems will imply that there was a singularity in the past." (see also Neil Turok's article Inflation and the Beginning of the Universe).

Causal Loop Theories

Let's imagine what sort of objects could appear out of nothingness. It would appear that only those objects which could create themselves could come into existence. But what sort of objects can create themselves? And, most importantly, could our very universe have created itself?

Theories which seem most successful at avoiding an initial singularity seem to those which involve some sort of looping in time, or "curving" of spacetime. These theories seem to have the advantage of theories involving linear time which inevitably lead back to an initial singularity. Also, some of these "curvy" theories seem to propose a universe which can, indeed, "create itself", thus providing an all-important motive for our existence.

John Wheeler has proposed a model for a participatory universe which would appear to belong to this category of "universes which could create themselves". The principle of the participatory universe is summed-up by Martin Rees: "The universe could only come into existence if someone observed it. It does not matter that the observers turned up several billion years later. The universe exists because we are aware of it." In this sense, the participatory universe borrows strongly from the quantum mechanics interpretation that "observation creates reality" - by observing the universe, we somehow bring it into existence. However, I (in common with the majority of physicists) do not subscribe to the controversial "conscious observation creates reality" interpretation of quantum mechanics (see the page on Quantum Reality for my reasoning).

What other possibilities are on the table? I prefer examples which create themselves through some sort of causal loop. For example, in the film The Terminator a robotic killing machine (Arnold Schwarzenegger) is sent back in time to kill a rebel's mother before the rebel is born. The terminator is defeated, but some of his advanced technological body parts are retained, and it is the technological advances which result from examining those body parts which allow future scientists to construct the Terminator. So nobody actually invented the technology used to construct the Terminator! (I'm convinced this is the technique Apple used to develop the amazing new touch-screen iPods!). An object such as this which is "invented out of nothing" by a causal loop is called a jinn.

As strange as this may sound, there is no logical inconsistency here. There is nothing in physics to say this is not allowed: nothing both happens and doesn't happen. And if the technique can produce something as complex as the Terminator from nothing then maybe it could also produce a universe from nothing.

J. Richard Gott has suggested a method for a universe to create itself if it formed a causal loop in spacetime (see here). His principle imagines an extreme curvature of spacetime forming a wormhole, allowing time travel to the past. The diagram below shows the loop in time:

There is another possible causal loop solution which would also explain anthropic fine-tuning. Maybe an advanced (human?) civilisation in the future could travel back in time to create the universe which they knew would evolve themselves into existence. Such a process would explain fine-tuning because they would have to set the parameters very precisely to ensure their particular civilisation and its inhabitants would result.

The civilisation would also have to produce spacetime itself: a flexible spacetime which could loop round to allow them to travel back in time. This idea is considered by J. Richard Gott on page 191 of his book Time Travel in Einstein's Universe: "With a time-travel loop, an intelligent civilisation could produce the trunk as well".

The "No Boundary" Proposal

Another "curvy time" solution is the "No Boundary" proposal of Stephen Hawking and James Hartle.

As we trace the universe back in time to the singularity we not only find our laws of physics breaking down, but we are also left with the apparently unanswerable question of "what happened before the Big Bang?"

In 1981, Stephen Hawking and James Hartle came up with an imaginative proposal which promised to avoid the singularity at the origin of the universe, and also gave a answer to the question of "what happened before the Big Bang?". But before we can consider the theory, we need to introduce a couple of concepts.

Firstly, we need to introduce the idea of a metric, which is a way of defining distance. In our (x, y, z) three-dimensional space, the formula for distance is provided by Pythagoras's theorem:

When we extend this notion to 4-dimensional spacetime, it might be imagined that the time axis is treated the same way (creating the Euclidean or Riemannian metric):

However, Einstein's theory of special relativity says that the clock which travels the furthest actually shows the smallest time measurement, not the largest. So we need to use the Lorentzian metric (the "time" element becomes negative):

Hartle and Hawking's proposal was to employ a mathematical transformation called a Wick rotation to modify the time axis to avoid the singularity. In the Wick rotation, the time axis is multiplied by the imaginary number i (the square root of minus one), in which case the Lorentzian metric is converted to a Euclidean metric:

As a result of the Wick rotation, the time axis is converted to a complex number. The time axis is rotated 90° anticlockwise from the original time axis to become the imaginary time axis:

The following diagram shows the resultant graph rotated by a further 90° (but this time clockwise) so that the imaginary time axis now points in the vertical direction - taking the place of the old time axis:

It is as though when we travel back in time we find time itself curving round so that spacetime forms a smooth surface, instead of coming to a point singularity:

So in the No Boundary proposal, there is no time before the Big Bang: time itself began with the Big Bang. Asking what came before the Big Bang is - in Hawking's words - like asking what lies north of the North Pole. The answer is nothing. And the question "what happened before the Big Bang?" is meaningless.

