Are inflationary observables plagued by large Infra-red corrections?

1. 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

2. ……Or the Phoenix reborn? 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

3. 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

4. A toy model 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 

5. Ways out? 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

6. The curvature perturbation 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

7. 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).

8. 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)

9. 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

10. 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

11. Infa-red divergences are associated with fluctuations of the background quantities on scales much larger than the presently observed region L M

12.  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)

13. 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)

14. Taking the averages (I) Insert the expansion Rewrite the loop corrections accounting for the variation in the background fields and the renormalized vertices

15. Taking the averages (II) 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

16. Running with the scale M 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)

17. 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

18. 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

19. Not the end of the story 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

20. Conclusions  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