A central assumption in our analysis of cosmic structure is that cosmological perturbations have zero ensemble mean. This property is one of the consequences of statistically homogeneity, the invariance of correlation functions under spatial translations. In this article we explore whether cosmological perturbations indeed have zero mean, and thus test one aspect of statistical homogeneity. We carry out a classical test of the zero mean hypothesis against a class of alternatives in which perturbations have non-vanishing means, but homogeneous and isotropic covariances. Apart from Gaussianity, our test does not make any additional assumptions about the nature of the perturbations and is thus rather generic and model-independent. The test statistic we employ is essentially Student's t statistic, applied to appropriately masked, foreground-cleaned cosmic microwave background anisotropy maps produced by the WMAP mission. We find evidence for a non-zero mean in a particular range of multipoles, but the evidence against the zero mean hypothesis goes away when we correct for multiple testing. We also place constraints on the mean of the temperature multipoles as a function of angular scale. On angular scales smaller than four degrees, a non-zero mean has to be at least an order of magnitude smaller than the standard deviation of the temperature anisotropies.
We present a self-contained summary of the theory of linear cosmological perturbations. We emphasize the effect of the six parameters of the minimal cosmological model, first, on the spectrum of Cosmic Microwave Background temperature anisotropies, and second, on the linear matter power spectrum. We briefly review at the end the possible impact of a few non-minimal dark matter and dark energy models.
The super-horizon second-order density perturbations corresponding to cosmological random fluctuations are considered, their non-vanishing spatial average is shown to be useful in solving the serious problem on the cosmological tension between measured Hubble constants at present and those at the early stage, and the difference from previous works on the backreaction is discussed.
We study the matter density fluctuations in the running cosmological constant (RCC) model using linear perturbations in the longitudinal gauge. Using this observable we calculate the growth rate of structures and the matter power spectrum, and compare them with the $SDSS$ data and other available data of the linear growth rate. The distribution of collapsed structures may also constraints models of dark energy. It is shown that RCC model enhances departures from the $ΛCDM$ model for both cluster number and cumulative cluster number predicted. In general increasing the characteristic parameter …
We discuss cosmological perturbation theory at third order, deriving the gauge transformation rules for metric and matter perturbations, and constructing third order gauge invariant quantities. We present the Einstein tensor components, the evolution equations for a perfect fluid, and the Klein-Gordon equation at third order, including scalar, vector and tensor perturbations. In doing so, we also give all second order tensor components and evolution equations in full, exhilarating generality.
We have derived fourth-order perturbative equations in Lagrangian perturbation theory for a cosmological dust fluid. These equations are derived under the supposition of Newtonian cosmology in the Friedmann-Lemaître-Robertson-Walker Universe model. Even if we consider the longitudinal mode in the first-order perturbation, the transverse mode appears in the third-order perturbation. Furthermore, in this case, six longitudinal-mode equations and four transverse-mode equations appear in the fourth-order perturbation. The application of the fourth-order perturbation leads to a precise prediction o…
A short pedagogical overview of cosmological perturbation theory, following the lectures given during the brazilian school of cosmology held in August 2012. Topics treated are: I. The background II. SVT decomposition and the gauge issue. III. The example of the tensor modes. IV. Density fluctuations, transfer function and power spectrum. V. Initial condition theory: quantum vacuum fluctuations.
It is shown that the serious problem on the cosmological tension between the direct measurements of the Hubble constant at present and the constant derived from the Planck measurements of the CMB anisotropies can be solved by considering the renormalized model parameters. They are deduced by taking the spatial average of second-order perturbations in the flat Lambda-CDM model, which includes random adiabatic fluctuations.
Within the framework of the interacting fluid formalism, we provide the numerical solution to the Boltzmann equation describing the evolution of an inflaton field coupled to radiation. We study the behavior of the system during the slow-roll regime, in the case in which an additional stochastic source term is included in the set of equations, and we recover the expression for the cosmological perturbations previously obtained in the Warm inflation scenarios.