This idea that there is no time before the Big Bang appears undoubtedly correct: "time" as we understand it is wholly contained within the universe, so time had to be created when the universe was created. We will return to this idea of "no time outside the universe" later on this page.

But can the No Boundary proposal answer the question: "How did the universe appear out of nothing?". Well, it would say that the question is flawed. According to the No Boundary proposal, there is no time before the start of the universe, so nothing could have created the universe (causes would occur at earlier times, but no cause could have occurred before the start of the universe). So according to Hawking, nothing created the universe: it just is. It just "exists", essentially for no reason. Here's a quote from Hawking's book "A Brief History of Time": "So long as the universe had a beginning, we could suppose it had a creator. But if the universe is really completely self-contained, having no boundary or edge, it would have neither beginning nor end: it would simply be. What place, then, for a creator?"

However, this "Nothing created the universe; the universe just is" approach of Hawking is effectively the same argument as someone saying "Nothing created God; God just exists". Hawking is proposing the unquestioned acceptance of the existence of the universe in just the same manner as the 13th century theologian Thomas Aquinas posited the existence of God as his First Cause. Both approaches seem to require a fundamental "bottom layer" whose existence must be assumed but is beyond deeper analysis. I don't see how the No Boundary proposal gets us anywhere new in that respect.

So, for me, the No Boundary proposal doesn't answer the question "Why?": "Why should the universe exist at all instead of not existing?". I feel this rather passive universe, lacking any form of motive its existence means the No Boundary proposal is not as compelling as deeper theories in which the universe is an object capable of creating itself (such as the causal loop theories presented above). In common with the No Boundary proposal, the universe is a self-contained object in causal loop theories. However, causal loop theories allow deeper analysis and provide an answer to the question "Why does the universe exist instead of not existing?".

For more details on the No Boundary proposal, see chapter 3 of Stephen Hawking's paper The Nature of Space and Time. There is also a series of video lectures on this subject by Stephen Hawking on YouTube:

• Origin of the Universe - Stephen Hawking (1 of 5)
• Origin of the Universe - Stephen Hawking (2 of 5)
• Origin of the Universe - Stephen Hawking (3 of 5)
• Origin of the Universe - Stephen Hawking (4 of 5)
• Origin of the Universe - Stephen Hawking (5 of 5)

Quantum Gravity: The Wheeler-DeWitt Equation

Quantum cosmology is the attempt to find a fully quantum mechanical description of the large-scale universe and to explain the origin of the universe. In order to study the universe in this way, we need to examine spacetime - the underlying substrate which seems to compose the universe. Because gravity is treated as a curvature of spacetime itself (in Einstein's theory of general relativity), most quantum cosmology theories are based around trying to find a quantum theory of gravity: quantum gravity.

A great puzzle in physics has been how to reconcile Einstein's theory of general relativity with quantum mechanics. General relativity remains our main theory for describing gravity, and is extremely accurate for with large objects (stars and planets, etc.). Quantum mechanics, on the other hand, is our main theory for dealing with microscopic objects, and the other three fundamental forces which act at the atomic scale (see the It's a Small World page for a description of those other three fundamental forces). General relativity describes space as being a smooth surface, but quantum mechanics reveals a discontinuous microscopic world with constant fluctuations and activity. So, each of these theories is accurate in its own right but they describe the nature of space and matter so differently that it has proven highly problematic to combine the theories into a single unified theory.

But perhaps there is an even more fundamental difference between the two theories. Quantum mechanics attempts to describe the behaviour of particles as they move within a universe composed of spacetime. However, a theory of quantum gravity would seek to describe the behaviour of that background spacetime itself. They seem to be two different entities, and we, as yet, have no evidence that it is valid to apply the principles of quantization to spacetime. This question will be considered later on this page.

