Initial Power Spectrum Research Articles

CMB quadrupole depression produced by early fast-roll inflation: Monte Carlo Markov chains analysis of WMAP and SDSS data

Generically, the classical evolution of the inflaton has a brief fast-roll stage that precedes the slow-roll regime. The fast-roll stage leads to a purely attractive potential in the wave equations of curvature and tensor perturbations (while the potential is purely repulsive in the slow-roll stage). This attractive potential leads to a depression of the CMB quadrupole moment for the curvature and $B$-mode angular power spectra. A single new parameter emerges in this way in the early universe model: the comoving wave number ${k}_{1}$ characteristic scale of this attractive potential. This mode ${k}_{1}$ happens to exit the horizon precisely at the transition from the fast-roll to the slow-roll stage. The fast-roll stage dynamically modifies the initial power spectrum by a transfer function $D(k)$. We compute $D(k)$ by solving the inflaton evolution equations. $D(k)$ effectively suppresses the primordial power for $k<{k}_{1}$ and possesses the scaling property $D(k)=\ensuremath{\Psi}(k/{k}_{1})$ where $\ensuremath{\Psi}(x)$ is a universal function. We perform a Monte Carlo Markov chain analysis of the WMAP and SDSS data including the fast-roll stage and find the value ${k}_{1}=0.266\text{ }\text{ }{\mathrm{Gpc}}^{\ensuremath{-}1}$. The quadrupole mode ${k}_{Q}=0.242\text{ }\text{ }{\mathrm{Gpc}}^{\ensuremath{-}1}$ exits the horizon earlier than ${k}_{1}$, about one-tenth of an $e$-fold before the end of fast roll. We compare the fast-roll fit with a fit without fast roll but including a sharp lower cutoff on the primordial power. Fast roll provides a slightly better fit than a sharp cutoff for the temperature--temperature, temperature--$E$ modes, and $E$ modes--$E$ modes. Moreover, our fits provide nonzero lower bounds for $r$, while the values of the other cosmological parameters are essentially those of the pure $\ensuremath{\Lambda}\mathrm{CDM}$ model. We display the real space two point ${C}^{\mathrm{TT}}(\ensuremath{\theta})$ correlator. The fact that ${k}_{Q}$ exits the horizon before the slow-roll stage implies an upper bound in the total number of $e$-folds ${N}_{\mathrm{tot}}$ during inflation. Combining this with estimates during the radiation dominated era we obtain ${N}_{\mathrm{tot}}\ensuremath{\sim}66$, with the bounds $62<{N}_{\mathrm{tot}}<82$. We repeated the same analysis with the WMAP-5, ACBAR-2007, and SDSS data confirming the overall picture.

Classical bounce: Constraints and consequences

We perform a detailed investigation of the simplest possible cosmological model in which a bounce can occur, namely, that where the dynamics is led by a simple massive scalar field in a general self-interacting potential and a background spacetime with positively curved spatial sections. By means of a phase space analysis, we give the conditions under which an initially contracting phase can be followed by a bounce and an inflationary phase lasting long enough (i.e., at least 60--70 e-folds) to suppress spatial curvature in today's observable universe. We find that, quite generically, this realization requires some amount of fine-tuning of the initial conditions. We study the effect of this background evolution on scalar perturbations by propagating an initial power-law power spectrum through the contracting phase, the bounce, and the inflationary phase. We find that it is drastically modified, both spectrally ($k$-mode mixing) and in amplitude. It also acquires, at leading order, an oscillatory component, which, once evolved through the radiation and matter dominated eras, happens to be compatible with observational data.

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 >4h -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.

The Nonlinear Matter Power Spectrum

We modify the public PM code developed by Anatoly Klypin and Jon Holtzman to simulate cosmology with arbitrary initial power spectrum and equation of state of dark energy. With this tool in hand, we perform the following studies on the matter power spectrum. With an artificial sharp peak at k~0.2 h/Mpc in the initial power spectrum, we find that the position of the peak is not shifted by nonlinear evolution. We also find that the existence of a peak in the linear power spectrum would boost the nonlinear power at all scales evenly. This is contrary to what HKLM scaling relation predicts, but roughly consistent with that of halo model. We construct two dark energy models with the same linear power spectra today but different linear growth histories. We demonstrate that their nonlinear power spectra differ at the level of the maximum deviation of the corresponding linear power spectra in the past. Similarly, two constructed dark energy models with the same growth histories result in consistent nonlinear power spectra. This is hinting, not a proof, that linear power spectrum together with linear growth history uniquely determine the nonlinear power spectrum. Based on these results, we propose that linear growth history be included in the next generation fitting formulas of nonlinear power spectrum. (abridged)

