Cosmological perturbations without the Boltzmann hierarchy

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” communicates 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.

Figure 1

Flowcharts for the perturbation calculation with (a) the Boltzmann hierarchy and (b) the integral-equation approach. An arrow points from an element that appears in the differential equation for the element it points to. Ingredients that appear in the integral equation for a given quantity are indicated in (b) with an integral sign. As both figures indicate, the higher moments ($l\ge 3$ for ${\mathrm{\Delta }}_l^{\mathrm{T}}$ and ${\mathrm{\Delta }}_l^{\nu }$ and $\ge 2$ for $E_l$) communicate to the rest of the system of equations only through the quadrupole ($l=2$). The diagrams also indicate that in both cases, the photon-intensity quadrupole ${\mathrm{\Delta }}_2^{\mathrm{T}}$ feeds into the rest of the system of equations only through the photon velocity ${\theta }_{\gamma }$, and similarly for the neutrino quadrupole.

Figure 2

The CMB visibility function $\dot{\kappa }({\tau }_0,\tau )$ as a function of conformal time. It is shown to indicate the range of conformal times, peaked at $\tau \simeq 280\mathrm{Mpc}$, that which contribute to the observed CMB power spectra from recombination.

Figure 3

The transfer function ${\mathrm{\Delta }}_2^{\mathrm{T}}(\tau )$ for the CMB photon-intensity quadrupole as a function of conformal time $\tau$ for a Fourier mode of wave number $k=0.2\mathrm{Mpc}$ (which corresponds roughly to a CMB multipole moment $l\sim 3000$). The black curve shows the results of the full Boltzmann hierarchy as a function of conformal time. The other curves show results of the iterative integral-equation solution, taking ${\mathrm{\Delta }}_2^{\mathrm{T}}(\tau )=0=\mathrm{\Pi }(\tau )$ as an initial ansatz. The yellowish curve shows the result for ${\mathrm{\Delta }}_2^{\mathrm{T}}(\tau )$ after the first iteration—i.e., after integrating the differential equations for all perturbation variables except ${\mathrm{\Delta }}_2^{\mathrm{T}}(\tau )$ and $\mathrm{\Pi }(\tau )$ and then integrating the integral equations for ${\mathrm{\Delta }}_2^{\mathrm{T}}(\tau )$ and $\mathrm{\Pi }(\tau )$ using the results of the differential equations. The red curve shows results after three iterations, and the blue curve after five iterations. The thickness of the curves is such that if two are indistinguishable, the agreement between the two is $O(0.1\%$).