Uniform Curvature Gauge Research Articles

Scalar Field Evolution at Background and Perturbation Levels for a Broad Class of Potentials

AbstractIn this paper, a non‐interacting scalar field cosmology with an arbitrary potential using the f‐deviser method that relies on the differentiability properties of the potential is investigated. Using this alternative mathematical approach, a unified dynamical system analysis at a scalar field's background and perturbation levels with arbitrary potentials is presented. For illustration, a monomial and double exponential potential is considered. These two classes of potentials comprise the asymptotic behavior of several classes of scalar field potentials, and, therefore, they provide the skeleton for the typical behavior of arbitrary potentials. Moreover, the linear cosmological perturbations in the matterless case by considering three scalar perturbations: the evolution of the Bardeen potentials, the comoving curvature perturbation, the so‐called Sasaki‐Mukhanov variable, or the scalar field perturbation in uniform curvature gauge is analyzed. Finally, an exhaustive dynamical system analysis for each scalar perturbation is presented, including the evolution of Bardeen potentials in the presence of matter.

Axion as a fuzzy dark matter candidate: proofs in different gauges

Axion as a coherently oscillating massive scalar field is known to behave as a zero-pressure irrotational fluid with characteristic quantum stress on a small scale. In relativistic perturbation theory, the case was proved in the axion-comoving gauge up to fully nonlinear and exact order. Our basic assumption is that the field is oscillating with Compton frequency and the Compton wavelength is smaller than the horizon scale. Here, we revisit the relativistic proof to the linear order in the other gauge conditions. We show that the same equation for density perturbation known in the non-relativistic treatment can be derived in two additional gauge conditions: the zero-shear gauge and the uniform-curvature gauge. The uniform-expansion gauge fails to get the aimed equation, and the quantum stress term is missing in the synchronous gauge. For comparison, we present the relativistic density perturbation equations in the zero-pressure fluid in these gauge conditions. Except for the comoving and the synchronous gauge, the equations strikingly differ from the axion case. We clarify that the relativistic analysis based on time averaging is valid for scales larger than the Compton wavelength. Below the Compton wavelength, the field is not oscillating, and our oscillatory ansatz does not apply. We suggest an equation valid in all scales in the comoving gauge. For comparison, we review the non-relativistic quantum hydrodynamics and present the Schrödinger equation to first-order post-Newtonian expansion in the cosmological context.

Effective anisotropic stresses of the relic gravitons

The effective anisotropic stresses induced by the scalar modes of the geometry depend on the coordinate system so that the comparison of the competing results is ultimately determined by the evolution of the pivotal variables in each particular gauge. After arguing that the only reasonable physical coordinate systems for this problem are the ones where the gauge freedom is completely fixed (like the longitudinal and the uniform curvature gauges), we propose a novel gauge-invariant strategy for the comparison of gauge-dependent results. Instead of employing the pivotal variables of a given coordinate system, the effective anisotropic stress is solely expressed in terms of the gravitating normal modes of the plasma and in terms of their conformal time derivatives. The new approach is explicitly gauge-invariant and when the wavelengths of the normal modes are either shorter or larger than the sound horizon, the physical limits of the anisotropic stresses are determined without relying on the specific details of the background evolution. The relevance of the proposed strategy is discussed in the general situation where the scalar anisotropic stress and the nonadiabatic pressure fluctuations are simultaneously present. We finally argue that the anisotropic stress can be most efficiently obtained from the second-order effective action of the curvature inhomogeneities.

Gauge transformation of scalar induced gravitational waves

The gauge dependence of the scalar induced gravitational waves (SIGWs) generated at the second order imposes a challenge to the discussion of the secondary gravitational waves generated by scalar perturbations. We provide a general formula that is valid in any gauge for the calculation of SIGWs and the relationship for SIGWs calculated in various gauges under the coordinate transformation. The formula relating SIGWs in the Newtonian gauge to other gauges is used to calculate SIGWs in six different gauges. We find that the Newtonian gauge, the uniform curvature gauge, the synchronous gauge and the uniform expansion gauge yield the same result for the energy density of SIGWs. We also identify and eliminate the pure gauge modes that exist in the synchronous gauge. In the total matter gauge and the comoving orthogonal gauge, the energy density of SIGWs increases as $\eta^2$. While in the uniform density gauge, the energy density of SIGWs increases as $\eta^6$.

