Clustering Dark Energy Research Articles

K-eμlator: emulating clustering effects of the k-essence dark energy

Abstract We build an emulator based on the polynomial chaos expansion (PCE) technique to efficiently model the non-linear effects associated with the clustering of the k-essence dark energy in the effective field theory (EFT) framework. These effects can be described through a modification of Poisson’s equation, denoted by the function μ(k, z), which in general depends on wavenumber k and redshift z. To emulate this function, we perform 200 high-resolution N-body simulations sampled from a seven-dimensional parameter space with the Latin hypercube method. These simulations are executed using the k-evolution code on a fixed mesh, containing 12003 dark matter particles within a box size of 400 Mpc/h. The emulation process has been carried out within UQLab, a MATLAB-based software specifically dedicated to emulation and uncertainty quantification tasks. Apart from its role in emulation, the PCE method also facilitates the measurement of Sobol indices, enabling us to assess the relative impact of each cosmological parameter on the μ function. Our results show that the PCE-based emulator efficiently and accurately reflects the behavior of the k-essence dark energy for the cosmological parameter space defined by $w_0 c_s^2 \text{CDM} +\sum m_{\nu }$. Compared against actual simulations, the emulator achieves sub-percent accuracy up to the wavenumber k ≈ 9.4 hMpc−1 for redshifts z ≤ 3. Our emulator provides an efficient and reliable tool for Markov chain Monte Carlo (MCMC) analysis, and its capability to closely mimic the properties of the k-essence dark energy makes it a crucial component in Bayesian parameter estimations. The code is publicly available at https://github.com/anourizo/k-emulator.

Clusternets: a deep learning approach to probe clustering dark energy

ABSTRACT Machine learning (ML) algorithms are becoming popular in cosmology for extracting valuable information from cosmological data. In this paper, we evaluate the performance of a convolutional neural network (CNN) trained on matter density snapshots to distinguish clustering dark energy (DE) from the cosmological constant scenario and to detect the speed of sound (cs) associated with clustering DE. We compare the CNN results with those from a Random Forest (RF) algorithm trained on power spectra. Varying the DE equation of state parameter wDE within the range of −0.7 to −0.99 while keeping $c_s^2 = 1$, we find that the CNN approach results in a significant improvement in accuracy over the RF algorithm. The improvement in classification accuracy can be as high as 40 per cent depending on the physical scales involved. We also investigate the ML algorithms’ ability to detect the impact of the speed of sound by choosing $c_s^2$ from the set {1, 10−2, 10−4, 10−7} while maintaining a constant wDE for three different cases: wDE ∈ {−0.7, −0.8, −0.9}. Our results suggest that distinguishing between various values of $c_s^2$ and the case where $c_s^2=1$ is challenging, particularly at small scales and when wDE ≈ −1. However, as we consider larger scales, the accuracy of $c_s^2$ detection improves. Notably, the CNN algorithm consistently outperforms the RF algorithm, leading to an approximate 20 per cent enhancement in $c_s^2$ detection accuracy in some cases.

High-accuracy emulators for observables in ΛCDM, Neff, Σmν, and w cosmologies

ABSTRACT We use the emulation framework CosmoPower to construct and publicly release neural network emulators of cosmological observables, including the cosmic microwave background (CMB) temperature and polarization power spectra, matter power spectrum, distance-redshift relation, baryon acoustic oscillation (BAO) and redshift-space distortion (RSD) observables, and derived parameters. We train our emulators on Einstein–Boltzmann calculations obtained with high-precision numerical convergence settings, for a wide range of cosmological models including ΛCDM, wCDM, ΛCDM + Neff, and ΛCDM + Σmν. Our CMB emulators are accurate to better than 0.5 per cent out to ℓ = 104, which is sufficient for Stage-IV data analysis, and our P(k) emulators reach the same accuracy level out to $k=50 \, \, \mathrm{Mpc}^{-1}$, which is sufficient for Stage-III data analysis. We release the emulators via an online repository (CosmoPower Organisation), which will be continually updated with additional extended cosmological models. Our emulators accelerate cosmological data analysis by orders of magnitude, enabling cosmological parameter extraction analyses, using current survey data, to be performed on a laptop. We validate our emulators by comparing them to class and camb and by reproducing cosmological parameter constraints derived from Planck TT, TE, EE, and CMB lensing data, as well as from the Atacama Cosmology Telescope Data Release 4 CMB data, Dark Energy Survey Year-1 galaxy lensing and clustering data, and Baryon Oscillation Spectroscopic Survey Data Release 12 BAO and RSD data.

