Fear of a Black Universe

by Stephon Alexander

  To delve deeper into this and other issues about the early universe, it’s useful to address a few common misconceptions about the big bang itself. Perhaps the biggest misunderstanding is that the big bang was some cataclysmic explosion that fueled the universe’s expansion, where galaxies went flinging away from each other. In the standard big bang theory predicted by general relativity, the universe is assumed to be filled with a hot and dense gas of matter and radiation, whose energy and pressure source the expansion of space-time itself. We’ve discussed before that general relativity provides an expanding solution to its equations; called the FRW model, for its discoverers Alexander Friedmann, Howard Robertson, and Arthur Walker, it was for a long time the standard model of cosmology. According to that solution to the equations of general relativity, the expansion of space is not an “explosion” but actually gradual in time. Even more interesting is that time ends in a singularity of infinite curvature and energy density. What is this singularity really trying to tell us?

  To try to answer the above question, Stephen Hawking and Roger Penrose provided a theorem that the FRW expanding solution will always suffer a singularity. At the singularity, the curvature goes to infinity, marking a breakdown of general relativity as a valid description of the space-time structure. The Hawking-Penrose theorem was based on a powerful equation discovered simultaneously by Indian theorist Amal Kumar Raychaudhuri and Soviet theorist Lev Landau. These Landau-Raychaudhuri equations relate paths of observers in a curved space-time, called geodesics, to singularities. Hawking and Penrose implemented these equations to prove that geodesics in an expanding space-time will exhibit pathologies as they approach the infinite curvature singularity at the beginning of the big bang. A useful set of geodesics to use as a diagnostic for singularities are light rays. In a flat space-time two light rays will follow parallel lines and never cross each other. In a curved space-time the paths of light rays can twist, focus, or even diverge, depending on the warping of the space-time. Hawking and Penrose showed that as light rays approach the earliest times toward the infinite past in an expanding space-time, their geodesics terminate. This geodesic incompleteness signals the infinite curvature and a singularity as time goes to zero.

  A similar story happens for black hole singularities with geodesics, and many interpret singularities as a breakdown in the validity of general relativity at the singularity. Of course, all theorems, including the Hawking-Penrose theorem, do come with assumptions, or axioms, and axioms aren’t necessarily true. Sometimes they can be relaxed; you may know that the parallel postulate in Euclidean geometry can be relaxed, and rather than breaking geometry, actually points the way to two other kinds, hyperbolic and elliptic. Perhaps if we can relax one of them this could be a way out of the inevitability of the cosmic singularity. One assumption of the Hawking-Penrose theorem is that matter is classical. Keep in mind, though, that the classical matter in the world is made up from quantum matter. And cosmic inflation uses quantum matter and the inflation field, and it may help resolve the singularity. But even the quantum state of inflation does not rescue the universe from the singularity.1 The onset of inflation still remains unresolved. Over the last four decades cosmologists have been investigating various mechanisms based on new physics that could alleviate the big bang singularity. There are a handful of promising approaches my colleagues and I have pursued.

  Aside from potentially satisfying the urge to understand the birth of the universe, pursuing the problems facing big bang physics provides an opportunity for physicists to guide themselves toward a theory of quantum gravity and to test the validity of those they find. At one level we have constructed theories of quantum gravity by attempting to make the principles of general relativity and quantum mechanics consistent with each other. But the issues that the early universe presents can serve as guideposts to a more fundamental theory—and it may not even be quantum gravity. As we go back in time, the universe approaches a length scale called the Planck length, which is thirty-five orders of magnitude smaller than a meter. Here, we expect gravity and quantum mechanics to speak to each other. In some quantum theories of gravity, space-time emerges from a pregeometric phase, a vision of reality that would force us to conceptualize matters where there is no space-time to refer to. In other models, the expanding universe must arise from primordial atoms of space-time. In those cases how these “atoms” interact to give us an expanding universe will be the prize hunt. As we explored, both string theory and loop quantum gravity have ingredients to avoid certain space-time singularities, and models of the early universe have been proposed and developed to transcend the singularity.

  Understanding the evolution of the earliest universe provides the foundation for a sequence of important events for other forms of evolution to take place, such as star formation, which is central to life as we know it. Therefore, understanding the earliest stages and ideally the origin of the universe may give us new insights into the universe we currently inhabit. Just like the genome reveals new secrets about an organism, could it be that the pre–big bang universe can shed light on our universe in its current epoch?

  Recall that a successful theory of the early universe must solve a handful of cosmological problems that currently have no confirmed solution. A suite of observations have made it clear what these conundrums are. In addition to the big bang singularity problem, these are the horizon and flatness problems (which João Magueijo, among others we’ve met, are working on), the fine-tuning problem, as well as the question of the origin of large-scale structure such as galaxies and the CMB. We have seen how cosmic inflation, without departing too much from the principles of general relativity and quantum field theory, has already been able to solve some of these problems, providing answers to the horizon and flatness problems, as well as giving an explanation for the universe’s large-scale structure. You may wonder, if cosmic inflation does such a great job of explaining our current universe, then why seek alternatives? One answer is that it can’t explain the singularity and fine-tuning problems. Another is that, even if an alternative theory turns out to be wrong or simply less successful than the explanations we already have, exploring it can still give us a new perspective that can improve the more conventional theory. What’s more, I believe that the singularity problem that plagues inflation points to the source of all the cosmological problems—if we can resolve singularities, we might resolve it all.

