Remarks on Inflation

It has been shown that sub-Planckian models of inflation require initial homogeneity on super-Hubble scales under certain commonly held assumptions. Here I remark on the possible implications of this result for inflationary cosmology. The observed homogeneity of the universe on superhorizon scales can be explained if there was a period of accelerated expansion (inflation) in the early universe that took a relatively small homogeneous patch of space and blew it up to encompass a volume larger than what we observe today. In a recent paper [1], Mark Trodden and I addressed the issue of how large the initial homogeneous patch has to be. The result we obtained is that there is a lower bound on the inflationary horizon, H inf , which depends on the pre-inflationary cosmology. In particular, if the pre-inflationary cosmology is a Friedman-Robertson-Walker (FRW) universe, we must have H inf > H −1 FRW . If we further assume that, to get inflation with Hubble constant H inf , the appropriate inflationary conditions (homogeneity, vacuum domination etc.) must be satisfied over a region of physical size L which is larger than H inf , then we obtain L > H −1 FRW . Hence the conditions for inflation need to be satisfied on cosmological scales specified by the pre-inflationary epoch. In the particular case of a radiation dominated FRW, the problem seems to be even more severe since the causal horizon coincides with H FRW . Hence, it is clear that to solve the homogeneity problem, inflationary models in which inflation emerges from a non-inflationary, classical epoch, must assume large-scale homogeneity as an initial condition. Perhaps more important than the result itself is the fact that we have identified the conditions under which such a derivation is possible. The key assumptions are that Einstein’s equations and the weak energy conditions are valid, and that spacetime topology is trivial. (We also assume that singularities apart from the big bang are absent.) If these conditions hold, one would conclude that sub-Planckian inflation alleviates the large-scale homogeneity problem but does not solve it. 1) The word “solve” may hold different meanings for different individuals. My view is that a solution should not assume the result it purports to obtain. Note that there are two related issues that are brought to the forefront in the above discussion. The first is do inflationary models (within the stipulated assumptions) solve the homogeneity problem? and my answer to this question is in the negative. The second is can inflation occur, given that it seems to require unlikely initial conditions? Here the answer is in the positive inflation can indeed occur as long as we have a suitable physical theory. However, one may wish to further consider the likelihood of having no inflation or the probabilities with which various kinds of inflation can occur, and this is where the questions become difficult. Hopefully some of the issues involved will become clearer by the end of this article. A way around the result in [1] is to consider inflationary models in which inflation starts at the Planck epoch (for example, chaotic inflation [2]) since these do not contain a classical pre-inflationary epoch. Hence, at least as far as classical physics goes, inflation is imposed as an initial condition in these models. These initial conditions are sometimes justified by using quantum cosmology – if the wavefunction of the universe is “highly peaked” around the inflationary initial conditions, one might say that these conditions are favored. However, what if the wavefunction only has a small tail around the inflationary initial conditions? I do not think that that would exclude an inflating epoch of our universe since, it could be argued, that most of the other non-inflating universes (where the peak of the wave-function is) would not be able to harbor observers such as us. Hence the argument appears to be inconclusive at present. On the other hand, anthropic arguments are essential to any theory in which the creation of universes is probabilistic. As our understanding of astrophysics and gravity improves, the arguments are likely to get sharpened. As a very simple example of possible forthcoming refinements, if we assume that life can only exist on planets, then an understanding of the cosmological conditions leading to maximal planetary formation will help in narrowing down a measure for calculating probabilities. Guth [3] has given a persuasive argument for believing that inflation took place in the early universe in spite of any required unlikely initial conditions. He likens inflationary cosmology to the evolution of life. Today we observe a rich variety of life forms, and also a large homogeneous universe. It is hard to imagine how all the miraculous forms of life could have been created directly. Similarly, it is hard to explain the direct creation of our universe. These wonders are easier to comprehend in terms of evolutionary theory life started out in the shape of some very simple molecules, which then inevitably evolved into the present forms of life. Similarly, a small patch of the universe that satisfied some suitable properties underwent inflation and inevitably evolved into our present universe. So Guth makes the correspondence shown in Fig. 1. I find this to be a very compelling argument for the existence of an inflationary phase of the universe. The discussion in Ref. [1] impacts on this correspondence in the first stage what is the chance of getting a small patch with the correct conditions? This is the same question that biologists may ask what is the chance of getting the first few molecules from which life can follow? DNA Small patch with suitable conditions