A generic instability in clustering dark energy?

In this paper, we study the effective field theory (EFT) of dark energy (DE) for the k-essence model beyond linear order. Using particle-mesh N-body simulations that consistently solve the DE evolution on a grid, we find that the next-to-leading order in the EFT expansion, which comprises the terms of the equations of motion that are quadratic in the field variables, gives rise to a generic instability in the regime of low speed of sound (high Mach number). We rule out the possibility of a numerical artefact by considering simplified cases in spherically and plane symmetric situations analytically. If the speed of sound vanishes exactly, the non-linear instability makes the evolution singular in finite time, signalling a breakdown of the EFT framework. The case of finite (but small) speed of sound is subtle, and the local singularity could be replaced by some other type of behaviour with strong non-linearities. While an ultraviolet completion may cure the problem in principle, there is no reason why this should be the case in general. As a result, for a large range of the effective speed of sound cs , a linear treatment is not adequate.

Spherical collapse of non-top-hat profiles in the presence of darkenergy with arbitrary sound speed

We study the spherical collapse of non-top-hat matter fluctuations in the presence of dark energy with arbitrary sound speed. The model is described by a system of partial differential equations solved using a pseudo-spectral method with collocation points. This method can reproduce the known analytical solutions in the linear regime with an accuracy better than 10-6% and better than 10-2% for the virialization threshold given by the usual spherical collapse model. We show the impact of nonlinear dark energy fluctuations on matter profiles, matter peculiar velocity and gravitational potential. We also show that phantom dark energy models with low sound speed can develop a pathological behaviour around matter halos, namely negative energy density. The dependence of the virialization threshold density for collapse on the dark energy sound speed is also computed, confirming and extending previous results in the limit for homogeneous and clustering dark energy.

Biased tracers as a probe of beyond-ΛCDM cosmologies

Cosmological models beyond ΛCDM, such as those featuring massive neutrinos or modifications of gravity, often display a characteristic change (scale-dependent suppression or enhancement) in the matter power spectrum when compared to a six-parameter ΛCDM baseline. It is therefore a widely held view that constraints on those models can be obtained by searching for such features in the clustering statistics of large-scale structure. However, when using biased tracers of matter in the analysis, the situation is complicated by the fact that the bias also depends on cosmology. Here we investigate how the selection of tracers affects the observed signatures for two examples of beyond-ΛCDM cosmologies: massive neutrinos and clustering dark energy (k-essence). We study the signatures in the monopole, quadrupole, and hexadecapole of the redshift-space power spectra for halo catalogues from large N-body simulations and argue that a fixed selection criterion based on local attributes, such as tracer mass, leads to a near loss of signal in most cases. Instead, the full signal is recovered only if the selection of tracers is done at fixed bias. This emphasises the need to model or measure the bias parameters accurately in order to get meaningful constraints on the cosmological model.

Tidal virialization of dark matter haloes with clustering dark energy

We extend the analysis of Pace et al. [1] by considering the virialization process in the extended spherical collapse model for clustering dark-energy models, i.e., accounting for dark-energy fluctuations. Differently from the standard approach, here virialization is naturally achieved by properly modelling deviations from sphericity due to shear and rotation induced by tidal interactions. We investigate the time evolution of the virial overdensity Δvir in seven clustering dynamical dark energy models and compare the results to the ΛCDM model and to the corresponding smooth dark-energy models.Taking into account all the appropriate corrections, we deduce the abundance of convergence peaks for Rubin Observatory-LSST and Euclid-like weak-lensing surveys, of Sunyaev-Zel'dovich peaks for a Simon Observatory-like CMB survey, and of X-ray peaks for an eROSITA-like survey. Despite the tiny differences in Δvir between clustering and smooth dark-energy models, owing to the large volumes covered by these surveys, five out of seven clustering dark-energy models can be statistically distinguished from ΛCDM. The contribution of dark-energy fluctuation cannot be neglected, especially for the Chevallier-Polarski-Limber and Albrecht-Skordis models, provided the instrumental configurations provide high signal-to-noise ratio. These results are almost independent of the tidal virialization model.

