Generalized Boltzmann hierarchy for massive neutrinos in cosmology

Boltzmann solvers are an important tool for the computation of cosmological observables in the linear regime. In the presence of massive neutrinos, they involve solving the Boltzmann equation followed by an integration in momentum space to arrive at the desired fluid properties, a procedure which is known to be computationally slow. In this work we introduce the so-called generalized Boltzmann hierarchy (GBH) for massive neutrinos in cosmology, an alternative to the usual Boltzmann hierarchy, where the momentum dependence is integrated out leaving us with a two-parameter infinite set of ordinary differential equations. Along with the usual expansion in multipoles, there is now also an expansion in higher velocity weight integrals of the distribution function. Using a toy code, we show that the GBH produces the density contrast neutrino transfer function to a $\lesssim 0.5\%$ accuracy at both large and intermediate scales compared to the neutrino free-streaming scale, thus providing a proof-of-principle for the GBH. We comment on the implementation of the GBH in a state of the art Boltzmann solver.

Figure 1

Evolution of the neutrino horizon, i.e., T for $q/T_0=3$, and $m=0.1\mathrm{eV}$. It grows like the conformal time $\tau$ up to the time of transition, when it effectively freezes as it approaches the nonrelativistic regime. The significant difference between T and $\tau$ in the nonrelativistic regime explains why one needs a much higher $l_{\max }$ for radiation than for massive neutrinos [22].

Figure 2

Choices of $l_{\max }$ and $n_{\max }$ as a function of $x$, as given by Eq. (12), to approximately ensure convergence, up to $x=30$. Note that $n_{\max }$ grows faster than linearly with $x$.

Figure 3

Relative difference in the density contrast neutrino transfer function from the GBH and CLASS (in high precision settings) for neutrino masses of $m=0.02\mathrm{eV}$, $m=0.1\mathrm{eV}$, and $m=0.5\mathrm{eV}$, and at intermediate scales defined by $k=x/\mathrm{T}$, with $x=15$ (top) and $x=30$ (bottom). The agreement is in the $\lesssim 0.5\%$ level for all redshift. We conclude that the GBH is accurately producing neutrino transfer functions.

Figure 4

Density contrast neutrino transfer functions, as a function of scale and at redshift $z=0$, from the GBH while switching to a fluid approximation at $x_{\max }=15$ ($\mathrm{GBH}+\mathrm{FA}$), CLASS in both default (CLASS-DPS), and high (CLASS-HPS) precision settings, and CAMB in both default (CAMB-DPS) and high (CAMB-HPS) precision settings as well. Here we set the neutrino mass to $m=0.1\mathrm{eV}$. We verified that choosing a value of $x_{\max }=30$ for the turning point between the GBH and the FA produced very similar results for the black dots. Also, the neutrino transfer function from CAMB is originally in the synchronous gauge, so we had to perform a gauge transformation to the Newtonian gauge to produce the curves in cyan.

Figure 5

Choices of $l_{\max }$ and $n_{\max }$, as given by Eq. (12) to approximately ensure convergence of the GBH, as a function of scale for varying neutrino mass, and up to $x=30$. For a given fixed scale, the dimension of the GBH increases for smaller neutrino mass. In fact, since $\mathrm{T}\sim 1/\sqrt{m}$, we can use Eq. (12) to conclude that $l_{\max }\sim 1/\sqrt{m}$, while $n_{\max }\sim 1/m^{0.8}$.

Figure 6

Left- (GBH) and right-hand (TS) sides of Eq. (b1) as obtained from the GBH. Here $x=15$, or $k=0.008{\mathrm{Mpc}}^{-1}$, $m=0.1\mathrm{eV}$, $l_{\max }=8$, and $n_{\max }=16$. The approximation reproduces the right features, but is not accurate.

Figure 7

Relative difference between sound speeds squared, coming from the GBH and the assumption of adiabaticity, when compared to the exact solution from CLASS. Here $x=15$, or $k=0.008{\mathrm{Mpc}}^{-1}$, $m=0.1\mathrm{eV}$, $l_{\max }=8$, and $n_{\max }=16$.

Figure 8

Relative difference between ${\mathrm{\Lambda }}_1$’s, coming from CLASS and GBH truncation schemes, when compared to the exact solution from the GBH. Here $x=15$, or $k=0.008{\mathrm{Mpc}}^{-1}$, $m=0.1\mathrm{eV}$, $l_{\max }=8$, and $n_{\max }=16$.

Figure 9

Relative difference between ${\mathrm{\Lambda }}_2$’s, coming from CLASS and GBH truncation schemes, when compared to the exact solution from the GBH. Here $x=15$, or $k=0.008{\mathrm{Mpc}}^{-1}$, $m=0.1\mathrm{eV}$, $l_{\max }=8$, and $n_{\max }=16$.

Figure 10

Relative difference between density contrast neutrino transfer functions coming from the fluid approximation with GBH and CLASS truncation schemes, when compared to the exact solution from CLASS. Here $x=15$, or $k=0.008{\mathrm{Mpc}}^{-1}$, and $m=0.1\mathrm{eV}$.