We live in the aftermath of a great explosion—the big bang—that occurred 13.7 billion years ago. At the time of the big bang, the universe was filled with a fireball, a dense mixture of energetic particles and radiation. For nearly a century, physicists have been studying how the fireball expanded and cooled, how particles combined to form atoms, and how galaxies and stars were gradually pulled together by gravity. This story is now understood in great quantitative detail and is supported by abundant observational data.^{1}

The question, however, remains whether the big bang was truly the beginning of the universe. A beginning in what? Caused by what? And determined by what, or whom? These questions have prompted physicists to make every attempt to avoid a cosmic beginning.

In this essay, I review where we now stand.

## The Penrose Singularity Theorem

The problem has been staring us in the face since the early days of scientific cosmology. In the 1920s, the Russian mathematician Alexander Friedmann provided a mathematical description of an expanding universe by solving Albert Einstein’s equations of general relativity. Friedmann assumed for the sake of simplicity that the distribution of matter in the universe was perfectly uniform. His solutions had a mystifying feature: as the evolution of the universe is followed backward in time, the density of matter and the curvature of space-time grow without bounds, becoming infinite a finite time ago.^{2} The moment of infinite density is a cosmological singularity. At that point, the mathematical expressions appearing in the equations of general relativity become ill defined, and evolution cannot be continued. This would seem to suggest that the universe did have a beginning, but not one describable by the laws of physics.

Physicists hoped initially that the singularity might be an artifact of Friedmann’s simplifying assumption of perfect uniformity, and that it would disappear in more realistic solutions of Einstein’s equations. Roger Penrose closed this loophole in the mid-1960s by showing that, under a very general assumption, the singularity was unavoidable.^{3} Under the null convergence condition, gravity always forces light rays to converge.^{4} This implies that the density of matter or energy measured by any observer cannot be negative. The conclusion holds for all familiar forms of classical matter.

Penrose’s proof hinges on the concept of an incomplete geodesic. In general relativity, the trajectories of matter are represented by straight lines in space-time, or geodesics. If space-time is free of singularities, all geodesics should have infinite extent. A geodesic encountering a singularity cannot be further extended. Such geodesics are incomplete. Penrose showed that a space-time satisfying the null convergence condition (and some additional mild assumptions) must contain incomplete geodesics. A singularity appears to be unavoidable.

Penrose’s argument was not entirely convincing. Even though the null convergence condition is satisfied by classical matter, quantum fluctuations can create regions of negative energy density.^{5} In the extreme conditions near the big bang, quantum fluctuations are of great importance. Penrose’s argument no longer applies.

This is where things stood until the early 1980s, when Alan Guth introduced the idea of cosmic inflation.^{6}

## Eternal Inflation

Inflation is a period of super-fast, accelerated expansion in the early history of the universe. In a fraction of a second, a tiny subatomic region blows up to dimensions greater than the entire currently observable universe. The expansion is driven by a false vacuum.

A vacuum is ordinarily thought of as empty space, but according to modern particle physics what is empty is not nothing. The vacuum is a physical object, endowed with energy density and pressure. It can be in a number of different states, or vacua. The properties and types of elementary particles differ from one vacuum to another.

The gravitational force induced by a false vacuum is unusual in that it is repulsive. The higher the energy of the vacuum, the stronger the repulsion. This kind of vacuum is unstable. It decays into a low-energy vacuum and the excess energy produces a fireball of particles and radiation. False vacua were not invented for the purposes of inflation. Their existence follows from particle physics and general relativity.

The theory of inflation assumes that at some early time in its history the universe occupied a high-energy false vacuum. Repulsive gravitational forces then caused a super-fast, exponential expansion of the universe. There is a characteristic time in which the size of the universe doubles. Depending on the model, the doubling times can be as short as 10^{-37} seconds. In about 330 doubling times, the universe grows by a factor of 10^{100}. No matter its initial size, the universe very quickly becomes huge. Because the false vacuum is unstable, it eventually decays, producing a fireball, marking the end of inflation. The fireball continues to expand by inertia and evolves along the lines of standard big bang cosmology.

Inflation explained some otherwise puzzling features of the universe, matters that big bang cosmology was forced to assume. It explained the expansion of the universe, its high temperature, and its observed homogeneity. Inflationary theory predicted that Euclidean geometry describes the universe on the largest scales. It also predicted a nearly scale-independent spectrum of small-density perturbations caused by quantum fluctuations during inflation. These predictions have been confirmed.