Before we consider quantum gravity, we need to introduce the concept of the Lagrangian. The Lagrangian is a reformulation of classical mechanics, i.e., it is a different approach to describing the motion of a system. The Lagrangian allows us to describe the complete motion of a system - however complex, however many objects it contains - by a single equation: the Lagrangian. The Lagrangian is defined as the kinetic energy of a system minus its potential energy. The Lagrangian is defined for all possible configurations of all elements of the system, so it is possible to plot the Lagrangian as a surface in space (more accurately, an N-dimensional manifold):

It can be shown that the state of the system as it moves from point A to point B along the Lagrangian follows the path of least action (in other words, the shortest distance between the two points - see here). This essentially means that nature wants to convert potential energy to kinetic energy (or vice versa) by the most efficient means possible with respect to time. In the diagram above, it can be seen that the state of the system moves along the base of the trough between the start and end points - the shortest distance between the two points. The Lagrangian is such a powerful tool as once you have managed to find the Lagrangian for a system you know precisely how it will behave: it will follow the path of least action. The Lagrangian alone is enough to describe the system. For this reason, in modern fundamental physics, when some new theory is proposed, it is almost invariably given in the form of a Lagrangian ("For reasons as yet utterly mysterious, this quantity stays as small as possible under all circumstances. Theorists are convinced that action must be incredibly important - so much so that the discovery of any new fundamental law prompts a race to work out the particular action needed to produce it. The trouble is that no one understands the principles behind nature's infatuation with action." - quoted from this New Scientist article).

The Hamiltonian is closely related to the Lagrangian, but instead of the difference between the kinetic and potential energy of a system, we now consider the sum of the kinetic and potential energy. To be precise, the Hamiltonian is the sum of the kinetic and potential energy of a closed system expressed in terms of momentum, position, and time (see here).

Now let's see how the Hamiltonian was used to produce the very first results of quantum mechanics. Consider the case of a single particle of mass m moving in some external field given by a potential energy function, V, which can depend on position: V=V(x,y,z).

The (classical) Hamiltonian is the sum of kinetic plus potential energy. Let's derive it:

where p_x, p_y, and p_z are the spatial momenta in the direction of the Cartesian x, y, and z axes.

Now, in order to convert this Hamiltonian to its quantum mechanical form, the momentum variable in this equation is replaced by the result for quantum momentum (which we derived earlier on the Quantum Casino page):

As this is a momentum operator, we need something for it to operate on. So we have to again introduce this strange concept of a wavefunction, , extending through space. Our equation for the Hamiltonian operator (there is now a circumflex over the H) now becomes:

The Hamiltonian, H, is really the total energy, E, so the Hamiltonian operator is the energy operator. But we already derived an expression for the energy operator on the Quantum Casino page:

So substituting for this value, our equation now becomes:

which is the famous Schrödinger equation! This Hamiltonian approach is how Erwin Schrödinger first derived the equation in his 1926 paper "Quantization as an Eigenvalue Problem" (original German paper here).

In 1965, John Wheeler and Bryce DeWitt wondered what would happen if you applied this "standard" method of quantization (called canonical quantization) to the force of gravity. Let's see what happens:

The first principle to note is that of general covariance (sometimes called diffeomorphism invariance). According to the principle of general covariance, a theory must give the same results not matter how you move its coordinate system around in the universe (general covariance arose from the requirements of general relativity). To put it another way, general covariance says that no set of coordinates is special: there is no absolute coordinate system, everything is relative. It can be seen that general covariance inevitably arises from another very simple principle: there is nothing outside the universe, or, to put it another way, there are no absolute axes of reference for space or time outside the universe by which we can make our measurements (this principle - that there is nothing outside the universe - is described as the "first principle of cosmology" by Lee Smolin in his book Three Roads to Quantum Gravity).

Hence, it is impossible to define a position for our "universe object" (and hence no potential energy) and it is impossible to define a speed (and hence no kinetic energy). The Hamiltonian allows a plot of the variation in total energy of a system as the components of a system take different configurations (e.g., galaxies moving around in our "universe object"). So in this case, the Hamiltonian (kinetic + potential energy) is zero (see this New Scientist article):

(You may find it hard to accept this idea that the total energy of the universe is zero. This is only possible if we consider gravity to provide "negative energy". Stephen Hawking explains it well in this extract from his book "A Brief History of Time": "The total energy of the universe is exactly zero. The matter in the universe is made out of positive energy. However, the matter is all attracting itself by gravity. Two pieces of matter that are close to each other have less energy than the same two pieces a long way apart, because you have to expend energy to separate them against the gravitational force that is pulling them together. Thus in a sense, the gravitational field has negative energy. In the case of a universe that is approximately uniform in space, one can show that this negative gravitational energy exactly cancels the positive energy represented by the matter. So the total energy of the universe is zero.")

