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Evolution of the Second-Order Perturbations

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The Intrinsic Bispectrum of the Cosmic Microwave Background

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

The Boltzmann equation dictates the evolution of the matter components of the Universe in an inhomogeneous Universe, while the Einstein equation describes how the curvature is affected by the distribution of matter, energy and momentum. In this chapter, we provide a numerical treatment of the Boltzmann-Einstein system of differential equations at second order in the cosmological perturbations. We first introduce our code, SONG, which numerically solves the system for photons, massless neutrinos, baryons and cold dark matter, including the effect of perturbed recombination (Sects. 5.2 and 5.3). This is a complex task that involves solving the inherent stiffness of the differential system and devising efficient sampling techniques for the time and wavemode grids. Before even solving the differential system, one has to carefully match the initial conditions with the analyical solution of the system in the early Universe, in order to avoid exciting the decaying mode (Sect. 5.4). Once suitable initial conditions are specified deep in the radiation dominated era, the second-order system can be solved all the way to today. In practice, however, the CMB anisotropies cannot be computed in this way because of the size of the differential system. Instead, we use the line of sight (LOS) formalism to directly compute the today’s transfer functions in a numerically efficient way (Sect. 5.5). Finally, we compare the numerical results of SONG against some analytical limits known in the literature and find an excellent agreement (Sect. 5.6).

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Notes

  1. 1.

    It is important to note that this property is not a consequence of the decomposition theorem, which holds only at first order, but of the fact that the second-order system shares the same linear structure with the first-order one. Mode details can be found in Sect. 3.3.2.

  2. 2.

    The CLASS code initially used Eq. 5.7 to evolve \(\varPhi \); this was changed in v1.4 after we communicated with the authors about the numerical instability. CLASS now uses the time-time equation, as SONG does.

  3. 3.

    There are obviously other ways to sample the triangular wavemode, \(k_3\). In fact, in CMBquick [45] a different technique is used where, for each \(k_1\) and \(k_2\), the \(k_3\) grid is chosen so that the angle between \({\varvec{k_1}} \) and \({\varvec{k_2}} \) is linearly sampled for a fixed number of time (16 in the latest version of CMBQuick).

  4. 4.

    The Jacobian is computed only for the purpose of accelerating the convergence of Newton’s method; it is not used in building the differentiation formulae. Therefore, reusing it does not imply a loss of precision, but just a slightly slower convergence.

  5. 5.

    The expression matches with Eq. 98 of Ma and Bertschinger [32], that is \(\,\theta =(k^2\,\tau )\,\varPsi \,\), once we realise that, at first order, \(\,\theta =i\,k_j\,v^j=i\,k\,v_{[0]}\,\).

  6. 6.

    In order to facilitate the comparison with the literature, we express \(\zeta \) in terms of the perturbation \(\mathcal {R}\) used in Pitrou et al. [45]. The two variables are unperturbatively related by \(e^{2\zeta }=1-2\mathcal {R} \), which translates to \(R=-\zeta -\zeta ^2\) up to second order. We also note that Eq. 5.67 is the same as Eq. 3.6b of Ref. [45], with \(\varPhi \leftrightarrow \varPsi \) and a multiplicative factor in the quadratic part, to account for the fact that we use the perturbative expansion \(X\approx X^{(1)} +X^{(2)} \) instead of \(X\approx X^{(1)} +\frac{1}{2}X^{(2)} \).

  7. 7.

    Note that our projection functions are related to those defined in [14] by

    figure a

    .

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  1. Astronomy Centre, University of Sussex, Brighton, UK

    Guido Walter Pettinari

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Pettinari, G.W. (2016). Evolution of the Second-Order Perturbations. In: The Intrinsic Bispectrum of the Cosmic Microwave Background. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-21882-3_5

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