### Inflation and the Initial Conditions of the Universe

Over at Preposterous Universe, Sean has been discussing an essay he's written about his recent work on eternal inflation and the arrow of time. In the paper, and in the comments on his post, there has been a discussion about what it means to require a sufficiently large, smooth, potential energy dominated patch of the universe in order for inflation to begin. Sean referred people to a paper I wrote with Tanmay Vachaspati and I thought that, given the current interest, it might be useful to describe that work here. This will be a little more technical than usual, but far less technical than the actual paper (hopefully).

As I first learned about inflation, the idea can be summarized as the following: the universe is born and one can say very little about it since quantum gravity (whatever that is) is undoubtedly important at extremely early times. However, after some time (approximately the Planck time), the semi-classical universe emerges, and we can begin to analyze meaningfully such things as the dynamics of field theories, and the response of gravity to them. There is no a priori reason for the universe to be homogeneous at this epoch. However, local, causal particle dynamics can act to homogenize patches of the universe. After some time, a small patch becomes homogeneous and dominated by the vacuum energy of a scalar field. This patch then undergoes inflation - a quasi-exponential period of expansion in which the original small patch expands to a size many orders of magnitude larger than the observable universe today. This expansion explains the flatness of the universe, and its homogeneity on large scales today.

Now, there are a number of models of inflation in which the above story is modified (in particular, chaotic inflation), and I'll get back to them later. For now let me focus on this claim of homogeneity in the theories I described above.

Why does inflation, as described, "solve" the homogeneity (or horizon) problem? Clearly, the idea is that the homogeneity of the initial pre-inflationary patch, explained by causal physics, is translated into the homogeneity of the larger space after the exponential expansion. At the risk of being pedantic (me? pedantic?), this can only be true if the original patch is made homogeneous by causal processes, otherwise homogeneity would once again be an assumption, albeit a less severe one.

What did we do in our paper? We first imagined that the early universe, emerging from the Planck epoch, was not inflating. To make progress we'll need a few definitions, which I'll define below in a more blog-friendly way than in the paper.

Let's focus on spherically symmetric space-times and pick an origin. Then examine spherical surfaces centered on this origin. Such surfaces can be divided into three categories in the following way. Imagine sitting on such a surface with two flashlights, both pointing radially and close together. The flashlights can both be pointing outwards (away from the origin), or both pointing inwards (towards the origin). The categories are then:

The main tool we used is called the Raychaudhuri equation. It describes the rate of change of divergence of close by pairs of light rays, as I described above. The equation is a little complicated but, by considering the types of light rays I mentioned above (perpendicular to spherical surfaces), and by making two further assumptions: that the Einstein equations are satisfied, and that the weak energy condition holds (matter isn't too weird), the most important consequence of the Raychaudhuri equation can be stated as

It is possible that inflation, if it begins, leads to "eternal inflation", in which one patch gives rise to an infinite expanding space, which produces an infinite number of regions of the universe that look like ours. A full understanding of the question of initial conditions (or probabilities) in that case is partly what Sean's paper is about. Tanmay and I viewed our work as explaining what is needed for such an inflationary model to work.

As I first learned about inflation, the idea can be summarized as the following: the universe is born and one can say very little about it since quantum gravity (whatever that is) is undoubtedly important at extremely early times. However, after some time (approximately the Planck time), the semi-classical universe emerges, and we can begin to analyze meaningfully such things as the dynamics of field theories, and the response of gravity to them. There is no a priori reason for the universe to be homogeneous at this epoch. However, local, causal particle dynamics can act to homogenize patches of the universe. After some time, a small patch becomes homogeneous and dominated by the vacuum energy of a scalar field. This patch then undergoes inflation - a quasi-exponential period of expansion in which the original small patch expands to a size many orders of magnitude larger than the observable universe today. This expansion explains the flatness of the universe, and its homogeneity on large scales today.

Now, there are a number of models of inflation in which the above story is modified (in particular, chaotic inflation), and I'll get back to them later. For now let me focus on this claim of homogeneity in the theories I described above.

Why does inflation, as described, "solve" the homogeneity (or horizon) problem? Clearly, the idea is that the homogeneity of the initial pre-inflationary patch, explained by causal physics, is translated into the homogeneity of the larger space after the exponential expansion. At the risk of being pedantic (me? pedantic?), this can only be true if the original patch is made homogeneous by causal processes, otherwise homogeneity would once again be an assumption, albeit a less severe one.

What did we do in our paper? We first imagined that the early universe, emerging from the Planck epoch, was not inflating. To make progress we'll need a few definitions, which I'll define below in a more blog-friendly way than in the paper.

Let's focus on spherically symmetric space-times and pick an origin. Then examine spherical surfaces centered on this origin. Such surfaces can be divided into three categories in the following way. Imagine sitting on such a surface with two flashlights, both pointing radially and close together. The flashlights can both be pointing outwards (away from the origin), or both pointing inwards (towards the origin). The categories are then:

- NORMAL: When the flashlights point inwards, the rays converge to the origin. When they point outwards, the rays diverge away from the origin. This is how regular parts of space-time behave; for example, points in our universe closer to us than the horizon.

- TRAPPED: When the flashlights point inwards, the rays converge to the origin. When they point outwards, the rays still converge to the origin. Such surfaces can be found inside the horizon of a black hole.

- ANTI-TRAPPED: When the flashlights point inwards, the rays nevertheless diverge away from the origin. When they point outwards, the rays diverge away from the origin. Such surfaces can be found, for example, beyond the horizon in our universe.

The main tool we used is called the Raychaudhuri equation. It describes the rate of change of divergence of close by pairs of light rays, as I described above. The equation is a little complicated but, by considering the types of light rays I mentioned above (perpendicular to spherical surfaces), and by making two further assumptions: that the Einstein equations are satisfied, and that the weak energy condition holds (matter isn't too weird), the most important consequence of the Raychaudhuri equation can be stated as

Light rays pointing inwards cannot emanate from a normal surface and cross an anti-trapped one.What does this mean? Well, if the original inflating patch is smaller than the Hubble size of the background space-time, then, it can be shown that light rays violating the above statement must exist. Thus, we conclude that the size of the initial inflating patch is at least as large as the Hubble size of the background space-time. But this size is large compared to typical particle physics processes that can act to homogenize a region (actually, if the background space-time is radiation-dominated FRW, the Hubble size IS the causal horizon). Thus, it is very hard to see how such an initially homogeneous inflating patch might form. This is our main result.

It is possible that inflation, if it begins, leads to "eternal inflation", in which one patch gives rise to an infinite expanding space, which produces an infinite number of regions of the universe that look like ours. A full understanding of the question of initial conditions (or probabilities) in that case is partly what Sean's paper is about. Tanmay and I viewed our work as explaining what is needed for such an inflationary model to work.

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