Spherical collapse and cluster counts in modified gravity models

Modifications to the gravitational potential affect the nonlinear gravitational evolution of large scale structures in the Universe. To illustrate some generic features of such changes, we study the evolution of spherically symmetric perturbations when the modification is of Yukawa type; this is nontrivial, because we should not and do not assume that Birkhoff’s theorem applies. We then show how to estimate the abundance of virialized objects in such models. Comparison with numerical simulations shows reasonable agreement: When normalized to have the same fluctuations at early times, weaker large scale gravity produces fewer massive halos. However, the opposite can be true for models that are normalized to have the same linear theory power spectrum today, so the abundance of rich clusters potentially places interesting constraints on such models. Our analysis also indicates that the formation histories and abundances of sufficiently low mass objects are unchanged from standard gravity. This explains why simulations have found that the nonlinear power spectrum at large $k$ is unaffected by such modifications to the gravitational potential. In addition, the most massive objects in models with normalized cosmic microwave background and weaker gravity are expected to be similar to the high-redshift progenitors of the most massive objects in models with stronger gravity. Thus, the difference between the cluster and field galaxy populations is expected to be larger in models with stronger large scale gravity.

Figure 1

Ratio of linear theory growth factor to that in standard $\Lambda \mathrm{CDM}$, at $a=1$, when $r_s=5h^{-1}\mathrm{Mpc}$ and $\alpha =1$ (top) and $\alpha =-1$ (bottom). Dashed lines show this ratio for the model in Eq. (1), and solid lines for Eq. (6). For the solid lines, $r_c=70h^{-1}\mathrm{Mpc}$, and dotted $r_c=350h^{-1}\mathrm{Mpc}$.

Figure 2

Convergence to solution as number of shells increases for two objects with mass ${10}^{14.5}{h}^{-1}{M}_{\odot }$, with $\alpha =0.5$ (left) and $\alpha -1.0$ (right). The thick solid lines are actually the positions of the shell on the edge of the density perturbation for 3, 10, 20, 40, 50, 60, 80, 100, and 200 shells; note they all lie nearly on top of each other. The dashed curves show the evolution when $\alpha =0$.

Figure 3

Ratio of initial density required for collapse at the present time to that in the standard gravity model, when the background cosmology is $\Lambda \mathrm{CDM}$, $r_s=5h^{-1}\mathrm{Mpc}$, and $r_c=70h^{-1}\mathrm{Mpc}$. From top to bottom, curves show models in which $\alpha =-1$, $-0.5$, 0, 0.5, and 1 (note that $\alpha =0$ is standard gravity).

Figure 4

Expected halo abundance, $\log dN/d\log m$, as a function of mass $m$ for models with $\alpha =1$ (short dashed) and $\alpha =-1$ (long dashed), the solid line is for $\alpha =0$. The rapid rise of the negative $\alpha$ barrier in Fig. 3 is why there is a larger effect on the abundance of high mass halos for negative $\alpha$ than positive $\alpha$.

Figure 5

Ratio of halo abundances when $\alpha =-1$ (left) and $\alpha =0.5$ (right) to that when $\alpha =0$. Solid curve shows this ratio when the excursion set method is correctly implemented, using the initial fluctuation field values $S_i(M)$ and a moving barrier ${\delta }_c(M)$, and $S_i(M)$ is independent of $\alpha$. Dashed curve uses $S_0(M)$ from the linearly evolved field and a constant barrier ${\delta }_c=1.686$. Dotted curve uses $S_i(M)$ and ${\delta }_c(M)$, but now $S_i(M)$ is modified so that $S_0(M)$ is the same for both values of $\alpha$. Note that all these approaches produce different results.

Figure 6

Halo abundances with $r_s=5\mathrm{Mpc}$ ratioed to $\alpha =0$. In all panels, data points are from simulation, whereas curves are from our excursion set calculation. In the top left panel, (blue) long dashed and (red) short dashed curves are for $\alpha =0.5$ and $-1$, respectively, and all models had the same initial fluctuation spectrum (so that $\alpha =0$ has ${\sigma }_8=1$ today). The top right panel shows the ratio of counts in two $\alpha =0$ runs, but with different initial conditions, tailored so that ${\sigma }_8$ at $z=0$ is the same as in the $\alpha =0.5$ (long dashed) and $-1$ (short dashed) runs. Short dashed curves in bottom panel show the ratio of the $\alpha =0.5$ counts to that in the $\alpha =0$ run when it has been normalized to have the same (linear theory) power spectrum at $z=0$ as the $\alpha =0.5$ run. Long dashed curve shows a similar analysis of the $\alpha =-1$ case. This panel shows the ratio of the numerators in the previous two panels.

Figure 7

The panel on the top is for $\alpha =-1.0$ vs standard; the bottom is $\alpha =0.5$ vs standard. The points come from selecting big halos in each realization of the modified gravity simulation and then finding the corresponding halo in the standard gravity simulation, and then the reverse. $N$ is the number of particles in the identified halos. The tilt shows that in stronger gravity massive halos are likely to have more particles. The “tails” in the lower left are unimportant.

Figure 8

Spatial distribution of a massive halo chosen from the $\alpha =0.5$ (top) and $-1$ (bottom) simulations. The black, green, blue, and red contours show the positions of the particles that make up the chosen halo in the $\alpha =0.5$, 0, $-0.5$, and $-1$ simulations, respectively. In the top panel, note that the central dense halo spreads out for weakening gravity, and in the bottom panel the upper left halo merely moves. Axes show the $(x,y)$ coordinates of the halos in grid coordinates, $100/256h^{-1}\mathrm{Mpc}$.