Dynamics of gravitational clustering II. Steepest-descent method for the quasi-linear regime

We develop a non-perturbative method to derive the probability distribution $P(\delta_R)$ of the density contrast within spherical cells in the quasi-linear regime. Indeed, since this corresponds to a rare-event limit a steepest-descent approximation can yield asymptotically exact results. We check that this is the case for Gaussian initial density fluctuations, where we recover most of the results obtained by perturbative methods from a hydrodynamical description. Moreover, we correct an error which was introduced in previous works for the high-density tail of the pdf. This feature, which appears for power-spectra with a slope $n<0$, points out the limitations of perturbative approaches which cannot describe the pdf $P(\delta_R)$ for \delta_R \ga 3 even in the limit $\sigma \to 0$. This break-up does not involve shell-crossing and it is naturally explained within our framework. Thus, our approach provides a rigorous treatment of the quasi-linear regime, which does not rely on the hydrodynamical approximation for the equations of motion. Besides, it is actually simpler and more intuitive than previous methods. Our approach can also be applied to non-Gaussian initial conditions.Comment: 18 pages, final version published in A&

Transients from Zel'dovich initial conditions

We investigate the error implied by the use of the Zel'dovich approximation to set up the initial conditions at a finite redshift zi in numerical simulations. Using a steepest-descent method developed in a previous work we derive the probability distribution P(delta_R) of the density contrast in the quasi-linear regime. This also provides its dependence on the redshift zi at which the simulation is started. Thus, we find that the discrepancy with the exact pdf (defined by the limit zi->infinity) is negligible after the scale factor has grown by a factor a/a_i>5, for scales which were initially within the linear regime with sigma_i>0.1. This shows that the use of the Zel'dovich approximation to implement the initial conditions is sufficient for practical purposes since these are not very severe constraints.Comment: 6 pages, final version published in A&

The phase-diagram of the IGM and the entropy floor of groups and clusters: are clusters born warm?

We point out that two problems of observational cosmology, the facts i) that > 60% of the baryonic content of the universe is not observed at z=0 and ii) that the properties of small clusters do not agree with simple expectations, could be closely related. As shown by recent studies, the shock-heating associated with the formation of large-scale structures heats the intergalactic medium (IGM) and leads to a ``warm IGM'' component for the gas. In the same spirit, we suggest the intracluster medium (ICM) to be a mixture of galaxy-recycled, metal enriched gas and intergalactic gas, shock-heated by the collapsing much larger scales. This could be obtained through two processes: 1) the late infalling gas from the external warm IGM is efficiently mixed within the halo and brings some additional entropy, or 2) the shocks generated by larger non-linear scales are also present within clusters and can heat the ICM. We show that if assumption (1) holds, the entropy brought by the warm IGM is sufficient to explain the observed properties of clusters, in particular the entropy floor and the LX-T relation. On the other hand, we briefly note that the scenario (2) would require a stronger shock-heating because of the larger density of the ICM as compared with filaments. Our scenario of clusters being "born warm" can be checked through the predicted redshift evolution of the entropy floor.Comment: 8 pages, final version published in MNRA

Dynamics of gravitational clustering IV. The probability distribution of rare events

Using a non-perturbative method developed in a previous article (paper II) we investigate the tails of the probability distribution $P(\rho_R)$ of the overdensity within spherical cells. We show that our results for the low-density tail of the pdf agree with perturbative results when the latter are finite (up to the first subleading term), that is for power-spectra with $-3<n<-1$. Over the range $-1<n<1$ some shell-crossing occurs (which leads to the break-up of perturbative approaches) but this does not invalidate our approach. In particular, we explain that we can still obtain an approximation for the low-density tail of the pdf. This feature also clearly shows that perturbative results should be viewed with caution (even when they are finite). We point out that our results can be recovered by a simple spherical model but they cannot be derived from the stable-clustering ansatz in the regime $\sigma \gg 1$ since they involve underdense regions which are still expanding. Second, turning to high-density regions we explain that a naive study of the radial spherical dynamics fails. Indeed, a violent radial-orbit instability leads to a fast relaxation of collapsed halos (over one dynamical time) towards a roughly isotropic equilibrium velocity distribution. Then, the transverse velocity dispersion stabilizes the density profile so that almost spherical halos obey the stable-clustering ansatz for $-3<n<1$. We again find that our results for the high-density tail of the pdf agree with a simple spherical model (which takes into account virialization). Moreover, they are consistent with the stable-clustering ansatz in the non-linear regime. Besides, our approach justifies the large-mass cutoff of the Press-Schechter mass function (although the various normalization parameters should be modified).Comment: 27 pages, final version published in A&

Combining perturbation theories with halo models

We investigate the building of unified models that can predict the matter-density power spectrum and the two-point correlation function from very large to small scales, being consistent with perturbation theory at low $k$ and with halo models at high $k$. We use a Lagrangian framework to re-interpret the halo model and to decompose the power spectrum into "2-halo" and "1-halo" contributions, related to "perturbative" and "non-perturbative" terms. We describe a simple implementation of this model and present a detailed comparison with numerical simulations, from $k \sim 0.02$ up to $100 h$Mpc$^{-1}$, and from $x \sim 0.02$ up to $150 h^{-1}$Mpc. We show that the 1-halo contribution contains a counterterm that ensures a $k^2$ tail at low $k$ and is important not to spoil the predictions on the scales probed by baryon acoustic oscillations, $k \sim 0.02$ to $0.3 h$Mpc$^{-1}$. On the other hand, we show that standard perturbation theory is inadequate for the 2-halo contribution, because higher order terms grow too fast at high $k$, so that resummation schemes must be used. We describe a simple implementation, based on a 1-loop "direct steepest-descent" resummation for the 2-halo contribution that allows fast numerical computations, and we check that we obtain a good match to simulations at low and high $k$. Our simple implementation already fares better than standard 1-loop perturbation theory on large scales and simple fits to the power spectrum at high $k$, with a typical accuracy of 1% on large scales and 10% on small scales. We obtain similar results for the two-point correlation function. However, there remains room for improvement on the transition scale between the 2-halo and 1-halo contributions, which may be the most difficult regime to describe.Comment: 29 page