Gauge dependence of gravitational waves generated at second order from scalar perturbations

We revisit and clarify the gauge dependence of gravitational waves generated at second order from scalar perturbations. In a universe dominated by a perfect fluid with a constant equation-of-state parameter $w$, we compute the energy density of such induced gravitational waves in the Newtonian, comoving, and uniform curvature gauges. Huge differences are found between the Newtonian and comoving gauge results for any $w \,(\ge 0)$. This is always caused by the perturbation of the shift vector. Interestingly, the Newtonian and uniform curvature gauge calculations give the same energy density for $w>0$. In the case of $w=0$, the uniform curvature gauge result differs only by a factor from that of the comoving gauge, but deviates significantly from that of the Newtonian gauge. Our calculation is done analytically for $w=0$ and $w=1/3$, and our result is consistent with the previous numerical one.

Scalar induced gravitational waves in different gauges

In this letter we calculate the scalar induced gravitational waves (SIGWs) accompanying the formation of primordial black hole during the radiation dominated era in three different gauges, i.e. synchronous gauge, Newton gauge and uniform curvature gauge, and we find that the energy density spectra of SIGWs, $\ogw(k)$, are identical in these three different gauges.

Second-order cosmological perturbations: New conserved quantities and the general solution at super-horizon scale

The study of long wavelength scalar perturbations, in particular the existence of conserved quantities when the perturbations are adiabatic, plays an important role in e.g. inflationary cosmology. In this paper we present some new conserved quantities at second order and relate them to the curvature perturbation in the uniform density gauge, $\zeta$, and the comoving curvature perturbation, ${\cal R}$. We also, for the first time, derive the general solution of the perturbed Einstein equations at second order, which thereby contains both growing and decaying modes, for adiabatic long wavelength perturbations for a stress-energy tensor with zero anisotropic stresses and zero heat flux. The derivation uses the total matter gauge, but results are subsequently translated to the uniform curvature and Poisson (longitudinal, zero shear) gauges.

Second order density perturbations for dust cosmologies

We present simple expressions for the relativistic first and second order fractional density perturbations for Friedmann-Lema\^{\i}tre cosmologies with dust, in four different gauges: the Poisson, uniform curvature, total matter and synchronous-comoving gauges. We include a cosmological constant and arbitrary spatial curvature in the background. A distinctive feature of our approach is our description of the spatial dependence of the perturbations using a canonical set of quadratic differential expressions involving an arbitrary spatial function that arises as a conserved quantity. This enables us to unify, simplify and extend previous seemingly disparate results. We use the primordial matter and metric perturbations that emerge at the end of the inflationary epoch to determine the additional arbitrary spatial function that arises when integrating the second order perturbation equations. This introduces a non-Gaussianity parameter into the expressions for the second order density perturbation. In the special case of zero spatial curvature we show that the time evolution simplifies significantly, and requires the use of only two nonelementary functions, the so-called growth suppression factor at the linear level, and one new function at the second order level. We expect that the results will be useful in applications, for example, studying the effects of primordial non-Gaussianity on the large scale structure of the Universe.

Simple expressions for second order density perturbations in standard cosmology

In this paper we present four simple expressions for the relativistic first and second order fractional density perturbations for ΛCDM cosmologies in different gauges: the Poisson, uniform curvature, total matter and synchronous gauges. A distinctive feature of our approach is the use of a canonical set of quadratic differential expressions involving an arbitrary spatial function, the so-called comoving curvature perturbation, to describe the spatial dependence, which enables us to unify, simplify and extend previous seemingly disparate results. The simple structure of the expressions makes the evolution of the density perturbations completely transparent and clearly displays the effect of the cosmological constant on the dynamics, namely that it stabilizes the perturbations. We expect that the results will be useful in applications, for example, studying the effects of primordial non-Gaussianity on the large scale structure of the universe.

