Order In Cosmological Perturbation Theory Research Articles

Gravitational waves induced from primordial black hole fluctuations: the effect of an extended mass function

The gravitational potential of initially Poisson distributed primordial black holes (PBH) can induce a stochastic gravitational-wave background (SGWB) at second order in cosmological perturbation theory. This SGWB was previously studied in the context of general relativity (GR) and modified gravity setups by assuming a monochromatic PBH mass function. Here we extend the previous analysis in the context of GR by studying the aforementioned SGWB within more physically realistic regimes where PBHs have different masses. In particular, starting from a power-law cosmologically motivated primordial curvature power spectrum with a running spectral index we extract the extended PBH mass function and the associated to it PBH gravitational potential which acts as the source of the scalar induced SGWB. At the end, by taking into account the dynamical evolution of the PBH gravitational potential during the transition from the matter era driven by PBHs to the radiation era we extract the respective GW signal today. Interestingly, in order to trigger an early PBH-dominated era and avoid the GW constraints at BBN we find that the running of the spectral index αs of our primordial curvature power spectrum should be within the narrow range αs ⋲ [3.316,3.355] × 10-3 while at the same time the GW signal is found to be potentially detectable by LISA.

Redshift weighted galaxy number counts

In this paper we introduce the 'redshift fluctuation' as a gauge-invariant cosmological observable and give its fully relativistic expression at first order in cosmological perturbation theory. We show that this corresponds effectively to number counts with a radial window function with vanishing mean which therefore resolve smaller scale radial modes than standard number counts. In a detailed analysis of the angular power spectrum of this new variable, we study the relevance of different relativistic contributions, and how it differs from the conventional observable galaxy number count fluctuations. In order to investigate its utility for future spectroscopic surveys, we perform Fisher forecasts for a Euclid-like and an SKAII-like configuration, as examples. Particular focus is placed on the dependence of the results on the size of the redshift bins and on the cutoff in ℓ adopted in the analysis.

Approximate gauge independence of the induced gravitational wave spectrum

Gravitational waves (GWs) induced by scalar curvature fluctuations are an important source of the cosmological GW background and a crucial counterpart of the primordial black hole scenario. However, doubts have been cast on the theoretically predicted induced GW spectrum due to its seeming gauge dependence. In this paper, we shed light on the gauge dependence issue of the induced GW spectrum in general cosmological backgrounds. First, inspired by the Hamiltonian formalism we provide very simple formulas for the tensor modes at second order in cosmological perturbation theory. We also emphasize the difference between observable and gauge invariant variables. Second, we argue that the Newton (or shear-free) gauge is suitable for both the calculation of induced GWs and the physical interpretation. We then show that, most notably, the induced GW spectrum is invariant under a set of reasonable gauge transformations, i.e. physically well behaved on small scales, once the source term has become inactive. This includes the commonly used flat, constant Hubble and synchronous gauges but excludes the comoving slicing gauge. We also show that a particular solution of the GW equation in a dust dominated universe while the source term is active can be gauged away by a small change of gauge.

Galaxy number counts at second order: an independent approach

Next generation surveys will be capable of determining cosmological parameters beyond percent level. To match this precision, theoretical descriptions should look beyond the linear perturbations to approximate the observables in large scale structure. A quantity of interest is the Number density of galaxies detected by our instruments. This has been focus of interest recently, and several efforts have been made to explain relativistic effects theoretically, thereby testing the full theory. However, the results at nonlinear level from previous works are in disagreement. We present a new and independent approach to computing the relativistic galaxy number counts to second order in cosmological perturbation theory. We derive analytical expressions for the full second order relativistic observed redshift, for the angular diameter distance and for the volume spanned by a survey. Finally, we compare our results with previous works which compute the general distance-redshift relation, finding that our result is in agreement at linear order.

