Second-order Perturbations Research Articles

Second-order perturbations of the Schwarzschild spacetime: Practical, covariant, and gauge-invariant formalisms

Second-order perturbations of the Schwarzschild spacetime: Practical, covariant, and gauge-invariant formalisms

Quadratic perturbations of the Schwarzschild black hole: the algebraically special sector

We investigate quadratic algebraically special perturbations (ASPs) of the Schwarzschild black hole. Their dynamics are derived from the expansion up to second order in perturbation of the most general algebraically special twisting vacuum solution of general relativity. Following this strategy, we present analytical expressions for the axial-axial, polar-polar and polar-axial source terms entering in the dynamical equations. We show that these complicated inhomogeneous equations can be solved analytically and we present explicit expressions for the profiles of the quadratic ASPs. As expected, they exhibit exponential growth both at the past and future horizons even in the non-linear regime. We further use this result to analyze the quadratic zero modes and their interpretation in terms of quadratic corrections to mass and spin of the Schwarzschild black hole. The present work provides a direct extension beyond the linear regime of the original work by Couch and Newman.

Primordial gravitational waves assisted by cosmological scalar perturbations

Primordial gravitational waves are a crucial prediction of inflation theory, and their detection through their imprints on the cosmic microwave background is actively being pursued. However, these attempts have not yet been successful. In this paper, we propose a novel approach to detect primordial gravitational waves by searching for a signal of second-order tensor perturbations. These perturbations were produced due to nonlinear couplings between the linear tensor and scalar perturbations in the early universe. We anticipate a blue-tilted tensor spectral index, and suggest that the tensor-to-scalar ratio can potentially be measured with high precision using a detector network composed of the ground-based Einstein Telescope and the space-borne LISA project on a decade timescale.

Primordial Gravitational Wave- and Curvature Perturbation-Induced Energy Density Perturbations

We study the second-order scalar and density perturbations generated by Gaussian curvature perturbations and primordial gravitational waves in the radiation-dominated era. After presenting all the possible second-order source terms, we obtain the explicit expressions of the kernel functions and the power spectra of the second-order scalar perturbations. We show that the primordial gravitational waves might affect second-order energy density perturbation δ(2)=δρ(2)/ρ(0) significantly. The effects of primordial gravitational waves are studied in terms of different kinds of primordial power spectra.

Scale-Invariant Mode in Collisionless Spherical Stellar Systems

An analytical solution for the perturbed equations, applicable to all ergodic models of collisionless spherical stellar systems with a single length parameter, has been derived. This solution corresponds to variations in this parameter, i.e., the expansion or contraction of the sphere while conserving total mass. During this process, the system maintains an equilibrium state. The simplicity of the solution allows for the explicit expression of the distribution function, potential, and density across all orders of perturbation theory. This, in turn, aids in clarifying the concept of perturbation energy, which, being of second order in amplitude, cannot be calculated using linear theory. It is demonstrated that the correct expression for perturbation energy, accounting for second-order perturbations, does not align with the well-known expression for perturbation energy via a quadratic form, derived from first-order perturbations within linear theory. However, both these energies are integrals of motion and differ only by a constant. The derived solution can be utilized to verify the correctness of codes and the precision of calculations in the numerical study of collisionless stellar models.

The effective field theory approach to the strong coupling issue in f(T) gravity

We investigate the scalar perturbations and the possible strong coupling issues of f(T) around a cosmological background, applying the effective field theory (EFT) approach. We revisit the generalized EFT framework of modified teleparallel gravity, and apply it by considering both linear and second-order perturbations for f(T) theory. We find that no new scalar mode is present in both linear and second-order perturbations in f(T) gravity, which suggests a strong coupling problem. However, based on the ratio of cubic to quadratic Lagrangians, we provide a simple estimation of the strong coupling scale, a result which shows that the strong coupling problem can be avoided at least for some modes. In conclusion, perturbation behaviors that at first appear problematic may not inevitably lead to a strong coupling problem, as long as the relevant scale is comparable with the cutoff scale M of the applicability of the theory.

Enhanced Gravitational Waves from Inflaton Oscillons.

