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Cosmological Perturbations Without the Boltzmann Hierarchy

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Cosmological Perturbations Without the Boltzmann Hierarchy Cosmological perturbations without the Boltzmann hierarchy Marc Kamionkowski Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218 (Dated: May 10, 2021) Calculations of the evolution of cosmological perturbations generally involve solution of a large number of coupled differential equations to describe the evolution of the multipole moments of the distribution of photon intensities and polarization. However, this \Boltzmann hierarchy" commu- nicates with the rest of the system of equations for the other perturbation variables only through the photon-intensity quadrupole moment. Here I develop an alternative formulation wherein this photon-intensity quadrupole is obtained via solution of two coupled integral equations|one for the intensity quadrupole and another for the linear-polarization quadrupole|rather than the full Boltzmann hierarchy. This alternative method of calculation provides some physical insight and a cross-check for the traditional approach. I describe a simple and efficient iterative numerical solution that converges fairly quickly. I surmise that this may allow current state-of-the-art cosmological- perturbation codes to be accelerated. PACS numbers: I. INTRODUCTION provide new insights into the physics and may perhaps provide tools that can be applied to other problems. Linear-theory calculations of the evolution of primor- It was realized that for primordial tensor perturbations dial density perturbations provide the foundation for (i.e., gravitational waves), the Boltzmann hierarchy can the interpretation of cosmic microwave background and be replaced by a small set of integral equations (IEs) large-scale-structure measurements. They are thus an [7, 8], an approach used in Refs. [9, 10] A similar ap- essential tool in the construction of our current cosmo- proach was discussed for scalar perturbations (primordial logical model and in the continuing quest for new cosmo- density perturbations) in Ref. [11], but not implemented logical physics. numerically. The calculations, which trace back over 50 years Here, I re-visit this integral-equation approach for pri- [1], involve time evolution of a set of coupled differ- mordial density perturbations. I discuss simplifications ential equations [2] for the metric perturbations and to the equations in Ref. [11] and describe a specific im- for the dark-matter, baryon, neutrino, and photon den- plementation where the Boltzmann hierarchy for all pho- sity and velocity perturbations. There is also a (nomi- ton intensity/polarization multipole moments from the nally infinite) \Boltzmann hierarchy" of differential equa- quadrupole (l = 2) and higher are replaced by two IEs, tions for the higher moments (quadrupole, octupole, one for the photon quadrupole, and another for the po- etc.) of the photon-intensity and photon-polarization larization quadrupole. I discuss the numerical solution and neutrino-momentum distributions. The photon hi- of these integral equations and how the initial conditions erarchies can be truncated at some maximum multi- for the IEs are set from an early-time solution obtained pole moment lmax ' 30 to provide sufficient precision with the TCA. I describe an iterative algorithm to solve for the monopole, dipole, and octupole from which the these integral equations simultaneously with the differ- higher-order moments (which provide the CMB tem- ential equations for the other perturbation variables. I perature/polarization power spectra) can be obtained show results from two simple numerical codes that are through a line-of-sight integral [3]. Higher-order exten- identical except for the replacement of the Boltzmann hi- sions to the tight-coupling approximation (TCA) [4, 5], erarchy in the first with the two integral equations in the improved numerical integrators, and novel approxima- second. Numerical experiments with these codes suggest that this iterative IE algorithm may, with further work, arXiv:2105.02887v1 [astro-ph.CO] 6 May 2021 tions to free-streaming relativistic particles [5]) have pro- vided incredible code acceleration to what is still a fairly allow current state-of-the-art codes to be accelerated. complicated numerical calculation. At present, virtually This paper is organized as follows. Section II, presents all work in cosmology now relies on two publicly available and discusses the integral equations. Section III pro- codes, CAMB [6] and CLASS [5], which combine speed and vides the differential equations for the other perturbation precision with model flexibility. variables (i.e., for neutrinos, dark matter, baryons, and These codes are now extremely efficient and reliable. the metric) and describe how