Our Universe and the Forward March of Time.-- Scientific thought and experiment normally starts out with a given set of initial conditions. No one ever has to worry about how the initial conditions got to be a certain way because the simple answer is that they were set up like that. In cosmology, this is not the case. Because cosmologists are responsible for the entire history of the universe, this includes its origins and how it may have gotten to that original state. We are quite familiar with studying the evolution of our universe given its conditions at some earlier time, but can we find a mechanism in which the universe dynamically evolves to have a matter-antimatter or time asymmetry instead of just imposing that as an initial condition? Here, we would like to address the latter question of why our universe exhibits an arrow of time.

The fundamental laws of physics exhibit a high degree of symmetry, and yet our world is in many ways very asymmetric. There are many processes that happen, for which the reverse process is never observed. You can mix two colors of paint, for example, but not un-mix them by swirling in the opposite direction. As time goes by, everything only ever gets messier and older, and sadly, nothing ever spontaneously cleans up. All of these unidirectional processes together dictate a direction in time. On a macroscopic scale, rewind is easily distinguishable from fast forward when microscopically there isn't a difference.

Entropy, the 2nd Law, and Initial Conditions.-- Why time exists and only flows in one direction is related to how our observable universe began. It has been known since Boltzmann's time, that in order to have a macroscopic concept of time, a system must start in an orderly state, which then progressively becomes more and more disordered via what is known as the Second Law of Thermodynamics. This evolution is what gives us a sense of time, and the 2nd Law dictates that this evolution only progresses in the direction of disorder. Thus, the mystery of why our universe exhibits this unidirectional evolution, or an arrow of time, reduces to the question of how our universe got to be in such an orderly state in the beginning. You see the difficulty -- if a system can only dynamically evolve to disorder, how did it become orderly in the first place? Your home or office cannot spontaneously clean itself, but the patch of universe we live in appears to have accomplished just that, more than 14 billion years ago.

In order to approach this question, we need to first lay down the basics on what is called entropy. Entropy is a precise measure of the degree of disorder in a system, and the 2nd Law states that entropy can never decrease. (Low entropy corresponds to order, and high entropy to disorder.) If you looked through a microscope and knew the position and momentum of each particle of your system, you can calculate the entropy of a certain macroscopic state (that is, the state of the system perceived by the naked eye) by counting the number of different microscopic configurations that look the same macroscopically. The entropy is related to the logarithm of the number of such configurations. As an example, considered a box of colored gas, where all of the molecules of gas start in one corner and evolve to uniformly fill the box. There are far fewer microscopic configurations for which all of the gas is constrained to be in a tiny corner of the box than for the gas being anywhere in the box, and thus the latter is higher entropy.

From the evolution of the gas we can identify a concept and direction of time, but that the gas particles can visibly evolve at all is due to the fact that they started in the particular state of all being in the corner. If all the gas particles started in the equilibrium distribution of uniformly filling the box, nothing would change on a macroscopic level, and there would be no macroscopic concept of time.

In order to prepare this special initial state for the box of gas, someone had to pump the gas into one corner. In this process, the entropy has decreased inside the box, but when the pump is on, the box is not a closed system and the entropy of the box plus its surroundings has increased. Suppose, for example, we had to burn fuel in order to produce the energy needed to operate the pump. Since we never witness the un-burning of fuel, we know the entropy has increased in its burning. Suppose further that this fuel came from decaying carbon-based prehistoric life, which survived off plants that grew from sunlight. All of the processes involved increased entropy, and they were allowed to occur because the entropy of the relevant subsystem was low. The low entropy of the box can then be traced to the low entropy of our solar system, and ultimately the low entropy of our universe. This is summarized in this figure from Andreas Albrecht's enlightening article on the arrow of time (astro-ph/0210527).

Thus, we see that we can pump the entropy out of a subsystem by raising the entropy of the subsystem plus surroundings (#1). Without invoking anything external to our universe, it would be difficult, if not impossible for our universe to dynamically evolve to an orderly state as observed. The fact that the universe is so orderly compared to what it could be, despite having presumably an eternity to equilibrate, is akin to happening upon an ice cube sitting in the open on a summer day, with no one around to have produced it that way. Observing the current universe evolve is like watching the ice cube melt, and wondering how it got frozen to begin with.

