Initial Power Spectrum Research Articles

Joint galaxy-lensing observables and the dark energy

Deep multicolor galaxy surveys with photometric redshifts will provide a large number of two-point correlation observables: galaxy-galaxy angular correlations, galaxy-shear cross correlations, and shear-shear correlations between all redshifts. These observables can potentially enable a joint determination of the dark-energy-dependent evolution of the dark matter and distances as well as the relationship between galaxies and dark matter halos. With recent cosmic microwave background determinations of the initial power spectrum, a measurement of the mass clustering at even a single redshift will constrain a well-specified combination of dark energy (DE) parameters in a flat universe; we provide convenient fitting formulas for such studies. The combination of galaxy-shear and galaxy-galaxy correlations can determine this amplitude at multiple redshifts. We illustrate this ability in a description of the galaxy clustering with 5 free functions of redshift which can be fitted from the data. The galaxy modeling is based on a mapping onto halos of the same abundance that models a flux-limited selection. In this context and under a flat geometry, a 4000 deg${}^{2}$ galaxy-lensing survey can achieve a statistical precision of $\ensuremath{\sigma}({\ensuremath{\Omega}}_{\mathrm{DE}})=0.005$ for the dark energy density, $\ensuremath{\sigma}{(w}_{\mathrm{DE}})=0.02$ and $\ensuremath{\sigma}{(w}_{a})=0.17$ for its equation of state and evolution, evaluated at dark energy matter equality $z\ensuremath{\approx}0.4,$ as well as constraints on the 5 halo functions out to $z=1.$ More importantly, a joint analysis can make dark energy constraints robust against systematic errors in the shear-shear correlation and halo modeling.

Statistical characteristics of large scale structure

We generalize and extend the statistical description of formation and evolution of Large Scale matter distribution in the Universe proposed in our previous papers. We investigate the impact of transverse motions – expansion and/or compression – on properties of the Large Scale Structure elements – walls, pancakes, filaments and clouds – and, generalizing the Press-Schechter formalism, derive their mass functions. Using the Zel'dovich theory of gravitational instability we show that these mass functions are approximately the same and the mass of each type of elements is found to be concentrated near the corresponding mean mass. At high redshifts, both the mass function and the mean mass of formed elements depend upon the small scale part of the initial power spectrum and, in particular, upon the mass of dominant fraction of dark matter (DM) particles. Using these results we obtain independent estimates of probable redshifts of the reionization and reheating periods of the Universe. We show that the transverse motions do not significantly change the redshift evolution of the observed mass function and the mean linear number density of low mass pancakes related to absorption lines in the spectra of the farthest quasars. Application of this approach to the observed Lyman–α forest allows one to directly estimate the shape of initial power spectrum on small scales.

Numerical study of the cosmological velocity field as a function of density

We report on a new study of the velocity distribution in N-body simulations. We investigate the center-of-mass and internal kinetic energies of coarsening cells as a function of time, cell size and cell mass. By using self-similar cosmological models, we are able to derive theoretical predictions for comparison and to assess the influence of finite-size and resolution effects. The most interesting result is the discovery of a polytropic-like relationship between the average velocity dispersion (internal kinetic energy) and the mass density in an intermediate range of densities, K ~ rho^{2-eta}. The exponent eta measures the deviations from the virial prediction, eta_virial=0. For self-similar models, eta depends only on the spectral index of the initial power spectrum. We also study CDM models and confirm a previous result that the same polytropic-like dependence exists (astro-ph/0103313), with a time and coarsening length dependent eta. The dependence K(rho) is an important input for a recently proposed theoretical model of cosmological structure formation which improves over the standard dust model (pressureless fluid) by regularizing the density singularities.