The Asymptotic Form of Cosmic Structure: Small‐Scale Power and Accretion History

We explore the effects of small-scale structure on the formation and equilibrium of dark matter halos in a universe dominated by vacuum energy. We present the results of a suite of four N-body simulations, two with a ΛCDM initial power spectrum and two with WDM-like spectra that suppress the early formation of small structures. All simulations are run into the far future when the universe is 64 h-1 Gyr old, long enough for halos to essentially reach dynamical equilibrium. We quantify the importance of hierarchical merging on the halo mass accretion history, the substructure population, and the equilibrium density profile. We modify the mass accretion history function of Wechsler and coworkers by introducing a parameter, γ, that controls the rate of mass accretion, d ln M/d ln a ∝ a-γ, and find that this form characterizes both hierarchical and monolithic formation. Subhalo decay rates are exponential in time with a much shorter timescale for WDM halos. At the end of the simulations, we find truncated Hernquist density profiles for halos in both the CDM and WDM cosmologies. There is a systematic shift to lower concentration for WDM halos, but both cosmologies lie on the same locus relating concentration and formation epoch. Because the form of the density profile remains unchanged, our results indicate that the equilibrium halo density profile is set independently of the halo formation process.

Cluster‐Supercluster Alignments

We study correlations in spatial orientation between galaxy clusters and their host superclusters using a Hubble Volume N-body realization of a concordance cosmology and an analytic model for tidally-induced alignments. We derive an analytic form for distributions of the alignment angle as functions of halo mass (M), ellipticity (epsilon), distance (r) and velocity (v) and show that the model, after tuning of three parameters, provides a good fit to the numerical results. The parameters indicate a high degree of alignment along anisotropic, collapsed filaments. The degree of alignment increases with M and epsilon while it decreases with r and is independent of v. We note the possibility of using the cluster-supercluster alignment effect as a cosmological probe to constrain the slope of the initial power spectrum.

Power transfer in non-linear gravitational clustering and asymptotic universality

Abstract We study the non-linear gravitational clustering of collisionless particles in an expanding background using an integro-differential equation for the gravitational potential. In particular, we address the question of how the non-linear mode–mode coupling transfers power from one scale to another in the Fourier space if the initial power spectrum is sharply peaked at a given scale. We show that the dynamical equation allows self-similar evolution for the gravitational potential φk(t) in Fourier space of the form φk(t) = F(t)D(k) where the function F(t) satisfies a second-order non-linear differential equation. We analyse the relevant solutions of this equation, thereby determining the asymptotic time evolution of the gravitational potential and density contrast. The analysis suggests that both F(t) and D(k) have well-defined asymptotic forms indicating that the power transfer leads to a universal power spectrum at late times. The analytic results are compared with numerical simulations, showing good agreement over the range at which we could test them.

Large-scale structure and the Cardassian fluid

In this paper, we confront the predictions of the power law cardassian model for the baryon power spectrum with the observations of the SDSS galaxy survey. We show that they fit only for very unusual values of the cold dark matter or baryon density parameters, the Hubble parameter or the spectral index of the initial power spectrum. Moreover, the best-fit Cardassian models turn out to be phantom models. If one wants to recover the usual values for these constants, as quoted by the WMAP team, the power law Cardassian model turns out to be indistinguishable from a LCDM model.

Population III Star Formation in a ΛWDM Universe

In this paper we examine aspects of primordial star formation in a gravitino warm dark matter universe with a cosmological constant (ΛWDM). We compare a set of simulations using a single cosmological realization but with a wide range of WDM particle masses that have not yet been conclusively ruled out by observations. The addition of a WDM component to the initial power spectrum results in a delay in the collapse of high-density gas at the center of the most massive halo in the simulation and, as a result, an increase in the virial mass of this halo at the onset of baryon collapse. Both of these effects become more pronounced as the WDM particle mass becomes smaller. A cosmology using a gravitino WDM power spectrum assuming a particle mass of mWDM 40 keV is effectively indistinguishable from the cold dark matter case, whereas the mWDM 15 keV case delays star formation in the parent halo by 108 yr. There is remarkably little scatter between simulations in the final properties of the protostar that forms at the center of the halo. The detailed evolution of the collapsing halo core in two representative WDM cosmologies is described. At low densities (nb 105 cm-3) the evolution of the two calculations is qualitatively similar, but occurs on significantly different timescales, with the halo in the lower particle mass calculation taking much longer to evolve over the same density range and reach runaway collapse. Once the gas in the center of the halo reaches relatively high densities (nb 105 cm-3), the overall evolution is essentially identical in the two calculations.

