We introduce a new mechanism for generating magnetic fields in the recombination era. This Harrison-like mechanism utilizes vorticity in baryons that is sourced through the Bose-Einstein condensate of axions via gravitational interactions. The magnet
ic fields generated are on the galactic scales $sim 10,{rm kpc}$ and have a magnitude of the order of $Bsim10^{-23},{rm G}$ today. The field has a greater magnitude than those generated from other mechanisms relying on second order perturbation theory, and is sufficient to provide a seed for battery mechanisms.
We consider a model of the early universe which consists of two scalar fields: the inflaton, and a second field which drives the stabilisation of the Planck mass (or gravitational constant). We show that the non-minimal coupling of this second field
to the Ricci scalar sources a non-adiabatic pressure perturbation. By performing a fully numerical calculation we find, in turn, that this boosts the amplitude of the primordial power spectrum after inflation.
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.
In this paper I review some recent, interlinked, work undertaken using cosmological perturbation theory -- a powerful technique for modelling inhomogeneities in the Universe. The common theme which underpins these pieces of work is the presence of no
n-adiabatic pressure, or entropy, perturbations. After a brief introduction covering the standard techniques of describing inhomogeneities in both Newtonian and relativistic cosmology, I discuss the generation of vorticity. As in classical fluid mechanics, vorticity is not present in linearized perturbation theory (unless included as an initial condition). Allowing for entropy perturbations, and working to second order in perturbation theory, I show that vorticity is generated, even in the absence of vector perturbations, by purely scalar perturbations, the source term being quadratic in the gradients of first order energy density and isocurvature, or non-adiabatic pressure perturbations. This generalizes Croccos theorem to a cosmological setting. I then introduce isocurvature perturbations in different models, focusing on the entropy perturbation in standard, concordance cosmology, and in inflationary models involving two scalar fields. As the final topic, I investigate magnetic fields, which are a potential observational consequence of vorticity in the early universe. I briefly review some recent work on including magnetic fields in perturbation theory in a consistent way. I show, using solely analytical techniques, that magnetic fields can be generated by higher order perturbations, albeit too small to provide the entire primordial seed field, in agreement with some numerical studies. I close with a summary and some potential extensions of this work.
Magnetic fields are present on all scales in the Universe. While we understand the processes which amplify the fields fairly well, we do not have a natural mechanism to generate the small initial seed fields. By using fully relativistic cosmological
perturbation theory and going beyond the usual confines of linear theory we show analytically how magnetic fields are generated. This is the first analytical calculation of the magnetic field at second order, using gauge-invariant cosmological perturbation theory, and including all the source terms. To this end, we have rederived the full set of governing equations independently. Our results suggest that magnetic fields of the order of $10^{-30}$ G can be generated (although this depends on the small scale cut-off of the integral), which is largely in agreement with previous results that relied upon numerical calculations. These fields are likely too small to act as the primordial seed fields for dynamo mechanisms.
Inflationary models involving more than one scalar field naturally produce isocurvature perturbations. However, while these are fairly well studied, less is known about their evolution through the reheating epoch, when the inflationary fields decay i
nto the standard constituents of the present universe. In this paper, by modelling reheating perturbatively, we calculate the power spectrum of the non-adiabatic pressure perturbation in three different inflationary models. We show that the isocurvature can grow large initially, but decays faster than the pressure perturbations. When reheating ends, the isocurvature is negligible for the double quadratic and double quartic inflationary models. For the product exponential potential, which features large isocurvature at the end of inflation, the isocurvature decays during reheating and is around five orders of magnitudes smaller than the pressure perturbation at the end of reheating.
Vorticity is ubiquitous in nature however, to date, studies of vorticity in cosmology and the early universe have been quite rare. In this paper, based on a talk in session CM1 of the 13th Marcel Grossmann Meeting, we consider vorticity generation fr
om scalar cosmological perturbations of a perfect fluid system. We show that, at second order in perturbation theory, vorticity is sourced by a coupling between energy density and entropy gradients, thus extending a well-known feature of classical fluid dynamics to a relativistic cosmological framework. This induced vorticity, sourced by isocurvature perturbations, may prove useful in the future as an additional discriminator between inflationary models.
Currently, most of the numerical simulations of structure formation use Newtonian gravity. When modelling pressureless dark matter, or `dust, this approach gives the correct results for scales much smaller than the cosmological horizon, but for scena
rios in which the fluid has pressure this is no longer the case. In this article, we present the correspondence of perturbations in Newtonian and cosmological perturbation theory, showing exact mathematical equivalence for pressureless matter, and giving the relativistic corrections for matter with pressure. As an example, we study the case of scalar field dark matter which features non-zero pressure perturbations. We discuss some problems which may arise when evolving the perturbations in this model with Newtonian numerical simulations and with CMB Boltzmann codes.
How much does the curvature perturbation change after it leaves the horizon, and when should one evaluate the power spectrum? To answer these questions we study single field inflation models numerically, and compare the evolution of different curvatu
re perturbations from horizon crossing to the end of inflation. In particular we calculate the number of efolds it takes for the curvature perturbation at a given wavenumber to settle down to within a given fraction of their value at the end of inflation. We find that e.g. in chaotic inflation, the amplitude of the comoving and the curvature perturbation on uniform density hypersurfaces differ by up to 180 % at horizon crossing assuming the same amplitude at the end of inflation, and that it takes approximately 3 efolds for the curvature perturbation to be within 1 % of its value at the end of inflation.
Cosmological perturbation theory is crucial for our understanding of the universe. The linear theory has been well understood for some time, however developing and applying the theory beyond linear order is currently at the forefront of research in t
heoretical cosmology. This thesis studies the applications of perturbation theory to cosmology and, specifically, to the early universe. Starting with some background material introducing the well-tested standard model of cosmology, we move on to develop the formalism for perturbation theory up to second order giving evolution equations for all types of scalar, vector and tensor perturbations, both in gauge dependent and gauge invariant form. We then move on to the main result of the thesis, showing that, at second order in perturbation theory, vorticity is sourced by a coupling term quadratic in energy density and entropy perturbations. This source term implies a qualitative difference to linear order. Thus, while at linear order vorticity decays with the expansion of the universe, the same is not true at higher orders. This will have important implications on future measurements of the polarisation of the Cosmic Microwave Background, and could give rise to the generation of a primordial seed magnetic field. Having derived this qualitative result, we then estimate the scale dependence and magnitude of the vorticity power spectrum, finding, for simple power law inputs a small, blue spectrum. The final part of this thesis concerns higher order perturbation theory, deriving, for the first time, the metric tensor, gauge transformation rules and governing equations for fully general third order perturbations. We close with a discussion of natural extensions to this work and other possible ideas for off-shooting projects in this continually growing field.
