Anatoly Klypin

Cold Dark Matter with cosmological constant (ΛCDM) cosmological models with early dark energy (EDE) have been proposed to resolve tensions between the Hubble constant |$H_0=100\, h$| km ṡ⁻¹Ṁpc⁻¹ measured locally, giving h ≈ 0.73, and H₀ deduced from Planck cosmic microwave background (CMB) and other early-Universe measurements plus ΛCDM, giving h ≈ 0.67. EDE models do this by adding a scalar field that temporarily adds dark energy equal to about 10 per cent of the cosmological energy density at the end of the radiation-dominated era at redshift z ∼ 3500. Here, we compare linear and non-linear predictions of a Planck-normalized ΛCDM model including EDE giving h = 0.728 with those of standard Planck-normalized ΛCDM with h = 0.678. We find that non-linear evolution reduces the differences between power spectra of fluctuations at low redshifts. As a result, at z = 0 the halo mass functions on galactic scales are nearly the same, with differences only 1–2 per cent. However, the differences dramatically increase at high redshifts. The EDE model predicts 50 per cent more massive clusters at z = 1 and twice more galaxy-mass haloes at z = 4. Even greater increases in abundances of galaxy-mass haloes at higher redshifts may make it easier to reionize the universe with EDE. Predicted galaxy abundances and clustering will soon be tested by the James Webb Space Telescope (JWST) observations. Positions of baryonic acoustic oscillations (BAOs) and correlation functions differ by about 2 per cent between the models – an effect that is not washed out by non-linearities. Both standard ΛCDM and the EDE model studied here agree well with presently available acoustic-scale observations, but the Dark Energy Spectroscopic Instrument and Euclid measurements will provide stringent new tests.

In order to capture as much information as possible large galaxy surveys have been increasing their volume and redshift depth. To face this challenge theory has responded by making cosmological simulations of huge computational volumes with equally increasing the number of dark matter particles and supercomputing resources. Thus, it is taken for granted that the ideal situation is when a single computational box encompasses the whole effective volume of the observational survey, e.g., ~50 Gpch^3 for the DESI and Euclid surveys. Here we study the effects of missing long-waves in a finite volume using several relevant statistics: the abundance of dark matter halos, the PDF, the correlation function and power spectrum, and covariance matrices. Finite volume effects can substantially modify the results if the computational volumes are less than ~(500Mpch)^3. However, the effects become extremely small and practically can be ignored when the box-size exceeds ~1Gpch^3. We find that the average power spectra of dark matter fluctuations show remarkable lack of dependence on the computational box-size with less than 0.1% differences between 1Gpch and 4Gpch boxes. No measurable differences are expected for the halo mass functions for these volumes. The covariance matrices are scaled trivially with volume, and small corrections due to super-sample modes can be added. We conclude that there is no need to make those extremely large simulations when a box-size of 1-1.5Gpch is sufficient to fulfil most of the survey science requirements.

Making cosmological inferences from the observed galaxy clustering requires accurate predictions for the mean clustering statistics and their covariances. Those are affected by cosmic variance -- the statistical noise due to the finite number of harmonics. The cosmic variance can be suppressed by fixing the amplitudes of the harmonics instead of drawing them from a Gaussian distribution predicted by the inflation models. Initial realizations also can be generated in pairs with 180 degrees flipped phases to further reduce the variance. Here, we compare the consequences of using paired-and-fixed vs Gaussian initial conditions on the average dark matter clustering and covariance matrices predicted from N-body simulations. As in previous studies, we find no measurable differences between paired-and-fixed and Gaussian simulations for the average density distribution function, power spectrum and bispectrum. Yet, the covariances from paired-and-fixed simulations are suppressed in a complicated scale- and redshift-dependent way. The situation is particularly problematic on the scales of Baryon Acoustic Oscillations where the covariance matrix of the power spectrum is lower by only 20% compared to the Gaussian realizations, implying that there is not much of a reduction of the cosmic variance. The non-trivial suppression, combined with the fact that paired-and-fixed covariances are noisier than from Gaussian simulations, suggests that there is no path towards obtaining accurate covariance matrices from paired-and-fixed simulations. Because the covariances are crucial for the observational estimates of galaxy clustering statistics and cosmological parameters, paired-and-fixed simulations, though useful for some applications, cannot be used for the production of mock galaxy catalogs.

Large-scale surveys of galaxies require accurate theoretical predictions of the dark matter clustering for thousands of mock galaxy catalogues. We demonstrate that this goal can be achieved with the new Parallel Particle-Mesh (PM) N-body code GLAM at a very low computational cost. We run ∼22 000 simulations with ∼2 billion particles that provide ∼1 per cent accuracy of the dark matter power spectra P(k) for wavenumbers up to k ∼ 1 h Mpc⁻¹. Using this large data set we study the power spectrum covariance matrix. In contrast to many previous analytical and numerical results, we find that the covariance matrix normalized to the power spectrum C(k, k′)/P(k)P(k′) has a complex structure of non-diagonal components: an upturn at small k, followed by a minimum at k ≈ 0.1–0.2 h Mpc⁻¹, and a maximum at k ≈ 0.5–0.6 h Mpc⁻¹. The normalized covariance matrix strongly evolves with redshift: C(k, k′) ∝ δ^α(t)P(k)P(k′), where δ is the linear growth factor and α ≈ 1–1.25, which indicates that the covariance matrix depends on cosmological parameters. We also show that waves longer than |$1\,h^{-1}\rm{\,Gpc}$| have very little impact on the power spectrum and covariance matrix. This significantly reduces the computational costs and complexity of theoretical predictions: relatively small volume |${\sim } (1\,h^{-1}\rm{\,Gpc})^3$| simulations capture the necessary properties of dark matter clustering statistics. As our results also indicate, achieving ∼1 per cent errors in the covariance matrix for k < 0.50 h Mpc⁻¹ requires a resolution better than |$\epsilon \sim 0.5\,h^{-1}\rm{\,Mpc}$|⁠.

