111. Talk in Munich. One interesting infrared scale in inflationary cosmology

Posted by Dmitry

November 26, 2008

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I am back to Helsinki, was not this visit really short? :-)

For those of you how were unable to come to the Sommerfeld Center in Munich to hear my talk and for those of you who were there but did not understand it ? I decided to put the outline of my talk on the blog.

The talk was about different infrared divergences seemingly present in correlators we calculate in inflationary perturbation theory and how to properly regularize them.

As a kind of introduction, let us first to identify some interesting infrared scales present in inflating Universe. In what follows, I shall particularly focus on chaotic inflation with quartic potential $\frac{1}{4}\lambda\phi^4$ .

We find one such scale when look at the behavior of the correlation function of the scalar perturbation (Newtonian potential) $\delta_\Phi=\frac{\delta\phi}{\phi}$ . As we know (see for example the excellent book by Mukhanov),

$\delta_\Phi^2\sim\lambda^{1/2}\log\left(H_k\lambda_{\rm phys}\right)\right)^{3/2}$ . (1)

The physical meaning of this result is very simple: the spectrum of inflationary perturbations has slightly red tilt, i.e., while we look at larger and larger scales, we find that characteristic amplitude of cosmological perturbations grows more and more.

Imagine that we live in the radiation-dominated Universe. The horizon size slowly grows, more and more inflationary modes reenter our Hubble volume. Since the spectrum tilt is red, we will see inevitably at some point that cosmological perturbations became of the order 1. Which scale does this happen at?

From (1) we immediately conclude that the corresponding physical scale (we will call it $L_\Phi$ ) is

$L_\Phi\sim{}H^{-1}\exp(\lambda^{-1/3})$ .(2)

This scale is huge compared to the present size of the cosmological horizon. Indeed, recall that $\lambda\sim{}10^{-15}\div{}10^{-14}$ from the Cobe normalization, so we have something like

$L_\Phi\sim{}H^{-1}\exp(10^5)$ .

Also observe that the value of the scale is non-perturbative w.r.t. the coupling $\lambda$ .

Let us now take the perturbation of the curvature of 3?dimensional slice $\zeta$ instead of the scalar potential and estimate its mean square. We find:

$\langle\zeta^2\rangle\sim\frac{H^4}{{\dot{\phi}}^2}\sim\frac{H^2}{\epsilon M_P^2}$ . (3)

At which scale $L_\zeta$ does $\langle\zeta^2\rangle$ become of the order 1?

From (3) we have

$H^2\sim\frac{\lambda\phi^4}{M_P^2}\sim\epsilon{}M_P^2$ ,

where $\epsilon\sim\frac{M_P^2}{\phi^2}\ll{}1$ is the slow roll parameter. Taking into account that

$a/a_i\sim\exp\left(\pi{}M_P^{-1}(\phi_i-\phi)^2\right)$

and

$L\sim{}H_f^{-1}\frac{a_f}{a}$ ,

we find that thу scale $L_\zeta$ where the perturbation of 3?dimensional curvature slice also becomes of the order 1 actually coincides with $L_\Phi$ , i.e.,

$L_\zeta\sim{}H^{-1}\exp(\lambda^{-1/3})$

Could you guess before I continue what physical situation or process does this scale correspond to?

Tags: cosmological+perturbation+theory, Cosmology, general+relativity, inflation, inflationary+perturbations

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

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