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We numerically solve the Klein-Gordon equation at second order in cosmological perturbation theory in closed form for a single scalar field, describing the method employed in detail. We use the slow-roll version of the second order source term and argue that our method is extendable to the full equation. We consider two standard single field models and find that the results agree with previous calculations using analytic methods, where comparison is possible. Our procedure allows the evolution of second order perturbations in general and the calculation of the non-linearity parameter f_NL to be examined in cases where there is no analytical solution available.
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We examine the covariant properties of generalized models of two-field inflation, with non-canonical kinetic terms and a possibly non-trivial field metric. We demonstrate that kinetic-term derivatives and covariant field derivatives do commute in a proper covariant framework, which was not realized before in the literature. We also define a set of generalized slow-roll parameters, using a unified notation. Within this framework, we study the most general class of models that allows for well-defined adiabatic and entropic sound speeds, which we identify as the models with parallel momentum and field velocity vectors. For these models we write the exact cubic action in terms of the adiabatic and isocurvature perturbations. We thus provide the tool to calculate the exact non-Gaussianity beyond slow-roll and at any scale for these generalized models. We illustrate our general results by considering their long-wavelength limit, as well as with the example of two-field DBI inflation.
Inflationary cosmology is the leading explanation of the very early universe. Many different models of inflation have been constructed which fit current observational data. In this work theoretical and numerical methods for constraining the parameter space of a wide class of such models are described. First, string-theoretic models with large non-Gaussian signatures are investigated. An upper bound is placed on the amplitude of primordial gravitational waves produced by ultra-violet Dirac-Born-Infeld inflation. In all but the most finely tuned cases, this bound is incompatible with a lower bound derived for inflationary models which exhibit a red spectrum and detectable non-Gaussianity. By analysing general non-canonical actions, a class of models is found which can evade the upper bound when the phase speed of perturbations is small. The multi-coincident brane scenario with a finite number of branes is one such model. For models with a potentially observable gravitational wave spectrum the number of coincident branes is shown to take only small values. The second method of constraining inflationary models is the numerical calculation of second order perturbations for a general class of single field models. The Klein-Gordon equation at second order, written in terms of scalar field variations only, is numerically solved. The slow roll version of the second order source term is used and the method is shown to be extendable to the full equation. This procedure allows the evolution of second order perturbations in general and the calculation of the non-Gaussianity parameter in cases where there is no analytical solution available.
We present simple expressions for the relativistic first and second order fractional density perturbations for FL cosmologies with dust, in four different gauges: the Poisson, uniform curvature, total matter and synchronous 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 non-elementary functions, the so-called growth supression 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.
We numerically calculate the evolution of second order cosmological perturbations for an inflationary scalar field without resorting to the slow-roll approximation or assuming large scales. In contrast to previous approaches we therefore use the full non-slow-roll source term for the second order Klein-Gordon equation which is valid on all scales. The numerical results are consistent with the ones obtained previously where slow-roll is a good approximation. We investigate the effect of localised features in the scalar field potential which break slow-roll for some portion of the evolution. The numerical package solving the second order Klein-Gordon equation has been released under an open source license and is available for download.
Motivated by the desire to understand the leading order nonlinear gravitational wave interactions around arbitrarily rapidly rotating Kerr black holes, we describe a numerical code designed to compute second order vacuum perturbations on such spacetimes. A general discussion of the formalism we use is presented in (arXiv:2008.11770); here we show how we numerically implement that formalism with a particular choice of coordinates and tetrad conditions, and give example results for black holes with dimensionless spin parameters $a=0.7$ and $a=0.998$. We first solve the Teukolsky equation for the linearly perturbed Weyl scalar $Psi_4^{(1)}$, followed by direct reconstruction of the spacetime metric from $Psi_4^{(1)}$, and then solve for the dynamics of the second order perturbed Weyl scalar $Psi_4^{(2)}$. This code is a first step toward a more general purpose second order code, and we outline how our basic approach could be further developed to address current questions of interest, including extending the analysis of ringdown in black hole mergers to before the linear regime, exploring gravitational wave turbulence around near-extremal Kerr black holes, and studying the physics of extreme mass ratio inspiral.
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