Spherical collapse in quintessence models with zero speed of sound

TL;DR

This work analyzes spherical collapse in the presence of quintessence with negligible speed of sound ($c_s^2\to 0$), showing that quintessence comoves with dark matter and the collapsing region forms an exact FLRW interior. The authors derive the collapse threshold $\delta_c(z)$ and demonstrate that the dominant impact on structure formation arises from modifications to the linear growth function $D(z)$, with a secondary effect from the threshold. A qualitatively new result is that clustering quintessence adds a mass component to halos, with $M_Q/M_m\sim(1+w)\Omega_Q/\Omega_m$ at late times, which alters the total halo mass function at low redshift and offers a potential observational signature. The analysis combines a local coordinate treatment (Fermi coordinates) with extended Press-Schechter/Sheth-Tormen formalisms and uses CAMB for the matter power spectrum, highlighting observable strategies in cluster counts and lensing to distinguish clustering quintessence from smooth dark energy.

Abstract

We study the spherical collapse model in the presence of quintessence with negligible speed of sound. This case is particularly motivated for w<-1 as it is required by stability. As pressure gradients are negligible, quintessence follows dark matter during the collapse. The spherical overdensity behaves as a separate closed FLRW universe, so that its evolution can be studied exactly. We derive the critical overdensity for collapse and we use the extended Press-Schechter theory to study how the clustering of quintessence affects the dark matter mass function. The effect is dominated by the modification of the linear dark matter growth function. A larger effect occurs on the total mass function, which includes the quintessence overdensities. Indeed, here quintessence constitutes a third component of virialized objects, together with baryons and dark matter, and contributes to the total halo mass by a fraction ~ (1+w) Omega_Q / Omega_m. This gives a distinctive modification of the total mass function at low redshift.

Figures (12)

• Figure 1: Spherical collapse
• Figure 2: Thick lines: time evolution of the radius for a spherical collapse. Thin lines: time evolution following the linearized solutions. The quintessence equation of state is $w=-0.7$ (above) and $w=-1.3$ (below). Starting with the same overdensity, a model with CDM only is the first to collapse. In the upper figure $\Lambda$CDM collapses before the quintessence models as dark energy with $w=-0.7$ is more important in the past. The situation is reversed for $w=-1.3$. For $w=-0.7$ the $c_s=0$ quintessence collapses before $c_s=1$ as positive energy clusters together with dark matter. For $w=-1.3$ the situation is reversed as negative energy clusters and hinders the collapse. Note that quintessence models with $c_s^2=1$ and $w<-1$ are plagued by ghost instabilities and are thus very pathological on short scales. In this figure and in the following ones we study this case only for comparison with the $c_s^2 =0$ case.
• Figure 3: Redshift of collapse as a function of the initial overdensity. In the upper figure quintessence models have $w=-0.7$, while they have $w=-1.3$ in the lower one. The behavior follows that explained in figure \ref{['fig:collapse']}.
• Figure 4: Linear overdensity at collapse as a function of the redshift of collapse. In the upper figure the quintessence models have $w=-0.7$, while they have $w=-1.3$ in the lower one.
• Figure 5: Ratio between the growth functions $D(z)$ for $c_s=0$ and $c_s=1$ as a function of the redshift $z$.