The limits of bound structures in the accelerating Universe

According to the latest evidence, the Universe is entering an era of exponential expansion, where gravitationally bound structures will get disconnected from each other, forming isolated 'island universes'. In this scenario, we present a theoretical criterion to determine the boundaries of gravitationally bound structures and a physically motivated definition of superclusters as the largest bound structures in the Universe. We use the spherical collapse model self-consistently to obtain an analytical condition for the mean density enclosed by the last bound shell of the structure (2.36 times the critical density in the present Universe, assumed to be flat, with 30 per cent matter and 70 per cent cosmological constant, in agreement with the previous, numerical result of Chiueh & He). N -body simulations extended to the future show that this criterion, applied at the present cosmological epoch, defines a sphere that encloses ≈99.7 per cent of the particles that will remain bound to the structure at least until the scale parameter of the Universe is 100 times its present value. On the other hand, (28 ± 13) per cent of the enclosed particles are in fact not bound, so the enclosed mass overestimates the bound mass, in contrast with the previous, less rigorous criterion of, e.g. Busha and collaborators, which gave a more precise mass estimate. We also verify that the spherical collapse model estimate for the radial infall velocity of a shell enclosing a given mean density gives an accurate prediction for the velocity profile of infalling particles, down to very near the centre of the virialized core.     

Cosmological parameter estimation and the spectral index from inflation

Accurate estimation of cosmological parameters from microwave background anisotropies requires high-accuracy understanding of the cosmological model. Normally, a power-law spectrum of density perturbations is assumed, in which case the spectral index n can be measured to around ± 0.004 using microwave anisotropy satellites such as MAP Planck . However, inflationary models generically predict that the spectral index n of the density perturbation spectrum will be scale-dependent. We carry out a detailed investigation of the measurability of this scale dependence by Planck , including the influence of polarization on the parameter estimation. We also estimate the increase in the uncertainty in all other parameters if the scale dependence has to be included. This increase applies even if the scale dependence is too small to be measured, unless it is assumed absent. We study the implications for inflation models, beginning with a brief examination of the generic slow-roll inflation situation, and then move to a detailed examination of a recently devised hybrid inflation model for which the scale dependence of n may be observable.     

Looking Down the Light Cone: Can Deep Redshift Surveys Alone Measure the Power Spectrum?

We analyse the window functions for the spherical harmonic mode estimators of all-sky, volume-limited surveys, considering evolutionary effects along the past light-cone which include the deviation of the distance scale from a linear relationship with redshift, linear peculiar velocity corrections, and linear evolution of the density perturbations. The spherical harmonic basis functions are considered, because they correspond most closely to the symmetries of typical survey geometries and of the light-cone effects we consider. Our results show substantial broadening of the windows over that expected by ignoring light-cone effects, indicating the difficulty of measuring the power spectrum independently from cosmology. We suggest that because of light-cone effects, deep redshift surveys should be analysed either in conjunction with CMBR data which determines the cosmological parameters, or by using a Bayesian likelihood scheme in which varying cosmological parameters and a simple parametrization of the primordial power spectrum are assumed as the priors, so that observed data can be mapped from redshift to real space. The derived power spectrum can then be compared with underlying models of fluctuation generation and growth in structure formation to evaluate both these models and the cosmological priors.     

Cosmic microwave background constraints on the epoch of reionization

We use a compilation of cosmic microwave anisotropy data to constrain the epoch of reionization in the Universe, as a function of cosmological parameters. We consider spatially flat cosmologies, varying the matter density Ω₀ (the flatness being restored by a cosmological constant), the Hubble parameter h and the spectral index n of the primordial power spectrum. Our results are quoted both in terms of the maximum permitted optical depth to the last-scattering surface, and in terms of the highest allowed reionization redshift assuming instantaneous reionization. For critical-density models, significantly tilted power spectra are excluded as they cannot fit the current data for any amount of reionization, and even scale-invariant models must have an optical depth to last scattering of below 0.3. For the currently favoured low-density model with Ω₀=0.3 and a cosmological constant, the earliest reionization permitted to occur is at around redshift 35, which roughly coincides with the highest estimate in the literature. We provide general fitting functions for the maximum permitted optical depth, as a function of cosmological parameters. We do not consider the inclusion of tensor perturbations, but if present they would strengthen the upper limits that we quote.     

