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[This is a version of the question that I've revised based on helpful comments from Dan.]

I haven't studied inflation at a technical level. My picture of the process is that we have an inflaton field $\phi$ which is a scalar and has a potential $V(\phi)$. Before inflation, the field starts at some $\phi_o$ that doesn't equal the value $\phi_m$ that minimizes $V$. It then rolls downhill to the minimum.

Assuming I have this right, what I don't understand is (1) why $\phi$ initially has a single value $\phi_o$ everywhere, and (2) what sets $\phi_o$. Before the onset of inflation, the universe's temperature was much higher than any scale set by $V$. When the age of the universe was on the order of the Planck time, presumably its temperature was on the Planck scale. If the temperature was that high, wouldn't thermal fluctuations cause the field to sample a wide range of values of $\phi$, contradicting #1? And if it could sample all those values, I would expect that thermodynamically, it would consist overwhelmingly of values of $\phi$ close to $\phi_m$, not some other value $\phi_o$.

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The theory of inflation I'm familiar with doesn't have a mexican hat potential, as such a potential doesn't have the proper criterion for slow roll and thus doesn't yield the proper amount of e-folds necessary. But I do know the mexican hat potential represents the higgs potential. –  Dan Jun 7 '13 at 23:19
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@Dan: Thanks for your comment. Looking at this review paper, arxiv.org/abs/0705.0164 by Linde, he says on p. 9, "The first models were based on the theories with polynomial potentials, such as $V(\phi)=\pm \frac{m^2}{2} \phi^2+\frac{\lambda}{4}\phi^4$." I guess the minus sign on the first term would be a Mexican hat, but clearly things are more complicated than I thought. He starts the paper with a plain old simple harmonic oscillator potential, with the field rolling downhill toward the center. –  Ben Crowell Jun 8 '13 at 0:19
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True I do recall that potential, I just know that any potential needs a gently sloping initial segment (for slow roll $ V (\phi) > \dot \phi $) then a rapid decline into a harmonic oscillator state –  Dan Jun 8 '13 at 0:39
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Ginsburg slide 19 has a useful picture here –  twistor59 Jun 12 '13 at 7:24
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Still thinking... this ~ eq 56, has pretty much convinced me that the gradients fade into nothingness. Even if initial inhomogeneities mean different inflation rates in different directions, the gradients still fade and approach zero, which is what we want. –  twistor59 Jun 17 '13 at 15:56

2 Answers 2

Let us assume single field inflation

1) $\phi$ does not necessarily have the some value of initial condition everywhere, $$\phi(t,\vec{x}) = \bar{\phi}(t)+\delta\phi(t,\vec{x}) $$ usually it is the $\bar{\phi}(t)$ having same initial condition, while $\delta\phi(t,\vec{x})$ is treated as perturbations. In addition to that, what is your definition of "initial"? If one chooses the synchronous gauge in which the time foliation is defined by constant $\phi$ everywhere (independent of $\vec{x}$), then $\phi$ is always the same everywhere.

2) The temperature during inflation is the Gibbons-Hawking temperature for de Sitter space, $T\propto H$ , which is of the order of hubble parameter and hence the order of $V$. People have worried about the fluctuation of the inflation field, hawking, Bardeen etc. They found that the fluctuation is small but I could not find the paper I have in mind. I will try to find it. It is safe to discuss inflation with generically small fluctuations.

You asked a good question which puzzles many people. The main concern of your question is the initial condition of inflation. It seems to some people that theorists are just hiding the problems in the initial condition. However, to many physicists, a theory explaining many features with one input is no doubt a successful theory. I guess you already know the utility of inflationary theories so I am not explaining it further. There are theories postulating that our universe is created by quantum creation in which the initial conditions are set by chance, there are other universe having very different initial conditions. String theorists have also calculated how likely universes like ours are created in the context of string theory.

Inflation is a huge research field and there are still many problems to be solved.

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I don't have enough reputation to leave comment on the question yet so I give my comments here. response to twistor59's comment, what you are saying is true, inflation dilutes almost everything, including gradient of the field but that just explains the flatness at the end of inflation instead of the initial conditions –  mastrok Jun 17 at 6:44
    
I forgot to mention one more thing. Even if the initial condition of $\phi$ was very different in different patch of space, the patches with $\phi$ staying at higher position of the potential $V(\phi)$ would inflate more rapidly such that those patches dominate in space, and as mentioned, the value of $\phi$ was smoothed out throughout those patches. –  mastrok Jun 18 at 7:20
    
Overall it's good, +1. However, I want to point out that the synchronous gauge does not require a foliation of time where $\phi$ is constant everywhere. One can easily define such a foliation in a comoving gauge, which when transformed to synchronous would have the value of $\phi$ becomes spatially dependent. Point is, one can just construct a gauge in which the desired properties of the inflaton field exist –  Jim Aug 1 at 19:14
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In fact, usually, the comoving gauge is taken to have the perturbations in the field as zero, where the synchronous gauge is where the same amount of proper time passes for all spatial coordinates. –  Jim Aug 1 at 19:16
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Thank you for pointing out that. What I have in mind was comoving gauge. I have to clarify here, synchronous gauge is the gauge choice such that the time component metric is unperturbed. –  mastrok Aug 2 at 2:55

I found this paper (http://arxiv.org/abs/1401.2253) that discusses pre-inflationary dynamics, and specifically talks about "Kinetic Dominance", where the $\dot\phi^2$ terms dominate over $V(\phi)$, and this ends up giving some initial conditions that are independent of the specific form of the potential.

This might not answer the question of "why just one initial value?" since it seems to posit that the actual value of the field at the onset of inflation is tunable.

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