Scale-dependent halo bias and the squeezed limit bispectrum in the presence of radiation

We investigate the gravitational effect of large-scale radiation perturbations on small-scale structure formation. In addition to making the growth of matter perturbations scale dependent, the free streaming of radiation also affects the coupling between structure formation at small and large scales. We study this using separate universe $N$-body simulations to compute the (isotropized) squeezed limit matter bispectrum and the linear halo bias. Our results show that the scale dependence in the growth of long-wavelength matter perturbations, caused by radiation, translates into these quantities acquiring a nontrivial scale dependence at $k\lesssim 0.05{\mathrm{Mpc}}^{-1}$. In a universe with radiation composed of cosmic microwave background photons and three species of massless neutrinos, the bias of halos with $b=2$ at high $k$ will decrease by $0.29\%,0.45\%$, and 0.8% between $k=0.05{\mathrm{Mpc}}^{-1}$ and $k=0.0005{\mathrm{Mpc}}^{-1}$ at redshifts $z=0$, 1, and 3, respectively. For objects with $b\gg 1$, these differences approach $0.43\%,0.68\%$, and 1.2%, respectively.

Figure 1

Top: Evolution of ${\delta }_c(k_L,a)/{\delta }_c(k_L,a=1)$ for different values of $k_L$. Bottom: Evolution of ${\delta }_c(k_L,a)/{\delta }_c(k_{L\downarrow }=5\times {10}^{-2}{\mathrm{Mpc}}^{-1},a)$ for different values of $k_L$. In this figure, the effects of radiation have been enhanced by setting ${\mathrm{\Omega }}_{\mathrm{rad}}=3.7\times {10}^{-4}$.

Figure 2

The response of the linear growth factor [Eq. (10)] to the long-wavelength modes shown in Fig. 1. The dashed line is the same quantity computed in an exactly matter-dominated separate universe (= $13/21$).

Figure 3

Relative linear growth factor response, Eq. (11), for neutrinos of different masses but common temperature. The location of the step feature depends on the particle mass, in this case implemented as neutrinos with the standard temperature, $T_{\nu }\approx 1.7\times {10}^{-4}\mathrm{eV}$, and degenerate masses 0.05, 0.025, 0.01, and 0.0 eV. These curves do not have a common number of species nor energy density of neutrinos, the number of states was adjusted to produce the same asymptotic value at high $k_L$.

Figure 4

The amplitude of the step in the linear growth factor response, Eq. (12) vs the number of neutrino species $N_{\nu }$ at $z=0$ for neutrinos of degenerate mass $m_{\nu }=0.05$, 0.025, 0.01, and 0.0 eV.

Figure 5

The amplitude of the step in the linear growth factor response, Eq. (12), as a function of neutrino masses for fixed neutrino energy densities at $z=0$.

Figure 6

Relative linear growth factor response, Eq. (11), for different massless neutrino densities with ${\mathrm{\Omega }}_{\mathrm{rad},0}=1.01\times {10}^{-3}$. Here $f_{\nu }$ is the fraction of the radiation energy density carried by massless neutrinos, $f_{\nu }={\mathrm{\Omega }}_{\nu }/({\mathrm{\Omega }}_{\nu }+{\mathrm{\Omega }}_{\gamma })$. The fractional difference between the different curves is $\lesssim 5\times {10}^{-4}$.

Figure 7

The step in the linear growth factor response normalized to unity at $z=0$ plotted as a function of redshift for 28 species of neutrinos with different masses.

Figure 8

The growth contribution to the power spectrum response, $R_{\text{growth}}$, for different long-wavelength modes at $z=1$. The points show results from $N$-body simulations in the separate universe, while the curves show predictions from one-loop perturbation theory.

Figure 9

The growth contribution to the power spectrum response, $R_{\text{growth}}$, for different long-wavelength modes at $z=3$. The points show results from $N$-body simulations in the separate universe, while the curves show predictions from one-loop perturbation theory.

Figure 10

Top: Cumulative Lagrangian bias as a function of halo mass at $z=0$. Bottom: Relative Lagrangian bias between $k_{L\downarrow }$ and $k_{L\uparrow }$ as a function of halo mass at $z=0$. The dashed orange line shows the ratio of the linear growth factor responses $R_D(k_{L\downarrow })/R_D(k_{L\uparrow })$.

Figure 11

Top: Cumulative Lagrangian bias as a function of halo mass at $z=1$. Bottom: Relative Lagrangian bias between $k_{L\downarrow }$ and $k_{L\uparrow }$ as a function of halo mass at $z=1$. The dashed orange line shows the ratio of the linear growth factor responses $R_D(k_{L\downarrow })/R_D(k_{L\uparrow })$.

Figure 12

Plotted is the scale dependence of the Eulerian halo bias caused by large-scale perturbations in radiation composed of CMB photons and three massless neutrinos. This quantity is shown at several different redshifts for a fixed Eulerian bias $b=2$ at $k_{\downarrow }=5\times {10}^{-2}{\mathrm{Mpc}}^{-1}$.

Figure 13

The overall amplitude of the change in the Eulerian halo bias between $k_{\downarrow }=5\times {10}^{-2}{\mathrm{Mpc}}^{-1}$ and $k_{\uparrow }=5\times {10}^{-4}{\mathrm{Mpc}}^{-1}$, plotted in Fig. 12, shown as a function of $b$, the Eulerian bias at $k_{\downarrow }$.