The impact of box size on the properties of dark matter haloes in cosmological simulations

We investigate the impact finite simulation box size has on the structural and kinematic properties of cold dark matter (CDM) haloes forming in cosmological simulations. Our approach involves generating a single realization of the initial power spectrum of density perturbations and studying how truncation of this power spectrum on scales larger than Lcut= 2π/kcut affects the structure of dark matter haloes at z= 0. In particular, we have examined the cases of Lcut=fcutLbox with fcut= 1 (i.e. no truncation), 1/2, 1/3 and 1/4. In common with previous studies, we find that the suppression of long wavelength perturbations reduces the strength of clustering, as measured by a suppression of the two-point correlation function ξ(r), and reduces the numbers of the most massive haloes, as reflected in the depletion of the high-mass end of the mass function n(M). However, we note that the number density of ‘intermediate’-mass haloes (M∼ 1013h−1 M⊙) is enhanced by the suppression of long wavelength perturbations – by as much as ∼20 per cent for fcut= 1/4– and that this effect becomes more pronounced as fcut is decreased. Such a trend is in good agreement with the predictions of the Sheth–Tormen prescription. Interestingly, we find that truncation of the initial power spectrum has little impact on the internal properties of haloes. The masses of high-mass haloes decrease in a systematic manner as Lcut is reduced, but the distribution of concentrations is unaffected. On the other hand, the median spin parameter is ∼15 per cent lower in runs with fcut < 1. We argue that this is an imprint of the linear growth phase of the halo's angular momentum by tidal torquing, and that the absence of any measurable trend in concentration or halo shape reflects the importance of virialization and complex mass-accretion histories for these quantities. These results are of interest for studies that require high-mass resolution and statistical samples of simulated haloes, such as simulations of the population of first stars. Our analysis shows that large-scale tidal fields have relatively little effect on the internal properties of CDM haloes and hence may be ignored in these studies.

Mass spectrum of primordial black holes from inflationary perturbation in theRandall–Sundrum braneworld: a limit on blue spectra

The mass spectrum of the primordial black holes formed by density perturbation in theradiation-dominated era of the Randall–Sundrum type-2 cosmology is given. Thespectrum coincides with the standard four-dimensional one on large scales but thedeviation is apparent on smaller scales. The mass spectrum is initially softer than thestandard four-dimensional one, while after accretion during the earliest era itbecomes harder than that. We also show expected extragalactic diffuse photonbackground spectra varying the initial perturbation power-law power spectrum and giveconstraints on the blue spectra and/or the reheating temperature. The most recentobservations on the small scale density perturbation from WMAP, SDSS andLyman-α Forest are used. What we get are interpreted as constraints on thesmaller-scale inflation on the brane connected to the larger one at the scale ofLyman-α Forest. If we set the bulk curvature radius to be 0.1 mm and assume the reheating temperature is higherthan 106 GeV, the scalar spectral index from the smaller scale inflation is constrained to be . Typically, the constraints are tighter than the standard four-dimensional one, which isalso revised by us using the most recent observations.

Challenges in nonlinear gravitational clustering

This article addresses some issues related to the statistical description of gravitating systems in an expanding backgrounds. In particular, I describe (a) how the nonlinear mode-mode coupling transfers power from one scale to another in the Fourier space if the initial power spectrum is sharply peaked at a given scale and (b) what are the asymptotic characteristics of gravitational clustering that are independent of the initial conditions. The analysis uses a new approach based on an integro-differential equation for the evolution of the gravitational potential in the Fourier space. I show how this equation allows one to understand several aspects of nonlinear gravitational clustering and provides insight in to the transfer of power from one scale to another through nonlinear mode coupling. Numerical simulations as well as analytic work shows that power transfer leads to a universal power spectrum at late times, somewhat reminiscent of the existence of Kolmogorov spectrum in fluid turbulence. To cite this article: T. Padmanabhan, C. R. Physique 7 (2006).

