Analytical Approximations to Galaxy Clustering

We discuss some recent progress in constructing analytic approximations to the galaxy clustering. We show that successful models can be constructed for the clustering of both dark matter and dark matter haloes. Our understanding of galaxy clustering and galaxy biasing can be greatly enhanced by these models.Comment: 10 pages, Latex, crckapb.sty, figure included, to appear in the proceedings of Ringberg Workshop on Large-Scale Structure (ed. D. Hamilton; Kluwer Academic Publishers

Large scale bias and the peak background split

Dark matter haloes are biased tracers of the underlying dark matter distribution. We use a simple model to provide a relation between the abundance of dark matter haloes and their spatial distribution on large scales. Our model shows that knowledge of the unconditional mass function alone is sufficient to provide an accurate estimate of the large scale bias factor. Then we use the mass function measured in numerical simulations of SCDM, OCDM and LCDM to compute this bias. Comparison with these simulations shows that this simple way of estimating the bias relation and its evolution is accurate for less massive haloes as well as massive ones. In particular, we show that haloes which are less/more massive than typical M* haloes at the time they form are more/less strongly clustered than formulae based on the standard Press-Schechter mass function predict.Comment: 8 pages, 6 figures, submitted to MNRAS corrected y-label for fig.4 (newlabel = 1 + oldlabel

On the Distribution of Haloes, Galaxies and Mass

The stochasticity in the distribution of dark haloes in the cosmic density field is reflected in the distribution function $P_V(N_h|\delta_m)$ which gives the probability of finding $N_h$ haloes in a volume $V$ with mass density contrast $\delta_m$. We study the properties of this function using high-resolution $N$-body simulations, and find that $P_V(N_n|\delta_m)$ is significantly non-Poisson. The ratio between the variance and the mean goes from $\sim 1$ (Poisson) at $1+\delta_m\ll 1$ to $<1$ (sub-Poisson) at $1+\delta_m\sim 1$ to $>1$ (super-Poisson) at $1+\delta_m\gg 1$. The mean bias relation is found to be well described by halo bias models based on the Press-Schechter formalism. The sub-Poisson variance can be explained as a result of halo-exclusion while the super-Poisson variance at high $\delta_m$ may be explained as a result of halo clustering. A simple phenomenological model is proposed to describe the behavior of the variance as a function of $\delta_m$. Galaxy distribution in the cosmic density field predicted by semi-analytic models of galaxy formation shows similar stochastic behavior. We discuss the implications of the stochasticity in halo bias to the modelling of higher-order moments of dark haloes and of galaxies.Comment: 10 pages, 6 figures, Latex using MN2e style. Minor changes. Accepted for publication in MNRA

Testing the Warm Dark Matter paradigm with large-scale structures

We explore the impact of a LWDM cosmological scenario on the clustering properties of large-scale structure in the Universe. We do this by extending the halo model. The new development is that we consider two components to the mass density: one arising from mass in collapsed haloes, and the second from a smooth component of uncollapsed mass. Assuming that the nonlinear clustering of dark matter haloes can be understood, then from conservation arguments one can precisely calculate the clustering properties of the smooth component and its cross-correlation with haloes. We then explore how the three main ingredients of the halo calculations, the mass function, bias and density profiles are affected by WDM. We show that, relative to CDM: the mass function is suppressed by ~50%, for masses ~100 times the free-streaming mass-scale; the bias of low mass haloes can be boosted by up to 20%; core densities of haloes can be suppressed. We also examine the impact of relic thermal velocities on the density profiles, and find that these effects are constrained to scales r<1 kpc/h, and hence of little importance for dark matter tests, owing to uncertainties in the baryonic physics. We use our modified halo model to calculate the non-linear matter power spectrum, and find significant small-scale power in the model. However, relative to the CDM case, the power is suppressed. We then calculate the expected signal and noise that our set of LWDM models would give for a future weak lensing mission. We show that the models should in principle be separable at high significance. Finally, using the Fisher matrix formalism we forecast the limit on the WDM particle mass for a future full-sky weak lensing mission like Euclid or LSST. With Planck priors and using multipoles l<5000, we find that a lower limit of 2.6 keV should be easily achievable.Comment: Replaced with version accepted for publication in PRD. Inclusion of: new figure showing dependence of predictions on cut-off mass; new discussion of mass function; updated refs. 18 pages, 10 Figure

Analysis of Laser ARPES from Bi2_2Sr2_2CaCu2_2O8+δ_{8+\delta} in superconductive state: angle resolved self-energy and fluctuation spectrum

