Cosmological Perturbation Theory Research Articles

Fluctuation-dissipation relation in cosmic microwave background

We study the fluctuation-dissipation relation for sound waves in the cosmic microwave background (CMB), employing effective field theory (EFT) for fluctuating hydrodynamics.Treating sound waves as the linear response to thermal radiation, we establish the fluctuation-dissipation relation within a cosmological framework.While dissipation is elucidated in established linear cosmological perturbation theory, the standard Boltzmann theory overlooks the associated noise, possibly contributing to inconsistencies in Lambda Cold Dark Matter (ΛCDM) cosmology. This paper employs EFT for fluctuating hydrodynamics in cosmological perturbation theory, deriving sound wave noise.Notably, the long-time limit of the noise spectrum is independent of viscosity details, resembling a Brownian motion bounded in a harmonic potential.The net energy transfer between the sound wave system and the radiation environment reaches a balance within Hubble time, suggesting the thermal equilibrium of the sound waves themselves.The induced density power spectrum is characterized as white noise dependent on the inverse of the entropy density, which is negligibly small on the CMB scale. The energy density of the entire sound wave system scales as a -4, akin to radiation. While the numerical factor is not determined in the present calculation, the back reaction of the sound wave system to the background radiation may not be negligible, serving as a potential source for various fitting issues in ΛCDM cosmology.

Single field inflation in the light of Pulsar Timing Array Data: quintessential interpretation of blue tilted tensor spectrum through Non-Bunch Davies initial condition

In this work, we present a quintessential interpretation of having a blue-tilted tensor power spectrum for canonical single-field slow-roll inflation to explain the recently observed Pulsar Timing Array (NANOGrav 15-year and EPTA) signal of Gravitational Waves (GW). We formulate the complete semi-classical description of cosmological perturbation theory in terms of scalar and tensor modes using the Non-Bunch Davies initial condition. We found that the existence of the blue tilt (nt)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n_t)$$\\end{document} within the favoured range 1.2<nt<2.5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1.2<n_t<2.5$$\\end{document} can be explained in terms of a newly derived consistency relation. Further, we compute a new field excursion formula using the Non-Bunch Davies initial condition, that validates the requirement of Effective Field Theory in the sub-Planckian regime, |Δϕ|≪Mpl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\Delta \\phi |\\ll M_{\ extrm{pl}}$$\\end{document} for the allowed value of the tensor-to-scalar ratio, r<0.06\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r<0.06$$\\end{document} from CMB observations. In our study, we refer to this result as Anti Lyth bound as it violates the well-known Lyth bound originally derived for Bunch Davies initial condition. Further, we study the behaviour of the spectral density of GW and the associated abundance with the frequency, which shows that within the frequency domain 10-9Hz<f<10-7Hz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-9}~{\ extrm{Hz}}<f<10^{-7}~{\ extrm{Hz}}$$\\end{document} the outcome obtained from our analysis is completely consistent with the Pulsar Timing Array (NANOGrav 15-year and EPTA) signal. Also, we found that the behaviour of GW spectra satisfies the CMB constraints at the low frequency, f∗∼7.7×10-17Hz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_*\\sim 7.7\ imes 10^{-17}~{\ extrm{Hz}}$$\\end{document} corresponding to the pivots scale wave number, k∗∼0.05Mpc-1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_*\\sim 0.05~{\ extrm{Mpc}}^{-1}.$$\\end{document} Finally, the sharp falling behaviour of the GW spectra within the frequency domain 10-7Hz<f<1Hz\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{-7}~{\ extrm{Hz}}<f<1~{\ extrm{Hz}}$$\\end{document} validates our theory in the comparatively high-frequency regime as well.

Temporal evolution of cosmological density perturbations of the Bose–Einstein condensate dark matter

Dark matter is assumed to be composed of scalar boson particles that form Bose–Einstein condensate during the cosmic evolution of the Universe when the temperature of the dark matter is below the critical temperature Tcr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{cr}$$\\end{document}. At around a redshift z∼1200\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\sim 1200$$\\end{document}, the normal dark matter converted to Bose–Einstein condensate dark matter through a first-order phase transition and continued to complete the condensation process for nearly 106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^6$$\\end{document} years, until then, both phases coexist. In this manuscript, considering Bose–Einstein condensate dark matter as a Gross–Pitaevskii–Poisson system, we study the time evolution of density contrast of Bose–Einstein condensate dark matter using cosmological linear perturbation theory following previous works on the subject. The evolution equation contains quantum pressure due to Heisenberg’s uncertainty principle and self-interaction pressure terms. We solve the temporal density contrast equation of Bose–Einstein condensate dark matter analytically for both the Thomas-Fermi limit and the non-interacting case where the expansion rule follows similarly to that of the Einstein–de Sitter Universe. In addition, we also numerically analyze the temporal nature of the evolution of density contrast of Bose–Einstein condensate dark matter for the complete equation without applying any approximation. We find that the Bose–Einstein condensate model of dark matter could modify the temporal nature of density contrast evolution due to the presence of self-interaction and quantum pressure terms, which shows significant differences with respect to the conventional standard cold dark matter, and successfully solves the small-scale problems in the context of cosmological structure formation.