Principal power of the CMB

We study the physical limitations placed on cosmic microwave background (CMB) temperature and polarization measurements of the initial power spectrum. These limitations include geometric projection, acoustic physics, gravitational lensing and degeneracies with cosmological parameters. Detailed information on the spectrum is greatly assisted by polarization information and localized to the acoustic regime $k=0.02\ensuremath{-}0.2{\mathrm{Mpc}}^{\ensuremath{-}1}$ with a fundamental resolution of $\ensuremath{\Delta}k/kg0.05.$ From this study we construct principal component based statistics, which are orthogonal to cosmological parameters including the initial amplitude and tilt of the spectrum, that best probe deviations from scale-free initial conditions. These statistics resemble Fourier modes confined to the acoustic regime and ultimately can yield $\ensuremath{\sim}50$ independent measurements of the power spectrum features to percent level precision. They are straightforwardly related to more traditional parametrizations, such as the running of the tilt, and in the future can provide many statistically independent measurements of it for consistency checks. Though we mainly consider physical limitations on the measurements, these techniques can be readily adapted to include instrumental limitations or other sources of power spectrum information.

Observational Estimates of the Initial Power Spectrum at Small Scale from Lyα Absorbers

We present a new method of measuring the power spectrum of initial perturbations to an unprecedentedly small scale of ~10 h-1 kpc. We apply this method to a sample of 4500 Lyα absorbers and recover the cold dark matter (CDM)-like power spectrum at scales ≥300 h-1 kpc with a precision of ~10%. However, at scales ~10-300 h-1 kpc, the measured and CDM-like spectra are noticeably different. This result suggests a complex inflation with generation of excess power at small scales. The magnitude and reliability of these deviations depend upon the possible incompleteness of our sample and poorly understood process of formation of weak absorbers. Confirmation of the CDM-like shape of the initial power spectrum or detection of its distortions at small scales are equally important for widely discussed problems of physics of the early universe, galaxy formation, and reheating of the universe. We use the Zeldovich theory of gravitational instability to derive statistical description of the properties of observed structure. Our method links the observed mass function of absorbers with the correlation function of the initial velocity field and therefore avoids the Nyquist restrictions limiting the investigations based on the smoothed flux or density fields. This approach is in general consistent with numerical simulations of the process of structure formation, describes reasonably well the large-scale structure observed in the galaxy distribution at small redshifts, and emphasizes the generic similarity of galaxies and absorbers. The physical model of absorbers adopted here asserts that they are formed in the course of both linear and nonlinear adiabatic or shock compression of dark matter (DM) and gaseous matter. It allows us to link the column density and overdensity of DM and gaseous components with observed characteristics of absorbers such as the column density of neutral hydrogen, redshifts, and Doppler parameter. At scales ≥1 h-1 Mpc, all characteristics of the DM component, and in particular, their redshift distribution, are found to be consistent with theoretical expectations for Gaussian initial perturbations with a CDM-like power spectrum.

Beam deconvolution in noisy CMB maps

The subject of this paper is beam deconvolution in small angular scale CMB experiments. The beam effect is reversed using the Jacobi iterative method, which was designed to solved systems of algebraic linear equations. The beam is a non circular one which moves according to the observational strategy. A certain realistic level of Gaussian instrumental noise is assumed. The method applies to small scale CMB experiments in general (cases A and B), but we have put particular attention on PLANCK mission at 100 GHz (cases C and D). In cases B and D, where noise is present, deconvolution allows to correct the main beam distortion effect and recover the initial angular power spectrum up to the end of the fifth acoustic peak. An encouraging result whose importance is analyzed in detail. More work about deconvolution in the presence of other systematics is in progress. This paper is related to the PLANCK LFI activities.

Reconstructing the primordial power spectrum

We reconstruct the shape of the primordial power spectrum from the latest cosmic microwave background data, including the new results from the Wilkinson microwave anisotropy probe (WMAP), and large-scale structure data from the two degree field galaxy redshift survey (2dFGRS). We take into account the uncertainties in four cosmological parameters. We use a model independent parameterization, with 16 bands in wavenumber, and find no obvious sign of deviation from a power law spectrum on the scales investigated. Furthermore, the values of the other cosmological parameters defining the model remain relatively well constrained despite the freedom in the shape of the initial power spectrum.

Correlated adiabatic and isocurvature cosmic microwave background fluctuations in the wake of the results from the wilkinson microwave anisotropy probe.