The theory of inflation has led to a revision in our view of the universe. Inflation does not come to an end everywhere at once. Regions where the false vacuum decays somewhat later are rewarded by a larger inflationary expansion, so false vacuum regions tend to multiply faster than they decay. In our cosmic neighborhood, inflation ended 13.7 billion years ago; in remote parts of the universe it is still continuing. Regions like ours are constantly being formed. This never-ending process is called eternal inflation. Eternal inflation is generic; and predicted by most models.

False vacuum decay is model-dependent. In this essay, I focus on models in which it occurs through bubble nucleation. Vacuum decay is similar to the boiling of water. Low-energy regions appear as microscopic bubbles and immediately start to grow at a rate rapidly approaching the speed of light. The bubbles are then driven apart by inflationary expansion, making room for more bubbles. We live in one of these bubbles, but we can observe only a small part of it. No matter how fast we travel, we cannot catch up to the expanding boundary of our universe.

Ours is a self-contained universe.

Eternal inflation raises an intriguing possibility. If inflation goes on and on into the future, could it also have gone on and on into the past?^{7} A universe without a beginning would make it unnecessary to ask how it began.

As is so often the case in physics, an irresistible force is now about to encounter an immovable obstruction.

## The Borde-Guth-Vilenkin Theorem

The obstruction may be found in the Borde-Guth-Vilenkin (BGV) theorem.^{8} Loosely speaking, our theorem states that if the universe is, on average, expanding, then its history cannot be indefinitely continued into the past. More precisely, if the average expansion rate is positive along a given world line, or geodesic, then this geodesic must terminate after a finite amount of time. Different geodesics, different times. The important point is that the past history of the universe cannot be complete. An outline of the proof is provided in the Appendix.

The BGV theorem allows for some periods of contraction, but on average expansion wins. The volume of the universe increases with time. Inflation cannot be eternal and must have some sort of a beginning.

The BGV theorem is sweeping in its generality. It makes no assumptions about gravity or matter. Gravity may be attractive or repulsive, light rays may converge or diverge, and even general relativity may decline into desuetude: the theorem would still hold.

A number of physicists have constructed models of an eternal universe in which the BGV theorem is no longer pertinent. George Ellis and his collaborators have suggested that a finite, closed universe, in which space closes upon itself like the surface of a sphere, could have existed forever in a static state and then burst into inflationary expansion.^{9} Averaged over infinite time, the expansion rate would then be zero, and the BGV theorem would not apply. Ellis constructed a classical model of a stable closed universe and provided a mechanism triggering the onset of expansion. Ellis made no claim that his model was realistic; it was intended as a proof of concept, showing that an eternal universe is possible. Not so. A static universe is unstable with respect to quantum collapse.^{10} It may be stable by the laws of classical physics, but in quantum physics a static universe might make a sudden transition to a state of vanishing size and infinite density. No matter how small the probability of collapse, the universe could not have existed for an infinite amount of time before the onset of inflation.

There is another way that the universe might be eternal in the past. It could have cycled through an infinite succession of expansions and contractions. This notion was briefly popular in the 1930s, but was then abandoned because of its apparent conflict with the second law of thermodynamics. The second law requires that entropy should increase in each cycle of cosmic evolution. If the universe had already completed an infinite number of cycles, it would have reached a state of thermal equilibrium, and so a state of maximum entropy. All the energy of ordered motion would have turned into heat, a uniform temperature prevailing throughout.

We do not find ourselves in such a state.

The idea of a cyclic universe was recently revived by Paul Steinhardt and Neil Turok.^{11} They suggested that in each cycle expansion is greater than contraction, so that the volume of the universe is increased. The entropy of the universe we can now observe could be the same as the entropy of some similar region in an earlier cycle; nonetheless, the total entropy of the universe would have increased because the volume of the universe is now greater than it was before. As time goes on, both the entropy and the total volume grow without bounds, and the state of maximum entropy is never reached. There is no maximum entropy.^{12}

The problem with this scenario is that, on average, the volume of the universe still grows, and thus the BGV theorem can be applied. This leads immediately to the conclusion that a cyclic universe cannot be past-eternal.

## God’s Proof

Theologians have welcomed any evidence for the beginning of the universe as evidence for the existence of God. “As to the first cause of the universe,” wrote the British astrophysicist Edward Milne, “this is left for the reader to insert, but our picture is incomplete without Him.”^{13} Some scientists feared that a cosmic beginning could not be described in scientific terms. “To deny the infinite duration of time,” asserted the Walter Nernst, “would be to betray the very foundations of science.”^{14}

Richard Dawkins, Lawrence Krauss and Victor Stenger have argued that modern science leaves no room for the existence of God. A series of science–religion debates has been staged, with atheists like Dawkins, Daniel Dennett, and Krauss debating theists like William Lane Craig.^{15} Both sides have appealed to the BGV theorem, both sides appealing to me—of all people!—for a better understanding.