Gravitational waves in New General Relativity

The models of New General Relativity have recently got attention of research community, and there are some works studying their dynamical properties. The formal aspects of this investigation have been mostly restricted to the primary constraints in the Hamiltonian analysis. However, it is by far not enough for counting their degrees of freedom or judging whether they are any good and viable. In this paper we study linearised equations in vacuum around the trivial Minkowski tetrad. By taking the approach of cosmological perturbation theory we show that the numbers of primary constraints are very easily seen without any need of genuine Hamiltonian techniques, and give the full count of linearised degrees of freedom in the weak field limit of each and every version of New General Relativity without matter.

Minimal basis for exact time dependent kernels in cosmological perturbation theory and application to ΛCDM and w 0 wa CDM

We derive a minimal basis of kernels furnishing the perturbative expansion of the density contrast and velocity divergence in powers of the initial density field that is applicable to cosmological models with arbitrary expansion history, thereby relaxing the commonly adopted Einstein-de-Sitter (EdS) approximation. For this class of cosmological models, the non-linear kernels are at every order given by a sum of terms, each of which factorizes into a time-dependent growth factor and a wavenumber-dependent basis function. We show how to reduce the set of basis functions to a minimal amount, and give explicit expressions up to order n = 5. We find that for this minimal basis choice, each basis function individually displays the expected scaling behaviour due to momentum conservation, being non-trivial at n ≥ 4. This is a highly desirable property for numerical evaluation of loop corrections. In addition, it allows us to match the density field to an effective field theory (EFT) description for cosmologies with an arbitrary expansion history, which we explicitly derive at order four. We evaluate the differences to the EdS approximation for ΛCDM and w 0 wa CDM, paying special attention to the irreducible cosmology dependence that cannot be absorbed into EFT terms for the one-loop bispectrum. Finally, we provide algebraic recursion relations for a special generalization of the EdS approximation that retains its simplicity and is relevant for mixed hot and cold dark matter models.

A clarification on prevailing misconceptions in unimodular gravity

The traditional presentation of Unimodular Gravity (UG) consists on indicating that it is an alternative theory of gravity that restricts the generic diffeomorphism invariance of General Relativity. In particular, as often encountered in the literature, unlike General Relativity, Unimodular Gravity is invariant solely under volume-preserving diffeomorphisms. That characterization of UG has led to some confusion and incorrect statements in various treatments on the subject. For instance, sometimes it is claimed (mistakenly) that only spacetime metrics such that |det gμν | = 1 can be considered as valid solutions of the theory. Additionally, that same (incorrect) statement is often invoked to argue that some particular gauges (e.g. the Newtonian or synchronous gauge) are not allowed when dealing with cosmological perturbation theory in UG. The present article is devoted to clarify those and other misconceptions regarding the notion of diffeomorphism invariance, in general, and its usage in the context of UG, in particular.

Synchronous coordinates and gauge-invariant observables in cosmological spacetimes

We consider the relational approach to construct gauge-invariant observables in cosmological perturbation theory using synchronous coordinates. We construct dynamical synchronous coordinates as non-local scalar functionals of the metric perturbation in the fully non-linear theory in an arbitrary gauge. We show that the observables defined in this dynamical coordinate system are gauge-independent, and that the full perturbed metric has the expected form in these coordinates. Our construction generalises the familiar synchronous gauge in linearised gravity, widely used in cosmological perturbation theory, to the non-linear theory. We also work out the expressions for the gauge-invariant Einstein equation, sourced either by an ideal fluid or a scalar field up to second order in perturbation theory, and give explicit expressions for the Hubble rate—as measured by synchronous observers or by observers co-moving with the matter field—up to that order. Finally, we consider quantised linear perturbations around Minkowski and de Sitter backgrounds, and compute the two-point function of the gauge-invariant metric perturbation in synchronous coordinates, starting with two-point function in a general linear covariant gauge. Although the gauge-fixed two-point function contains gauge modes, we show that the resulting gauge-invariant two-point function only contains the physical tensor modes and it is thus positive, i. e. it has a spectral representation.

