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Register a new userThe curvature perturbation at second order
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We give an explicit relation, up to second-order terms, between scalar-field fluctuations defined on spatially-flat slices and the curvature perturbation on uniform-density slices. This expression is a necessary ingredient for calculating observable quantities at second-order and beyond in multiple-field inflation. We show that traditional cosmological perturbation theory and the `separate universe approach yield equivalent expressions for superhorizon wavenumbers, and in particular that all nonlocal terms can be eliminated from the perturbation-theory expressions.
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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.
Working with perturbations about an FLRW spacetime, we compute the gauge-invariant curvature perturbation to second order solely in terms of scalar field fluctuations. Using the curvature perturbation on uniform density hypersurfaces as our starting point, we give our results in terms of field fluctuations in the flat gauge, incorporating both large and small scale behaviour. For ease of future numerical implementation we give our result in terms of the scalar field fluctuations and their time derivatives.
We derive the evolution equation for the second order curvature perturbation using standard techniques of cosmological perturbation theory. We do this for different definitions of the gauge invariant curvature perturbation, arising from different splits of the spatial metric, and compare the expressions. The results are valid at all scales and include all contributions from scalar, vector and tensor perturbations, as well as anisotropic stress, with all our results written purely in terms of gauge invariant quantities. Taking the large-scale approximation, we find that a conserved quantity exists only if, in addition to the non-adiabatic pressure, the transverse traceless part of the anisotropic stress tensor is also negligible. We also find that the version of the gauge invariant curvature perturbation which is exactly conserved is the one defined with the determinant of the spatial part of the inverse metric.
We study the structure of scalar-tensor theories of gravity based on derivative couplings between the scalar and the matter degrees of freedom introduced through an effective metric. Such interactions are classified by their tensor structure into conformal (scalar), disformal (vector) and extended disformal (traceless tensor), as well as by the derivative order of the scalar field. Relations limited to first derivatives of the field ensure second order equations of motion in the Einstein frame and hence the absence of Ostrogradski ghost degrees of freedom. The existence of a mapping to the Jordan frame is not trivial in the general case, and can be addressed using the Jacobian of the frame transformation through its eigenvalues and eigentensors. These objects also appear in the study of different aspects of such theories, including the metric and field redefinition transformation of the path integral in the quantum mechanical description. Although sane in the Einstein frame, generic disformally coupled theories are described by higher order equations of motion in the Jordan frame. This apparent contradiction is solved by the use of a hidden constraint: the contraction of the metric equations with a Jacobian eigentensor provides a constraint relation for the higher field derivatives, which allows one to express the dynamical equations in a second order form. This signals a loophole in Horndeskis theorem and allows one to enlarge the set of scalar-tensor theories which are Ostrogradski-stable. The transformed Gauss-Bonnet terms are also discussed for the simplest conformal and disformal relations.
We discuss the difference between various gauge-invariant quantities typically used in single-field inflation, namely synchronous $zeta_s$, comoving $zeta_c$, and unitary $zeta_u$ curvatures. We show that conservation of $zeta_c$ outside the horizon is quite restrictive on models as it leads to conservation of $zeta_s$ and $zeta_u$, whereas the reverse does not hold. We illustrate the consequence of these differences with two inflationary models: ultra-slow-roll (USR) and braiding-ultra-slow-roll (BUSR). In USR, we show that out of the three curvatures, only $zeta_s$ is conserved outside the horizon, and we connect this result to the concepts of separate universe and the usage of the $delta N$ formalism. We find that even though $zeta_s$ is conserved, there is still a mild violation of the separate universe approximation in the continuity equation. Nevertheless, the $delta N$ formalism can still be applied to calculate the primordial power spectrum of some gauge-invariant quantities such as $zeta_u$, although it breaks down for others such as the uniform-density curvature. In BUSR, we show that both $zeta_u$ and $zeta_s$ are conserved outside the horizon, but take different values. Additionally, since $zeta_u ot=zeta_c$ we find that the prediction for observable curvature fluctuations after inflation does not reflect $zeta_c$ at horizon crossing during inflation and moreover involves not just $zeta_u$ at that epoch but also the manner in which the braiding phase ends.
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