Spherical collapse in generalized dark matter models

The influence of considering a generalized dark matter (GDM) model, which allows for a non-pressure-less dark matter and a nonvanishing sound speed in the nonlinear spherical collapse model is discussed for the Einstein-de Sitter-like and $\mathrm{\Lambda }\mathrm{GDM}$ models. By assuming that the vacuum component responsible for the accelerated expansion of the Universe is not clustering and therefore behaving similarly to the cosmological constant $\mathrm{\Lambda }$, we show how the change in the GDM characteristic parameters affects the linear density threshold for collapse of the nonrelativistic component (${\delta }_{\mathrm{c}}$) and its virial overdensity (${\mathrm{\Delta }}_{\mathrm{V}}$). We found that a positive GDM equation of state parameter, $w_{\mathrm{gdm}}$, is responsible for lower values of ${\delta }_{\mathrm{c}}$ as compared to the standard spherical collapse model and that this effect is much stronger than the one induced by a change in the GDM sound speed, $c_{s,\mathrm{gdm}}^2$. We also found that ${\mathrm{\Delta }}_{\mathrm{V}}$ is only slightly affected and mostly sensitive to $w_{\mathrm{gdm}}$. These effects could be relatively enhanced for lower values of the matter density. We found that the effects of the additional physics on ${\delta }_{\mathrm{c}}$ and ${\mathrm{\Delta }}_{\mathrm{V}}$, when translated to nonlinear observables such as the halo mass function, induce an overall deviation of about 40% with respect to the standard $\mathrm{\Lambda }\mathrm{CDM}$ model at late times for high mass objects. However, within the current constraints for $c_{s,\mathrm{gdm}}^2$ and $w_{\mathrm{gdm}}$, we found that these changes are the consequence of properly taking into account the correct linear matter power spectrum for the GDM model while the effects coming from modifications in the spherical collapse model remain negligible. Using a phenomenologically motivated approach, we also study the nonlinear matter power spectrum and found that the additional properties of the dark matter component lead, in general, to a strong suppression of the nonlinear power spectrum with respect to the corresponding $\mathrm{\Lambda }\mathrm{CDM}$ one. Finally, as a practical example, we compare $\mathrm{\Lambda }\mathrm{GDM}$ and $\mathrm{\Lambda }\mathrm{CDM}$ using galaxy cluster abundance measurements, and found that these small scale probes will allow us to put more stringent constraints on the nature of dark matter.

Figure 1

The linear critical density contrast ${\delta }_{\mathrm{c}}$ as a function of the collapse redshift $z$ for different values of the generalized dark matter sound speed $c_{s,\mathrm{gdm}}^2$ for an EdSGDM and flat $\mathrm{\Lambda }\mathrm{GDM}$ cosmology. For reference, we consider also the standard $\mathrm{\Lambda }\mathrm{CDM}$ model, where $c_{s,\mathrm{gdm}}^2=0$.

Figure 2

Top (bottom) panel: The linear critical density contrast ${\delta }_{\mathrm{c}}$ (virial overdensity ${\mathrm{\Delta }}_{\mathrm{V}}$) as a function of the collapse redshift $z$ for different values of the generalized dark matter equation of state parameter $w_{\mathrm{gdm}}$ for an EdSGDM and flat $\mathrm{\Lambda }\mathrm{GDM}$ cosmology.

Figure 3

Relative differences between the $\mathrm{\Lambda }\mathrm{CDM}$ and $\mathrm{\Lambda }\mathrm{GDM}$ linear critical density parameter ${\delta }_{\mathrm{c}}$ (left) and virial overdensity ${\mathrm{\Delta }}_{\mathrm{V}}$ (right) as a function of the collapse redshift $z$ for different values of the matter density parameter ${\mathrm{\Omega }}_{\mathrm{m}}$. We assume $w_{\mathrm{gdm}}=5\times {10}^{-4}$ and $c_{s,\mathrm{gdm}}^2=5\times {10}^{-7}$ as reference values for the parameters of the GDM model.

Figure 4

In the top panel we show the differential mass function as a function of mass for different redshifts using the appropriate linear matter power spectrum for each model ($\mathrm{\Lambda }\mathrm{CDM}$ and $\mathrm{\Lambda }\mathrm{GDM}$). In the middle panel we present the evolution of the square root of the mass variance as a function of mass for different redshifts and for both models. In the bottom panel we show the differential mass function as in the top panel but using the $\mathrm{\Lambda }\mathrm{CDM}$ matter power spectrum for both $\mathrm{\Lambda }\mathrm{CDM}$ and $\mathrm{\Lambda }\mathrm{GDM}$, while the rest of GDM model-dependant quantities are kept. In all panels we assume $w_{\mathrm{gdm}}=5\times {10}^{-4}$ and $c_{s,\mathrm{gdm}}^2=5\times {10}^{-7}$ for the $\mathrm{\Lambda }\mathrm{GDM}$ model.

Figure 5

Number of objects above a given mass, as explained in the text, in a given redshift bin. We consider 25 bins between $z=0$ and $z=1$ and three different cosmological models: $\mathrm{\Lambda }\mathrm{CDM}$ (black line), GDM with $w_{\mathrm{gdm}}=5\times {10}^{-4}$ and $c_{s,\mathrm{gdm}}^2=5\times {10}^{-7}$ (orange line), and GDM with $w_{\mathrm{gdm}}=5\times {10}^{-4}$ and $c_{s,\mathrm{gdm}}^2=5\times {10}^{-6}$ (purple line).

Figure 6

Evolution of the linear (solid lines) and nonlinear (dot-dashed lines) matter power spectra for the $\mathrm{\Lambda }\mathrm{CDM}$ (purple lines) and $\mathrm{\Lambda }\mathrm{GDM}$ (red lines) models at $z=0$. The blue dot-dashed line represents the nonlinear matter power spectrum using the halo-model prescription of [26]. For the GDM model, we assumed the following parameters: $w_{\mathrm{gdm}}={10}^{-4}$ and $c_{\mathrm{s,\; gdm}}^2={10}^{-6}$.