The cosmological argument for the existence of God consists of two parts. The first is straightforward:

- everything that begins to exist has a cause;
- the universe began to exist;
- therefore, the universe has a cause.
^{16}

The second part affirms that the cause must be God.

I would now like to take issue with the first part of the argument. Modern physics can describe the emergence of the universe as a physical process that does not require a cause.

Nothing can be created from nothing, says Lucretius, if only because the conservation of energy makes it impossible to create nothing from nothing. For any isolated system, energy is proportional to mass and must be positive. Any initial state, prior to the creation of the system, must have the same energy as the state after its creation.

There is a loophole in this reasoning. The energy of the gravitational field is negative;^{17} it is conceivable that this negative energy could compensate for the positive energy of matter, making the total energy of the cosmos equal to zero. In fact, this is precisely what happens in a closed universe, in which the space closes on itself, like the surface of a sphere. It follows from the laws of general relativity that the total energy of such a universe is necessarily equal to zero. Another conserved quantity is the electric charge, and once again it turns out that the total charge must vanish in a closed universe.

I will illustrate these statements for the case of an electric charge, using a two-dimensional analogy. Imagine a two-dimensional closed universe, which we can picture as a surface of a globe. Suppose we place a positive charge at the north pole of this universe. Then the lines of the electrical field emanating from the charge will wrap around the sphere and converge at the south pole. This means that a negative charge of equal magnitude should be present there. Thus, we cannot add a positive charge to a closed universe without adding an equal negative charge at the same time. The total charge of a closed universe must therefore be equal to zero.

If all the conserved numbers of a closed universe are equal to zero, then there is nothing to prevent such a universe from being spontaneously created out of nothing. And according to quantum mechanics, any process which is not strictly forbidden by the conservation laws will happen with some probability.^{18}

A newly-born universe can have a variety of different shapes and sizes and can be filled with different kinds of matter. As is usual in quantum theory, we cannot tell which of these possibilities is actually realized, but we can calculate their probabilities. This suggests that there could be a multitude of other universes.

Quantum creation is similar to quantum tunneling through energy barriers in quantum mechanics. An elegant mathematical description of this process can be given in terms of a Wick rotation. Time is expressed using imaginary numbers, introduced only for computational convenience. The distinction between the dimensions of time and space disappears. This description is very useful, since it provides a convenient way to determine tunneling probabilities. The most probable universes are the ones with the smallest initial size and the highest vacuum energy. Once a universe is formed, it immediately starts expanding due to the high energy of the vacuum.

This provides a beginning for the story of eternal inflation.

One might imagine that closed universes are popping out of nothing like bubbles in a glass of champagne, but this analogy would not be quite accurate. Bubbles pop out in liquid, but in the case of universes, there is no space out of which they might pop. A nucleated closed universe is all the space there is, aside from the disconnected spaces of other closed universes. Beyond it, there is no space, and no time.

What causes the universe to pop out of nothing? No cause is needed. If you have a radioactive atom, it will decay, and quantum mechanics gives the decay probability in a given interval of time, say, a minute. There is no reason why the atom decayed at this particular moment and not another. The process is completely random. No cause is needed for the quantum creation of the universe.

The theory of quantum creation is no more than a speculative hypothesis. It is unclear how, or whether, it can be tested observationally. It is nonetheless the first attempt to formulate the problem of cosmic origin and to address it in a quantitative way.^{19}

## An Unaddressable Mystery

The answer to the question, “Did the universe have a beginning?” is, “It probably did.” We have no viable models of an eternal universe. The BGV theorem gives us reason to believe that such models simply cannot be constructed.

When physicists or theologians ask me about the BGV theorem, I am happy to oblige. But my own view is that the theorem does not tell us anything about the existence of God. A deep mystery remains. The laws of physics that describe the quantum creation of the universe also describe its evolution. This seems to suggest that they have some independent existence.

What exactly this means, we don’t know.

And why are these laws the ones we have? Why not other laws?

We have no way to begin to address this mystery.

## Appendix: Mathematical Details

In this Appendix, I outline a proof of the BGV theorem.

Start with a homogeneous, isotropic, and spatially flat universe with the metric:

$$d{s}^{2}=d{t}^{2}-{a}^{2}\left(t\right)d{\overrightarrow{x}}^{2}.$$

The Hubble expansion rate is $H=\dot{a}/a$, where the dot denotes a derivative with respect to time $t$. We can imagine that the universe is filled with comoving particles, moving along the timelike geodesics $\overrightarrow{x}=const$. Consider an inertial observer, whose world line is ${x}^{\mu}\left(\tau \right)$, parametrized by the proper time $\tau $. For an observer of mass $m$, the 4-momentum is ${P}^{\mu}=md{x}^{\mu}/d\tau $, so that $d\tau =(m/E)dt$, where $E={P}^{0}=\sqrt{{p}^{2}+{m}^{2}}$ denotes the energy, and $p$, the magnitude of the 3-momentum. It follows from the geodesic equation of motion that $p\propto 1/a\left(t\right)$, so that $p\left(t\right)=\left[a\left({t}_{f}\right)/a\left(t\right)\right]{p}_{f}$, where ${p}_{f}$ designates the momentum at some reference time ${t}_{f}$.

