On the Magnitude of Dark Energy Voids and Overdensities

We investigate the clustering of dark energy within matter overdensities and voids. In particular, we derive an analytical expression for the dark energy density perturbations, which is valid both in the linear, quasi-linear and fully non-linear regime of structure formation. We also investigate the possibility of detecting such dark energy clustering through the ISW effect. In the case of uncoupled quintessence models, if the mass of the field is of order the Hubble scale today or smaller, dark energy fluctuations are always small compared to the matter density contrast. Even when the matter perturbations enter the non-linear regime, the dark energy perturbations remain linear. We find that virialised clusters and voids correspond to local overdensities in dark energy, with $\delta_{\phi}/(1+w) \sim \Oo(10^{-5})$ for voids, $\delta_{\phi}/(1+w) \sim \Oo(10^{-4})$ for super-voids and $\delta_{\phi}/(1+w) \sim \Oo(10^{-5})$ for a typical virialised cluster. If voids with radii of $100-300 {\rm Mpc}$ exist within the visible Universe then $\delta_{\phi}$ may be as large as $10^{-3}(1+w)$. Linear overdensities of matter and super-clusters generally correspond to local voids in dark energy; for a typical super-cluster: $\delta_{\phi}/(1+w) \sim \Oo(-10^{-5})$. The approach taken in this work could be straightforwardly extended to study the clustering of more general dark energy models.