In broad classes of inflationary models the period of accelerated expansion is followed by fragmentation of the inflaton scalar field into localized, long-lived, and massive oscillon excitations. We demonstrate that matter dominance of oscillons, followed by their rapid decay, significantly enhances the primordial gravitational wave (GW) spectrum. These oscillon-induced GWs, sourced by second-order perturbations, are distinct and could be orders of magnitude lower in frequency than the previously considered GWs associated with oscillon formation. We show that detectable oscillon-induced GW signatures establish direct tests independent from cosmic microwave background radiation for regions of parameter space of monodromy, and logarithmic and pure natural (plateau) potential classes of inflationary models, among others. We demonstrate that oscillon-induced GWs in a model based on pure natural inflation could be directly observable with the Einstein Telescope, Cosmic Explorer, and DECIGO. These signatures offer a new route for probing the underlying inflationary physics.

Measuring the primordial curvature perturbations from the scalar induced gravitational waves

The scalar induced gravitational waves are produced from primordial curvature perturbations in the second order of perturbations. We constrain the fractional energy density of scalar induced gravitational waves from gravitational waves observations. If there is no detection of the scalar induced gravitational waves, the fractional energy density of scalar induced gravitational waves is constrained by some upper limits. Depends on these upper limits, we can obtain the constraints on the power spectrum of the primordial curvature perturbations. For a power-law scalar power spectrum, the constraints on the power spectrum are affected by adding the upper limit of scalar induced gravitational waves from Square Kilometer Array (SKA). In the standard model, the mean values of the scalar amplitude and the spectral index shift to lower values when SKA is added to the combination of Cosmic Microwave Background (CMB) and Baryon Acoustic Oscillation (BAO) datasets, namely $\ln(10^{10}A_s)=3.038\pm0.013$ and $n_s=0.9589^{+0.0021}_{-0.0011}$ at $68\%$ confidence level. We also consider the effects of the existing ground-based gravitational-wave detectors, the existing Pulsar Timing Arrays (PTAs) and Five-hundred-meter Aperture Spherical radio Telescope (FAST), while the constraints from CMB+BAO datasets are totally within their upper limits of scalar induced gravitational waves. Furthermore, we characterize the scalar fluctuation spectrum in terms of the spectral index $n_s$ and its first two derivatives. We calculate corresponding power spectrum of scalar induced gravitational waves theoretically and give the constraints on the running of the spectral index and the running of the running of the spectral index.

Backreaction in cosmic screening approach

We investigate the backreaction of nonlinear perturbations on the global evolution of the Universe within the cosmic screening approach. To this end, we have considered the second-order scalar perturbations. An analytical study of these perturbations followed by a numerical evaluation shows that, first, the corresponding average values have a negligible backreaction effect on the Friedmann equations and, second, the second-order correction to the gravitational potential is much less than the first-order quantity. Consequently, the expansion of perturbations into orders of smallness in the cosmic screening approach is correct.

Generation and propagation of nonlinear quasinormal modes of a Schwarzschild black hole

In the analysis of a binary black hole coalescence, it is necessary to include gravitational self-interactions in order to describe the transition of the gravitational wave signal from the merger to the ringdown stage. In this paper we study the phenomenology of the generation and propagation of nonlinearities in the ringdown of a Schwarzschild black hole, using second-order perturbation theory. Following earlier work, we show that the Green's function and its causal structure determines how both first-order and second-order perturbations are generated, and hence highlight that both of these solutions share some physical properties. In particular, we discuss the sense in which both linear and quadratic quasi-normal modes (QNMs) are generated in the vicinity of the peak of the gravitational potential barrier (loosely referred to as the light ring). Among the second-order perturbations, there are solutions with linear QNM frequencies (whose amplitudes are thus renormalized from their linear values), as well as quadratic QNM frequencies with a distinct spectrum. Moreover, we show using a WKB analysis that, in the eikonal limit, waves generated inside the light ring propagate towards the black hole horizon, and only waves generated outside propagate towards an asymptotic observer. These results might be relevant for recent discussions on the validity of perturbation theory close to the merger. Finally, we argue that even if nonlinearities are small, quadratic QNMs may be detectable and would likely be useful for improving ringdown models of higher angular harmonics and future tests of gravity.

Dynamic Instabilities and Pattern Formation in Diffusive Epidemic Spread

The COVID-19 pandemic has refocused research on mathematical modeling and analysis of epidemic dynamics. We analytically investigate a partial differential equation (PDE) based, compartmental model of spatiotemporal epidemic spread, marked by strongly nonlinear infection forces representing the infection transmission mechanism. Employing higher-order perturbation analysis and computing the local Lyapunov exponent, we observe the emergence of dynamic instabilities induced by stochastic environmental forces driving the epidemic spread. Notably, the instabilities are uncovered using third-order perturbations whilst they are not observed under second-order perturbations. Moreover, the onset of instability is more likely with increasing noise strength of the stochastic environmental forces.