We investigate the linear growth rate of cosmological matter density perturbations of a viable f(R) model both numerically and analytically. We find that the growth rate in the scalar-tensor regime can be characterized by a simple analytic formula. We also investigate a prospect of constraining the Compton wavelength scale of the f(R) model with a future weak lensing survey.
We show that at second order in cosmological perturbation theory vorticity generation is sourced by entropy gradients. This is an extension of Crocco's theorem to a cosmological setting.
We prove that in the infinite speed-of-light limit (i.e., non-relativistic and subhorizon limits), the relativistic fully nonlinear cosmological perturbation equations in two gauge conditions, the zero-shear gauge and the uniform-expansion gauge, exactly reproduce the Newtonian hydrodynamic perturbation equations in the cosmological background; as a consequence, in the same two gauge conditions, the Newtonian hydrodynamic equations are exactly recovered in the Minkowsky background.
We present a comprehensive analysis of classical scalar, vector and tensor cosmological perturbations in ghost-free massive bigravity. In particular, we find the full evolution equations and analytical solutions in a wide range of regimes. We show that there are viable cosmological backgrounds but, as has been found in the literature, these models generally have exponential instabilities in linear perturbation theory. However, it is possible to find stable scalar cosmological perturbations for a very particular choice of parameters. For this stable subclass of models we find that vector and te…
The theory of cosmological perturbations is extended to spacetimes displaying isotropic expansion but anisotropic curvature. The perturbed Einstein equation and Boltzmann equations for massless and massive particles are derived in a general gauge and a decomposition of perturbations into harmonic modes and moments is proposed. Generalization to the case where anisotropic expansion is also present in the background is discussed.
We investigate the cosmological perturbations in the context of a Scalar-Tensor-Vector theory of Gravity known as MOG in the literature. Recent investigations show that MOG reproduces a viable background cosmological evolution comparable to $Λ$CDM. However, the matter-dominated era is slightly different. In this paper, we study the linear matter perturbations and estimate the relevant modified gravity parameters. We show that MOG reduces the growth rate of the perturbations and comparing with the RSD data reveals that MOG suggests a higher value for $σ_8$, compare to $Λ$CDM. This point, consti…
The constant mean extrinsic curvature on a spacelike slice may constitute a physically preferred time coordinate, `York time'. One line of enquiry to probe this idea is to understand processes in our cosmological history in terms of York time. Following a review of the theoretical motivations, we focus on slow-roll inflation and the freezing and Hubble re-entry of cosmological perturbations. We show how the mathematical account of these processes is distinct from the conventional account in terms of standard cosmological or conformal time. We also consider the cosmological York-timeline more b…
Cosmological perturbations of FRW solutions in ghost free massive bigravity, including also a second matter sector, are studied in detail. At early time, we find that sub horizon exponential instabilities are unavoidable and they lead to a premature departure from the perturbative regime of cosmological perturbations.
We study the gauge invariant cosmological perturbations up to second order. We show that there are infinite families of gauge invariant variables at both of the first and second orders. The conversion formulae among different families are shown to be described by a finite number of bases that are gauge invariant. For the second order cosmological perturbations induced by the first order scalar perturbations, we explicitly represent the equations of motion of them in terms of the gauge invariant Newtonian, synchronous and hybrid variables, respectively.
It is known that some cosmological perturbations are conformal invariant. This facilitates the studies of perturbations within some gravitational theories alternative to general relativity, for example the scalar-tensor theory, because it is possible to do equivalent analysis in a certain frame in which the perturbation equations are simpler. In this paper we revisit the problem of conformal invariances of cosmological perturbations in terms of the covariant approach in which the perturbation variables have clear geometric and physical meanings. We show that with this approach the conformal in…
In the present work, we analyze the evolution of the scalar and tensorial perturbations and the quantities relevant for the physical description of the Universe, as the density contrast of the scalar perturbations and the gravitational waves energy density during the Bose-Einstein condensation of dark matter. The behavior of these parameters during the Bose-Einstein phase transition of dark matter is analyzed in details. To study the cosmological dynamics and evolution of scalar and tensorial perturbations in a Universe with and without cosmological constant we use both analytical and numerica…
We derive the basic equations of the cosmological first-order post-Newtonian approximation from the recently formulated fully nonlinear and exact cosmological perturbation theory in Einstein's gravity. Apparently the latter, being exact, should include the former, and here we use this fact as a new derivation of the former. The complete sets of equations in both approaches are presented without fixing the temporal gauge conditions so that we can use the gauge choice as an advantage. Comparisons between the two approaches are made. Both are potentially important in handling relativistic aspects…
Cosmological perturbations due to statistical thermal fluctuations in a single fluid characterized by an arbitrary equation of state are computed. Formulas to predict the scalar and tensor perturbation spectra and nongaussianity parameters at a given temperature are derived. These results are relevant to any cosmological scenario where cosmic structures may have been seeded thermally instead of originating purely from quantum vacuum fluctuations.