In general correlated models, in addition to the usual adiabatic component with a spectral index n(ad1) there is another adiabatic component with a spectral index n(ad2) generated by entropy perturbation during inflation. We extend the analysis of a correlated mixture of adiabatic and isocurvature cosmic microwave background fluctuations of the Wilkinson Microwave Anisotropy Probe (WMAP) group, who set the two adiabatic spectral indices equal. Allowing n(ad1) and n(ad2) to vary independently we find that the WMAP data favor models where the two adiabatic components have opposite spectral tilts. Using the WMAP data only, the 2sigma upper bound for the isocurvature fraction f(iso) of the initial power spectrum at k(0)=0.05 Mpc(-1) increases somewhat, e.g., from 0.76 of n(ad2)=n(ad1) models to 0.84 with a prior n(iso)<1.84 for the isocurvature spectral index.

The effect of (strong) discrete absorption systems on the Lyman α forest flux power spectrum

We demonstrate that the Lyman a forest flux power spectrum of 'randomized' quasi-stellar object (QSO) absorption spectra is comparable in shape and amplitude to the flux power spectrum of the original observed spectra. In the randomized spectra a random shift in wavelength has been added to the observed absorption lines as identified and fitted with VPFIT. At 0.03 s km -1 15 contribute at large scales, k < 0.03 s km -1 . We further show that a fraction of ≥ 15 per cent of the mean flux decrement is contributed by strong absorbers at z ≥ 2.1. Analysis of the flux power spectrum which use numerical simulations with too few strong absorption systems calibrated with the observed mean flux may underestimate the inferred rms fluctuation amplitude and the slope of the initial dark matter power spectrum.

Dependence of the inner dark matter profile on the halo mass

I compare the density profile of dark matter (DM) haloes in cold dark matter (CDM) N-body simulations with 1, 32, 256 and 1024 h - 1 Mpc box sizes. I compare the profiles when the most massive haloes are composed of about 10 5 DM particles. The DM density profiles of the haloes in the 1 h - 1 Mpc box at redshift z ∼ 10 show systematically shallower cores with respect to the corresponding haloes in the 32 h - 1 Mpc simulation at z ∼ 3 that have masses, M d m , typical of the Milky Way and are fitted by a Navarro-Frenk-White (NFW) profile. The DM density profiles of the haloes in the 256 h - 1 Mpc box at z ∼ 0 are consistent with having steeper cores than the corresponding haloes in the 32 h - 1 Mpc simulation, but higher mass resolution simulations are needed to strengthen this result. Combined, these results suggest that the density profile of DM haloes is not universal, presenting shallower cores in dwarf galaxies and steeper cores in clusters. More work is needed to validate this finding at z = 0. Physically, the result sustains the hypothesis that the mass function of the accreting satellites determines the inner slope of the DM profile. However, the result can also be interpreted as a trend with the dynamical state in the assembly process of haloes of different mass. In comoving coordinates, r, the profile ρ d m with X = c Δ r/r Δ and a = (9 + 3n)/(5 + n) 1.3 + (M 1 / 6 dm,14 - 1)/(M 1 / 6 dm,14 + 1), provides a good fit to all the DM haloes from dwarf galaxies to clusters at any redshift with the same concentration parameter c Δ ∼ 7. Here, r Δ (M d m ) is the virial radius, n is the effective spectral index of the initial power spectrum of density perturbations and M d m , 1 4 = M d m /(3 × 10 1 4 M O .). The slope, y, of the outer parts of the halo appears to depend on the acceleration of the universe; when the scale parameter is a = (1+z) - 1 ≤ 1, the slope is γ 3 as in the NEW profile, but γ 4 at a > 1 when Ω Λ ∼ 1 and the universe is inflating. The shape of the DM profiles presents a significant scatter around the mean. It is therefore important to analyse a significant statistical sample of haloes in order to determine the mean profile. I compare the DM profiles in the 1 h - 1 Mpc box with the same simulation including stars, baryons and radiative transfer presented by Ricotti, Gnedin and Shull. Radiative feedback effects produce a larger scatter in the density profile shapes but, on average, do not affect the shape of the DM profiles significantly.

