On Soft Limits of Large-Scale Structure Correlation Functions

Sagunski, Laura; Sigl, Guenter; Konstandin, Thomas

Book/Dissertation / PhD Thesis

PUBDB-2016-03361

On Soft Limits of Large-Scale Structure Correlation Functions

Sagunski, L. (Corresponding author)DESY* ; Konstandin, T. (Thesis advisor)DESY* ; Sigl, G. (Thesis advisor)Extern*

2016
Verlag Deutsches Elektronen-Synchrotron Hamburg

Hamburg : Verlag Deutsches Elektronen-Synchrotron, DESY-THESIS 247 pp. (2016) [10.3204/PUBDB-2016-03361] = Dissertation, Universität Hamburg, 2016

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Please use a persistent id in citations: doi:10.3204/PUBDB-2016-03361

Report No.: DESY-THESIS-2016-023

Abstract: Large-scale structure surveys have the potential to become the leading probe for precision cosmology in the next decade. To extract valuable information on the cosmological evolution of the Universe from the observational data, it is of major importance to derive accurate theoretical predictions for the statistical large-scale structure observables, such as the power spectrum and the bispectrum of (dark) matter density perturbations. Hence, one of the greatest challenges of modern cosmology is to theoretically understand the non-linear dynamics of large-scale structure formation in the Universe from first principles. While analytic approaches to describe the large-scale structure formation are usually based on the framework of non-relativistic cosmological perturbation theory, we pursue another road in this thesis and develop methods to derive generic, non-perturbative statements about large-scale structure correlation functions. We study unequal- and equal-time correlation functions of density and velocity perturbations in the limit where one of their wavenumbers becomes small, that is, in the soft limit. In the soft limit, it is possible to link $\left(\mathcal{N}+1\right)$-point and $\mathcal{N}$-point correlation functions to non-perturbative `consistency conditions'. These provide in turn a powerful tool to test fundamental aspects of the underlying theory at hand. In this work, we first rederive the (resummed) consistency conditions at unequal times by using the so-called eikonal approximation. The main appeal of the unequal-time consistency conditions is that they are solely based on symmetry arguments and thus are universal. Proceeding from this, we direct our attention to consistency conditions at equal times, which, on the other hand, depend on the interplay between soft and hard modes. We explore the existence and validity of equal-time consistency conditions within and beyond perturbation theory. For this purpose, we investigate the predictions for the soft limit of the bispectrum of density and velocity perturbations in two different approaches, namely in the perturbative time-flow approach and in a non-perturbative background method. This background method, which relies on absorbing a spherically symmetric soft mode into a locally curved background cosmology, has recently inspired a proposal for an (allegedly non-perturbative) angular-averaged equal-time consistency condition for the bispectrum of density perturbations (henceforth referred to as VKPR proposal). We demonstrate explicitly for an Einstein-de Sitter universe that the time-flow relations as well as the VKPR proposal are only fulfilled at leading order in perturbation theory, but are not exact beyond it. Since the VKPR proposal still leads to qualitatively accurate predictions for the bispectrum of density perturbations beyond the linear perturbative order, it can nevertheless be regarded as a reasonable empirical approximation in this case. However, transferring the VKPR proposal to the velocity perturbations significantly fails beyond linear order in perturbation theory. In consequence, we generalize the background method to properly account for the effect of local curvature both in the density and velocity perturbations on short distance scales. This allows us not only to identify the discrepancies of the VKPR proposal, but also to formulate a proper generalization of it which includes both the density and velocity perturbations. In addition, we use the background method to deduce a generic, non-perturbative angular-averaged bispectrum consistency condition, which depends on the density power spectrum of hard modes in the presence of local curvature.Building upon this, we proceed by deriving a non-perturbative equation for the power spectrum in the soft limit. To this end, we perform an operator product expansion, on the one hand, and deduce a non-perturbative angular-dependent bispectrum consistency condition, on the other hand. We obtain the latter from extending the background method to the case of a directional soft mode, being absorbed into a locally curved anisotropic background cosmology. The resulting non-perturbative power spectrum equation encodes the coupling to ultraviolet (UV) modes in two time-dependent coefficients. These can most generally be inferred from response functions to geometrical parameters, such as spatial curvature, in the locally curved anisotropic background cosmology. However, we can determine one coefficient by use of the angular-averaged bispectrum consistency condition together with the generalized VKPR proposal, and we show that the impact of the other one is subleading. Neglecting the latter in consequence, we confront the non-perturbative power spectrum equation against numerical simulations and find indeed a very good agreement within the expected error bars. Moreover, we argue that both coefficients and thus the non-perturbative power spectrum in the soft limit depend only weakly on UV modes deep in the non-linear regime. This non-perturbative finding allows us in turn to derive important implications for perturbative approaches to large-scale structure formation. First, it leads to the conclusion that the UV dependence of the power spectrum found in explicit computations within standard perturbation theory is an artifact. Second, it implies that in the Eulerian (Lagrangian) effective field theory (EFT) approach, where UV divergences are canceled by counter-terms, the renormalized leading-order coefficient(s) receive most contributions from modes close to the non-linear scale. The non-perturbative approach we developed can in principle be used to precisely infer the size of these renormalized leading-order EFT coefficient(s) by performing small-volume numerical simulations within an anisotropic `separate universe' framework. Our results suggest that the importance of these coefficient(s) is a $\sim 10 \%$ effect at most.

Note: Dissertation, Universität Hamburg, 2016

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