543 research outputs found
A completeness-like relation for Bessel functions
Completeness relations are associated through Mercer's theorem to complete
orthonormal basis of square integrable functions, and prescribe how a Dirac
delta function can be decomposed into basis of eigenfunctions of a
Sturm-Liouville problem. We use Gegenbauer's addition theorem to prove a
relation very close to a completeness relation, but for a set of Bessel
functions not known to form a complete basis in
CMB in a box: causal structure and the Fourier-Bessel expansion
This paper makes two points. First, we show that the line-of-sight solution
to cosmic microwave anisotropies in Fourier space, even though formally defined
for arbitrarily large wavelengths, leads to position-space solutions which only
depend on the sources of anisotropies inside the past light-cone of the
observer. This happens order by order in a series expansion in powers of the
visibility , where is the optical depth to Thompson
scattering. We show that the CMB anisotropies are regulated by spacetime window
functions which have support only inside the past light-cone of the point of
observation. Second, we show that the Fourier-Bessel expansion of the physical
fields (including the temperature and polarization momenta) is an alternative
to the usual Fourier basis as a framework to compute the anisotropies. In that
expansion, for each multipole there is a discrete tower of momenta
(not a continuum) which can affect physical observables, with the
smallest momenta being . The Fourier-Bessel modes take into
account precisely the information from the sources of anisotropies that
propagates from the initial value surface to the point of observation - no
more, no less. We also show that the physical observables (the temperature and
polarization maps), and hence the angular power spectra, are unaffected by that
choice of basis. This implies that the Fourier-Bessel expansion is the optimal
scheme with which one can compute CMB anisotropies. (Abridged)Comment: 23 pages, 7 figure
Back Reaction And Local Cosmological Expansion Rate
We calculate the back reaction of cosmological perturbations on a general
relativistic variable which measures the local expansion rate of the Universe.
Specifically, we consider a cosmological model in which matter is described by
a single field. We analyze back reaction both in a matter dominated Universe
and in a phase of scalar field-driven chaotic inflation. In both cases, we find
that the leading infrared terms contributing to the back reaction vanish when
the local expansion rate is measured at a fixed value of the matter field which
is used as a clock, whereas they do not appear to vanish if the expansion rate
is evaluated at a fixed value of the background time. We discuss possible
implications for more realistic models with a more complicated matter sector.Comment: 7 pages, No figure
One Loop Back Reaction On Power Law Inflation
We consider quantum mechanical corrections to a homogeneous, isotropic and
spatially flat geometry whose scale factor expands classically as a general
power of the co-moving time. The effects of both gravitons and the scalar
inflaton are computed at one loop using the manifestly causal formalism of
Schwinger with the Feynman rules recently developed by Iliopoulos {\it et al.}
We find no significant effect, in marked contrast with the result obtained by
Mukhanov {\it et al.} for chaotic inflation based on a quadratic potential. By
applying the canonical technique of Mukhanov {\it et al.} to the exponential
potentials of power law inflation, we show that the two methods produce the
same results, within the approximations employed, for these backgrounds. We
therefore conclude that the shape of the inflaton potential can have an
enormous impact on the one loop back-reaction.Comment: 28 pages, LaTeX 2 epsilo
CMB and Random Flights: temperature and polarization in position space
The fluctuations in the temperature and polarization of the cosmic microwave
background are described by a hierarchy of Boltzmann equations. In its integral
form, this Boltzmann hierarchy can be converted from the usual Fourier-space
base into a position-space and causal description. We show that probability
densities for random flights play a key role in this description. The integral
system can be treated as a perturbative series in the number of steps of the
random flights, and the properties of random flight probabilities impose
constraints on the domains of dependence. We show that, as a result of these
domains, a Fourier-Bessel decomposition can be employed in order to calculate
the random flight probability densities. We also illustrate how the H-theorem
applies to the cosmic microwave background: by using analytical formulae for
the asymptotic limits of these probability densities, we show that, as the
photon distribution approaches a state of equilibrium, both the temperature
anisotropies and the net polarization must vanish.Comment: Minor revisions; matches version published in JCAP06(2013)04
Energy-Momentum Tensor of Cosmological Fluctuations during Inflation
We study the renormalized energy-momentum tensor (EMT) of cosmological scalar
fluctuations during the slow-rollover regime for chaotic inflation with a
quadratic potential and find that it is characterized by a negative energy
density which grows during slow-rollover. We also approach the back-reaction
problem as a second-order calculation in perturbation theory finding no
evidence that the back-reaction of cosmological fluctuations is a gauge
artifact. In agreement with the results on the EMT, the average expansion rate
is decreased by the back-reaction of cosmological fluctuations.Comment: 19 pages, no figures.An appendix and references added, conclusions
unchanged, version accepted for publication in PR
The back reaction and the effective Einstein's equation for the Universe with ideal fluid cosmological perturbations
We investigate the back reaction of cosmological perturbations on the
evolution of the Universe using the renormalization group method. Starting from
the second order perturbed Einstein's equation, we renormalize a scale factor
of the Universe and derive the evolution equation for the effective scale
factor which includes back reaction due to inhomogeneities of the Universe. The
resulting equation has the same form as the standard Friedman-Robertson-Walker
equation with the effective energy density and pressure which represent the
back reaction effect.Comment: 16 pages, to appear in Phys. Rev.
Back Reaction Problem in the Inflationary Universe
We investigate the back reaction of cosmological perturbations on an
inflationary universe using the renormalization-group method. The second-order
zero mode solution which appears by the nonlinearity of the Einstein equation
is regarded as a secular term of a perturbative expansion, we renormalized a
constant of integration contained in the background solution and absorbed the
secular term to this constant in a gauge-invariant manner. The resultant
renormalization-group equation describes the back reaction effect of
inhomogeneity on the background universe. For scalar type classical
perturbation, by solving the renormalization-group equation, we find that the
back reaction of the long wavelength fluctuation works as a positive spatial
curvature, and the short wavelength fluctuation works as a radiation fluid. For
the long wavelength quantum fluctuation, the effect of back reaction is
equivalent to a negative spatial curvature.Comment: 17 page
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