Scalar-tensor mode mixing in higher-order cosmological perturbations

The theory of cosmological perturbations has become a very important subject of modern Cosmology because it allows to link the models of the very early Universe, such as the inflationary scenario, with the massive high-precision data on the Cosmic Mi- crowave Background radiation, on large-scale structures and future data on the primordial Gravitational-Wave stochastic background. When working with perturbations within General Relativity a difficulty arises: we have After an introduction, I start recovering the first-order results: scalar and tensor perturbations evolve independently, thus I can easily study and write Einstein’s equations for scalars and then for tensors. First order vectors are neglected due to the fact that, if generated, they are fast redshifted away with the expansion of the Universe. So, the equations governing the evolution of these quantities are obtained, recovering the results in the literature. When one goes to second order, computations start to be more complex, revealing the underlying non-linearity of Einstein’s equations. For the first time, the second-order perturbed metric is obtained directly in the Poisson gauge, with scalars at first and second order, vectors at second order, tensors at first and second order. With this choice, I show that we have second-order mixed terms which source scalar and tensor perturbations. Namely, second-order scalar modes are sourced by first-order scalars coupled with first-order tensors, by two coupled first-order scalar and two first- order tensors. The same problem has been discussed in other papers but in a different gauge. Now, Einstein’s equations start to be more complicated and finding solutions is not so easy. Starting from computing the second-order perturbed quantities such as the Ricci tensor and the Ricci scalar, the second order energy-momentum tensor, from the traceless i − j component of the Einstein field equation I can find the difference between the second order scalar perturbations, which as a first result is non zero even for a perfect fluid in absence of an anisotropic stress tensor, in contrast with the first order result, which gives ψ = φ for a perfect fluid. With this equation in hand, I derived the other Einstein equations and the continuity equation to close the system for the scalar quantities. The following part is focused on writing the Boltzmann equations at second order in order to study the evolution of particle species, such as photons, baryons and cold dark matter. In this section I write the Boltzmann equation accounting for first and second order scalar perturbation, second order vector perturbation, first and second order tensor perturbation. The same formalism can be later applied to derive the Boltzmann equation for other particle species, such as baryons and cold dark matter (CDM). Studying the evolution of cold dark matter is a very important topic because it plays a fundamental role in structure formation. In conclusion the main goal of this project is to add an original contribution to the second-order evolution of scalar quantities, in Poisson gauge, such as the gravitational potentials, the density contrast and the velocity of baryons and CDM, starting from the perturbed expression of the metric through the Einstein and Boltzmann equations, con- sidering as non-negligible the contributions from first-order tensor modes which can be coupled to themselves and to other first-order scalar modes