Reconstructing the primordial power spectrum

We reconstruct the shape of the primordial power spectrum from the latest cosmic microwave background data, including the new results from the Wilkinson Microwave Anisotropy Probe (WMAP), and large scale structure data from the two degree field galaxy redshift survey (2dFGRS). We tested four parameterizations taking into account the uncertainties in four cosmological parameters. First we parameterize the initial spectrum by a tilt and a running spectral index, finding marginal evidence for a running spectral index only if the first three WMAP multipoles (l = 2,3,4) are included in the analysis. Secondly, to investigate further the low CMB large scale power, we modify the conventional power-law spectrum by introducing a scale above which there is no power. We find a preferred position of the cut at kc � 3 × 10 4 Mpc 1 although kc = 0 (no cut) is not ruled out. Thirdly we use a model independent parameterization, with 16 bands in wavenumber, and find no obvious sign of deviation from a power law spectrum on the scales investigated. Furthermore the values of the other cosmological parameters defining the model remain relatively well constrained despite the freedom in the shape of the initial power spectrum. Finally we investigate a model motivated by double inflation, in which the power spectrum has a break between two characteristic wavenumbers. We find that if a break is required to be in the range 0.01 < k/Mpc 1 < 0.1 then the ratio of amplitudes across the break is constrained to be 1.23±0.14. Our results are consistent with a power law spectrum that is featureless and close to scale invariant over the wavenumber range 0.005 < � k/Mpc 1 < � 0.15, with a hint of a decrease in power on the largest scales.

Galactic densities, substructure and the initial power spectrum

Although the currently favored cold dark matter plus cosmological constant model for structure formation assumes an n = 1 scale-invariant initial power spectrum, most inflation models produce at least mild deviations from n = 1. Because the lever arm from the CMB normalization to galaxy scales is long, even a small “tilt” can have important implications for galactic observations. Here we calculate the COBS-normalized power spectra for several well-motivated models of inflation and compute implications for the substructure content and central densities of galaxy halos. Using an analytic model, normalized against N-body simulations, we show that while halos in the standard ( n = 1) model are overdense by a factor of ∼ 6 compared to observations, several of our example inflation+LCDM models predict halo densities well within the range of observations, which prefer models with n ∼ 0.85. We go on to use a semi-analytic model (also normalized against N-body simulations) to follow the merger histories of galaxy-sized halos and track the orbital decay, disruption, and evolution of the merging substructure. Models with n ∼ 0.85 predict a factor of ∼ 3 fewer subhalos at a fixed circular velocity than the standard n = 1 case. Although this level of reduction does not resolve the “dwarf satellite problem’, it does imply that the level of feedback required to match the observed number of dwarfs is sensitive to the initial power spectrum. Finally, the fraction of galaxy-halo mass that is bound up in substructure is consistent with limits imposed by multiply imaged quasars for all models considered: f sat > 0.01 even for an effective tilt of n ∼ 0.8. We conclude that, at their current level, lensing constraints of this kind do not provide an interesting probe of the primordial power spectrum.

Statistical characteristics of the observed Ly α forest and the shape of initial power spectrum

Properties of approximately 4500 observed Ly α absorbers are investigated using the model of formation and evolution of dark matter (DM) structure elements based on the modified Zel’dovich theory. This model is generally consistent with simulations of absorber formation, describes the large-scale structure (LSS) observed in the galaxy distribution at small redshifts reasonably well and emphasizes the generic similarity of the LSS and absorbers. The simple physical model of absorbers asserts that they are composed of DM and gaseous matter. It allows us to estimate the column density and overdensity of DM and gaseous components and the entropy of the gas trapped within the DM potential wells. The parameters of the DM component are found to be consistent with theoretical expectations for the Gaussian initial perturbations with the warm dark matter-like power spectrum. The basic physical factors responsible for the evolution of the absorbers are discussed. The analysis of redshift distribution of absorbers confirms the self-consistency of the adopted physical model, Gaussianity of the initial perturbations and allows one to estimate the shape of the initial power spectrum at small scales that, in turn, restricts the mass of the dominant fraction of DM particles to M DM 1.5‐5 keV. Our results indicate possible redshift variations of intensity of the ultraviolet background by approximately a factor of 2‐3 at redshifts z ∼ 2‐3. One of the most promising methods for studying the processes responsible for the formation and evolution of the structure of the Universe is the analysis of properties of absorbers observed in the spectra of the furthest quasars. The great potential of such investigations has been discussed by Oort (1981, 1984) just after Sargent et al. (1980) established the intergalactic nature of the Ly α forest. The available Keck and VLT high-resolution observations of the forest provide a reasonable data base and allow one to apply statistical methods for such investigations. The essential progress achieved recently through numerous highresolution simulations of absorber formation and evolution confirms that this process is closely connected with the initial power spectrum of perturbations. These results allow us to consider the properties of absorbers in the context of the non-linear theory of gravitational instability (Zel’dovich 1970; Shandarin & Zel’dovich 1989) and to apply the statistical description of structure formation and evolu