Scalar statistics on the sphere: application to the cosmic microwave background

A method to compute several scalar quantities of cosmic microwave background (CMB) maps on the sphere is presented. We consider here four type of scalars: the Hessian matrix scalars, the distortion scalars, the gradient-related scalars and the curvature scalars. Such quantities are obtained directly from the spherical harmonic coefficients   a _{ℓ m}   of the map. We also study the probability density function of these quantities for the case of a homogeneous and isotropic Gaussian field, which are functions of the power spectrum of the initial field. From these scalars it is possible to construct a new set of scalars which are independent of the power spectrum of the field. We test our results using simulations and find good agreement between the theoretical probability density functions and those obtained from simulations. Therefore, these quantities are proposed to investigate the presence of non-Gaussian features in CMB maps. Finally, we show how to compute the scalars in the presence of anisotropic noise and realistic masks.     

Redshift-space distortions in the PSCz galaxy catalogue

We apply a spherical harmonic analysis to the Point Source Redshift Survey (PSC z ), to compute the real-space galaxy power spectrum and the degree of redshift distortion caused by peculiar velocities. We employ new parameter eigenvector and hierarchical data compression techniques, allowing a much larger number of harmonic modes to be included, and correspondingly smaller error bars. Using 4644 harmonic modes, compressed to 2278, we find that the IRAS redshift-space distortion parameter is     and the amplitude of galaxy clustering on a scale of     is     . Combining these we find the amplitude of mass perturbations is     . While this is compatible with results from the cosmic microwave background (CMB), with a small degree of tilt, it disagrees with the amplitude of matter perturbations estimated from the abundance of clusters by a factor of 2, independent of cosmology. A preliminary model fitting analysis combining the CMB with either the PSC z or cluster abundances shows that the cosmological matter density parameter     , and the IRAS bias parameter     . However, the cluster abundances suggest that     and     , while the PSC z requires     and     . Given the physics of galaxy formation is poorly constrained, we conclude that IRAS galaxies and mass are only partially correlated.     

Extended Press–Schechter theory and the density profiles of dark matter haloes

An inside–out model for the formation of haloes in a hierarchical clustering scenario is studied. The method combines the picture of the spherical infall model and a modification of the extended Press–Schechter theory. The mass accretion rate of a halo is defined to be the rate of its mass increase due to minor mergers. The accreted mass is deposited at the outer shells without changing the density profile of the halo inside its current virial radius. We applied the method to a flat Λ-cold dark matter universe. The resulting density profiles are compared with analytical models proposed in the literature, and a very good agreement is found. A trend is found of the inner density profile to become steeper for larger halo mass, which also results from recent N -body simulations. Additionally, present-day concentrations as well as their time evolution are derived and it is shown that they reproduce the results of large cosmological N -body simulations.     

Markov chain reconstruction of the 2dF Galaxy Redshift Survey real-space power spectrum

The real-space power spectrum of L _* galaxies measured from the 2dF Galaxy Redshift Survey (2dFGRS) is presented. Markov chain Monte Carlo (MCMC) sampling was used to fit radial and angular modes resulting from a spherical harmonics decomposition of the 2dFGRS overdensity field (described in a previous paper) with 16 real-space power spectrum values and linear redshift-space distortion parameter  β( L _*, 0)  . The recovered marginalized band powers are compared to previous estimates of galaxy power spectra. Additionally, we provide a simple model for the 17-dimensional likelihood hypersurface in order to allow the likelihood to be quickly estimated given a set of model band powers and β( L _*, 0). The likelihood surface is not well approximated by a multivariate Gaussian distribution with model-independent covariances. Instead, a model is presented in which the distribution of each band power has a Gaussian distribution in a combination of the band power and its logarithm. The relative contribution of each component was determined by fitting the MCMC output. Using these distributions, we demonstrate how the likelihood of a given cosmological model can be quickly and accurately estimated, and we use a simple set of models to compare estimated likelihoods with likelihoods calculated using the full spherical harmonics procedure. All of the data are made publicly available (from http://www.roe.ac.uk/~wjp/ ), enabling the spherical harmonics decomposition of the 2dFGRS of Percival et al. to be easily used as a cosmological constraint.     