Entropy with Gravity.-- To understand why our universe is in such a special state, we must carefully examine just what constitutes "special" and "generic" for an open system with gravity. In the previous example, we have chosen a closed system in which gravitational interactions are negligible, and in such cases it's easy to formally define the entropy -- it's the logarithm of the number of microscopic configurations for a given macroscopic state. As always, "special" is low entropy and "generic" is high. But to be able to count microstates, they must be discrete, and that means quantum. Thus, lacking a quantum description of gravity, we are unable to define the microscopic states of a gravitational system, much less count them. The best we can do is to use the fact that low entropy systems evolve to high entropy systems and say that a "special" state is one that evolves, and a "generic" state is what systems typically evolve to.

We know that in astrophysical systems, stars can collapse into black holes, and thanks to Bekenstein and Hawking, we miraculously know the formula for the entropy of a black hole. It's quite large -- our universe has an entropy of 10 exp(88) in normal matter (in units of Boltzmann's constant), but should we stuff it all into one giant black hole, it would have an entropy of 10 exp (121), a whopping factor of 10 exp (33) larger. Supposing that the box above is large enough for gravity to be important, the most entropic state for the box of gas is in fact a black hole.

The fact that our universe is homogenous and flat, a far cry from a lumpy space filled with black holes, indicates that the observable universe is at a much lower entropy state than can be expected for an infinitely, or just very old universe. Since inflation is a mechanism that could bring about homogeneity and flatness, it is sometimes invoked as the responsible agent for generating an arrow of time. This is a partial answer, however, as inflation itself requires an even lower entropy state for its onset, and really, exacerbates the arrow of time problem as we are better off entropically just by simply invoking a flat, homogeneous universe to begin with.

Generic States.-- If our universe is in a special state, what then is a generic state for the universe? We know that: 1. Matter either dilutes away with expansion or collapses into black holes. 2. Space with a collection of black holes will either expand or contract (it's not stable). 3. Black holes radiate away, and given a sufficiently large universe, the radiation can escape to infinity.

There are thus two possibilities for a final generic state of the universe: either the space will expand and the black holes will radiate away leaving nearly empty space, or the entire universe will crunch into one big black hole -- a Big Crunch. The latter case requires that the universe be contracting everywhere -- should the universe be expanding anywhere, we'd have a universe outside of the big black hole, and then the black hole would radiate away once again leaving nearly empty space. Such a restrictive condition where we'd have to specify the state of a presumably infinite universe to be everywhere contracting renders this latter possibility less likely than a universe that is allowed to expand or contract in any part as it wishes. Thus a universe that is everywhere collapsing is fine-tuned and we will not consider it further here.

In absence of a cosmological constant, the final generic state of an infinitely large universe is nearly empty flat space. A special state can then be defined as a state that bears no resemblance to generic states.

Deviations from Generic Conditions: Fluctuations.-- Our universe is fortunately much more interesting than nearly empty flat space, and the question is how our universe got to be so full of energy and matter. It turns out that with a small vacuum energy, such as a cosmological constant, the picture above changes. Quantum fluctuations, which normally happen everywhere and all the time, can become real, and given the right kind of fluctuation (in energy and wavelength) a small patch of generic state can undergo inflation, which could give rise to the universe we observe. In addition to producing a large flat, homogeneous and monopole-free universe and to seeding the growth of structure, inflation, upon its demise, converts its energy into normal energy and matter, generating all the stuff we have.

Starting with a patch of empty flat space with a small vacuum energy, we calculated the probability of the spontaneous onset of inflation for its simplest model. The probability that inflation will start up in an inflationary Hubble volume (#2) is 10 exp (-10; 10; 56). It's interesting that this number is so small, but the important thing is that it's nonzero. Given an infinitely large universe and an infinite amount of time to wait, such a fluctuation into inflation will occur infinitely many times. Occasionally, a pocket universe will exit inflation in some of these inflating regions, and form universes that look like ours. Thus, in this picture, the initial conditions for our universe could naturally arise from a generic state, and we'd have a mechanism for generating an arrow of time.