  Over the years my colleagues and I tried to alleviate some conceptual and mathematical problems that plagued cosmic inflation by constructing a superstring theory inflationary model, and we kept hitting roadblocks. It was only when we researched alternatives to inflation within the context of superstring theory that we got new insights into moving forward with the inflationary roadblocks—an example of the value of gaining an outsider’s perspective. The fact that the universe is expanding actually limits the options as to what may have occurred before it started to expand. This boils down to the question of whether the universe had a beginning or is eternal.

  We will now venture into discussing promising theories for understanding how the earliest stages of the universe could have emerged. Keep in mind that the theories that we will consider may or may not need inflation. And as we venture to get a sense of the future of the physics of the early universe, we should take stock of the two general and divergent hypotheses behind approaching a theory of quantum gravity.

  Hypothesis 1: We should quantize gravity the same way that proved to be successful in other systems, such as quantum electrodynamics and quantum chromodynamics. In this approach, one identifies the classical system, such as the state space (or phase space) and applies rules for quantization. For example, in particle mechanics the phase space will comprise all positions and momenta of the dynamical particles. These measures commute with each other, which means that the order in which we multiply with them doesn’t change the outcome.2 Quantization rules impose that position and momenta don’t commute, however, and the classical phase space gets
promoted to a Hilbert space. In Hilbert space, the central objects are not finite vectors (of the kind that described momentum in our discussion of phase space early in the book), but rather infinite dimensional vectors. These are otherwise known as wave functions and correspond to a probability distribution.

  Hypothesis 2: General relativity is a classical theory that should not be directly quantized according to hypothesis 1. Instead, at shorter distance scales there are more fundamental quantum degrees of freedom that give general relativity as a long-distance, low-energy classical theory: this is the quantum emergent principle at work. This is similar to how fluid dynamics emerges as a long-range theory of interacting atoms. As we discussed, string theory does not quantize general relativity, but general relativity instead emerges as a low-energy effective theory (that is, a theory that describes effects but not causes of an observed phenomenon—in this case, gravity), albeit in ten dimensions.

  We’ve explored both so far in this book. But let’s turn now to a promising approach to the early universe that pursues new directions. One of the interesting things about this approach is that it addresses primarily the big bang singularity and the other cosmological problems that cosmic inflation fails to address. This isn’t unique to cosmic inflation: We will see that all approaches are limited and require concepts and tools that go beyond the current theories that we are considering.

  In a pioneering publication in 1989, two theoretical physicists, Robert Brandenberger and Cumrun Vafa, presented a new approach to the early universe by attempting to solve three questions in one fell swoop. The key insight into the BV mechanism, as their result came to be called, harkens back to the concept of duality in quantum mechanics. They not only questioned the initial spatial singularity and infinite temperature at the bang but also related the solutions to those problems with something we take for granted: the dimensionality of space. The basic laws as we know them would be different if space were not three-dimensional. Electric and gravitational fields would not fall off inversely proportional to the square of the distance, which would affect all the chemistry necessary for the world as we know it. If we lived in two spatial dimensions, for example, life would not exist as we know it; for the function of carbon-based life depends on three-dimensional structure of folded proteins. In the spirit of relativity, we can think that the dimensionality of space-time is not absolute and treat the fact that we live in three dimensions as a physical condition that emerged in the early universe.

  Of course we’ve seen that string theory is a quantum theory of gravity that requires nine spatial dimensions in order to be quantum mechanically consistent. What Brandenberger and Vafa realized was that string theory had the correct ingredients to address the other cosmological problems in one fell swoop by first asking how the unique properties and symmetries of quantum strings address these cosmological conundrums. Consistent with superstring theory—a theory that fuses supersymmetry with string theory—Brandenberger and Vafa considered an early universe in nine spatial dimensions. Instead of the early universe being occupied with a thermal state of particles and radiation, it is filled with a thermal state of strings. Because strings are extended objects, and unlike particles, they have the special property that they can wind around a spatial direction. So, string theory has both oscillatory strings, and wound-up strings called winding strings. To get a feel for the differences between those two types of strings versus particles, consider the geometry of a two-dimensional torus, which looks like a doughnut. A particle can only move along the surface of the doughnut, but a string can both move along the surface of the torus and wrap around one of the cycles of the torus. These winding states, like rubber bands, have tension energy; if they are left to their devices they will cause the torus to collapse. This is also true if you are speaking about the kind of hypertorus you’d find in a nine-dimensional world such as Brandenberger and Vafa were considering. But according to general relativity the radiation that lives in space makes space expand. So, in a stringy universe there exist three types of matter: winding strings, string loops, and radiation. In such a universe space will expand and the winding energy will force the space to collapse after it expands too much.