Some Robertson-Walker models with time dependent <Emphasis Type="Italic">G</Emphasis> and Λ

Einstein field equations with variable gravitational and cosmological constants are considered in the presence of perfect fluid for Robertson-Walker universe by assuming the cosmological term proportional to the Hubble parameter. This variation law for vacuum density has recently been proposed by Schützhold on the basis of quantum field estimations in the curved and expanding background. The cosmological term tends asymptotically to a genuine cosmological constant and the model tends to a deSitter universe. We obtain that the present universe is accelerating with a large fraction of cosmological density in the form of cosmological term.     

Peaks in the cosmological density field: sensitivity to initial power spectrum, redshift distortions and galaxy halo occupation

We investigate the number density of maxima in the cosmological galaxy density field smoothed with a filter as a probe of clustering. In previous work it has been shown that this statistic is closely related to the slope of the linear power spectrum, even when the directly measured power spectrum is non-linear. In the present paper we investigate the sensitivity of the peak number density to various models with differing power spectra, including rolling index models, cosmologies with massive neutrinos and different baryon densities. We find that rolling index models which have given an improved fit to CMB/LSS (cosmic microwave background/large scale structure) data yield a ∼10 per cent difference in peak density compared to the scale invariant case. Models with 0.3 eV neutrinos have effects of similar magnitude and it should be possible to constrain them with data from current galaxy redshift surveys. Baryon oscillations in the power spectrum also give rise to distinctive features in the peak density. These are preserved without modification when measured from the peak density in fully non-linear N -body simulations. Using the simulations, we also investigate how the peak density is modified in the presence of redshift distortions. Redshift distortions cause a suppression of the number of peaks, largely due to fingers of God overlapping in redshift space. We find that this effect can be modelled by using a modification of the input power spectrum. We also study the results when the simulation density field is traced by galaxies obtained by populating haloes with a halo occupation distribution consistent with observations. The peak number density is consistent with that in the dark matter for filter scales  >4  h ⁻¹ Mpc  , for which we find good agreement with the linear theory predictions. In a companion paper we analyse data from the 2dF Galaxy Redshift Survey.     

Fast and reliable Markov chain Monte Carlo technique for cosmological parameter estimation

Markov chain Monte Carlo (MCMC) techniques are now widely used for cosmological parameter estimation. Chains are generated to sample the posterior probability distribution obtained following the Bayesian approach. An important issue is how to optimize the efficiency of such sampling and how to diagnose whether a finite-length chain has adequately sampled the underlying posterior probability distribution. We show how the power spectrum of a single such finite chain may be used as a convergence diagnostic by means of a fitting function, and discuss strategies for optimizing the distribution for the proposed steps. The methods developed are applied to current cosmic microwave background and large-scale structure data interpreted using both a pure adiabatic cosmological model and a mixed adiabatic/isocurvature cosmological model including possible correlations between modes. For the latter application, because of the increased dimensionality and the presence of degeneracies, the need for tuning MCMC methods for maximum efficiency becomes particularly acute.     

Bianchi type-I cosmological models with time dependent G and Λ

Einstein field equations with variable gravitational and cosmological constants are considered in the presence of perfect fluid for Bianchi type-I universe by assuming the cosmological term proportional to the Hubble parameter. This variation law for vacuum density has recently been proposed by Schützhold on the basis of quantum field estimations in the curved and expanding background. The model obtained approaches isotropy. The cosmological term tends asymptotically to a genuine cosmological constant, and the model tends to a deSitter universe. We obtain that the present universe is accelerating with a large fraction of cosmological density in the form of cosmological term.     

Density–potential pairs for spherical stellar systems with Sérsic light profiles and (optional) power-law cores

Popular models for describing the luminosity-density profiles of dynamically hot stellar systems (e.g. Jaffe, Hernquist, Dehnen) were constructed with the desire to match the deprojected form of an   R ^1/4  light profile. Real galaxies, however, are now known to have a range of different light-profile shapes that scale with mass. Consequently, although highly useful, the above models have implicit limitations, and this is illustrated here through their application to a number of real galaxy density profiles. On the other hand, the analytical density profile given by Prugniel & Simien closely matches the deprojected form of Sérsic   R ^{1/ n}   light profiles – including deprojected exponential light profiles. It is thus applicable for describing bulges in spiral galaxies, dwarf elliptical galaxies, and both ordinary and giant elliptical galaxies. Moreover, the observed Sérsic quantities define the parameters of the density model. Here we provide simple equations, in terms of elementary and special functions, for the gravitational potential and force associated with this density profile. Furthermore, to match galaxies with partially depleted cores, and better explore the supermassive black hole/galaxy connection, we have added a power-law core to this density profile and derived similar expressions for the potential and force of this hybrid profile. Expressions for the mass and velocity dispersion, assuming isotropy, are also given. These spherical models may also prove appropriate for describing the dark matter distribution in haloes formed from ΛCDM cosmological simulations.     