Fluctuations of inflationary magnetogenesis

This analysis aims at exploring what can be said about the growth rate of magnetized inhomogeneities under two concurrent hypotheses: a phase of quasi---de Sitter dynamics driven by a single inflaton field and the simultaneous presence of a spectator field coupled to gravity and to the gauge sector. Instead of invoking ad hoc correlations between the various components, the system of scalar inhomogeneities is diagonalized in terms of two gauge-invariant quasinormal modes whose weighted sum gives the curvature perturbations on comoving orthogonal hypersurfaces. The predominance of the conventional adiabatic scalar mode implies that the growth rate of magnetized inhomogeneities must not exceed 2.2 in Hubble units if the conventional inflationary phase is to last about 70 e-folds and for a range of slow roll parameters between 0.1 and 0.001. Longer and shorter durations of the quasi---de Sitter stage lead, respectively, either to tighter or to looser bounds which are anyway more constraining than the standard backreaction demands imposed on the gauge sector. Since a critical growth rate of order 2 leads to a quasiflat magnetic energy spectrum, the upper bounds on the growth rate imply a lower bound on the magnetic spectral index. The advantages of the uniform curvature gauge are emphasized and specifically exploited throughout the treatment of the multicomponent system characterizing this class of problems.

Backreaction: Gauge and frame dependences

The cosmological backreaction from perturbations is clearly gauge-dependent, and obviously depends on the choice of averaged Hubble rate. We consider two common choices of Hubble rate and advocate the use of comoving volume-preserving gauges. We highlight two examples valid to an appropriate order in perturbation theory, uniform curvature gauge, which is as close to volume-preserving as possible, and a spatially-traceless uniform cold dark matter gauge which preserves the volume to linear order. We demonstrate the strong gauge- and frame-dependences in averaging. In traceless uniform CDM gauge the backreaction exhibits a strong ultra-violet divergence and can be tuned to an arbitrary magnitude with an appropriate choice of smoothing scale. In uniform curvature gauge we find that for a choice of Hubble rate locked to the spatial surface the backreaction vanishes identically, while for a Hubble rate defined from a fluid's expansion scalar the effective energy density at the current epoch in an Einstein-de Sitter universe is Omega_eff~5e-4, slightly bigger than but in broad agreement with previous results in conformal Newtonian gauge.

A simplified structure for the second order cosmological perturbation equations

Increasingly accurate observations of the cosmic microwave background and the large scale distribution of galaxies necessitate the study of nonlinear perturbations of Friedmann-Lemaitre cosmologies, whose equations are notoriously complicated. In this paper we present a new derivation of the governing equations for second order perturbations within the framework of the metric-based approach that is minimal, as regards amount of calculation and length of expressions, and flexible, as regards choice of gauge and stress-energy tensor. Because of their generality and the simplicity of their structure our equations provide a convenient starting point for determining the behaviour of nonlinear perturbations of FL cosmologies with any given stress-energy content, using either the Poisson gauge or the uniform curvature gauge.

SECOND-ORDER SOLUTIONS OF COSMOLOGICAL PERTURBATION IN THE MATTER-DOMINATED ERA

We present the growing mode solutions of cosmological perturbations to the second order in the matter dominated era. We also present several gauge-invariant combinations of perturbation variables to the second order in most general fluid context. Based on the solutions we study the Newtonian correspondence of relativistic perturbations to the second order. In addition to the previously known exact relativistic/Newtonian correspondence of density and velocity perturbations to the second order in the comoving gauge, here we show that in the sub-horizon limit we have the correspondences for density, velocity and potential perturbations in the zero-shear gauge and in the uniform-expansion gauge to the second order. Density perturbation in the uniform curvature gauge also shows the correspondence to the second order in the sub-horizon scale. We also identify the relativistic gravitational potential which shows exact correspondence to the Newtonian one to the second order.

Comparing two different formulations of metric cosmological perturbation theory

We compare and contrast two different metric-based formulations of nonlinear cosmological perturbation theory: the MW2009 approach in Malik and Wands (2009 Phys. Rep. 475 1) following Bardeen and the recent approach of the paper KN2010 (Nakamura 2010 Adv. Astron. 2010 576273). We present each formulation separately. In the MW2009 approach, one considers the gauge transformations of perturbative quantities, choosing a gauge by requiring that certain quantities vanish, rendering all other variables gauge invariant. In the KN2010 formalism, one decomposes the metric tensor into a gauge variant and gauge invariant part from the outset. We compare the two approaches in both the longitudinal and uniform curvature gauges. In the longitudinal gauge, we find that Nakamura’s gauge invariant variables correspond exactly to those in the longitudinal gauge (i.e. for scalar perturbations to the Bardeen potentials), and in the uniform curvature gauge we obtain the usual relationship between gauge invariant variables in the flat and longitudinal gauge. Thus, we show that these two approaches are equivalent.