Statistical characteristics of the observed Ly forest and the shape of the initial power spectrum

Properties of $\sim$ 5000 observed Ly-$\alpha$ absorbers are investigated using the model of formation and evolution of DM structure elements based on the Zel'dovich theory. This model is generally consistent with simulations of absorbers formation, accurately describes the Large Scale Structure observed in the galaxy distribution at small redshifts and emphasizes the generic similarity of the LSS and absorbers. The simple physical model of absorbers asserts that they are composed of DM and gaseous matter and it allows us to estimate the column density and overdensity of DM and gaseous components and the entropy of the gas trapped within the DM potential wells. The parameters of DM component are found to be consistent with theoretical expectations for the Gaussian initial perturbations with the WDM--like power spectrum. We demonstrate the influence of the main physical factors responsible for the absorbers evolution. The analysis of redshift distribution of absorbers confirms the self consistence of the assumed physical model and allows to estimate the shape of the initial power spectrum at small scales what in turn restricts the mass of dominant fraction of DM particles to $M_{DM}\approx 0.6 - 2$ keV. Our results verify the redshift variations of intensity of the UV background by about $4 - 6$ times and indicate that, perhaps, the available observations underestimate the intensity of this background.

Non-Gaussian tails of cosmological density distribution function from dark halo approach

We present a simple model based on the dark halo approach which provides a useful way to understand key points that determine the shape of the non-Gaussian tails of the dark matter one-point probability distribution function (PDF). In particular, using scale-free models with a power-law profile of dark haloes, we derive a simple analytic expression for the one-point PDF. It is found that the shape of the PDF changes at a characteristic value of δ*, which is defined by the smoothed density of a halo with characteristic mass M* at the epoch. In cold dark matter models with top-hat smoothing filters, the characteristic smoothed density at the present time typically takes the value δ*≫ 1 for a small smoothing scale Rth∼ 1 h−1 Mpc and conversely δ*≪ 1 for a large smoothing scale Rth > 10 h−1 Mpc. In the range δ/δ* 1 basically follow the steep exponential tails of the halo mass function, which exhibit a strong sensitivity on both the outer slope of the halo profile and the initial power spectrum. Based on these results, a discussion on the PDF of galaxy distribution and application to weak lensing statistics are also presented.

Inflation, cold dark matter, and the central density problem

A problem with high central densities in dark halos has arisen in the context of LCDM cosmologies with scale-invariant initial power spectra. Although n=1 is often justified by appealing to the inflation scenario, inflationary models with mild deviations from scale-invariance are not uncommon and models with significant running of the spectral index are plausible. Even mild deviations from scale-invariance can be important because halo collapse times and densities depend on the relative amount of small-scale power. We choose several popular models of inflation and work out the ramifications for galaxy central densities. For each model, we calculate its COBE-normalized power spectrum and deduce the implied halo densities using a semi-analytic method calibrated against N-body simulations. We compare our predictions to a sample of dark matter-dominated galaxies using a non-parametric measure of the density. While standard n=1, LCDM halos are overdense by a factor of 6, several of our example inflation+CDM models predict halo densities well within the range preferred by observations. We also show how the presence of massive (0.5 eV) neutrinos may help to alleviate the central density problem even with n=1. We conclude that galaxy central densities may not be as problematic for the CDM paradigm as is sometimes assumed: rather than telling us something about the nature of the dark matter, galaxy rotation curves may be telling us something about inflation and/or neutrinos. An important test of this idea will be an eventual consensus on the value of sigma_8, the rms overdensity on the scale 8 h^-1 Mpc. Our successful models have values of sigma_8 approximately 0.75, which is within the range of recent determinations. Finally, models with n>1 (or sigma_8 > 1) are highly disfavored.