A Short Review on Clustering Dark Energy

We review dark energy models that can present non-negligible fluctuations on scales smaller than Hubble radius. Both linear and nonlinear evolutions of dark energy fluctuations are discussed. The linear evolution has a well-established framework, based on linear perturbation theory in General Relativity, and is well studied and implemented in numerical codes. We highlight the main results from linear theory to explain how dark energy perturbations become important on the scales of interest for structure formation. Next, we review some attempts to understand the impact of clustering dark energy models in the nonlinear regime, usually based on generalizations of the Spherical Collapse Model. We critically discuss the proposed generalizations of the Spherical Collapse Model that can treat clustering dark energy models and their shortcomings. Proposed implementations of clustering dark energy models in halo mass functions are reviewed. We also discuss some recent numerical simulations capable of treating dark energy fluctuations. Finally, we summarize the observational predictions based on these models.

Do we need soft cosmology?

We examine the possibility of “soft cosmology”, namely small deviations from the usual framework due to the effective appearance of soft-matter properties in the Universe sectors. One effect of such a case would be the dark energy to exhibit a different equation-of-state parameter at large scales (which determine the universe expansion) and at intermediate scales (which determine the sub-horizon clustering and the large scale structure formation). Concerning soft dark matter, we show that it can effectively arise due to the dark-energy clustering, even if dark energy is not soft. We propose a novel parametrization introducing the “softness parameters” of the dark sectors. As we see, although the background evolution remains unaffected, due to the extreme sensitivity and significant effects on the global properties even a slightly non-trivial softness parameter can improve the clustering behavior and alleviate e.g. the fσ8 tension. Lastly, an extension of the cosmological perturbation theory and a detailed statistical mechanical analysis, in order to incorporate complexity and estimate the scale-dependent behavior from first principles, is necessary and would provide a robust argumentation in favour of soft cosmology.

Clustering dark energy imprints on cosmological observables of the gravitational field

ABSTRACT We study cosmological observables on the past light-cone of a fixed observer in the context of clustering dark energy. We focus on observables that probe the gravitational field directly, namely the integrated Sachs–Wolfe and non-linear Rees–Sciama effect (ISW-RS), weak gravitational lensing, gravitational redshift, and Shapiro time delay. With our purpose-built N-body code ‘k-evolution’ that tracks the coupled evolution of dark matter particles and the dark energy field, we are able to study the regime of low speed of sound cs where dark energy perturbations can become quite large. Using ray tracing, we produce two-dimensional sky maps for each effect and we compute their angular power spectra. It turns out that the ISW-RS signal is the most promising probe to constrain clustering dark energy properties coded in $w-c_\mathrm{ s}^2$, as the linear clustering of dark energy would change the angular power spectrum by ${\sim}30{{\ \rm per\ cent}}$ at low ℓ when comparing two different speeds of sound for dark energy. Weak gravitational lensing, Shapiro time delay, and gravitational redshift are less sensitive probes of clustering dark energy, showing variations of only a few per cent. The effect of dark energy non-linearities in all the power spectra is negligible at low ℓ, but reaches about $2{{\ \rm per\ cent}}$ and $3{{\ \rm per\ cent}}$, respectively, in the convergence and ISW-RS angular power spectra at multipoles of a few hundred when observed at redshift ∼0.85. Future cosmological surveys achieving per cent precision measurements will allow us to probe the clustering of dark energy to a high degree of confidence.