We analyze the ultra high resolution laser angle resolved photo-emission spectroscopy (ARPES) intensity from the slightly underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ in the superconductive (SC) state. The momentum distribution curves (MDC) were fitted at each energy \w employing the SC Green's function along several cuts perpendicular to the Fermi surface with the tilt angle $\theta$ with respect to the nodal cut. The clear observation of particle-hole mixing was utilized such that the complex self-energy as a function of $\omega$ is directly obtained from the fitting. The obtained angle resolved self-energy is then used to deduce the Eliashberg function \alpha^2 F^{(+)}(\th,\w) in the diagonal channel by inverting the d-wave Eliashberg equation using the maximum entropy method. Besides a broad featureless spectrum up to the cutoff energy $\omega_c$, the deduced $\alpha^2 F$ exhibits two peaks around 0.05 eV and 0.015 eV. The former and the broad feature are already present in the normal state, while the latter emerges only below $T_c$. Both peaks become enhanced as $T$ is lowered or the angle $\th$ moves away from the nodal direction. The implication of these findings are discussed.Comment: 7 pages, 5 figures, summited to PR

Halo bias in the excursion set approach with correlated steps

In the Excursion Set approach, halo abundances and clustering are closely related. This relation is exploited in many modern methods which seek to constrain cosmological parameters on the basis of the observed spatial distribution of clusters. However, to obtain analytic expressions for these quantities, most Excursion Set based predictions ignore the fact that, although different k-modes in the initial Gaussian field are uncorrelated, this is not true in real space: the values of the density field at a given spatial position, when smoothed on different real-space scales, are correlated in a nontrivial way. We show that when the excursion set approach is extended to include such correlations, then one must be careful to account for the fact that the associated prediction for halo bias is explicitly a real-space quantity. Therefore, care must be taken when comparing the predictions of this approach with measurements in simulations, which are typically made in Fourier-space. We show how to correct for this effect, and demonstrate that ignorance of this effect in recent analyses of halo bias has led to incorrect conclusions and biased constraints.Comment: 7 pages, 3 figures; v2 -- minor clarifications, accepted in MNRA

Scale-dependent bias and the halo model

We use a simplified version of the halo model with a power law power spectrum to study scale dependence in galaxy bias at the very large scales relevant to baryon oscillations. In addition to providing a useful pedagogical explanation of the scale dependence of galaxy bias, the model provides an analytic tool for studying how changes in the Halo Occupation Distribution (HOD) impact the scale dependence of galaxy bias on scales between 10 and 1000 Mpc/h, which is useful for interpreting the results of complex N-body simulations. We find that changing the mean number of galaxies per halo of a given mass will change the scale dependence of the bias, but that changing the way the galaxies are distributed within the halo has a smaller effect on the scale dependence of bias at large scales. We use the model to explain the decay in amplitude of the baryon oscillations as k increases, and generalize the model to make predictions about scale dependent galaxy bias when redshift space distortions are introduced.Comment: 13 pages, 2 figures; corrected typos, extended discussion of redshift space distortions, matches published versio

The abundance and clustering of dark haloes in the standard Lambda CDM cosmogony

Much evidence suggests that we live in a flat Cold Dark Matter universe with a cosmological constant. Accurate analytic formulae are now available for many properties of the dark halo population in such a Universe. Assuming current ``concordance'' values for the cosmological parameters, we plot halo abundance against redshift as a function of halo mass, of halo temperature, of the fraction of cosmic matter in haloes, of halo clustering strength, and of the clustering strength of the z=0 descendants of high redshift haloes. These plots are useful for understanding how nonlinear structure grows in the model. They demonstrate a number of properties which may seem surprising, for example: 10^9 solar mass haloes are as abundant at z=20 as L_* galaxies are today; 10^6K haloes are equally abundant at z=8 and at z=0; 10% of all matter is currently in haloes hotter than 1 keV, while more than half is in haloes too cool to trap photo-ionized gas; 1% of all matter at z=15 is in haloes hot enough to ionise hydrogen; haloes of given mass or temperature are more clustered at higher redshift; haloes with the abundance of present-day L_* galaxies are equally clustered at all z10 are more clustered at z=0 than are L_* galaxies.Comment: 10 pages, 2 ps figures, version to be published in MNRA

The environmental dependence of clustering in hierarchical models

In hierarchical models, density fluctuations on different scales are correlated. This induces correlations between dark halo masses, their formation histories, and their larger-scale environments. In turn, this produces a correlation between galaxy properties and environment. This correlation is entirely statistical in nature. We show how the observed clustering of galaxies can be used to quantify the importance of this statistical correlation relative to other physical effects which may also give rise to correlations between the properties of galaxies and their surroundings. We also develop a halo model description of this environmental dependence of clustering.Comment: 11 pages, 6 figures, MNRAS in pres

On the equivalence between the effective cosmology and excursion set treatments of environment

In studies of the environmental dependence of structure formation, the large scale environment is often thought of as providing an effective background cosmology: e.g. the formation of structure in voids is expected to be just like that in a less dense universe with appropriately modified Hubble and cosmological constants. However, in the excursion set description of structure formation which is commonly used to model this effect, no explicit mention is made of the effective cosmology. Rather, this approach uses the spherical evolution model to compute an effective linear theory growth factor, which is then used to predict the growth and evolution of nonlinear structures. We show that these approaches are, in fact, equivalent: a consequence of Birkhoff's theorem. We speculate that this equivalence will not survive in models where the gravitational force law is modified from an inverse square, potentially making the environmental dependence of clustering a good test of such models.Comment: 4 pages, 0 figures, accepted to MNRA