Redshift-space distortions with the halo occupation distribution - I. Numerical simulations

We show how redshift-space distortions of the galaxy correlation function or power spectrum can constrain the matter density parameter Ω m and the linear matter fluctuation amplitude σ 8 . We improve on previous treatments by adopting a fully non-linear description of galaxy clustering and bias, which allows us to achieve the accuracy demanded by larger galaxy redshift surveys and to break parameter degeneracies by combining large-scale and small-scale distortions. Given an observationally motivated choice of the initial power spectrum shape, we consider different combinations of Ω m and σ 8 and find parameters of the galaxy halo occupation distribution (HOD) that yield nearly identical galaxy correlation functions in real space. We use these HOD parameters to populate the dark matter haloes of large N-body simulations, from which we measure redshift-space distortions on small and large scales. We include a velocity bias parameter α v that allows the velocity dispersions of satellite galaxies in haloes to be systematically higher or lower than those of dark matter. Large-scale distortions are determined by the parameter combination β ≡ Ω 0.6 m /b g , where bg is the bias factor defined by the ratio of galaxy and matter correlation functions, in agreement with the linear theory prediction of parameter degeneracy. However, linear theory does not accurately describe the distortions themselves on scales accessible to our simulations. We provide fitting formulas to estimate β from measurements of the redshift-space correlation function or power spectrum, and we show that these formulas are significantly more accurate than those in the existing literature. On small scales, the 'FOG' distortions at projected separations ∼0.1 h -1 Mpc depend on Ω m α 2 v but are independent of as, while at intermediate separations they depend on σ 8 as well. One can thus use measurements of redshift-space distortions over a wide range of scales to separately determine Ω m , σ 8 , and α v .

Non-parametric reconstruction of the primordial power spectrum at horizon scales from WMAP data

We extend to large scales a method proposed in previous work that reconstructs non-parametrically the primordial power spectrum from cosmic microwave background data at high resolution. The improvement is necessary to account for the non-Gaussianity of the Wilkinson Microwave Anisotropy Probe (WMAP) likelihood primarily due to cosmic variance. We assume the concordance Λ cold dark matter (ΛCDM) cosmology, utilize a smoothing prior and perform Monte Carlo simulations around an initial power spectrum that is scale free and with spectral index ns= 0.97, very close to the concordance spectrum. The horizon scale for the model we are considering corresponds to the wavenumber kh= 4.52 × 10−4 Mpc−1. We find some evidence for the presence of features and we quantify the probabilities of exceeding the observed deviations in WMAP data with respect to the fiducial models. We detect the following marginal departures from a scale-free (spectral index ns= 0.97) initial spectrum: a cut-off at 0.0001 < k < 0.001 Mpc−1 at 79.5 per cent (92 per cent), a dip at 0.001 < k < 0.003 Mpc−1 at 87.2 per cent (98 per cent) and a bump at 0.003 < k < 0.004 Mpc−1 at 90.3 per cent (55.5 per cent) confidence level. These frequentist confidence levels are calculated by integrating over the distribution of the Monte Carlo reconstructions built around the fiducial models. The frequentist analysis finds the low-k cut-off of the estimated power spectrum to be about 2.5σ away from the ns= 0.97 model, while in the Bayesian analysis the model is about 1.5σ away from the estimated spectrum. (The σ values are different for the two different methods.)

Three‐dimensional Numerical Simulations of Thermal‐Gravitational Instability in the Protogalactic Halo Environment

We study thermal-gravitational instability in simplified models of protogalactic halos using three-dimensional hydrodynamic simulations. The simulations started with isothermal density perturbations of various power spectra and followed the evolution of gas with radiative cooling down to T = 104 K, background heating, and self-gravity for up to ~20 cooling times. Then cooled and condensed clouds were identified and their physical properties were examined in detail. In our models, the cooling timescale is several times shorter than the gravitational timescale. Hence, during the early stage clouds start to form around initial density peaks as a result of thermal instability. Small clouds appear first, and they are pressure-bound. Subsequently, the clouds grow through compression by the background pressure, as well as through gravitational infall. During the late stage, cloud-cloud collisions become important, and clouds grow mostly through gravitational merging. Gravitationally bound clouds with mass Mc 6 × 106 M☉ are found in the late stage. They are approximately in virial equilibrium and have radii Rc 150-200 pc. Those clouds have gained angular momentum through tidal torque, as well as merging, so they have large angular momentum with a spin parameter ~ 0.3. The clouds formed in a denser background tend to have smaller spin parameters, since self-gravity, compared to radiative cooling, is relatively less important at higher density. H2 cooling to below T = 104 K does not drastically change the evolution and properties of clouds, since it is much less efficient than the H Lyα cooling. The slope of the initial density power spectrum affects the morphology of the cloud distribution, but the properties of individual clouds do not sensitively depend on it. We point out limitations of our study and mention briefly the implications of our results for the formation of protoglobular cluster clouds in protogalactic halos.