Virial shocks in galactic haloes?

We investigate the conditions for the existence of an expanding virial shock in the gas falling within a spherical dark matter halo. The shock relies on pressure support by the shock-heated gas behind it. When the radiative cooling is efficient compared with the infall rate, the post-shock gas becomes unstable; it collapses inwards and cannot support the shock. We find for a monatomic gas that the shock is stable when the post-shock pressure and density obey     . When expressed in terms of the pre-shock gas properties at radius r it reads as  ρ r Λ( T )/ u ³ < 0.0126  , where ρ is the gas density, u is the infall velocity and Λ( T ) is the cooling function, with the post-shock temperature   T ∝ u ²  . This result is confirmed by hydrodynamical simulations, using an accurate spheri-symmetric Lagrangian code. When the stability analysis is applied in cosmology, we find that a virial shock does not develop in most haloes that form before   z ∼ 2  , and it never forms in haloes less massive than a few  10¹¹ M_⊙  . In such haloes, the infalling gas is not heated to the virial temperature until it hits the disc, thus avoiding the cooling-dominated quasi-static contraction phase. The direct collapse of the cold gas into the disc should have non-trivial effects on the star formation rate and on outflows. The soft X-ray produced by the shock-heated gas in the disc is expected to ionize the dense disc environment, and the subsequent recombination would result in a high flux of Lα emission. This may explain both the puzzling low flux of soft X-ray background and the Lα emitters observed at high redshift.     

Power spectrum distortions in CMB map one-dimensional cross-sections depending on the cosmological model. II

We examine the effect produced by the variation of cosmological parameters on the power spectra of one-dimensional cross-sections of the cosmic microwave background maps in a narrow range of spatial frequencies. Variation of the Ω _b and Ω_Λ density parameters has little effect on the power spectrum deviation from the one expected within the ΛCDM model. At the same time, variations in the spectral index of primordial fluctuations significantly affect the amplitude of the power spectrum of one-dimensional cross-sections. We observe a lack of signal generated by the even harmonics in the ILC map as compared with model expectations.     

The impact of non-Gaussian errors on weak lensing surveys

The weak lensing power spectrum carries cosmological information via its dependence on the growth of structure and on geometric factors. Since much of the cosmological information comes from scales affected by non-linear clustering, measurements of the lensing power spectrum can be degraded by non-Gaussian covariances. Recently, there have been conflicting studies about the level of this degradation. We use the halo model to estimate it and include new contributions related to the finite size of lensing surveys, following Rimes and Hamilton's study of three-dimensional simulations. We find that non-Gaussian correlations between different multipoles can degrade the cumulative signal-to-noise ratio (S/N) for the power spectrum amplitude by up to a factor of 2 (or 5 for a worst-case model that exceeds current N -body simulation predictions). However, using an eight-parameter Fisher analysis, we find that the marginalized errors on individual parameters are degraded by less than 10 per cent (or 20 per cent for the worst-case model). The smaller degradation in parameter accuracy is primarily because: individual parameters in a high-dimensional parameter space are degraded much less than the volume of the full Fisher ellipsoid; lensing involves projections along the line of sight, which reduce the non-Gaussian effect; some of the cosmological information comes from geometric factors which are not degraded at all. We contrast our findings with those of Lee and Pen who suggested a much larger degradation in information content. Finally, our results give a useful guide for exploring survey design by giving the cosmological information returns for varying survey area, depth and the level of some systematic errors.     

Properties of spherical galaxies and clusters with an NFW density profile

Using the standard dynamical theory of spherical systems, we calculate the properties of spherical galaxies and clusters whose density profiles obey the universal form first obtained in high-resolution cosmological N -body simulations by Navarro, Frenk & White (NFW). We adopt three models for the internal kinematics: isotropic velocities, constant anisotropy and increasingly radial OsipkovMerritt anisotropy. Analytical solutions are found for the radial dependence of the mass, gravitational potential, velocity dispersion, energy and virial ratio and we test their variability with the concentration parameter describing the density profile and amount of velocity anisotropy. We also compute structural parameters, such as half-mass radius, effective radius and various measures of concentration. Finally, we derive projected quantities, the surface mass density and line-of-sight as well as aperture-velocity dispersion, all of which can be directly applied in observational tests of current scenarios of structure formation. On the mass scales of galaxies, if constant mass-to-light is assumed, the NFW surface density profile is found to fit HubbleReynolds laws well. It is also well fitted by Sérsic R ^{1/ m} laws, for     but in a much narrower range of m and with much larger effective radii than are observed. Assuming in turn reasonable values of the effective radius, the mass density profiles imply a mass-to-light ratio that increases outwards at all radii.     