Cosmological parameter estimation and the inflationary cosmology

We consider approaches to cosmological parameter estimation in the inflationary cosmology, focusing on the required accuracy of the initial power spectra. Parametrizing the spectra, for example by power laws, is well suited to testing the inflationary paradigm but will only correctly estimate cosmological parameters if the parametrization is sufficiently accurate, and we investigate conditions under which this is achieved both for present data and for upcoming satellite data. If inflation is favored, reliable estimation of its physical parameters requires an alternative approach adopting its detailed predictions. For slow-roll inflation, we investigate the accuracy of the predicted spectra at first and second order in the slow-roll expansion (presenting the complete second-order corrections for the tensors for the first time). We find that, within the presently allowed parameter space, there are regions where it will be necessary to include second-order corrections to reach the accuracy requirements of MAP and Planck satellite data. We end by proposing a data analysis pipeline appropriate for testing inflation and for a cosmological parameter estimation from high-precision data.

Features in the primordial power spectrum: constraints from the cosmic microwave background and the limitation of the 2dF and SDSS redshift surveys to detect them

We allow a more general (step-function) form of the primordial power spectrum than the usual featureless power-law Harrison-Zeldovich (with spectral index n = 1) power spectrum, and fit it to the latest cosmic microwave background data sets. Although the best-fitting initial power spectrum can differ significantly from the power-law shape, and contains a dip at scales k ∼ 0.003 h Mpc - 1 , we find that Ω m 0.24, consistent with previous analyses that assume power-law initial fluctuations. We also explore the feasibility of the early releases of the 2dF and Sloan Digital Sky Survey (SDSS) galaxy redshifts surveys to see these features, and we find that even if features exist in the primordial power spectrum, they are washed out by the window functions of the redshift surveys on scales k < 0.03 h Mpc - 1 .

The Formation of Cosmic Structure with Modified Newtonian Dynamics

I consider the growth of inhomogeneities in a low-density baryonic, vacuum energy-dominated universe in the context of modified Newtonian dynamics (MOND). I first write down a two-field Langrangian-based theory of MOND (non-relativistic), which embodies several assumptions such as constancy of the MOND acceleration parameter, association of a MOND force with peculiar accelerations only, and the deceleration of the Hubble flow as a background field which influences the dynamics of a finite size region. In the context of this theory, the equation for the evolution of spherically symmetric over-densities is non-linear and implies very rapid growth even in a low-density background, particularly at the epoch when the putative cosmological constant begins to dominate the Hubble expansion. Small comoving scales enter the MOND regime earlier than larger scales and therefore evolve to large over-densities sooner. Taking the initial COBE-normalized power spectrum provided by CMBFAST (Seljak & Zeldarriaga 1996), I find that the final power-spectrum resembles that of the standard LCDM universe and thus retains the empirical successes of that model.

A primordial feature at the scale of superclusters of galaxies

We investigate a spatially flat cold dark matter model (with the matter density parameter with a primordial feature in the initial power spectrum. We assume that there is a bump in the power spectrum of density fluctuations at wavelengths , which corresponds to the scale of superclusters of galaxies. There are indications for such a feature in the power spectra derived from redshift surveys and also in the power spectra derived from peculiar velocities of galaxies. We study the mass function of clusters of galaxies, the power spectrum of the cosmic microwave background (CMB) temperature fluctuations, the rms bulk velocity and the rms peculiar velocity of clusters of galaxies. The baryon density is assumed to be consistent with the big bang nucleosynthesis value. We show that with an appropriately chosen feature in the power spectrum of density fluctuations at the scale of superclusters, the mass function of clusters, the CMB power spectrum, the rms bulk velocity and the rms peculiar velocity of clusters are in good agreement with the observed data.