On the spherical collapse model in dark energy cosmologies

TL;DR

This paper addresses how dark energy modeled as a scalar field affects non-linear spherical collapse and structure formation. It adopts four quintessence potentials and two clustering prescriptions, computing observables such as the nonlinear density contrast $\Delta_c$, the turnaround-to-virialization radius ratio $R_{ta}/R_V$, and the linear growth factor $D(z)$. Results show strong sensitivity to the potential, initial conditions, and dark-energy clustering, with inhomogeneous quintessence potentially shifting $\Delta_c$ by factors up to four at low virialisation redshift, while constant $w$ models exhibit milder differences. The work underscores the need for a more complete non-linear/relativistic treatment of dark energy (e.g., energy flux parameter $\Gamma$ or swiss-cheese approaches) to make robust predictions for cluster statistics and lensing in dark-energy cosmologies.

Abstract

We study the spherical collapse model in dark energy cosmologies, in which dark energy is modelled as a minimally coupled scalar field. We first follow the standard assumption that dark energy does not cluster on the scales of interest. Investigating four different popular potentials in detail, we show that the predictions of the spherical collapse model depend on the potential used. We also investigate the dependence on the initial conditions. Secondly, we investigate in how far perturbations in the quintessence field affect the predictions of the spherical collapse model. In doing so, we assume that the field collapses along with the dark matter. Although the field is still subdominant at the time of virialisation, the predictions are different from the case of a homogeneous dark energy component. This will in particular be true if the field is non--minimally coupled. We conclude that a better understanding of the evolution of dark energy in the highly non--linear regime is needed in order to make predictions using the spherical collapse model in models with dark energy.

Figures (10)

• Figure 1: Quintessence model $V=M(\exp(\beta\phi) + \exp(\gamma\phi))$copel. Top left panel: Evolution of $w_{\phi}$ in the background (dashed line) and inside an overdensity (solid line) for $\Gamma=0$, as a function of $\log (1+z)$ (overdensity virialises at $z=0$). Top right panel: Evolution of $\rho_{\phi}/\rho_{\phi_c}$ (solid line) and $\rho_{m}/\rho_{cdm}$ (dashed line) in the case $\Gamma=0$ as a function of $\log(1+z)$ (overdensity virialises at $z=0$). Middle left panel: The ratio $R_{ta}/R_V$ in the case of an inhomogeneous ($\Gamma=0$) (solid line) and homogeneous scalar field (dashed line). The ratio $\rho_\phi/\rho_{\rm matter}$ inside the overdensity (solid line) ($\Gamma=0$) and in the background (dashed line). Bottom left: $\Delta_{c}$ as a function of $z_V$, considering the effect of an inhomogeneous quintessence field ($\Gamma=0$) (solid line). The results for a homogeneous quintessence field are shown as well (dashed line). Bottom right: Predictions for the linear growth factor for this potential (solid line). The dashed line represents the $\Lambda$CDM case.
• Figure 2: Quintessence model $V= M(\exp(\gamma/\phi)-1)$steinhardt. Top left panel: Evolution of $w_{\phi}$ in the background (dashed line) and inside the overdensity (solid line) for $\Gamma=0$ as a function of $\log (1+z)$ (overdensity virialises at $z=0$). Top right panel: Evolution of $\rho_{\phi}/\rho_{\phi_c}$ (solid line) and $\rho_{m}/\rho_{cdm}$ (dashed line) in the case $\Gamma=0$ as a function of $\log(1+z)$ (overdensity virialises at $z=0$). Middle left panel: The ratio $R_{ta}/R_V$ in the case of an inhomogeneous ($\Gamma=0$) (solid line) and homogeneous scalar field (dashed line). The ratio $\rho_\phi/\rho_{\rm matter}$ inside the overdensity (solid line) ($\Gamma=0$) and in the background (dashed line). Bottom left: $\Delta_{c}$ as a function of $z_V$, considering the effect of an inhomogeneous quintessence field ($\Gamma=0$) (solid line). The results for a homogeneous quintessence field are shown as well (dashed line). Bottom right: Predictions for the linear growth factor for this potential (solid line). The dashed line represents the $\Lambda CDM$ case.
• Figure 3: Quintessence model $V= M(A+(\phi-B)^2)\exp(-\gamma\phi)$skordis. Top left panel: Evolution of $w_{\phi}$ in the background (dashed line) and inside the overdensity (solid line) for $\Gamma=0$ as a function of $\log (1+z)$ (overdensity virialises at $z=0$). Top right panel: Evolution of $\rho_{\phi}/\rho_{\phi_c}$ (solid line) and $\rho_{m}/\rho_{cdm}$ (dashed line) in the case $\Gamma=0$ as a function of $\log(1+z)$ (overdensity virialises at $z=0$). Middle left panel: The ratio $R_{ta}/R_V$ in the case of an inhomogeneous ($\Gamma=0$) (solid line) and homogeneous scalar field (dashed line). The ratio $\rho_\phi/\rho_{\rm matter}$ inside the overdensity (solid line) ($\Gamma=0$) and in the background (dashed line). Bottom left: $\Delta_{c}$ as a function of $z_V$, considering the effect of an inhomogeneous quintessence field ($\Gamma=0$) (solid line). The results for a homogeneous quintessence field are shown as well (dashed line). Bottom right: Predictions for the linear growth factor for this potential (solid line). The dashed line represents the $\Lambda CDM$ case.
• Figure 4: Quintessence model $V= M\exp(\phi^2)/\phi^{\gamma}$brax. Top left panel: Evolution of $w_{\phi}$ in the background (dashed line) and inside the overdensity (solid line) for $\Gamma=0$ as a function of $\log (1+z)$ (overdensity virialises at $z=0$). Top right panel: Evolution of $\rho_{\phi}/\rho_{\phi_c}$ (solid line) and $\rho_{m}/\rho_{cdm}$ (dashed line) in the case $\Gamma=0$ as a function of $\log(1+z)$ (overdensity virialises at $z=0$). Middle left panel: The ratio $R_{ta}/R_V$ in the case of an inhomogeneous ($\Gamma=0$) (solid line) and homogeneous scalar field (dashed line). The ratio $\rho_\phi/\rho_{\rm matter}$ inside the overdensity (solid line) ($\Gamma=0$) and in the background (dashed line). Bottom left: $\Delta_{c}$ as a function of $z_V$, considering the effect of an inhomogeneous quintessence field ($\Gamma=0$) (solid line). The results for a homogeneous quintessence field are shown as well (dashed line). Bottom right: Predictions for the linear growth factor for this potential (solid line). The dashed line represents the $\Lambda CDM$ case.
• Figure 5: Model with constant equation of state $w=-0.8$. The upper panel shows the predictions for $\Delta_c$ for the inhomogeneous case (dashed line) and homogeneous case (solid line). The middle panel shows the predictions for $R_{\rm V}/R_{\rm ta}$ for the inhomogeneous case (dashed line) and homogeneous case (solid line). The lower panel shows the evolution of the density contrast in matter (dashed line) and dark energy (solid line) for the case of an inhomogeneous dark energy component (cluster virialises at $z=0$).