13,061 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.
A constructive commutative quantum Lovasz Local Lemma, and beyond
The recently proven Quantum Lovasz Local Lemma generalises the well-known
Lovasz Local Lemma. It states that, if a collection of subspace constraints are
"weakly dependent", there necessarily exists a state satisfying all
constraints. It implies e.g. that certain instances of the kQSAT quantum
satisfiability problem are necessarily satisfiable, or that many-body systems
with "not too many" interactions are always frustration-free.
However, the QLLL only asserts existence; it says nothing about how to find
the state. Inspired by Moser's breakthrough classical results, we present a
constructive version of the QLLL in the setting of commuting constraints,
proving that a simple quantum algorithm converges efficiently to the required
state. In fact, we provide two different proofs, one using a novel quantum
coupling argument, the other a more explicit combinatorial analysis. Both
proofs are independent of the QLLL. So these results also provide independent,
constructive proofs of the commutative QLLL itself, but strengthen it
significantly by giving an efficient algorithm for finding the state whose
existence is asserted by the QLLL. We give an application of the constructive
commutative QLLL to convergence of CP maps.
We also extend these results to the non-commutative setting. However, our
proof of the general constructive QLLL relies on a conjecture which we are only
able to prove in special cases.Comment: 43 pages, 2 conjectures, no figures; unresolved gap in the proof; see
arXiv:1311.6474 or arXiv:1310.7766 for correct proofs of the symmetric cas
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