Lagrangian theory for cosmic structure formation with vorticity: Newtonian and post-Friedmann approximations

We study the nonlinear gravitational dynamics of a universe filled with a pressureless fluid and a cosmological constant $\Lambda$ in the context of Newtonian gravity, and in the relativistic post-Friedmann approach proposed in paper I [I. Milillo et al., Phys. Rev. D 92, 023519 (2015).]. The post-Friedmann approximation scheme is based on the $1/c$ expansion of the space-time metric and the energy-momentum tensor, and includes nonlinear Newtonian cosmology. Here we establish the nonlinear post-Friedmann framework in the Lagrangian-coordinates approach for structure formation. For this we first identify a Lagrangian gauge which is suitable for incorporating nonzero vorticity. We analyze our results in two limits: at the leading order we recover the fully nonlinear Newtonian cosmological equations in the Lagrangian formulation, and we provide a space-time metric consistent from the perspective of general relativity. We then linearize our expressions and recover the relativistic results at first order in cosmological perturbation theory. Therefore, the introduced approximation scheme provides a unified treatment for the two leading-order regimes, from the small scales described by Newtonian gravity to the large linear scale, where first-order relativistic cosmological perturbation theory gives a very good description of structure formation.

The intrinsic matter bispectrum in ΛCDM

We present a fully relativistic calculation of the matter bispectrum at second order in cosmological perturbation theory assuming a Gaussian primordial curvature perturbation. For the first time we perform a full numerical integration of the bispectrum for both baryons and cold dark matter using the second-order Einstein-Boltzmann code, SONG. We review previous analytical results and provide an improved analytic approximation for the second-order kernel in Poisson gauge which incorporates Newtonian nonlinear evolution, relativistic initial conditions, the effect of radiation at early times and the cosmological constant at late times. Our improved kernel provides a percent level fit to the full numerical result at late times for most configurations, including both equilateral shapes and the squeezed limit. We show that baryon acoustic oscillations leave an imprint in the matter bispectrum, making a significant impact on squeezed shapes.

Relativistic perturbations in ΛCDM: Eulerian & Lagrangian approaches

We study the relativistic dynamics of a pressure-less and irrotational fluid of dark matter (CDM) with a cosmological constant (Λ), up to second order in cosmological perturbation theory. In our analysis we also account for vector and tensor perturbations and include primordial non-Gaussianity. We consider three gauges: the synchronous-comoving gauge, the Poisson gauge and the total matter gauge, where the first is the unique relativistic Lagrangian frame of reference, and the latters are convenient gauge choices for Eulerian frames. Our starting point is the metric and fluid variables in the Poisson gauge up to second order. We then perform the gauge transformations to the synchronous-comoving gauge and subsequently to the total matter gauge. Our expressions for the metrics, densities, velocities, and the gauge generators are novel and coincide with known results in the limit of a vanishing cosmological constant.

The bispectrum of relativistic galaxy number counts

We discuss the dominant terms of the relativistic galaxy number counts to second order in cosmological perturbation theory on sub-Hubble scales and on intermediate to large redshifts. In particular, we determine their contribution to the bispectrum. In addition to the terms already known from Newtonian second order perturbation theory, we find that there are a series of additional `lensing-like' terms which contribute to the bispectrum. We derive analytical expressions for the full leading order bispectrum and we evaluate it numerically for different configurations, indicating how they can be measured with upcoming surveys. In particular, the new `lensing-like' terms are not negligible for wide redshift bins and even dominate the bispectrum at well separated redshifts. This offers us the possibility to measure them in future surveys.

A new approach to the propagation of light-like signals in perturbed cosmological backgrounds

We present a new method to compute the deflection of light rays in a perturbed FLRW geometry. We exploit the properties of the Geodesic Light Cone (GLC) gauge where null rays propagate at constant angular coordinates irrespectively of the given (inhomogeneous and/or anisotropic) geometry. The gravitational deflection of null geodesics can then be obtained, in any other gauge, simply by expressing the angular coordinates of the given gauge in terms of the GLC angular coordinates. We apply this method to the standard Poisson gauge, including scalar perturbations, and give the full result for the deflection effect in terms of the direction of observation and observed redshift up to second order, and up to third order for the leading lensing terms. We also compare our results with those presently available in the literature and, in particular, we provide a new non trivial check of a previous result on the luminosity-redshift relation up to second order in cosmological perturbation theory.

