We derive the Layzer-Irvine equation in the presence of a homogeneous (or quasi-homogeneous) dark energy component with an arbitrary equation of state. We extend the Layzer-Irvine equation to homogeneous and isotropic universes with an arbitrary numb
er of dimensions and obtain the corresponding virial relation for sufficiently relaxed objects. We find analogous equations describing the dynamics of cosmic string loops and other p-branes of arbitrary dimensionality, discussing the corresponding relativistic and non-relativistic limits. Finally, we generalize the Layzer-Irvine equation to account for a non-minimal interaction between dark matter and dark energy, discussing its practical use as a signature of such an interaction.
In this paper we compare the performances of the chi-square and median likelihood analysis in the determination of cosmological constraints using type Ia supernovae data. We perform a statistical analysis using the 307 supernovae of the Union 2 compi
lation of the Supernova Cosmology Project and find that the chi-square statistical analysis yields tighter cosmological constraints than the median statistic if only supernovae data is taken into account. We also show that when additional measurements from the Cosmic Microwave Background and Baryonic Acoustic Oscillations are considered, the combined cosmological constraints are not strongly dependent on whether one applies the chi-square statistic or the median statistic to the supernovae data. This indicates that, when complementary information from other cosmological probes is taken into account, the performances of the chi-square and median statistics are very similar, demonstrating the robustness of the statistical analysis.
In this paper we investigate possible solutions to the coincidence problem in flat phantom dark energy models with a constant dark energy equation of state and quintessence models with a linear scalar field potential. These models are representative
of a broader class of cosmological scenarios in which the universe has a finite lifetime. We show that, in the absence of anthropic constraints, including a prior probability for the models inversely proportional to the total lifetime of the universe excludes models very close to the $Lambda {rm CDM}$ model. This relates a cosmological solution to the coincidence problem with a dynamical dark energy component having an equation of state parameter not too close to -1 at the present time. We further show, that anthropic constraints, if they are sufficiently stringent, may solve the coincidence problem without the need for dynamical dark energy.