Thus:

$${\int}_{{t}_{i}}^{{t}_{f}}H\left(\tau \right)d\tau ={\int}_{a\left({t}_{i}\right)}^{a\left({t}_{f}\right)}\frac{mda}{\sqrt{{m}^{2}{a}^{2}+{p}_{f}^{2}{a}^{2}\left({t}_{f}\right)}}=F\left({\gamma}_{f}\right)-F\left({\gamma}_{i}\right)\le F\left({\gamma}_{f}\right),$$

where ${t}_{i}<{t}_{f}$ is some initial moment.

Note that:

$$F\left(\gamma \right)=\frac{1}{2}\mathrm{l}\mathrm{n}\left(\frac{\gamma +1}{\gamma -1}\right),$$

where $\gamma =1/\sqrt{1-{\nu}_{rel}^{2}}$ is the Lorentz factor, and ${\nu}_{rel}=p/E$ is the observer’s speed relative to the comoving particles.

For any non-comoving observer, $\gamma >1$ and $F\left(\gamma \right)>0$.

The expansion rate averaged over the observer’s world line is:

$${H}_{av}=\frac{1}{{\tau}_{f}-{\tau}_{i}}{\int}_{{t}_{i}}^{{t}_{f}}H\left(\tau \right)d\tau .$$

Assuming that

$${H}_{av}>0,$$

and using the first equation, it follows that

$${\tau}_{f}-{\tau}_{i}\le \frac{F\left({\gamma}_{f}\right)}{{H}_{av}}.$$

This implies that any non-comoving past-directed timelike geodesic satisfying the condition ${H}_{av}>0$, must have a finite proper length, and so must be past-incomplete.

There is no appealing to homogeneity and isotropy in an arbitrary space-time. Imagine that the universe is filled with a congruence of comoving geodesics, representing test particles and consider a non-comoving geodesic observer described by a world line ${x}^{\mu}\left(\tau \right)$.^{20} Let ${u}^{\mu}$ and ${\nu}^{\mu}$ designate the 4-velocities of test particles and the observer.

Then the Lorentz factor of the observer relative to the particles is

$$\gamma ={u}_{\mu}{\nu}^{\mu}.$$

To characterize the expansion rate in general space-time, it suffices to focus on test particle geodesics that cross the observer’s world line.

Consider two such geodesics encountering the observer at times $\tau $ and $\tau +\u2206\tau $.

Define the parameter

$$H={\mathrm{l}\mathrm{i}\mathrm{m}}_{\u2206\tau \to 0}\frac{\u2206{u}_{r}}{\u2206r},$$

where $\u2206{u}_{r}$ is the relative velocity of the particles in the observer’s direction of motion, and $\u2206r$ is the particle’s separation. Both quantities are computed in the rest frame of one of the particles.

For a homogeneous and isotropic universe, this definition reduces to the Hubble parameter.

The expansion parameter can be expressed as a total derivative,

$$H=\frac{d}{d\tau}F\left(\gamma \left(\tau \right)\right).$$

The integral of $H$ along the observer’s world line is still given by the difference of $F\left(\gamma \right)$ at its endpoints. Conclusions about homogeneous and isotropic universes carry over immediately to generic universes.

There remains the null observer described by a null geodesic. The role of the proper time $\tau $ is then played by an affine parameter. BGV showed that, with a suitable normalization of $\tau $, the expansion rate is given by

$$H=\frac{d}{d\tau}F\left(\gamma \left(\tau \right)\right),$$

with $F\left(\gamma \right)=1/\gamma $, and $\gamma $ defined by

$$H={\mathrm{l}\mathrm{i}\mathrm{m}}_{\u2206\tau \to 0}\frac{\u2206{u}_{r}}{\u2206r}.$$

Clearly, $F\left(\gamma \right)>0$, and the argument goes through as before.

A rigorous formulation of the BGV theorem is now possible. Let $\lambda $ be a timelike or null geodesic maximally extended to the past, and let $C$ be a timelike geodesic congruence defined along $\lambda $.

If the expansion rate of $C$ averaged along $\lambda $ is positive, then $\lambda $ must be past-incomplete.