If we are going to quantize gravity using the canonical quantization method, we need to convert this Hamiltonian to an operator (as described in the derivation of the Schrödinger equation above). But if we're converting the Hamiltonian to an operator, we once again need something for it to operate on: we need a wavefunction. So another new concept must now be introduced: the wavefunction of the universe. The principle of the "wavefunction of the universe" imagines the entire universe as a single object, a quantum object. Michio Kaku explains it well: "When the universe was born, it was smaller than an electron, which is a quantum object that can exist simultaneously in many states. So the universe must also be a quantum object and exist in many states." (see here). Whether it is valid to consider the entire universe as a single object subject to the laws of quantum mechanics we shall consider in a later section on this page, but for now we can apply our Hamiltonian operator to our "wavefunction of the universe":

This is the Wheeler-DeWitt equation - a sort of Schrödinger equation for the gravitational field. It is the most famous equation in quantum gravity.

(A variation on this canonical quantization of gravity eventually leads to the recent, cutting-edge theory of loop quantum gravity - see this Physics World article by Carlo Rovelli, or Lee Smolin's Scientific American article Atoms of Space and Time).

Time and the Wheeler-DeWitt Equation

There's something remarkable about the Wheeler-DeWitt equation, and it can be seen if we expand the Hamiltonian operator:

Or, expressed in words, the rate of change of the state of the universe with respect to time is zero. The universe isn't changing with time! But we look around us and we see things changing all the time: people are walking, birds are flying. So is the equation wrong? Well, no. What the equation is once again telling us is that there is no external time reference by which we can measure the progress of time within the universe: there is no clock outside the universe! As Andrei Linde explains: "The notion of evolution is not applicable to the universe as a whole since there is no external observer with respect to the universe, and there is no external clock that does not belong to the universe" (see page 25 of Andre Linde's paper Inflation, Quantum Cosmology and the Anthropic Principle).

The Wheeler-DeWitt equation suggests a model in which all of time is laid-out (just as the space dimension is laid-out), and all times are equally real: there is no special "now", no distinction between past and future. In fact, "past" and "present" do not exist - the movement of time is considered to be just an illusion of human perception (the Wheeler-DeWitt equation reveals how the universe does not change with time as there is no external time reference by which we can measure the progress of time within the universe: there is no clock outside the universe).

Most physicists would favour this model as the most accurate representation of time. It is also called block time because all of spacetime can be viewed as being laid-out as an unchanging four-dimensional block:

For more information about this, see this Scientific American article by Paul Davies.

Eternal Life

It might come as a surprise that this orthodox "block universe" view of time in fact leads us to conclude that we possess a form of eternal life! This is a consequence of the principle that in the block time model all periods of time are equally real. If a loved one dies, you might take some comfort from the knowledge that this period of time in which your loved one is dead has, in fact, no greater reality than the time when your loved one was alive. According to physics, it is just as valid to consider your loved one as alive as it is to consider them dead!

Einstein took comfort from this knowledge when his lifelong friend Michele Besso died. He wrote a letter consoling Besso's family: "Now he has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present, and future is only a stubbornly persistent illusion."

Of course, the flip-side is that you're already dead!

The Wavefunction of the Universe

In our discussion on quantum gravity we have introduced the concept of a "wavefunction of the universe". The idea is that the universe has a corresponding wavefunction which describes the probabilities of it having different states (forms). This is meant to be analogous to the well-established formulation in quantum mechanics whereby a quantum particle has a corresponding wavefunction which describes the probabilities of it appearing in a certain location, say. Michio Kaku explains it well: "When the universe was born, it was smaller than an electron, which is a quantum object that can exist simultaneously in many states. So the universe must also be a quantum object and exist in many states." (see here). The idea of a wavefunction for the universe was first proposed in the Hartle and Hawking "No Boundary" proposal, though it has since been used in many models of quantum cosmology.

However, it is possible to imagine several objections to this belief that the entire universe can be treated as if it was a quantum particle. In his online physics textbook Motion Mountain, Christoph Schiller draws attention to the many differences between the universe and a particle. He comes to the conclusion that there can be no such thing as a wavefunction of the universe: "Beware of anybody who claims to know something about the wavefunction of the universe. Just ask him: If you know the wavefunction of the universe, why aren't you rich?" (see page 835 of Motion Mountain).

Schiller's first complaint is that the universe cannot have a "state" because a "state" is defined as being the form of a system at a given time. However, that would appear to require some aspect of the universe to be able to change with time when the universe is viewed "from outside", i.e., the universe could appear to change when viewed by some form of external observer. However, as we discovered earlier in our discussion on quantum gravity, there is nothing outside the universe, certainly no external clock by which changes in the universe might be measured (this unchanging universe was revealed by the Wheeler-DeWitt equation in the previous section).