Galaxy number counts to second order and their bispectrum

We determine the number counts to second order in cosmological perturbation theory in the Poisson gauge and allowing for anisotropic stress. The calculation is performed using an innovative approach based on the recently proposed ``geodesic light-cone" gauge. This allows us to determine the number counts in a purely geometric way, without using Einstein's equation. The result is valid for general dark energy models and (most) modified gravity models. We then evaluate numerically some relevant contributions to the number counts bispectrum. In particular we consider the terms involving the density, redshift space distortion and lensing.

Nonlinear relativistic corrections to cosmological distances, redshift and gravitational lensing magnification: I. Key results

The next generation of telescopes will usher in an era of precision cosmology, capable of determining the cosmological model to beyond the percent level. For this to be effective, the theoretical model must be understood to at least the same level of precision. A range of subtle relativistic effects remain to be explored theoretically, and offer the potential for probing general relativity in this new regime. We present the distance–redshift relation to second order in cosmological perturbation theory for a general dark energy model. This relation determines the magnification of sources at high precision, as well as redshift space distortions in the mildly non-linear regime. We identify a range of new lensing effects, including: double-integrated and nonlinear-integrated Sachs–Wolfe contributions, transverse Doppler effects, lensing from the induced vector mode and gravitational wave backgrounds, in addition to lensing from the second-order potential. Modifications to Doppler lensing from redshift space distortions are identified. Finally, we find a new double-coupling between the density fluctuations integrated along the line of sight, and gradients in the density fluctuations coupled to transverse velocities along the line of sight. These can be large and thus offer important new probes of gravitational lensing and general relativity. This paper accompanies paper II (Umeh, Clarkson and Maartens 2014 Class. Quantum Grav. 31 205001), where a comprehensive derivation is given.

Tensor mode backreaction during slow-roll inflation

We consider the backreaction of the long wavelength tensor modes produced during a slow-roll inflationary regime driven by a single scalar field in a spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker background geometry. We investigate the effects on nonlocal observables such as the effective (averaged) expansion rate and equation of state at second order in cosmological perturbation theory. The coupling between scalar and tensor perturbations induces at second-order new tensor backreaction terms beyond the one already present in a de Sitter background. We analyze in detail the effects seen by the class of observers comoving with the inflaton field (taken as a clock) and the class of free-falling observers. In both cases the quantum backreaction is at least $1/\ensuremath{\epsilon}$ (with $\ensuremath{\epsilon}$ the slow-roll parameter) larger than the one which can be naively inferred from a de Sitter background. In particular, we compute the effect for a free massive inflaton model and obtain in both cases a quantum correction on the background expansion rate of the order of ${H}^{4}/({m}^{2}{M}_{\text{Pl}}^{2})$. A short discussion on the issue of the breakdown of perturbation theory is given.

The Poisson equation at second order in relativistic cosmology

We calculate the relativistic constraint equation which relates the curvature perturbation to the matter density contrast at second order in cosmological perturbation theory. This relativistic ''second order Poisson equation'' is presented in a gauge where the hydrodynamical inhomogeneities coincide with their Newtonian counterparts exactly for a perfect fluid with constant equation of state. We use this constraint to introduce primordial non-Gaussianity in the density contrast in the framework of General Relativity. We then derive expressions that can be used as the initial conditions of N-body codes for structure formation which probe the observable signature of primordial non-Gaussianity in the statistics of the evolved matter density field.

Gauge invariant backreaction in general single field models of inflation

In a general single field inflationary model, we consider the effects of long wavelength scalar fluctuations on the effective expansion rate and equation of state seen by a class of free-falling observers, using a physical gauge invariant formulation. In a previous work we showed that for a free massive inflaton no backreaction is observed within some constraints. In this paper we extend the validity of our previous results to the case of an arbitrary self-interacting inflation potential, working to second order in cosmological perturbation theory and to all order in slow-roll approximation. For these general inflationary models, we also show the equivalence of the free-falling observers to the ones comoving with the inflaton field.