In this paper we investigate the scalar mode of first-order metric perturbations over spatially flat FRW spacetime when the holonomy correction is taken into account in the semi-classical framework of loop quantum cosmology. By means of the Hamiltonian derivation, the cosmological perturbation equations is obtained in longitudinal gauge. It turns out that in the presence of metric perturbation the holonomy effects influence both background and perturbations, and contribute the non-trivial terms $S_{h1}$ and $S_{h2}$ in the cosmological perturbation equations.
We compare the cosmological first-order post-Newtonian (1PN) approximation with the relativistic cosmological linear perturbation theory in a zero-pressure medium with the cosmological constant. We compare equations and solutions in several different gauge conditions available in both methods. In the PN method we have perturbation equations for density, velocity and gravitational potential independently of the gauge condition to 1PN order. However, correspondences with these 1PN equations are available only in certain gauge conditions in the perturbation theory. Equations of perturbed velocity…
We study a class of almost scale-invariant modified gravity theories, using a particular form of $f(R, G) = αR^2 + βG \log G$ where $R$ and $G$ are the Ricci and Gauss-Bonnet scalars, respectively and $α$, $β$ are arbitrary constants. We derive the Einstein-like field equations to first order in cosmological perturbation theory in longitudinal gauge.
We consider linear perturbations about a homogeneous and isotropic cosmological background in the projectable version of Hořava-Lifshitz gravity. Starting from the action for cosmological perturbations, we identify the canonically normalized fluctuation variables. We find that - in contrast to what happens in the non-projectable version of the theory - the extra scalar cosmological perturbation mode is already dynamical at the level of linear perturbations. For values of the parameter $λ$ in the range $1/3 < λ< 1$, the extra mode is ghost-like, for values $1 < λ$ and $λ< 1/3$ it is tachyonic. …
The origin of galactic and extra-galactic magnetic fields is an unsolved problem in modern cosmology. A possible scenario comes from the idea of these fields emerged from a small field, a seed, which was produced in the early universe (phase transitions, inflation, ...) and it evolves in time. Cosmological perturbation theory offers a natural way to study the evolution of primordial magnetic fields. The dynamics for this field in the cosmological context is described by a cosmic dynamo like equation, through the dynamo term. In this paper we get the perturbed Maxwell's equations and compute th…
We use gauge-invariant cosmological perturbation theory to calculate the displacement field that sets the initial conditions for $N$-body simulations. Using first and second-order fully relativistic perturbation theory in the synchronous-comoving gauge, allows us to go beyond the Newtonian predictions and to calculate relativistic corrections to it. We use an Einstein--de Sitter model, including both growing and decaying modes in our solutions. The impact of our results should be assessed through the implementation of the featured displacement in cosmological $N$-body simulations.
Newtonian cosmological perturbation equations valid to full nonlinear order are well known in the literature. Assuming the absence of the transverse-tracefree part of the metric, we present the general relativistic counterpart valid to full nonlinear order. The relativistic equations are presented without taking the slicing (temporal gauge) condition. The equations do have the proper Newtonian and first post-Newtonian limits. We also present the relativistic pressure correction terms in the Newtonian hydrodynamic equations.