  Under ordinary circumstances, according to general relativity, such a universe will not be able to avoid a singularity. But string theory has a new ingredient that tells a different story: a symmetry we previously discussed (I used it in my thesis) called target space duality, or T-duality. This symmetry states that physics in the smallest possible region of space is equivalent to physics in the largest possible region of space. This happens because string theory has both matter and winding configurations present. If we perform a transformation that swaps the roles of these two states, it is equivalent to swapping motion in a large region of space and a small region of space. For example, if the torus universe has a radius R then the transformation will take R to 1/R; nevertheless, string theory remains unchanged. If the radius is cosmologically large, then strings moving will experience the same physics as if they were in a small space 1/R.

  FIGURE 26: This diagram shows how the BV mechanism avoids the singularity as the radius goes to zero. We enter the T-dual phase instead of diverging with infinite curvature.

  Here is an intuitive way to understand this. Let’s imagine that the universe shrinks to distance scales that are microscopically small, such as the string length. In the standard big bang picture, because only particles were present, the temperature would tend to infinity as the universe got smaller—exactly the kind of singularity we’d like to avoid. But in a stringy universe as the temperature gets high enough as the scale shrinks, the energy of particles starts getting fed into the string oscillations. Eventually all these oscillation states get occupied and the universe approaches not an infinite temperature but a maximum possible temperature, known as the Hagedorn temperature. At the same time as we approached the R=0 singularity, there is a dual description of the universe in which the physics is the same as having winding modes moving in a large nonsingular universe. In a nutshell, the Brandenberger-Vafa mechanism prevents the universe from reaching the singularity by having a symmetry in the physics between winding string modes and particle modes and swapping small with large distances. If the universe can never become infinitely large, then by symmetry it can never go to zero length and will avoid the spatial singularity found in standard big bang cosmology.

  But by using the same physics of winding states, the BV mechanism goes beyond this by predicting why we live in three spatial dimensions. Again, the argument uses the properties special to strings. Understanding why we live in three dimensions is as simple as understanding the question: In what dimension are two-point particles most likely to collide? In three dimensions if we set two particles (particles have zero dimension; strings have one dimension) off in a random direction, they are less likely to run into each other than in two dimensions. But in one dimension then the particles are guaranteed to run into each other.3 A similar argument applies to strings, which are one-dimensional objects. It turns out that in three dimensions strings are most likely to collide with each other without the collisions being unavoidable; in anything larger than three spatial dimensions, strings are likely to never meet. In the Brandenberger-Vafa model there are also strings that wind in opposite directions relative to each other. If these strings run into each other they, like particles and antiparticles, will annihilate into radiation. Consider a three-dimensional torus of both winding strings, which keep the space from expanding due to their tension exerted on the space. If these winding modes annihilate then the space will expand according to general relativity. In a ten-dimensional universe, Brandenberger and Vafa found that the strings in the extra six dimensions never get to annihilate. As a result, three dimensions become our observable expanding universe while the other six remain wound up by the winding modes.

  So now the BV mechanism tells us that our universe is a ten-dimensional, string-dominated space-time with all dimensions starting out micr
oscopic. Applying the principle of maximal symmetry, at the beginning all space-time dimensions and string states are on equal footing.4 Therefore, strings and antistrings will wind around all nine spatial dimensions. These dimensions will remain tiny until winding strings annihilate each other in three spatial dimensions. This will cause a large three-dimensional space to expand. Because of T-duality and the finite Hagedorn temperature, there is no big bang singularity. This suggests the universe did not emerge from a big bang singularity (that is, from a situation where the radius was zero and then began to increase).

  FIGURE 27: A schematic representation of a dimension being compactified by winding strings.

  The BV mechanism also provides a way of seeding the large-scale structure in the universe that is different from the way that inflation describes the process. In inflation cosmology it is quantum vacuum fluctuations that initiate cosmic structure. In the BV mechanism it is the thermal waves generated by the undulating gas of strings that give the observed nearly scale-invariant fluctuations seen in the CMB. Both mechanisms generate a spectrum of gravitational ripples of space-time that leave an imprint on the polarization of light in the CMB. These gravitational waves are predicted to leave in their wake a pattern on the CMB photons by creating an overall curling, like a pinwheel, of the light’s polarization, called B-mode. Observational cosmologists have been on a hunt to find this faint signal, pushing the envelope of the most advanced detection technology known. Currently the Simons Array telescope led by my colleague and friend Brian Keating is being built to finally detect the primordial gravitational waves. But although both models predict the waves, they don’t predict the same pattern in the B-mode polarization; observing the pattern could tell us if our universe underwent inflation or something more akin to BV. The predictions of gravitational waves between inflation and BV differ in that the shape of the power spectrum tilts in the opposite direction. In inflation the power spectrum is said to be red. This simply means that the longer wavelength perturbations have slightly less power than the shorter wavelength ones. BV has an opposite blue power spectrum.