the two integral equations However, modern cosmological analyses, which employ are combined with these other equations. Section IV Markov chain Monte Carlos to map the likelihood in a describes a simple algorithm to solve the integral equa- multi-dimensional parameter space, require these codes tions numerically and how the initial conditions for the to be run repeatedly, thus employing signficant compu- IE solver are obtained from the tight-coupling approx- tational resources. It is thus worthwhile to explore new imation at early times. This Section also describes an numerical approaches. New approaches can also often iterative algorithm to solve them in tandem with the dif- 2 LL 1 00 ferential equations. Section VI describes the two rudi- Also, Rl (x) = − 2 [jl(x) + 3jl (x)] [12, 13] in terms of mentary codes to evolve the Boltzmann hierarchy and the spherical Bessel functions jl(x), and θbk(τ) is the baryon IE equations. I then present and discuss results of the velocity. It is related to the photon velocity (suppressing calculation. Section VII concludes with a discussion of hereafter the subscript k for notational economy) θγ (τ) = T possible concerns and ideas for further steps. Appendix A 3k∆k1(τ) through provides the photon Boltzmann equations in the notation κ_ used here, and Appendix B provides details of the algo- _ 2 2 θb = −Hθb + csk δb + (θγ − θb); (4) rithm to solve the integral equation. The codes are pro- R vided at https://github.com/marckamion/IE for read- where H(τ) ≡ a=a_ and R(τ) ≡ (3=4)ρb(τ)/ργ (τ), the ers interested to follow up on calculational details that scale factor in units of 3=4 of that at matter-baryon equal- cannot be inferred from the presentation here. ity (ρb(τ) and ργ (τ) are mean baryon and photon energy densities, respectively). The baryon sound speed cs is in- creasingly important on small scales but has little effect II. FORMALISM on the larger distance/angular scales relevant for CMB fluctuations. Here, h(τ) is the standard synchronous- If we have a spectrum of initial curvature fluctuations gauge perturbation variable, and α(τ) = h(τ) + 6η(τ) in 2 with power spectrum PR(k) = jR~kj , then the CMB terms of the commonly used synchronous-gauge variable temperature/polarization power spectra are η(τ). Z The function Π(τ) is a linear combination of the XX0 2 −1 2 X X0 Cl = (2π ) k dk PR(k)∆kl(τ0)∆kl (τ0); (1) photon-intensity and polarization quadrupoles; for sim- plicity, I refer to it here as the polarization quadrupole. for X,X'=T,E with \T" the temperature and \E" the E- It can also be written as an IE, mode of the polarization. The transfer functions ∆X (τ) kl T are obtained through solution of differential equations Π(τ) = ∆2 (τ) + 9E2(τ); (5) for the time evolution of the relativistic gravitational po- with tentials, the baryon, dark-matter, photon, and neutrino Z τ 0 densities and bulk velocities, and the higher moments of 0 0 jl(k(τ − τ )) 0 El(τ) = dτ g(τ; τ ) Π(τ ): (6) the photon and neutrino momentum distributions. The 0 2 τ (k(τ − τ )) moments of the intensity distribution of photon momenta i are the transfer functions ∆T (τ) and the moments of the E kl The CMB E-mode transfer function is then ∆l (τ) = E p distribution of photon polarizations are ∆kl(τ). (3=4) (l + 2)!=(l − 2)!E (τ). 1 l The temperature transfer functions can be written as A derivation of Eqs. (2) and (5) will be provided in (" # Ref. [14] using the total-angular-momentum formalism Z τ 1 h_ (τ 0) ∆T (τ) = dτ 0 g(τ; τ 0) − k + ∆T (τ 0) j (x) [13], but it is easily verified that they agree with Eq. (18) kl 6 κ_ (τ 0) k0 l τi in Ref. [15], Eqs. (74) and (77) in Ref. [12], and with the 0 0 IEs in Ref. [7]. It can also be verified, using the relation, 1 α_ k(τ ) 1 0 LL 0 jl(x) − + Πk(τ ) R (x) + θbk(τ ) ; 0 LL 0 l (2l + 1)jl(x) = ljl−1(x) − (l + 1)jl+1(x) (which Rl (x) 3 κ_ (τ ) 2 k 0 and jl(x) also satisfy), that differentiation of these two (2) IEs recovers the usual Boltzmann hierarchy as given, for where x = k(τ − τ 0); a dot denotes a partial deriva- example, in Eqs. (2.4) of Ref. [5] or Eq. (63) of Ref. [16]. 0 tive with respect to τ; and g(τ; τ 0) = (d/dτ 0)e−κ(τ,τ ) = Thus, these two IEs are formally equivalent to the Boltz- 0 mann hierarchy. For completeness, the Boltzmann hier- κ_ (τ 0) e−κ(τ,τ ) is the visibility function. The initial con- archy is provided in the notation/conventions used here formal time τ must be taken to be deep in the tight- i in Appendix A. coupling regime and will be discussed more below. Here, κ_ (τ) = dκ/dτ is the opacity, the derivative of the Thomson-scattering optical depth with respect to con- III. IMPLEMENTATION formal time, and Z τ 0 The left flowchart in Fig. 1 shows the interdependency κ(τ; τ ) = dτ1 κ_ (τ1): (3) τ 0 between the different perturbation variables in the dif- ferential equations for their evolution.
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