However, fluctuations into pretty much anything are possible (with appropriately small probabilities), and the universe may be a crazy circus of rare fluctuations in which most do not lead to suitable conditions for life. We clearly arose from those that do, and the remaining issue to address is why we believe our universe originated from a period of inflation rather than some other fluctuation allowing for a habitable universe. We imagine that the set of anthropically viable fluctuations could include: 1. an expanding homogeneous, flat, monopole-free universe (that is, our current FRW universe) but which never underwent inflation, 2. a single galaxy containing our solar system, 3. a simple sun and earth system, 4. or even just an evanescent brain floating in space, where the rest of the universe is a figment of its imagination.

Clearly scenarios 2. and 3. are observationally ruled out -- our universe is in no sense minimal -- there is so much more out there than what is needed for our survival. If we believe that our current universe came through the most likely channel from a generic state, this dominant channel exhibits no principle of parsimony. That is, consider the set of all possible fluctuations that will give rise to universes with a set of characteristics. It does not seem true that the most likely fluctuations are the ones that give rise to minimal universes just exhibiting those characteristics. The fact that there is so much more to our universe than say, our cozy Milky Way is one indication that inflation could have been involved.

There are two more compelling reasons why it may be more likely that our universe arose out of a fluctuation into inflation over anything else. The first is the exponentially large volumes inflation generates. One fluctuation into inflation (or rather the eternal version of it) may give birth to zillions of universes like ours, whereas a fluctuation into a FRW universe or a brain only generates one. The other factor is that a small patch of generic universe more closely resembles a patch of inflationary universe than a patch of our current FRW universe. A universe of nearly empty space may have high total entropy, but like an inflationary region, it has low entropy density. The inflating patch differs in that it has high energy density, and the fluctuation required to start inflation is a high-energy fluctuation but over a very small region. Thus, if inflation arose out of a rare fluctuation and the FRW universe subsequently evolved from a part of that, a small inflationary universe is entropically closer to a small patch of generic state than a larger FRW universe is to a larger patch of generic state.

A fluctuation into a solipsistic brain is not necessarily more likely than a fluctuation into inflation, and most fluctuations into brains don't have an internal universe that makes physical sense (our dream-worlds never seem to follow a rigorous set of rules). Since our brains perceive a remarkably sensible world, it's hard to imagine that our brains made it up.

Ultra-Large-Scale Picture.-- We have argued that pretty much any state will evolve to a generic state of nearly empty flat space, and with a cosmological constant, fluctuations from generic states into inflationary universes are possible, which could then produce universes like ours. A remarkable result of this picture is that on ultra-large scales, one again recovers a time symmetry. Starting with a generic hypersurface, space can evolve in one direction to smooth out, with patches inflating and smaller patches converting to universes that may form galaxies and such, but in the other direction, it can also evolve to produce a similar fractal-like structure on ultra-large scales.

Conclusions.-- In this picture, we are supposing that there does not exist a maximal entropy state for the universe -- the universe is infinitely large and eternal and the entropy can be infinite and continue to grow. With the help of a little vacuum energy, and because energy is not globally conserved, fluctuations from the generic state of nearly empty flat space can occur. It could be that a fluctuation into inflation is the dominant channel into universes like ours; however, a rigorous calculation of this is still needed. From any state, the universe can continue to expand to increase its total entropy, creating low entropy density regions from which inflation can start, and we know how the story goes from there.

#1. We are in fact, such living pumps ourselves.

#2. A Hubble volume is the volume of spacetime in which everything is in causal contact. There are about 10 exp (214) inflationary Hubble volumes in our observable universe.

Reference: Spontaneous Inflation and the Origin of the Arrow of Time, Sean M. Carroll and Jennifer Chen, hep-th/0410270

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KICP Students: Jennifer Chen