7,001 research outputs found
The precision of slow-roll predictions for the CMBR anisotropies
Inflationary predictions for the anisotropy of the cosmic microwave
background radiation (CMBR) are often based on the slow-roll approximation. We
study the precision with which the multipole moments of the temperature
two-point correlation function can be predicted by means of the slow-roll
approximation. We ask whether this precision is good enough for the forthcoming
high precision observations by means of the MAP and Planck satellites. The
error in the multipole moments due to the slow-roll approximation is
demonstrated to be bigger than the error in the power spectrum. For power-law
inflation with the error from the leading order slow-roll
approximation is for the amplitudes and for the
quadrupoles. For the next-to-leading order the errors are within a few percent.
The errors increase with . To obtain a precision of 1% it is
necessary, but in general not sufficient, to use the next-to-leading order. In
the case of power-law inflation this precision is obtained for the spectral
indices if and for the quadrupoles if
only. The errors in the higher multipoles are even larger than those for the
quadrupole, e.g. for l=100, with at the
next-to-leading order. We find that the accuracy of the slow-roll approximation
may be improved by shifting the pivot scale of the primordial spectrum (the
scale at which the slow-roll parameters are fixed) into the regime of acoustic
oscillations. Nevertheless, the slow-roll approximation cannot be improved
beyond the next-to-leading order in the slow-roll parameters.Comment: 3 important additions: 1. discussion of higher multipoles, 2.
comparison of error from the slow-roll approximation with the error from the
cosmic variance, 3. suggestion for improvement of slow-roll approximation;
two figures and a table added; 15 pages, 14 figures, RevTeX; accepted for
publication in Phys. Rev.
Finite index subgroups without unique product in graphical small cancellation groups
We construct torsion-free hyperbolic groups without unique product whose
subgroups up to some given finite index are themselves non-unique product
groups. This is achieved by generalising a construction of Comerford to
graphical small cancellation presentations, showing that for every subgroup
of a graphical small cancellation group there exists a free group such that
admits a graphical small cancellation presentation.Comment: 8 pages, 1 figur
Random Moment Problems under Constraints
We investigate moment sequences of probability measures on
under constraints of certain moments being fixed. This
corresponds to studying sections of -th moment spaces, i.e. the spaces of
moment sequences of order . By equipping these sections with the uniform or
more general probability distributions, we manage to give for large precise
results on the (probabilistic) barycenters of moment space sections and the
fluctuations of random moments around these barycenters. The measures
associated to the barycenters belong to the Bernstein-Szeg\H{o} class and show
strong universal behavior. We prove Gaussian fluctuations and moderate and
large deviations principles. Furthermore, we demonstrate how fixing moments by
a constraint leads to breaking the connection between random moments and random
matrices.Comment: 43 page
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