The power spectrum of flux-limited X-ray galaxy cluster surveys

We compute the redshift space power spectrum of two X-ray cluster samples: the X-ray Brightest Abell Cluster Sample (XBACS) and the Brightest Cluster Sample (BCS) using the method developed by Feldman, Kaiser & Peacock. The power spectra derived for these samples are in agreement with determinations of other optical and X-ray cluster samples. For XBACS we find the largest power spectrum amplitude expected, given the high richness of this sample ( R ≥2) . In the range 0.05< k <0.4  h  Mpc⁻¹ the power spectrum shows a power-law behaviour P ( k )∝ k ⁿ with an index n ≃−1.2 . In a similar range, 0.04< k <0.3  h  Mpc⁻¹ , the BCS power spectrum has a smaller amplitude with index n ≃−1.0 . We do not find significant evidence for a peak at k ≃0.05  h  Mpc⁻¹ , suggesting that claims such of feature detections in some cluster samples could rely on artificial inhomogeneities of the data. We compare our results with power spectrum predictions derived by Moscardini et al. within current cosmological models (LCDM and OCDM). For XBACS we find that both models underestimate the amplitude of the power spectrum but for BCS there is reasonably good agreement at k ≳0.03  h  Mpc⁻¹ for both models.     

Magnetized, mass-loaded, rotating accretion flows

We present a semi-analytical investigation of a simple one-dimensional, steady-state model for a mass-loaded, rotating, magnetized, hydrodynamical flow. Our approach is analogous to one used in early studies of magnetized winds. The model represents the infall towards a central point mass of the gas generated in a cluster of stars surrounding it, as is likely to occur in some active nuclei and starburst galaxies. We describe the properties of the different classes of infall solutions. We find that the flow becomes faster than the fast-mode speed, and hence decoupled from the centre, only for a limited range of parameter values, and when magnetic stresses are ineffective. Such flow is slowed as it approaches a centrifugal barrier, implying the existence of an accretion disc. When the flow does not become super-fast and the magnetic torque is insufficient, no steady solution extending inward to the centre exists. Finally, with a larger magnetic torque, solutions representing steady sub-Alfvénic flows are found, which can resemble spherical hydrodynamical infall. Such solutions, if applicable, would imply that rotation is not important and that any accretion disc formed would be of very limited size.     

The Ω dependence in equations of motion

The equations of motion governing the evolution of a collisionless gravitating system of particles in an expanding universe can be cast in a form which is almost independent of the cosmological density parameter, Ω, and the cosmological constant, Λ. The new equations are expressed in terms of a time variable τ=ln D , where D is the linear rate of growth of density fluctuations. The dependence on the density parameter is proportional to ε=Ω^−0.2−1 times the difference between the peculiar velocity (with respect to τ) of particles and the gravity field (minus the gradient of the potential); or, before shell-crossing, times the sum of the density contrast and the velocity divergence. In a one-dimensional collapse or expansion, the equations are fully independent of Ω and Λ before shell crossing. In the general case, the effect of this weak Ω dependence is to enhance the rate of evolution of density perturbations in dense regions. In a flat universe with Λ7ne;0, this enhancement is less pronounced than in an open universe with Λ=0 and the same Ω. Using the spherical collapse model, we find that the increase of the rms density fluctuations in a low-Ω universe relative to that in a flat universe with the same linear normalization is ∼0.01ε(Ω)〈δ³〉, where δ is the density field in the flat universe. The equations predict that the smooth average velocity field scales like Ω^0.6, while the local velocity dispersion (rms value) scales, approximately, like Ω^0.5. High-resolution N -body simulations confirm these results and show that density fields, when smoothed on scales slightly larger than clusters, are insensitive to the cosmological model. Haloes in an open model simulation are more concentrated than haloes of the same M /Ω in a flat model simulation.