Using the covariant formalism, we derive the equations of motion for adiabatic and entropy perturbations at third order in perturbation theory for cosmological models involving two scalar fields. We use these equations to calculate the trispectrum of ekpyrotic and cyclic models in which the density perturbations are generated via the entropic mechanism. In these models, the conversion of entropy into curvature perturbations occurs just before the big bang, either during the ekpyrotic phase or during the subsequent kinetic energy dominated phase. In both cases, we find that the non-linearity pa…
We study cosmological perturbations around self-accelerating solutions to two extensions of nonlinear massive gravity: the quasi-dilaton theory and the mass-varying theory. We examine stability of the cosmological solutions, and the extent to which the vanishing of the kinetic terms for scalar and vector perturbations of self-accelerating solutions in massive gravity is generic when the theory is extended. We find that these kinetic terms are in general non-vanishing in both extensions, though there are constraints on the parameters and background evolution from demanding that they have the co…
Cosmological perturbations are considered in $f(T)$ and in scalar-torsion $f(φ)T$ teleparallel models of gravity. Full sets of linear perturbation equations are accurately derived and analysed at the relevant limits. Interesting features of generalisations to other teleparallel models, spatially curved backgrounds, and rotated tetrads are pointed out.
The celebrated Weinberg theorem in cosmological perturbation theory states that there always exist two adiabatic scalar modes in which the comoving curvature perturbation is conserved on super-horizon scales. In particular, when the perturbations are generated from a single source, such as in single field models of inflation, both of the two allowed independent solutions are adiabatic and conserved on super-horizon scales. There are few known examples in literature which violate this theorem. We revisit the theorem and specify the loopholes in some technical assumptions which violate the theor…
It is shown that a first-order cosmological perturbation theory for the open, flat and closed Friedmann-Lemaître-Robertson-Walker universes admits one, and only one, gauge-invariant variable which describes the perturbation to the energy density and which becomes equal to the usual Newtonian energy density in the non-relativistic limit. The same holds true for the perturbation to the particle number density. Using these two new variables, a new manifestly gauge-invariant cosmological perturbation theory has been developed. Density perturbations evolve diabatically. Perturbations in the total e…
We study cosmological perturbations in models with a single non-local scalar field originating from the string field theory description of the rolling tachyon dynamics. We construct the equation for the energy density perturbations of the non-local scalar field and explicitly prove that for the free field it is identical to a system of local cosmological perturbation equations in a particular model with multiple (maybe infinitely many) local free scalar fields.
I consider the generic model independent predictions of the theory of quantum cosmological perturbations. To describe the stage of cosmic inflation, where these perturbations are amplified, the hydrodynamical approch is used. The inflationary stage is completely characterized by the deviation of the equation of state from cosmological constant which is a smooth function of the number of e-folds until the end of inflation. It is shown that in this case the spectral index should deviate from the flat one at least by 3 percent irrespective of any particular scenario. Given the value of the spectr…
We investigate the cosmological perturbations around FLRW solutions to non- linear massive gravity with a new effective coupling to matter proposed recently. Unlike the case with minimal matter coupling, all five degrees of freedom in the gravity sector propagate on generic self-accelerating FLRW backgrounds. We study the stability of the cosmological solutions and put constraints on the parameters of the theory by demanding the correct sign for the kinetic terms for scalar, vector and tensor perturbations.
Recently, the present authors studied perturbations in the Lemaitre-Tolman-Bondi cosmological model by applying the second-order perturbation theory in the dust Friedmann-Lemaitre-Robertson-Walker universe model. Before this work, the same subject was studied in some papers by analyzing linear perturbations in the Lemaitre-Tolman-Bondi cosmological model under the assumption proposed by Clarkson, Clifton and February, in which two of perturbation variables are negligible. However, it is a non-trivial issue in what situation the Clarkson-Clifton-February assumption is valid. In this paper, we i…
An inflationary universe scenario in the context of tachyon field is studied. This study is done from an ansantz for the effective potential of cosmological perturbations $U(η)$. We describe in great detail the analytical solutions of the scalar and tensor perturbations for two different ansantz in the effective potential of cosmological perturbations; the Easther's model and an effective potential similar to power law inflation. Also we find from the background equations that the effective tachyonic potential $V(φ)$, in both models satisfy the properties of a tachyonic potential. We consider …
We present fully nonlinear and exact cosmological perturbation equations in the presence of multiple components of fluids and minimally coupled scalar fields. We ignore the tensor-type perturbation. The equations are presented without taking the temporal gauge condition in the Friedmann background with general curvature and the cosmological constant. For each fluid component we ignore the anisotropic stress. The multiple component nature, however, introduces the anisotropic stress in the collective fluid quantities. We prove the Newtonian limit of multiple fluids in the zero-shear gauge and th…
We provide a concise review on multi-field inflation and cosmological perturbations. We discuss convenient and physically meaningful bases in terms of which perturbations can be systematically studied. We give formal accounts on the gauge fixing conditions and present the perturbation action in two gauges. We also briefly review non-linear perturbations.
