2,149 research outputs found
Cosmological perturbations
We review the study of inhomogeneous perturbations about a homogeneous and
isotropic background cosmology. We adopt a coordinate based approach, but give
geometrical interpretations of metric perturbations in terms of the expansion,
shear and curvature of constant-time hypersurfaces and the orthogonal timelike
vector field. We give the gauge transformation rules for metric and matter
variables at first and second order. We show how gauge invariant variables are
constructed by identifying geometric or matter variables in physically-defined
coordinate systems, and give the relations between many commonly used
gauge-invariant variables. In particular we show how the Einstein equations or
energy-momentum conservation can be used to obtain simple evolution equations
at linear order, and discuss extensions to non-linear order. We present
evolution equations for systems with multiple interacting fluids and scalar
fields, identifying adiabatic and entropy perturbations. As an application we
consider the origin of primordial curvature and isocurvature perturbations from
field perturbations during inflation in the very early universe.Comment: 96 pages, submitted to Phys. Rep; v2: minor changes, typos corrected,
references added, 1 figure added, corresponds to published versio
Second Order Perturbations During Inflation Beyond Slow-roll
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.Comment: v2: version published in JCAP, references added; v1: 21 pages, 11
figures, numerical package available at http://pyflation.ianhuston.ne
Numerical calculation of second order perturbations
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.Comment: 18 pages, 12 figures; v2 version published by JCA
Double power series method for approximating cosmological perturbations
We introduce a double power series method for finding approximate analytical
solutions for systems of differential equations commonly found in cosmological
perturbation theory. The method was set out, in a non-cosmological context, by
Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases
where perturbations are on sub-horizon scales. The FSN method is essentially an
extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding
approximate analytical solutions for ordinary differential equations. The FSN
method we use is applicable well beyond perturbation theory to solve systems of
ordinary differential equations, linear in the derivatives, that also depend on
a small parameter, which here we take to be related to the inverse wave-number.
We use the FSN method to find new approximate oscillating solutions in linear
order cosmological perturbation theory for a flat radiation-matter universe.
Together with this model's well known growing and decaying M\'esz\'aros
solutions, these oscillating modes provide a complete set of sub-horizon
approximations for the metric potential, radiation and matter perturbations.
Comparison with numerical solutions of the perturbation equations shows that
our approximations can be made accurate to within a typical error of 1%, or
better. We also set out a heuristic method for error estimation. A Mathematica
notebook which implements the double power series method is made available
online.Comment: 22 pages, 10 figures, 2 tables. Mathematica notebook available from
Github at https://github.com/AndrewWren/Double-power-series.gi
Cosmological Perturbations in an Inflationary Universe
After introducing gauge-invariant cosmological perturbation theory we give an
improved set of governing equations for multiple fluids including energy
transfer. Having defined adiabatic and entropic perturbations we derive the
``conservation law'' for the curvature perturbation on large scales using only
the energy conservation equation. We then investigate the dynamics of assisted
inflation. By choosing an appropriate rotation in field space we can write down
explicitly the potential for the weighted mean field along the scaling solution
and for fields orthogonal to it. This allows us to present analytic solutions
describing homogeneous and inhomogeneous perturbations about the attractor
solution without resorting to slow-roll approximations. Finally we analyze the
simplest model of preheating analytically, and show that in linear perturbation
theory the effect of preheating on the amplitude of the curvature perturbation
on large scales is negligible. We end with some concluding remarks, possible
extensions and an outlook to future work.Comment: PhD thesis, University of Portsmouth, 82 pages, 5 figure
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