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# Gauge invariant Boltzmann equation and the fluid limit

(Impact Factor: 3.17). 07/2007; 24(24). DOI: 10.1088/0264-9381/24/24/001
Source: arXiv

ABSTRACT

This article investigates the collisionless Boltzmann equation up to second order in the cosmological perturbations. It describes the gauge dependence of the distribution function and the construction of a gauge invariant distribution function and brightness, and then derives the gauge invariant fluid limit.

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• "The remaining sources of B-polarisation from non-linear evolution can be treated using second-order perturbation theory. The Einstein and Boltzmann equations at second order have been studied in great detail [28] [29] [30] [31] [32] [33]. They are significantly more complicated than at first order and solving them numerically is a daunting task. "
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ABSTRACT: We estimate the B-polarisation induced in the Cosmic Microwave Background by the non-linear evolution of density perturbations. Using the second-order Boltzmann code SONG, our analysis incorporates, for the first time, all physical effects at recombination. We also include novel contributions from the redshift part of the Boltzmann equation and from the bolometric definition of the temperature in the presence of polarisation. The remaining line-of-sight terms (lensing and time-delay) have previously been studied and must be calculated non-perturbatively. The intrinsic B-mode polarisation is present independent of the initial conditions and might contaminate the signal from primordial gravitational waves. We find this contamination to be comparable to a primordial tensor-to-scalar ratio of $r\simeq10^{-7}$ at the angular scale $\ell\simeq100\,$, where the primordial signal peaks, and $r\simeq 5 \cdot 10^{-5}$ at $\ell\simeq700\,$, where the intrinsic signal peaks. Therefore, we conclude that the intrinsic B-polarisation from second-order effects is not likely to contaminate future searches of primordial gravitational waves.
Full-text · Article · Jan 2014 · Journal of Cosmology and Astroparticle Physics
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• "The non-linear signal can be quantified theoretically by using second-order perturbation theory; this is the leading order of non-Gaussianity since linear evolution cannot generate non-Gaussian features that are not already present in the initial conditions. The Einstein and Boltzmann equations at second order have been studied in great detail [26] [27] [28] [29] [30]. They are significantly more complicated than at first order and solving them numerically is a daunting task. "
##### Article: The intrinsic bispectrum of the Cosmic Microwave Background
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ABSTRACT: We develop a new, efficient code for solving the second-order Einstein-Boltzmann equations, and use it to estimate the intrinsic CMB non-Gaussianity arising from the non-linear evolution of density perturbations. The full calculation involves contributions from recombination and less tractable contributions from terms integrated along the line of sight. We investigate the bias that this intrinsic bispectrum implies for searches of primordial non-Gaussianity. We find that the inclusion or omission of certain line of sight terms can make a large impact. When including all physical effects but lensing and time-delay, we find that the local-type f_nl would be biased by f_nl ~ 0.5, below the expected sensitivity of the Planck satellite. The speed of our code allows us to confirm the robustness of our results with respect to a number of numerical parameters.
Full-text · Article · Feb 2013 · Journal of Cosmology and Astroparticle Physics
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• "(k) i = ∂ i ω (k) + ω (k)⊥ i , (4.10) with ∂ i ω (k)⊥ i = 0. Similarly for a tensor quantity, 7 χ (k) ij = D ij χ (k) + ∂ i χ (k)⊥ j + ∂ j χ (k)⊥ i + χ (k)⊤ ij . (4.11) Now, following [72] "
##### Article: Evolution of magnetic fields through cosmological perturbation theory
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ABSTRACT: The origin of galactic and extra-galactic magnetic fields is an unsolved problem in modern cosmology. A possible scenario comes from the idea of these fields emerged from a small field, a seed, which was produced in the early universe (phase transitions, inflation, ...) and it evolves in time. Cosmological perturbation theory offers a natural way to study the evolution of primordial magnetic fields. The dynamics for this field in the cosmological context is described by a cosmic dynamo like equation, through the dynamo term. In this paper we get the perturbed Maxwell's equations and compute the energy momentum tensor to second order in perturbation theory in terms of gauge invariant quantities. Two possible scenarios are discussed, first we consider a FLRW background without magnetic field and we study the perturbation theory introducing the magnetic field as a perturbation. The second scenario, we consider a magnetized FLRW and build up the perturbation theory from this background. We compare the cosmological dynamo like equation in both scenarios.
Full-text · Article · Apr 2011 · Physical review D: Particles and fields