Properties and errors of second-order parabolic and hyperbolic perturbations of a first-order symmetric hyperbolic system

The Cauchy problems are studied for a first-order multidimensional symmetric linear hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter $\tau>0$ multiplying the second derivatives with respect to $x$ and $t$. The existence and uniqueness of weak solutions of all three systems and $\tau$-uniform estimates for solutions of systems with perturbations are established. Estimates for the difference of solutions of the original system and the systems with perturbations are given, including ones of order $O(\tau^{\alpha/2})$ in the norm of $C(0,T;L^2(\mathbb{R}^n))$, for an initial function $\mathbf w_0$ in the Sobolev space $H^\alpha(\mathbb{R}^n)$, $\alpha=1,2$, or the Nikolskii space $H_2^{\alpha}(\mathbb{R}^n)$, $0<\alpha<2$, $\alpha\neq 1$, and under appropriate assumptions on the free term of the first-order system. For ${\alpha=1/2}$ a wide class of discontinuous functions $\mathbf w_0$ is covered. Estimates for derivatives of any order with respect to $x$ for solutions and of order $O(\tau^{\alpha/2})$ for their differences are also deduced. Applications of the results to the first-order system of gas dynamic equations linearized at a constant solution and to its perturbations, namely, the linearized second-order parabolic and hyperbolic quasi-gasdynamic systems of equations, are presented. Bibliography: 34 titles.

Asymptotic behavior of solutions of a system of nonlinear differential equations with small parameter

The present paper addresses a qualitative pattern of the behavior of solutions of a system of ordinary differential equations when small parameter tends to zero at a finite amount of time where slow variable passes through a certain point that corresponds to a bifurcation in the system of fast motions: stable limit cycle merges with unstable one and disappears. The problem of generation of an asymptotic approximation of the solution of a perturbed differential equation system is considered in the case where a bifurcation occurs in the “fast motions” equation when the parameter changes: two equilibrium positions merge, followed by a change in stability. The results of the article are used to determine second-order perturbations in rectangular coordinates and components of the velocity of the body under study. The coefficients at the projections of the perturbing acceleration are integral functions of the independent regularizing variable. Singular points are used to reduce the degree of approximating polynomials and to choose regularizing variables.

Second-order dust perturbations of the non-flat FLRW model with the positive cosmological constant

In this paper, a specific solution to the second-order cosmological perturbation theory is given. Perturbations are performed around any Friedmann–Lemaître–Robertson–Walker spacetime filled with dust and with a positive cosmological constant. In particular, with a possibly non-vanishing spatial curvature. The adopted symmetry condition allows us to simplify the equations, leaving us with a great deal of freedom to choose the density distribution. In the result, we get a relatively simple metric of an inhomogeneous cosmological model, which will give a perfect tool for studying the influence of the local inhomogeneities onto the cosmological observables.

Pointwise decay for the wave equation on nonstationary spacetimes

This first article in a two-part series (the second article being [arXiv:2205.13197]) assumes a weak form of the integrated local energy decay estimate and proves that solutions to the linear wave equation with variable coefficients in R1+3, first-order terms, and a potential decay at a rate depending on how rapidly the vector fields of these functions decay at spatial infinity. Roughly speaking, given metric perturbation terms and first-order terms whose vector fields decay at least as rapidly as the rate r−1−a, and potentials whose vector fields decay at least as rapidly as the rate r−2−a, our results imply that |ZJϕ(t,x)|≤Ct−2−a for bounded |x|, where Z denotes vector fields and J is a multi-index. We expect that this is a sharp decay rate. We prove results for both stationary and nonstationary metrics and coefficients. The proof uses integrated local energy decay to achieve an initial decay rate, and then uses an iteration involving the one-dimensional reduction (due to the positivity of the fundamental solution of the wave equation in three spatial dimensions) to achieve the full decay rate. For second-order perturbations of the Minkowski metric with coefficients that are not necessarily spherically symmetric, we analyze these coefficients' decay rates in a novel way to obtain a higher rate of pointwise decay for the solution than might be expected, assuming only weak pointwise decay on up to two time derivatives of the coefficients (and vector fields thereof); the latter weak decay is satisfied if, for example, the metric is stationary.

