54. Eternal inflation: stochastic approach 2 (Inflationary perturbations 7)

Posted by Dmitry
June 4, 2008

In the previous post we have started to discuss the regime of eternal inflation realized when classical displacement of the inflaton field becomes comparable with the average amplitude of fluctuations generated at super-Hubble scale. The latter in practice means that the gravitational perturbations become of the same order as the background. How to treat theory in this regime?

Let us for simplicity consider a scalar field \phi in the potential V(\phi)=V_{0}+\delta{}V such that |\delta V|\ll V_{0}. We will focus on what an observer leaving within a given Hubble patch sees. It is convenient to divide the quantum field \phi into subhubble and superhubble scales

\hat{\phi}=\Phi(t)+\frac{1}{(2\pi)^{3/2}}\int d^{3}k\Theta(k-\delta a(t)H_{0})\times
\times\left(\hat{a}_{k}\phi_{k}(t)\exp(-ikx)+\hat{a}_{k}^{\dagger}\phi_{k}^{*}(t)\exp(ikx)\right), (1)

where the second term satisfies the free scalar field equation, i.e.,

\phi_{k}=\frac{H_{0}}{\sqrt{2k}}\left(\eta-\frac{i}{k}\right)\exp(-ik\eta) (2)

and \eta=\int\frac{dt}{a(t)}=-\frac{1}{a(t)H_{0}} is conformal time. The Heisenberg field \hat{\phi} satisfies classical equation of motion

\Box\hat{\phi}+\frac{\partial V}{\partial\phi}=0, (3)

and after substituting the decomposition (1) into it we have for the IR part

\dot{\Phi}=-\frac{1}{3H_{0}}\frac{\partial V(\Phi)}{\partial\Phi}+f(t,x), (4)

where

f(t,x)=\frac{\delta aH_{0}^{2}}{(2\pi)^{3/2}}\int d^{3}k\delta(k-\delta aH_{0})\frac{(-i)H_{0}}{\sqrt{2}k^{3/2}}\times
\times\left(\hat{a}_{k}\exp(-ikx)-\hat{a}_{k}^{\dagger}\exp(ikx)\right). (5)

Although both \Phi and f are complicated composites of operators a_{k}, one can immediately check that all terms in the Eq. (4) are commuting with each other and therefore both f and \Phi can be considered classical quantitites.

Exercise 7.1. Check it explicitly.

On the other hand, calculating the correlation function of f in the Bunch-Davies vacuum we have

\langle0|f(t_{1})f(t_{2})|0\rangle=\frac{H_{0}^{3}}{4\pi^{2}}\delta(t_{1}-t_{2}). (6)

Therefore, equation (4) is nothing but a Langevin equation describing random walk (Brownian motion) of the variable \Phi under the action of the random Gaussian force f with correlation properties (6).

The physical picture related to this random walk is the following. As long as the classical displacement of the inflaton \Delta\phi becomes of the same order of magnitude as the average quantum fluctuation amplitude \delta\phi, deterministic classical description of the inflaton dynamics breaks down. Quantum modes constantly leave the Hubble scale and their amplitude becomes classical at t\gg H^{-1}. The quantum phase of the modes also freezes, and its value after leaving the Hubble scale is essentially random. The averaging in (6) is essentially averaging over this random phase. After modes leave the Hubble scale, they start to contribute into \Phi (it is simple impossible for an onserver inside a given Huble patch to make a distinction between superhubble fluctuation and the fluctuation of the background \phi).

As a result of this randomness, the overall spacetime becomes divided into Hubble patches with own dynamics of the IR inflaton field \Phi inside each Hubble patch.

  • Digg
  • StumbleUpon
  • Technorati
Cosmological perturbations and large scale structure, Cosmology, Lectures on de Sitter spacetime, Master Mason, Quantum field theory

If you enjoyed this post, please consider to leave a comment, subscribe to the RSS feed or get new posts sent directly to your Inbox. If you are interested to know what I am doing right now, follow me on Twitter. The posts below are probably related to the subject of this one:

99. Eternal inflation with many light scalar fields
53. Eternal inflation: stochastic approach 1 (Inflationary perturbations 7)
55. Eternal inflation: stochastic approach 3 (Inflationary perturbations 7)
Best posts
40. Inflation: field-theoretic description (Inflationary perturbations 4)

RSS feed | Trackback URI

Comments »

No comments yet.

Name
E-mail
URI
Subscribe to comments via email
Your Comment (smaller size | larger size)
You may use <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong> in your comment.

Trackback responses to this post