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The Boltzmann Equation

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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 Einstein equations are insufficient to describe the Universe unless we complement them with a model of its content. In this chapter we shall introduce the Boltzmann formalism, a statistical treatment of the Universe matter content that naturally coexists with general relativity. Each matter component (photons, baryons, cold dark matter, dark energy and neutrinos) is described via a distribution function, while interactions are modelled as collisions. Centerstage in the formalism is the Boltzmann equation, which, together with the Einstein equations derived in the previous chapter, will allow us to consistently describe (and numerically compute) the evolution of both the metric and matter of the Universe. To do so, we first use tetrads to introduce the local inertial frame as a convenient tool to derive the collision term and to express the energetics of the system (Sect. 4.2). We then show how to expand the distribution function of the cosmic microwave background around its equilibrium form, the blackbody spectrum; we shall also treat the issue of defining a temperature at second order (Sect. 4.3). The distribution function is then used to derive the Liouville term (Sect. 4.4) and the collision term (Sect. 4.5), that is the two parts of the Boltzmann equation encoding respectively the geodesic motion of the CMB photons and their scattering with the electrons during recombination. Finally, we join the two terms to obtain a form of the Boltzmann equation which is apt for a numerical treatment (Sect. 4.6), which will be the subject of the next chapter.

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Notes

  1. 1.

    Note that some mistakes in their equations were reported and corrected by Pitrou [38] and Senatore et al. [44].

  2. 2.

    Note that Senatore et al. [44] (Sect. 4.1) and Beneke and Fidler [3] (Sect. I) use the same convention, while Pitrou [38] (Sect. 4.2.2) and Naruko et al. [34] (Sect. 2.1), instead, choose the tetrad to be orthogonal to constant time hypersurfaces, that is \({e^{\underline{0}}} \,\propto \,\text {d}\tau \). See Sect. 5.3.1 of Pitrou [38] for further details.

  3. 3.

    The expression coincides with the one in Eq. 4.4 of Ref. [44] once we convert our potentials to the “exponential” ones using Eq. 3.23, but differs from the one in Ref. [34] due to the different choice of tetrads.

  4. 4.

    The expansion is obtained by following the procedure in Sect. 3.6.2, with the difference that now we are adopting the local intertial frame and, therefore, the metric is Minkowskian. In particular, we have defined \({v^{\underline{i}}} \equiv {U^{\underline{i}}}/a\) and we have used \(U^0=(1+U^iU_i)/a\).

  5. 5.

    It should be noted that the cosmic expansion not altering the CMB spectrum is not a coincidence; in fact, the spectral distortions cannot be induced by the geodesic motion encoded in the Liouville operator, for the simple reason that a photon follows the same geodesic trajectory regardless of its energy. Therefore, we expect the spectral distortions to arise only at the level of the collision term.

  6. 6.

    This can be proven by integrating \(M_m[f_{BB}]=4\pi \int \text {d}p\,p^{2+m}\,f_{BB}\) by parts and using the fact that \(\partial f_{BB}/\partial p = -T/p\;\partial f_{BB}/\partial T\,\).

  7. 7.

    Note that Pitrou et al. [40] proposed another definition of temperature, the occupation number temperature , \(\,T_{\#}\,\), which is the temperature associated to the blackbody spectrum with the same number density as the CMB,

    $$\begin{aligned} \left( \,\frac{T_{\#}}{\overline{T}}\,\right) ^3 \;\equiv \; \frac{n}{\overline{n}} \;. \end{aligned}$$
    (4.62)

    For a more detailed discussion on temperature moments and on their relation to what is measured by CMB experiment, refer to Pitrou and Stebbins [40].

  8. 8.

    Our expression for \(\text {d}p/\text {d}\tau \) matches the one given in Eq. 4.14 by Senatore et al. [44] but is different form the one in Eq. 3.14 of Bartolo et al. [1]. The reason for this discrepancy is explained in the footnote 11 of the former paper.

  9. 9.

    Note that, with respect to what we have written in [37], we have corrected a typo in the sign of \(\dot{\omega }_i\).

  10. 10.

    It should be noted, however, that reionisation does play a role at second-order as it generates spectral distortions in the CMB [40]; we have investigated the effect of reionisation spectral distortions on the CMB spectrum both in temperature and polarisation with SONG in Ref. [42].

  11. 11.

    It is interesting to note that, even if the energy transfer is very small, \(p-p'=E_{q'}-E_q=\mathcal {O} (T\,q/m_e)\), it is still possible for a photon to scatter with a large angle, \(|{\varvec{p}} '-{\varvec{p}} |=\mathcal {O} (T)\), so that \(\frac{p'-p}{|{\varvec{p}} '-{\varvec{p}} |}=\mathcal {O} (q/m_e)\).

  12. 12.

    At zero order in the energy transfer, neither the momentum nor the direction of propagation of an electron is changed by the scattering (\({\varvec{q}} '={\varvec{q}} \)) because the electrons have a large mass compared to the energy of the incident photon. This is reflected in Eq. 4.119 by the fact that, at zero order, \(g({\varvec{q}} ')=g({\varvec{q}})\). This is not the case for the scattering photon, whose direction can change even if the momentum stays constant (see previous footnote).

  13. 13.

    It should be noted that the perturbative expansion in the energy transfer is different from the one in the metric variables. For more details on this topic, refer to the discussion in Sect. 7.2 of Pitrou [38].

  14. 14.

    The below equations slightly differ from the ones in Dodelson and Jubas [14] in that we have merged the purely second-order terms into \(c^{(2)}\) and we have implemented the corrections that were pointed out in Appendix C of Senatore et al. [44]. For an alternative splitting strategy, refer to Eq. 6 of Hu et al. [24], where the photon distribution function is left unperturbed and the integrand function is expressed in terms of 7 contributions.

  15. 15.

    The expression obtained in Ref. [1] is not correct because it assumes that the first-order distribution function only has scalar components, i.e. \(f^{(1)}_{\ell m} ({\varvec{k_1}})\propto \delta _{m0}\). This is the case only if the polar axis is chosen to coincide with the wavemode \({\varvec{k_1}} \). In a second-order expression, however, the first-order quantities are evaluated in the convolution wavevectors, \({\varvec{k_1}}\) and \({\varvec{k_2}}\); since the polar axis was already chosen to be aligned with \({\varvec{k}} \), one cannot assume \(f^{(1)}_{\ell m} ({\varvec{k_1}})\propto \delta _{m0}\); as explained in Appendix B, the angular dependence of \(f^{(1)}({\varvec{k_1}})\) is given by \(f^{(1)}_{\ell m} ({\varvec{k_1}})\propto \tilde{f}^{(1)}_{\ell 0}(k_1)\,Y_{\ell m} ({\varvec{k}})\).

  16. 16.

    Note that this notation is the same that we have adopted in Pettinari et al. [37] and in Fidler et al. [19].

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

    Guido Walter Pettinari

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Pettinari, G.W. (2016). The Boltzmann Equation. In: The Intrinsic Bispectrum of the Cosmic Microwave Background. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-21882-3_4

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