Multi-field inflation with large scalar fluctuations: non-Gaussianity and perturbativity

Recently multi-field inflation models that can produce large scalar fluctuations on small scales have drawn a lot of attention, primarily because they could lead to primordial black hole production and generation of large second-order gravitational waves. In this work, we focus on models where the scalar fields responsible for inflation live on a hyperbolic field space. In this case, geometrical destabilisation and non-geodesic motion are responsible for the peak in the scalar power spectrum. We present new results for scalar non-Gaussianity and discuss its dependence on the model's parameters. On scales around the peak, we typically find that the non-Gaussianity is large and close to local in form. We validate our results by employing two different numerical techniques, utilising the transport approach, based on full cosmological perturbation theory, and the δN formalism, based on the separate universe approximation. We discuss implications of our results for the perturbativity of the underlying theory, focusing in particular on versions of these models with potentially relevant phenomenology at interferometer scales.

The Hawking energy in a perturbed Friedmann-Lemaître universe

Hawking's quasi-local energy definition quantifies the energy enclosed by a spacelike 2-sphere in terms of the amount of lightbending on the sphere caused by the energy distribution inside the sphere. This paper establishes for the first time a direct connection between the formal mathematical definition of a quasi-local energy and observations, in the context of cosmological perturbation theory. This is achieved by studying the Hawking Energy of spherical sections of the past lightcone of a cosmic observer in a perturbed Friedmann-Lemaître spacetime. We express the Hawking Energy in terms of gauge-invariant perturbation variables and comment on the cosmic observables needed to in principle measure it. We then calculate its angular power spectrum and interpret its contributions.

Effect of dissipation on chromo-natural inflation

In this work, we examined chromo-natural inflation in the context of warm inflation with constant and variable dissipation coefficients. We studied the cosmological perturbation theory in this model and obtained the scalar and tensor power spectra in each case. The sources of density fluctuations in this model are mainly the thermal fluctuations of the inflaton field like a general warm inflationary model. Finally, cosmological observables, namely, the spectral index and tensor-to-scalar ratio are calculated. It is found that the cosmological observables are consistent with observational Planck 2018 and BICEP bounds for sub-Planckian values of axion decay constant [Formula: see text] in case of temperature–inflaton-dependent dissipation coefficient. In case of the constant and temperature-dependent dissipation coefficient, the values of axion decay constant [Formula: see text] can be lowered below the GUT scale.

Testing growth rate dependence in cosmological perturbation theory using scale-free models

Testing growth rate dependence in cosmological perturbation theory using scale-free models

Cosmological perturbations engendered by discrete relativistic species

Within the extension of the varLambda CDM model, allowing for the presence of neutrinos or warm dark matter, we develop the analytical cosmological perturbation theory. It covers all spatial scales where the weak gravitational field regime represents a valid approximation. Discrete particles – the sources of the inhomogeneous gravitational field – may be relativistic. Similarly to the previously investigated case of nonrelativistic matter, the Yukawa interaction range is naturally incorporated into the first-order scalar metric corrections.

Using Cosmological Perturbation Theory to Distinguish between GR and Unimodular Gravity

Unimodular gravity is one of the oldest geometric gravity theories and alternatives to general relativity. Essentially, it is based on the Einstein–Hilbert Lagrangian with an additional constraint on the determinant of the metric. It can be explicitly shown that unimodular gravity can be recast as general relativity in the presence of a cosmological constant. This fact has led to many discussions on the equivalence of both theories at the classical and quantum levels. Here, we present an analysis focused on the classical scalar perturbations around a cosmological background. We focus on the unusual situation in which the typical conservation laws are not adopted. The discussion is extended to the case where a non-minimal coupled scalar field is introduced. We also present a gauge-invariant analysis showing that perturbations in unimodular gravity display instabilities. Our results reinforce that the equivalence is not verified completely at a cosmological perturbative level.

Effective description of generalized disformal theories

Generalized disformal transformations enable us to construct the generalized disformal Horndeski theories, which form the most general class of ghost-free scalar-tensor theories to this date. We extend the effective field theory (EFT) of cosmological perturbations to incorporate these generalized disformal Horndeski theories. The main difference from the conventional EFT is that our extended EFT involves operators with higher spatial derivatives of the lapse function. Our EFT also accommodates the generalized disformal transformation of U-DHOST theories.