Aether scalar tensor theory: Linear stability on Minkowski space

We have recently proposed a simple relativistic theory which reduces to modified Newtonian dynamics for the weak-field quasistatic situations applied to galaxies, and to cosmological behavior as in the $\Lambda$CDM model, yielding a realistic cosmology in line with observations. A key requirement of any such model is that Minkowski space is stable against linear perturbations. We expand the theory action to second order in perturbations on a Minkowski background and show that it leads to healthy dispersion relations involving propagating massive modes in the vector and the scalar sector. We use Hamiltonian methods to eliminate constraints present, demonstrate that the massive modes have Hamiltonian bounded from below and show that a nonpropagating mode with a linear time dependence may have unbounded Hamiltonian for wave numbers $k< \mu$ and bounded otherwise. The scale $\mu$ is estimated to be $\lesssim \mathrm{Mpc}^{-1}$ so that the low momenta instability may only play a role on cosmological scales.

On Second-Order Parabolic and Hyperbolic Perturbations of a First-Order Hyperbolic System

On Second-Order Parabolic and Hyperbolic Perturbations of a First-Order Hyperbolic System

Close relationship between the dressed metric and the hybrid approach to perturbations in effective loop quantum cosmology

The dressed metric and the hybrid approach to perturbations are the two main approaches to capture the effects of quantum geometry in the primordial power spectrum in loop quantum cosmology. Both consider Fock quantized perturbations over a loop quantized background and result in very similar predictions except for the modes which exit the horizon in the effective spacetime in the Planck regime. Understanding precise relationship between both approaches has so far remained obscured due to differences in construction and technical assumptions. We explore this issue at the classical and effective spacetime level for linear perturbations, ignoring backreaction, which is the level at which practical computations of the power spectrum in both of the approaches have so far been performed. We first show that at the classical level both the approaches lead to the same Hamiltonian up to the second order in perturbations and result in the same classical mass functions in the Mukhanov-Sasaki equation on the physical solutions. At the effective spacetime level, the difference in phenomenological predictions between the two approaches in the Planck regime can be traced to whether one uses the Mukhanov-Sasaki variable $Q_{\vec k}$ (the dressed metric approach) or its rescaled version $\nu_{\vec k}=aQ_{\vec k}$ (the hybrid approach) to write the Hamiltonian of the perturbations, and associated polymerization ambiguities. It turns out that if in the dressed metric approach one chooses to work with $\nu_{\vec{k}}$, the effective mass function can be written exactly as in the hybrid approach, thus leading to identical phenomenological predictions in all regimes. Our results explicitly show that the dressed metric and the hybrid approaches for linear perturbations, at a practical computational level, can be seen as two sides of the same coin.

Scalar induced gravitational waves from warm inflation

Stochastic gravitational waves can be induced from the primordial curvature perturbations generated during inflation, through scalar–tensor mode coupling at the second order of cosmological perturbation theory. Here we discuss a model of warm inflation in which large curvature perturbations are generated at the small scales because of inflaton dissipation. These overdense perturbations then collapse at later epoch to form primordial black holes, as was studied in our earlier work (Ref. Arya (2020)), and therefore may also act as a source to the second-order tensor perturbations. In this study, we calculate the spectrum of these secondary gravitational waves from our warm inflationary model. We find that our model leads to a generation of scalar induced gravitational waves (SIGW) over a frequency range ∼ ( 1 – 1 0 6 ) Hz. Further, we discuss the detection possibilities of these SIGWs, taking in account the sensitivity of different ongoing and future gravitational wave experiments.

Signatures of Primordial Gravitational Waves on the Large-Scale Structure of the Universe.

We study the generation and evolution of second-order energy-density perturbations arising from primordial gravitational waves. Such "tensor-induced scalar modes" approximately evolve as standard linear matter perturbations and may leave observable signatures in the large-scale structure of the Universe. We study the imprint on the matter power spectrum of some primordial models which predict a large gravitational-wave signal at high frequencies. This novel mechanism, in principle, allows us to constrain or detect primordial gravitational waves by looking at specific features in the matter or galaxy power spectrum, thereby allowing us to probe them on a range of scales unexplored so far.