112. Talk in Munich. Other two interesting infrared scales

Posted by Dmitry

November 27, 2008

Save This Post as PDF

As Instanton figured out in comments to the previous post, the scale $L_\Phi\sim{}L_\zeta$ is related to the self-reproduction scale. How to show this? Well, recall that the classical displacement of the inflaton field during one efolding is

$\Delta\phi\sim\frac{1}{3H^2}\frac{\partial{}V}{\partial{}\phi}\sim\sqrt{\epsilon}M_P$ , (1)

where $\epsilon$ is the slow roll parameter. On the other hand, during the same amount of time the fluctuations of $\phi$ are generated with characteristic amplitude

$\sqrt{\langle\phi^2\rangle}\sim{}H$ . (2)

The latter become important for the evolution of the inflaton background, when (1) becomes of the same order as (2). We have then

$H\sim\sqrt{\epsilon}M_P$ ,

which coincides with the condition for $\langle\zeta^2\rangle$ to become of the order 1, as I derived in the previous post (see the expression after Eq. (3) there). In other words, both $L_\Phi$ and $L_\zeta$ are given by the scale of self-reproduction.

Let me now mention two other infrared scales which are of interest for cosmologists.

First one appears if we analyze the loop expansion for the correlators of $\phi$ in the $\lambda\phi^4$ theory on de Sitter background. It turns out that every term in the loop expansion diverges in the infrared, and if we keep only leading IR divergences, we find

$\langle\phi^2\rangle=\frac{H^2}{(2\pi)^2}\log{}a\sum_{n=0}^\infty{}c_n(\lambda\log^2{}a)^n$ . (3)

As we note, the actual small parameter of the perturbation theory is $\lambda\log^2{}a$ and not $\lambda$ . Also, it seems that the expansion breaks down when

$\log{}a\sim\lambda^{-1/2}$ ,

i.e., the breakdown happens at the scale

[Unparseable or potentially dangerous latex formula. Error 2 ].

This scale is quite a bit longer than the scale of self-reproduction $L_\zeta\sim{}H^{-1}\exp(\lambda^{-1/3})$ .

What actually happens at this scale? Does the series (3) diverge or not? I will answer to this question in the next post.

Finally, analyzing loop expansion in the $\delta{}N$ formalism in order to calculate corrections to the correlation functions of the curvature perturbation $\langle\zeta(x_1)\zeta(x_2)\rangle$ , we find

$\langle\zeta(x_1)\zeta(x_2)\rangle=(N')^2{}P_\phi\log\left(\frac{|x_1-x_2|}{L_{\delta{}N}}\right)+\cdots$ ,

where is the number of efoldings, $P_\phi$ is the power spectrum of the inflaton field, dots denote terms of higher power w.r.t. the parameter $P_\phi\log\left(\frac{|x_1-x_2|}{L_{\delta{}N}}}\right)$ , and $L_{\delta{}N}}$ is some infrared scale where we need to cut the logarithm.

Experts in $\delta{}N$ formalism call this scale the size of the box. What is the size of the box, is it arbitrary and if not, how can it и related to other infrared scales we discussed? Does the $\delta{}N$ expansion diverge? This was the subject of my talk in Munich and will be the subject of a couple of the next posts.

Cosmological perturbations and large scale structure, Cosmology, Fellow Craft, Lectures on de Sitter spacetime, Quantum field theory

If you liked the post, please kindly consider to leave a comment, subscribe to the RSS feed or get new posts sent directly to your Inbox. If you want to chat with me in real time, you can find me on Twitter. The posts below are probably related to the subject of this one:

112. Talk in Munich. Other two interesting infrared scales

Comments »

Trackback responses to this post

Search

Readability

Topic

Authors

Lectures

Recent comments

Top commentators

Polls