Information content of the non-linear matter power spectrum

Abstract We use an ensemble of N-body simulations of the currently favoured (concordance) cosmological model to measure the amount of information contained in the non-linear matter power spectrum about the amplitude of the initial power spectrum. Two surprising results emerge from this study: (i) that there is very little independent information in the power spectrum in the translinear regime (k≃ 0.2 –0.8 h Mpc−1 at the present day) over and above the information at linear scales; and (ii) that the cumulative information begins to rise sharply again with increasing wavenumber in the non-linear regime. In the fully non-linear regime, the simulations are consistent with no loss of information during translinear and non-linear evolution. If this is indeed the case then the results suggest a picture in which translinear collapse is very rapid, and is followed by a bounce prior to virialization, impelling a wholesale revision of the HKLM–Peacock &amp; Dodds formalism.

Cosmic Data Fusion

We compare and combine likelihood functions of the cosmological parameters Ωm, h and σ8 from the CMB, type Ia supernovae and from probes of large scale structure. We include the recent results from the CMB experiments BOOMERANG and MAXIMA-1. Our analysis assumes a flat ACDM cosmology with a scale-invariant adiabatic initial power spectrum. First we consider three data sets that directly probe the mass in the Universe, without the need to relate the galaxy distribution to the underlying mass via a “biasing” relation: peculiar velocities, CMB and supernovae. We assume a baryonic fraction as inferred from Big-Bang Nucleosynthesis and find that all three data sets agree well, overlapping significantly at the 2σ level. This therefore justifies a joint analysis, in which we find a joint best fit point and 95% confidence limits of Ωm = 0.28 (0.17, 0.39), h = 0.74 (0.64, 0.86), and σ8 = 1.17 (0.98,1.37). Secondly we extend our earlier work on combining CMB, supernovae, cluster number counts, IRAS galaxy redshift survey data to include BOOMERANG and MAXIMA-1 data and to allow a free Ωbh2. We find that, given our assumption of a scale invariant initial power spectrum (n = 1), we obtain the robust result of Ωbh2 = 0.031 ± 0.03, which is dominated by the CMB constraint.

Making waves on CMB power spectrum and inflaton dynamics

We discuss cosmic microwave background anisotropies in models with an unconventional primordial power spectrum. In particular, we consider an initial power spectrum with some “spiky” corrections. Interestingly, such a primordial power spectrum generates “wavy” structure in the CMB angular power spectrum.

Phase-space structure of dark matter haloes: scale-invariant probability density function driven by substructure

We present a method for computing the six-dimensional coarse-grained phase-space density f(x, v) in an N-body system, and derive its distribution function v(f). The method is based on Delaunay tessellation, where v(f) is obtained with an effective fixed smoothing window over a wide f range. The errors are estimated, and v(f) is found to be insensitive to the sampling resolution or the simulation technique. We find that in gravitationally relaxed haloes built by hierarchical clustering, v(f) is well approximated by a robust power law, v(f) ∝f−2.5±0.05, over more than four decades in f, from its virial level to the numerical resolution limit. This is tested to be valid in the Λ cold dark matter cosmology for haloes with masses 109–1015 M⊙, indicating insensitivity to the slope of the initial fluctuation power spectrum. By mapping the phase-space density in position space, we find that the high-f end of v(f) is dominated by the ‘cold’ subhaloes rather than the parent-halo central region and its global spherical profile. The value of f in subhaloes near the virial radius is typically >100 times higher than its value at the halo centre, and it decreases gradually from the outside in toward its value at the halo centre. This seems to reflect phase mixing due to mergers and tidal effects involving puffing up and heating. The phase-space density can thus provide a sensitive tool for studying the evolution of subhaloes during the hierarchical build-up of haloes. It remains to be understood why the evolved substructure adds up to the actual universal power law of v(f) ∝f−5/2. It seems that this behaviour results from the hierarchical clustering process and is not a general result of violent relaxation.

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.