The growth of DM and DE perturbations in DBI non-canonical scalar field scenario

We study the effect of varying sound speed on clustering dark energy in the Dirac–Born–Infeld (DBI) scenario. The DBI action is included in the class of k-essence models, and it has an important role in describing the effective degrees of freedom of D-branes in the string theory. In the DBI setup, we take the anti-de Sitter (AdS) warp factor f(ϕ)=f0ϕ−4, and investigate the self-interacting quartic potential V(ϕ)=λϕ4∕4. We calculate the full expression of the effective sound speed for our model, and show that it can evolve with time during the cosmological evolution. Besides, the adiabatic sound speed evolves with time here, and this influences the background dynamics to some extent. We show that the effective sound speed is very close to the adiabatic sound speed. We examine the effect of the variable sound speed on growth of the perturbations in both the linear and non-linear regimes. In the linear regime, we apply the Pseudo-Newtonian formalism, and show that dark energy suppresses the growth of perturbations at low redshifts. From study of the Integrated Sachs–Wolfe (ISW) effect in our setup, we see that the model manifests some deviation from the concordance ΛCDM model. In the non-linear regime, we follow the approach of spherical collapse model, and calculate the linear overdensity δc(zc), the virial overdensity Δvir(zc), overdensity at the turn around ζ(zc) and the rate of expansion of collapsed region hta(z). Our results imply that the provided values of δc(zc), Δvir(zc), ζ(zc) and hta(z) in our clustering DBI dark energy model approach the fiducial value in the EdS universe at high enough redshifts. We further compute relative number density of halo objects above a given mass in our setting, and show that the number of structures with respect to the ΛCDM model is reduced more in the high mass tail at high redshifts.

Time dependent dark energy and the thermodynamics of many-body systems

In this paper, we study the thermodynamics and statistics of the galaxies clustering affected by the dynamical dark energy. We consider two important dark energy models based on time dependent equation of state to evaluate the gravitational partition function. In the first model, we consider barotropic dark energy with time dependent equation of state. However, in the second model, we consider various kinds of Chaplygin gas equation of state which originally introduced by string theory. We calculate, analytically and numerically, the thermodynamic quantities in canonical and grand canonical ensembles. We investigate validity of the second law of thermodynamics for the total system of clustering of galaxies and dynamical dark energy. We finally evaluate the galaxy-galaxy correlation function and compare our model with Peebles's power law and find that the model based on generalized Chaplygin gas may yields to more agreement with observations.

Skewness of matter distribution in clustering dark energy cosmologies

We calculate the skewness (the third moment $S_3$) of matter distribution in dynamical dark energy cosmologies. We pay particular attention to the impact of dark energy perturbations on this quantity. There is indeed a clear signature of dark energy perturbations on this quantity. By properly allowing dark energy perturbations we show that their impact on $S_3$ is strong enough (a factor $\sim 3$ greater) to easily discriminate between clustering and non-clustering dark energy cosmologies. This indicates that high order statistics of the cosmic density field are useful to the study of dark energy models and are potentially able to rule out clustering dark energy cosmologies.

Looking for interactions in the cosmological dark sector

We study observational signatures of non-gravitational interactions between the dark components of the cosmic fluid, which can be either due to creation of dark particles from the expanding vacuum or an effect of the clustering of a dynamical dark energy. In particular, we analyse a class of interacting models (Λ(t)CDM), characterised by the parameter α, that behaves at background level like cold matter at early times and tends to a cosmological constant in the asymptotic future. Our analysis improves previous works, where only the position of the Cosmic Microwave Background (CMB) first peak and supernova and LSS data where used, considering for the first time both background and primordial perturbations evolutions of the model. We use full spectra of the CMB Planck data together with late time observations, such as the Joint Light-curve Analysis (JLA) supernovae data, the Hubble Space Telescope (HST) measurement of the local value of the Hubble-Lemaȋtre parameter, and primordial deuterium abundance from Lyα systems to test the observational viability of the model and some of its extensions, adapting an Einstein-Boltzmann solver code for our purpose. We found that there is no preference for values of α different from zero (characterising interaction), even if there are some indications for positive values when the minimal Λ(t)CDM model is analysed. When extra degrees of freedom in the relativistic component of the cosmic fluid are considered, the data favour negative values of α, which means an energy flux from dark energy to dark matter.