Consistent cosmological structure formation on all scales in relativistic extensions of MOND

General relativity manifests very similar equations in different regimes, notably in large scale cosmological perturbation theory, non-linear cosmological structure formation, and in weak field galactic dynamics. The same is not necessarily true in alternative gravity theories, in particular those that possess MONDian behaviour (“relativistic extensions” of MOND). In these theories different regimes are typically studied quite separately, sometimes even with the freedom in the theories chosen differently in different regimes. If we wish to properly and fully test complete cosmologies containing MOND against the ΛCDM paradigm then we need to understand cosmological structure formation on all scales, and do so in a coherent and consistent manner. We propose a method for doing so and apply it to generalised Einstein-Aether theories as a case study. We derive the equations that govern cosmological structure formation on all scales in these theories and show that the same free function (which may contain both Newtonian and MONDian branches) appears in the cosmological background, linear perturbations, and non-linear cosmological structure formation. We show that MONDian behaviour on galactic scales does not necessarily result in MONDian behaviour on cosmological scales, and for MONDian behaviour to arise cosmologically, there will be no modification to the Friedmann equations governing the evolution of the homogeneous cosmological background. We comment on how existing N-body simulations relate to complete and consistent generalised Einstein-Aether cosmologies. The equations derived in this work allow consistent cosmological N-body simulations to be run in these theories whether or not MONDian behaviour manifests on cosmological scales.

Flows for the masses: A multi-fluid non-linear perturbation theory for massive neutrinos

Velocity dispersion of the massive neutrinos presents a daunting challenge for non-linear cosmological perturbation theory. We consider the neutrino population as a collection of non-linear fluids, each with uniform initial momentum, through an extension of the Time Renormalization Group perturbation theory. Employing recently-developed Fast Fourier Transform techniques, we accelerate our non-linear perturbation theory by more than two orders of magnitude, making it quick enough for practical use. After verifying that the neutrino mode-coupling integrals and power spectra converge, we show that our perturbation theory agrees with N-body neutrino simulations to within 10% for neutrino fractions Ω ν,0 h 2 ≤ 0.005 up to wave numbers of k = 1 h/Mpc, an accuracy consistent with ≤ 2.5% errors in the neutrino mass determination. Non-linear growth represents a > 10% correction to the neutrino power spectrum even for density fractions as low as Ω ν,0 h 2 = 0.001, demonstrating the limits of linear theory for accurate neutrino power spectrum predictions. Our code FlowsForTheMasses is avaliable online at github.com/upadhye/FlowsForTheMasses.

Manifestly unitary cosmological perturbation theory

The next decade will feature an abundance of novel cosmological data, while many fundamental questions about inflation remain. Given this, there is ample need for maximally efficient calculations, especially in non-standard scenarios for the early Universe. In inflationary cosmology, observables are computed within the framework of in-in perturbation theory. Weinberg introduced a now-widely used re-organization of perturbation theory for in-in calculations. There is a subtle difference in the iϵ prescriptions of Weinberg's perturbation series with traditional in-in, which could interfere with the projection onto the interacting vacuum. In this work, we show that a small adjustment to Weinberg's perturbation series yields agreement with standard in-in at every order of perturbation theory for commonly studied spins and masses in de Sitter spacetime. We then generalize the result to a large class of cosmological spacetimes, including slow roll spacetimes.

Perturbations in non-flat cosmology for f(T) gravity

The study of cosmological perturbation theory in f(T) gravity is a topic of great interest in teleparallel gravity since this is one of the simplest generalizations of the theory that modifies the teleparallel equivalent of general relativity. In this work, we explore the possibility of a non-flat FLRW background solution and perform perturbations for positively as well as negatively curved spatial geometries, together with a comparison to the flat case. We determine the generalized behaviour of the perturbative modes for this non-flat FLRW setting for arbitrary f(T) models, when the most general homogeneous and isotropic background tetrads are used. We also identify propagating modes in this setup, and relate this with the case of a flat cosmology.

Renormalization group and UV completion of cosmological perturbations: Gravitational collapse as a critical phenomenon

Cosmological perturbation theory is known to converge poorly for predicting the spherical collapse and void evolution of collisionless matter. Using the exact parametric solution as a testing ground, we develop two asymptotic methods in spherical symmetry that resolve the gravitational evolution to much higher accuracy than Lagrangian perturbation theory (LPT), which is the current gold standard in the literature. One of the methods selects a stable fixed-point solution of the renormalization-group flow equation, thereby predicting already at the leading order the critical exponent of the phase transition to collapsed structures. The other method completes the truncated LPT series far into the UV regime, by adding a non-analytic term that captures the critical nature of the gravitational collapse. We find that the UV method most accurately resolves the evolution of the nonlinear density as well as its one-point probability distribution function. Similarly accurate predictions are achieved with the renormalization-group method, especially when paired with Pad\'e approximants. Further, our results yield new, very accurate, formulas to relate linear and nonlinear density contrasts. Finally, we chart possible ways on how to adapt our methods to the case of cosmological random field initial conditions.

The Gauge Issue and the Hamiltonian Theory of Cosmological Perturbations

The Gauge Issue and the Hamiltonian Theory of Cosmological Perturbations