In this paper the implications of a recently proposed phenomenological model of cosmology, the Asymptotic Cosmological Model (ACM), on the behavior of scalar perturbations are studied. Firstly we discuss new fits of the ACM at the homogeneous level, including fits to the Type Ia Supernovae UNION dataset, first CMB peak of WMAP5 and BAOs. The linearized equations of scalar perturbations in the FRW metric are derived. A simple model is used to compute the CMB temperature perturbation spectrum. The results are compared with the treatment of perturbations in other approaches to the problem of the …
We implement the method developed in [1] to construct the most general parametrised action for linear cosmological perturbations of bimetric theories of gravity. Specifically, we consider perturbations around a homogeneous and isotropic background, and identify the complete form of the action invariant under diffeomorphism transformations, as well as the number of free parameters characterising this cosmological class of theories. We discuss, in detail, the case without derivative interactions, and compare our results with those found in massive bigravity.
We review recent developments in the theory of inflation and cosmological perturbations produced from inflation. After a brief introduction of the standard, single-field slow-roll inflation, and the curvature and tensor perturbations produced from it, we discuss possible sources of nonlinear, non-Gaussian perturbations in other models of inflation. Then we describe the so-called $δN$ formalism, which is a powerful tool for evaluating nonlinear curvature perturbations on super Hubble scales.
We present a covariant formulation for constructing general quadratic actions for cosmological perturbations, invariant under a given set of gauge symmetries for a given field content. This approach allows us to analyse scalar, vector and tensor perturbations at the same time in a straightforward manner. We apply the procedure to diffeomorphism invariant single-tensor, scalar-tensor and vector-tensor theories and show explicitly the full covariant form of the quadratic actions in such cases, in addition to the actions determining the evolution of vector and tensor perturbations. We also discus…
By making a suitable generalization of the Starobinsky stochastic inflation, we propose a classical phase space formulation of stochastic inflation which may be used for a quantitative study of decoherence of cosmological perturbations during inflation. The precise knowledge of how much cosmological perturbations have decohered is essential to the understanding of acoustic oscillations of cosmological microwave background (CMB) photons. In order to show how the method works, we provide the relevant equations for a self-interacting inflaton field. For pedagogical reasons and to provide a link t…
The parity violating gravity models based on the symmetric teleparallel gravity have been considered in the literature, but their applications in cosmology and especially the modifications to cosmological perturbations have not been fully explored. In this paper we consider such a simple model which modifies general relativity by a parity non-conserved coupling, within the framework of the symmetric teleparallel gravity. We study in detail its cosmological applications and focus on its cosmological perturbation theory. Besides the already known parity violation in the tensor perturbations, we …
We briefly review the need to perform renormalization of inflationary perturbations to properly work out the physical power spectra. We also summarize the basis of (momentum-space) renormalization in curved spacetime and address several misconceptions found in recent literature on this subject.