Gauges and cosmological backreaction

We present a formalism for spatial averaging in cosmology applicable to general spacetimes and coordinates, and allowing the easy incorporation of a wide variety of matter sources. We apply this formalism to a Friedmann-Lemaȋtre-Robertson-Walker universe perturbed to second-order and present the corrections to the background in an unfixed gauge. We then present the corrections that arise in uniform curvature and conformal Newtonian gauges.

Non-Gaussianity from the trispectrum in general single field inflation

We compute the fourth order action in perturbation theory for scalar and second order tensor perturbations for a minimally coupled single field inflationary model, where the inflaton's Lagrangian is a general function of the field's value and its kinetic energy. We obtain the fourth order interaction Hamiltonian in two gauges, the comoving gauge and the uniform curvature gauge. Using the comoving gauge action we calculate the trispectrum at leading order in slow roll, finding agreement with a previously known result in the literature. We point out that, in general, to obtain the correct leading order trispectrum one cannot ignore second order tensor perturbations as previously done by others. The next-to-leading order corrections may become detectable depending on the shape, and we provide the necessary formalism to calculate them.

A not so short note on the Klein–Gordon equation at second order

We give the governing equations for multiple scalar fields in a flat Friedmann–Robertson–Walker(FRW) background spacetime on all scales, allowing for metric and field perturbations upto second order. We then derive the Klein–Gordon equation at second order in closed formin terms of gauge-invariant perturbations of the fields in the uniform curvature gauge. Wealso give a simplified form of the Klein–Gordon equation using the slow-roll approximation.

Cosmological structure problem of the ekpyrotic scenario

We address the perturbation power spectrum generated in the recently proposed ekpyrotic scenario by Khoury et al. The issue has been raised recently by Lyth who used the conventional method based on a conserved variable in the large-scale limit, and derived different results from Khoury et al. The calculation is straightforward in the uniform-curvature gauge where the generated blue spectrum with a suppressed amplitude survives as the final spectrum. Whereas, although the metric fluctuations become unimportant and a scale-invariant spectrum is generated in the zero-shear gauge, the mode does not survive the bounce, but has the same final result. Therefore, an exponential potential leads to a power-law expansion or contraction $a\ensuremath{\propto}|t{|}^{p},$ and the power p dictates the final power spectra of both the scalar and tensor structures. If $p\ensuremath{\ll}1$ as one realization of the ekpyrotic scenario suggests, the results are ${n}_{S}\ensuremath{-}1\ensuremath{\simeq}2\ensuremath{\simeq}{n}_{T}$ and the amplitude of the scalar perturbation is suppressed relative to that of the gravitational wave by a factor $\sqrt{p}/2.$ Both results confirm Lyth's. An observation is made on the constraint on the dynamics of the seed generating stage from the requirement of a scale-invariant spectrum.

Perturbation evolution in cosmologies with a decaying cosmological constant

Structure formation models with a cosmological constant are successful in explaining large-scale structure data, but are threatened by the magnitude-redshift relation for type Ia supernovae. This has led to discussion of models where the cosmological ``constant'' decays with time, which might anyway be better motivated in a particle physics context. The simplest such models are based on scalar fields, and general covariance demands that a time-evolving scalar field also supports spatial perturbations. We consider the effect of such perturbations on the growth of adiabatic energy density perturbations in a cold dark matter component. We study two types of model, one based on an exponential potential for the scalar field and the other on a pseudo Nambu Goldstone boson. For each potential, we study two different scenarios, one where the scalar field presently behaves as a decaying cosmological constant and one where it behaves as dust. The initial scalar field perturbations are fixed by the adiabatic condition, as expected from the inflationary cosmology, though in fact we show that the choice of initial condition is of little importance. Calculations are carried out in both the zero-shear (conformal Newtonian) and uniform-curvature gauges. We find that both potentials allow models which can provide a successful alternative to cosmological constant models.

Cosmological perturbations in generalized gravity theories: conformal transformation

A broad class of generalized gravity theories can be cast into Einstein gravity with a minimally coupled scalar field using a suitable conformal rescaling of the metric. Using this conformal equivalence between the theories, we derived the equations for the background and the perturbations, and the general asymptotic solutions for the perturbations in the generalized gravity from the simple results known in the minimally coupled scalar field. Results for the scalar and tensor perturbations can be presented in unified forms. The large-scale evolutions for both perturbations are characterized by corresponding conserved quantities. The simple result for the scalar perturbation is possible mainly due to our proper choice of a gauge-invariant combination which corresponds to the perturbed scalar field in the uniform-curvature gauge.