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Are inflationary observables plagued by large Infra-red corrections?

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### Are inflationary observablesplagued by large Infra-red corrections?

Nicola Bartolo

Galileo Galilei Physics Dept., Padova

based on N.B., Matarrese, Pietroni, Riotto and Seery, JCAP 2008

Effects on the inflationary observables from the so-called

loops generated by

i) self-interactions of any scalar field during inflation

ii) and/or gravitational interactions (e.g for the inflaton field)

In the past various studies and discussions

(e.g. Mukhanov et al.’97; Boyanovsky et al. ‘05; Sasaki et al ‘93)

More recently a renewed interest seeded by two papers by Weinberg

in 2005/2006 followed by others containing interesting claims

(van der Meulen/Smit ‘07 ; Sloth 06/07; Seery 07)

Loop corrections can scale like (powers of)

NTOT= total number of-efolds of inflation

The existence of these large infra-red corrections may have

a dramatic impact on our ability to make a precise

comparison between models of inflation and high precision

CMB observations

A scalar field with cubic self-interactions 3/3! in a fixed de-Sitter

background (H=const).

use, e.g., the CPT (in-in) formalism

Infra-red correction

: horizon crossing;

: infrared momentum cut-off

If one chooses

then

which is the n. of e-folds from the beginning of inflation till the mode k exits

the horizon .

At n-th order the power spectrum gets corrections

A possibility is to try to resum these potentially large

logarithmic corrections, using, e.g., techniques of the

Renormalization Group, as in Matarrese/Pietroni ‘07.

(but in our case serious problems)

Or, first of all,

Ask whether these IR effects are really present in any

physical observable which can be measured

The answer: NO

To include gravitational interactions use the curvature perturbation

(Salopek/Bond ‘91 using ADM formalism)

in the perturbed Universe the number of e-folds of inflation vary

from place to place.

Choosing the initial slice to be flat (=0) and the final one to be

uniform density one has

The curvature perturbation (II)

Note that

this is a fully non-linear definition for the curvature perturbation

(Salopek/Bond’91; Kolb et al. ‘05; Lyth at al. ‘05) and at linear order it

reduces to the usual definition

b)

for single field models of slow-roll inflation remains constant

on scales bigger than the comsological horizon

c) When dealing with the interactions of , the terms ln(k) in the loop

correction can always be reabsorbed in a negligible coefficient; such

a time dependence is there just to guarantee that is conserved

(see discussion in Seery JCAP ‘07).

Evolution of perturbations on superhorizon scales

Smoothed over the horizon the non-linear evolution of the scalar and gravitational

fields ‘point by point’ are just those of homogeneous patches (Salopek/Bond 1991)

The shift between their expansion history is determined by the variation in the initial

conditions

field at horizon crossing

(now called N formalism; see

Starobinsky ‘82; Salopek/Bond ‘91;

Lyth et al. ‘05)

Correlation functions

Bispectrum

(intrinsic

non-Gaussianity)

Where to evaluate these correlation functions?

To predict the power-spectrum of make a computation within a comoving region of present size M not much bigger than the present horizon ( minimal box)

On the other hand one has to face the problem arising from the loops

(with integrals over all momenta) to consider also a superlarge box of size L leaving the horizon at the beginning of inflation

Physical origin of the Infrared divergences

What is the relation between the correlation functions within the

minimal and the superlarge-boxes?

The correlation functions within the superlarge box are averages of the

correlation functions within a horizon-sized box, taking the average

over all the ways the small box fits within the superlarge box

If the background quantities defined within a small box show large

variations within the superlarge box then the averages within the

superlarge box will develop significant contributions in the infrared

(on scales between M and L).

So one should demonstrate that the average within the superlarge box over the

various small boxes is equivalent to make the computation of the correlation

functions within the superlarge box

Infa-red divergences are associated with fluctuations of

the background quantities on scales much larger than the

presently observed region

L

M

Consider a minimal box of size M << L within the supelarge box.

We want to show that the averaged correlator P does not depend

on the size M .

Recipe: relate the expansion of the curvature perturbation within the

box M, with the corresponding quantities relative to the superlarge

box, accounting for the variation in the background field (Lyth ‘07)

Loop corrections to the power spectrum of

Let us focus on the contribution from the bispectrum of the scalar

field (for the case of Gaussian fields see Lyth ‘07).

(Byrnes et al. 2007)

where we are using the renormlized vertices

In this way loops that start and finish at the same vertex are

automatically taken into account (Byrnes et al. 2007)

Insert the expansion

Rewrite the loop corrections accounting for the variation in the background fields

and the renormalized vertices

The variation of the background fields is (x) smoothed with

a top-hat window function

What is crucial is that the average (within the superlarge box)

makes appear in P1-loopM terms corresponding to the next-order loop.

The running with M of this average is

here

Including also the contribution from the renormalized vertices brings

As a kind of magic the dependence of PM on the size M vanishes

( the M-dependence of P2-loopM comes form the running of the integral)

This shows that PM does not depend on the size of the

box M, and that it coincides with the power spectrum P

computed in the superlarge-box:

large infra-red divergences inevitably arise because what

we are computing is in fact PM , that is the power

spectrum on the superlarge box.

These fluctuations are not of observational interest: large

infra-red corrections are associated with large fluctuations

of the background fields on scales very much larger than

the presently observable universe

Infa-red divergences are associated with fluctuations of

the background quantities and

provide the level of uncertainty in the theoretical predictions,

measuring the variance for the background values of the inflaton

field

L

M

Loop corrections for the observablecorrelators are under control

One can try to ask more general questions: our box might be

untypical; what is the probability to find an inflaton field

homogeneous enough to lead the correct CMB anisotropy?

use the approach of stochastic inflation to derive a Fokker-Planck

equation for the evolution of the probability of the inflaton value

Such a probability will be high highly non-Gaussian because of the

large IR corrections (Salopek/Bond ‘91; Mollerach, Matarrese, Ortolan,

Lucchin, ‘91):

in the stochastic approach one trades the uncertainty in the predictions

due to the IR corrections with a probability distribution for background

quantities

It’s definitely worth to explore if loop corrections can have any

revealing signatures in the cosmological observables

Recent studies claim that large infra-red corrections can have a

significant impact on the comparison between model predicitons

and observables

However it has been shown that these infrared divergences actually

are not about observable quantities; rather their presence signals

that we are considering fluctuations on ultralarge (unobservable)

scales (Lyth ‘07, Bartolo et al. ‘08; but see also past discussions in

Salopek/Bond’91;)

Such infra-red corrections become relevant for other types of

questions, like the evolution of the probablity for the value of the

inflaton field to understand the underlying inflationary theory

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