We discuss the effect of super-Hubble cosmological fluctuations on the locally measured Hubble expansion rate. We consider a large bare cosmological constant in the early universe in the presence of scalar field matter (the dominant matter component), which would lead to a scale-invariant primordial spectrum of cosmological fluctuations. Using the leading order gradient expansion we show that the expansion rate measured by a (secondary) clock field which is not comoving with the dominant matter component obtains a negative contribution from infrared fluctuations, a contribution whose absolute …
Starting from the energy-momentum of matter and gravitational field in the framework of General Relativity and Teleparallel Gravity, we obtain the energy-momentum complex in flat FRW spacetime. We show that the complex vanishes at background level considering the various prescriptions, that is the Einstein, Møller, Landau-Lifshitz and Bergmann ones. On the other hand, at level of linear cosmological perturbations, the energy-momentum complex is different from zero and coincides in the various prescriptions. Finally, we evaluate the gravitational energy for different cosmological epochs governe…
A mechanism for generating metric perturbations in inflationary models is considered. Long-wavelength inhomogeneities of light scalar fields in a decoupled sector may give rise to superhorizon fluctuations of couplings and masses in the low-energy effective action. Cosmological phase transitions may then occur that are not simultaneous in space, but occur with time lags in different Hubble patches that arise from the long-wavelength inhomogeneities. Here an interesting model in which cosmological perturbations may be created at the electroweak phase transition is considered. The results show t…
Subhorizon approximation is often used in cosmological perturbation theory. In this paper, however, it is shown that the subhorizon approximation is not always a good approximation at least in case of $k$-essence model. We also show that the sound speed given by $k$-essence model exerts a huge influence on the time evolution of the matter density perturbation, and the future observations could clarify the differences between the $Λ$CDM model and $k$-essence model.
We investigate the linear cosmological perturbations in Hořava-Lifshitz gravity with a scalar field. Starting from the most general expressions of the metric perturbations as well as that of a canonical scalar field, we decompose the scalar, vector and tensor parts of the perturbed action. By reducing the Hamiltonian, we find that there are two independent degrees of freedom for the tensor perturbations while none for the vector perturbations. For the scalar perturbations, the remaining number of degrees of freedom, which are all gauge invariant, depends on whether the projectable condition is…
This article lays out a complete framework for an effective theory of cosmological perturbations with corrections from canonical quantum gravity. Since several examples exist for quantum-gravity effects that change the structure of space-time, the classical perturbative treatment must be rethought carefully. The present discussion provides a unified picture of several previous works, together with new treatments of higher-order perturbations and the specification of initial states.
The effective field theory of cosmological perturbations stems from considering a cosmological background solution as a state displaying spontaneous breaking of time translations and (adiabatic) perturbations as the related Nambu-Goldstone modes. With this insight, one can systematically develop a theory for the cosmological perturbations during inflation and, with minor modifications, also describe in full generality the gravitational interactions of dark energy, which are relevant for late-time cosmology. The formalism displays a unique set of Lagrangian operators containing an increasing nu…
We point out a weak side of the commonly used determination of scalar cosmological perturbations lying in the fact that their average values can be nonzero for some matter distributions. It is shown that introduction of the finite-range gravitational potential instead of the infinite-range one resolves this problem. The concrete illustrative density profile is investigated in detail in this connection.
We study linear cosmological perturbations in the ``healthy extension'' of Horava-Lifshitz gravity which has recently been analyzed \cite{BPS2}. We find that there are two degrees of freedom for scalar metric fluctuations, but that one of them decouples in the infrared limit. Also, for appropriate choices of the parameters defining the Lagrangian, the extra mode can be made well-behaved even in the ultraviolet.
We discuss scalar-tensor realizations of the Anamorphic cosmological scenario recently proposed by Ijjas and Steinhardt. Through an analysis of the dynamics of cosmological perturbations we obtain constraints on the parameters of the model. We also study gravitational Parker particle production in the contracting Anamorphic phase and we compute the fraction between the energy density of created particles at the end of the phase and the background energy density. We find that, as in the case of inflation, a new mechanism is required to reheat the universe.
Effective field theory provides a perturbative framework to study the evolution of cosmological large-scale structure. We investigate the underpinnings of this approach, and suggest new ways to compute correlation functions of cosmological observables. We find that, in contrast with quantum field theory, the appropriate effective theory of classical cosmological perturbations involves interactions that are nonlocal in time. We describe an alternative to the usual approach of smoothing the perturbations, based on a path-integral formulation of the renormalization group equations. This technique…