The one-point probability distribution function (PDF) of the matter density field in the universe is a fundamental property that plays an essential role in cosmology for estimates such as gravitational weak lensing, non-linear clustering, massive production of mock galaxy catalogs, and testing predictions of cosmological models. Here we make a comprehensive analysis of the dark matter PDF using a suite of 7000 N-body simulations that covers a wide range of numerical and cosmological parameters. We find that the PDF has a simple shape: it declines with density as a power-law P~rho**(-2), which is exponentially suppressed on both small and large densities. The proposed double-exponential approximation provides an accurate fit to all our N-body results for small filtering scales R< 5Mpc/h with rms density fluctuations sigma>1. In combination with the spherical infall model that works well for small fluctuations sigma<1, the PDF is now approximated with just few percent errors over the range of twelve orders of magnitude -- a remarkable example of precision cosmology. We find that at 5-10% level the PDF explicitly depends on redshift (at fixed sigma) and on cosmological density parameter Omega_m. We test different existing analytical approximations and find that the often used log-normal approximation is always 3-5 times less accurate than either the double-exponential approximation or the spherical infall model.

Upcoming and existing large-scale surveys of galaxies require accurate theoretical predictions of the dark matter clustering statistics for thousands of mock galaxy catalogs. We demonstrate that this goal can be achieve with our new Parallel Particle-Mesh (PM) Nbody code (PPM-GLAM) at a very low computational cost. We run about 15,000 simulations with ~2 billion particles that provide ~1% accuracy of the dark matter power spectra P(k) for wave-numbers up to k~ 1h/Mpc. Using this large data-set we study the power spectrum covariance matrix, the stepping stone for producing mock catalogs. In contrast to many previous analytical and numerical results, we find that the covariance matrix normalised to the power spectrum C(k,k')/P(k)P(k') has a complex structure of non-diagonal components. It has an upturn at small k, followed by a minimum at k=0.1-0.2h/Mpc. It also has a maximum at k=0.5-0.6h/Mpc. The normalised covariance matrix strongly evolves with redshift: C(k,k')~delta(t)^alpha P(k)P(k'), where delta is the linear growth factor and alpha ~ 1-1.25, which indicates that the covariance matrix depends on cosmological parameters. We also show that waves longer than 1Gpc have very little impact on the power spectrum and covariance matrix. This significantly reduces the computational costs and complexity of theoretical predictions: relatively small volume ~ (1Gpc)^3 simulations capture the necessary properties of dark matter clustering statistics. All the power spectra obtained from many thousands of our simulations are publicly available.

Accurately predicting structural properties of dark matter halos is one of the fundamental goals of modern cosmology. We use the new suite of MultiDark cosmological simulations to study the evolution of dark matter halo density profiles, concentrations, and velocity anisotropies. The MultiDark simulations cover a large range of masses 1e10-1e15Msun and volumes upto 50Gpc**3. The total number of dark matter halos in all the simulations exceeds 60 billion. We find that in order to understand the structure of dark matter halos and to make ~1% accurate predictions for density profiles, one needs to realize that halo concentration is more complex than the traditional ratio of the virial radius to the core radius in the NFW profile. For massive halos the averge density profile is far from the NFW shape and the concentration is defined by both the core radius and the shape parameter alpha in the Einasto approximation. Combining results from different redshifts, masses and cosmologies, we show that halos progress through three stages of evolution. (1) They start as rare density peaks that experience very fast and nearly radial infall. This radial infall brings mass closer to the center producing a high concentrated halo. Here, the halo concentration increases with the increasing halo mass and the concentration is defined by the alpha parameter with nearly constant core radius. Later halos slide into (2) the plateau regime where the accretion becomes less radial, but frequent mergers still affect even the central region. Now the concentration does not depend on halo mass. (3) Once the rate of accretion slows down, halos move into the domain of declining concentration-mass relation because new accretion piles up mass close to the virial radius while the core radius is staying constant. We provide accurate analytical fits to the numerical results for halo density profiles and concentrations.

Accurate predictions of the abundance and clustering of dark matter haloes play a key role in testing the standard cosmological model. Here, we investigate the accuracy of one of the leading methods of connecting the simulated dark matter haloes with observed galaxies– the halo abundance matching (HAM) technique. We show how to choose the optimal values of the mass and force resolution in large volume N-body simulations so that they provide accurate estimates for correlation functions and circular velocities for haloes and their subhaloes – crucial ingredients of the HAM method. At the 10 per cent accuracy, results converge for ∼50 particles for haloes and ∼150 particles for progenitors of subhaloes. In order to achieve this level of accuracy a number of conditions should be satisfied. The force resolution for the smallest resolved (sub)haloes should be in the range (0.1–0.3)r_s, where r_s is the scale radius of (sub)haloes. The number of particles for progenitors of subhaloes should be ∼150. We also demonstrate that the two-body scattering plays a minor role for the accuracy of N-body simulations thanks to the relatively small number of crossing-times of dark matter in haloes, and the limited force resolution of cosmological simulations.