This idea of a state as being the form of a system at a particular moment in time is even used (perhaps unwittingly) by Stephen Hawking in a description of the wavefunction of the universe on page 43 of his paper The Nature of Space and Time: "It is useful to introduce a concept that can describe the state of the universe at one time" (emphasis added). But we know that the wavefunction is not dependent on time! There's no external time axis. So when we talk about a "wavefunction of the universe" we are really considering a "block universe" state in which all of time is laid-out and all times are equally real (see the previous section and the section on "Time" on page 11 of Carlo Rovelli's paper Quantum Spacetime: What do we know? which considers this time-independent notion of "state").

Michio Kaku considers the quantum state of the universe:

The basic idea behind Michio Kaku's logic is that the universe, being composed of particles which obey the laws of quantum mechanics, could itself be considered a quantum object. This is fair enough. But an analogy which treats the universe as though it is an actual particle is unfortunate: the entire universe can never behave in the same way as a particle. As was revealed earlier, there is nothing outside the universe, no external axes, so a "universe particle" can never have a location, or a momentum - both of which are mainstays in the quantum behaviour of a particle. Likewise, we cannot think of the universe as having a "time" (as shown by the Wheeler-DeWitt equation) or an energy (for more on this idea of the energy of the universe being zero, see the quote from Stephen Hawking's A Brief History of Time further up this page). So we should never think of the universe as a particle, in a quantum superposition (as Michio Kaku suggests). If you need proof, try using that "universe particle" in the double-slit experiment! Where would you get the slits from?!

So if we shouldn't think of the universe as a particle, how should we imagine it? Surely when we are considering the wavefunction of the entire universe as a whole then we should not be just considering it as a single point-particle moving around in a background of spacetime. Surely we should consider it as "the collection of all matter and radiation, plus all of spacetime" (see page 835 of Motion Mountain). But is it valid to apply quantization techniques to spacetime itself? I think an answer can be found if we remember that before we observe a particle's position, we have to consider it to be in superposition of many different possible positions. We should then remember John Wheeler's description of general relativity: "Matter tells spacetime how to curve, spacetime tells matter how to move". If that is the case, then when a particle is in a superposition state we should also consider spacetime (the shape of which is determined by the position of the particle) to be in a superposition state! This implies that it is valid to apply the standard quantum mechanical formulation to spacetime (and, therefore, gravity), and it hence is also valid to consider a "wavefunction of the universe". When we quantize gravity we are quantizing the gravitational field and spacetime which forms the very essence of the universe itself (for more on this, see Carlo Rovelli's paper Quantum Spacetime: What do we know?). After all, it is the relationships between objects which define "space" - space does not exist as a separate entity on its own.

Is it meaningful to talk of an "expanding universe"?

But if there is no external time reference axis outside the universe, how can it ever be meaningful to talk of an "expanding universe"? After all, expansion means an increase in size with respect to some time reference. With no external time reference axes, there is no absolute directional reference axis for time for you to say "the universe is expanding" rather than "the universe is contracting" - one is obviously just the reverse of the other, and with no external time reference axis how could you possibly prefer one statement over the other?

It seems to my mind (and to John Cruickshank - see below) that we are relying far to heavily on the psychological arrow of time to determine our time directionality, and hence decide whether the universe is expanding or contracting (see the Arrow of Time page). We "perceive" the universe to be expanding because our brains determine our feeling of directional time flow in the forward time direction. But that psychological arrow of time is always going to align itself from a low entropy universe state to a high entropy universe state. So it's not really satisfactory to define the forward time direction on that basis because you are then always going to be defining the forward time direction as the direction in which entropy increases. It means our current justification for saying "The universe is expanding" is based on nothing more than saying "a larger universe has a larger entropy, a smaller universe has a smaller entropy". Any universe which satisifies that statement could be said to be "expanding". But there's nothing in that statement which describes the change in the size of the universe with respect to any external "time axis". So that seems a very unsatisfactory description of any "expanding universe" to my mind. I don't see how we can possibly say we live in an expanding universe on that basis.

Considering a universe in which entropy is always greater in a larger universe (hence, the psychological arrow of time will always point in the direction of increased universe size). a) An expanding universe, which is perceived to be expanding. b) A contracting universe which is ALSO perceived to be expanding.

So you can't ever really say "The universe is expanding" rather than "The universe is contracting". It's just not valid to make that distinction, because you would be implicitly using the psychological arrow of time to provide your forward time direction, and that arrow of time is intimately related to the expansion of the universe in an incestuous manner which results in a circular definition. I think John Cruickshank revealed the absurdity of that situation when he realised that if the universe operates in such a way that a larger universe always has a larger entropy then we would always perceive the universe as expanding! In that case, there could be no such thing as a contracting universe!

It is more accurate to say the universe is neither expanding or contracting. It just has a state. It just "is".

(This section was suggested by a comment posted by John Cruickshank on the Arrow of Time page).

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