Non-Gaussianity in the cosmic microwave background induced by dipolar dark matter

In previous work [L. Blanchet and A. Le Tiec, Phys. Rev. D 80 (2009) 023524], motivated by the phenomenology of dark matter at galactic scales, a model of dipolar dark matter (DDM) was introduced. At linear order in cosmological perturbations, the dynamics of the DDM was shown to be identical to that of standard cold dark matter (CDM). In this paper, the DDM model is investigated at second order in cosmological perturbation theory. We find that the internal energy of the DDM fluid modifies the curvature perturbation generated by CDM with a term quadratic in the dipole field. This correction induces a new type of non-Gaussianity in the bispectrum of the curvature perturbation with respect to standard CDM. Leaving unspecified the primordial amplitude of the dipole field, which could in principle be determined by a more fundamental description of DDM, we find that, in contrast with usual models of primordial non-Gaussianities, the non-Gaussianity induced by DDM increases with time after the radiation-matter equality on super-Hubble scales. This distinctive feature of the DDM model, as compared with standard CDM, could thus provide a specific signature in the CMB and large-scale structure probes of non-Gaussianity.

Estimating the amount of vorticity generated by cosmological perturbations in the early universe

We estimate the amount of vorticity generated at second order in cosmological perturbation theory from the coupling between first order energy density and nonadiabatic pressure, or entropy, perturbations. Assuming power law input spectra for the source terms, and working in a radiation background, we calculate the wave number dependence of the vorticity power spectrum and its amplitude. We show that the vorticity generated by this mechanism is non-negligible on small scales, and hence should be taken into consideration in current and future cosmic microwave background experiments.

Nonlinear matter spectra in growing neutrino quintessence

We investigate the nonlinear power spectra of density perturbations and acoustic oscillations in growing neutrino quintessence. In this scenario, the neutrino mass has a strong dependence on the quintessence field. The induced coupling stops the evolution of the field when the neutrinos become nonrelativistic, and triggers the transition to the accelerating phase of the cosmological expansion. For the calculation of the nonlinear spectra we employ the time renormalization group, which resums subsets of diagrams of arbitrarily high order in cosmological perturbation theory. At redshifts around five, the neutrino fluctuations are still linear and acoustic oscillations are present in the neutrino power spectrum, induced by the acoustic oscillations in the baryonic and dark-matter sectors. The neutrino perturbations become nonlinear at redshifts around three. The mode coupling generated by the nonlinearities erases the oscillations in the neutrino spectrum at some redshift above two. There is a potential danger that at later times the influence of the gravitational potentials induced by the neutrino inhomogeneities could erase the oscillations from the baryonic and dark-matter spectra, making the scenario incompatible with observations. For the scenario to be viable, the neutrino-induced gravitational potentials in the range of baryon acoustic oscillations should not grow to average values much larger than 10−4. The magnitude of the expected potentials is still not known reliably, as the process of structure formation is poorly understood in growing neutrino quintessence. The time renormalization group cannot describe the effects of nonlinear clustering. Alternative methods, such as hydrodynamic simulations, must be empoloyed for the calculation of the spectra at low redshifts.

Boltzmann hierarchy for the cosmic microwave background at second order including photon polarization

Non-gaussianity and B-mode polarization are particularly interesting features of the cosmic microwave background, as -- at least in the standard model of cosmology -- their only sources to first order in cosmological perturbation theory are primordial, possibly generated during inflation. If the primordial sources are small, the question arises how large is the non-gaussianity and B-mode background induced in second-order from the initially gaussian and scalar perturbations. In this paper we derive the Boltzmann hierarchy for the microwave background photon phase-space distributions at second order in cosmological perturbation theory including the complete polarization information, providing the basis for further numerical studies. As an aside we note that the second-order collision term contains new sources of B-mode polarization and that no polarization persists in the tight-coupling limit.

Numerical calculation of second order perturbations

We numerically solve the Klein-Gordon equation at second order incosmological perturbation theory in closed form for a single scalarfield, describing the method employed in detail. We use the slow-rollversion of the second order source term and argue that our method isextendable to the full equation. We consider two standard single fieldmodels and find that the results agree with previous calculationsusing analytic methods, where comparison is possible. Our procedureallows the evolution of second order perturbations in general and thecalculation of the non-linearity parameter fNL to be examined incases where there is no analytical solution available.

Vorticity generation at second order in cosmological perturbation theory

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.