K-evolution: a relativistic N-body code for clustering dark energy

We introduce k-evolution, a relativistic N-body code based on gevolution, which includes clustering dark energy among its cosmological components. To describe dark energy, we use the effective field theory approach. In particular, we focus on k-essence with a speed of sound much smaller than unity but we lay down the basis to extend the code to other dark energy and modified gravity models. We develop the formalism including dark energy non-linearities but, as a first step, we implement the equations in the code after dropping non-linear self-coupling in the k-essence field. In this simplified setup, we compare k-evolution simulations with those of CLASS and gevolution 1.2, showing the effect of dark matter and gravitational non-linearities on the power spectrum of dark matter, of dark energy and of the gravitational potential. Moreover, we compare k-evolution to Newtonian N-body simulations with back-scaled initial conditions and study how dark energy clustering affects massive halos.

Effects of completeness and purity on cluster dark energy constraints

The statistical properties of galaxy clusters can only be used for cosmological purposes if observational effects related to cluster detection are accurately characterized. These effects include the selection function associated to cluster finder algorithms and survey strategy. The importance of the selection becomes apparent when different cluster finders are applied to the same galaxy catalog, producing different cluster samples. We consider parametrized functional forms for the observable-mass relation, its scatter as well as the completeness and purity of cluster samples, and study how prior knowledge on these function parameters affects dark energy constraints derived from cluster statistics. Under the assumption that completeness and purity reach 50 % at masses around 10^{13.5} Msun/h, we find that self-calibration of selection parameters in current and upcoming cluster surveys is possible, while still allowing for competitive dark energy constraints. We consider a fiducial survey with specifications similar to those of the Dark Energy Survey (DES) with 5000 deg^2, maximum redshift of zmax ~ 1.0 and threshold observed mass M_{th} ~ 10^{13.8} Msun/h, such that completeness and purity ~ 60 % - 80 % at masses around M_{th}. Perfect knowledge of all selection parameters allows for constraining a constant dark energy equation of state to sigma(w)=0.033. Employing a joint fit including self-calibration of the effective selection degrades constraints to sigma(w)=0.046. External calibrations at the level of 1 % in the parameters of the observable-mass relation and completeness/purity functions are necessary to improve the joint constraints to sigma(w)=0.041. In the lack of knowledge of selection parameters, future experiments probing larger areas and greater depths suffer from stronger relative degradations on dark energy constraints compared to current surveys.

Model-independent reconstruction of the linear anisotropic stress η

In this work, we use recent data on the Hubble expansion rate H(z), the quantity fσ8(z) from redshift space distortions and the statistic Eg from clustering and lensing observables to constrain in a model-independent way the linear anisotropic stress parameter η. This estimate is free of assumptions about initial conditions, bias, the abundance of dark matter and the background expansion. We denote this observable estimator as ηobs. If ηobs turns out to be different from unity, it would imply either a modification of gravity or a non-perfect fluid form of dark energy clustering at sub-horizon scales. Using three different methods to reconstruct the underlying model from data, we report the value of ηobs at three redshift values, z=0.29, 0.58, 0.86. Using the method of polynomial regression, we find ηobs=0.57±1.05, ηobs=0.48±0.96, and ηobs=−0.11±3.21, respectively. Assuming a constant ηobs in this range, we find ηobs=0.49±0.69. We consider this method as our fiducial result, for reasons clarified in the text. The other two methods give for a constant anisotropic stress ηobs=0.15±0.27 (binning) and ηobs=0.53 ± 0.19 (Gaussian Process). We find that all three estimates are compatible with each other within their 1σ error bars. While the polynomial regression method is compatible with standard gravity, the other two methods are in tension with it.

Dark energy stars: Stable configurations

In present paper a spherically symmetric stellar configuration has been analyzed by assuming the matter distribution of the stellar configuration is anisotropic in nature and compared with the realistic objects, namely, the low mass X-ray binaries and X-ray pulsars. The analytic solution has been obtained by utilizing the dark energy equation of state for the interior solution corresponding to the Schwarzschild exterior vacuum solution at the junction interface. Several physical properties, like energy conditions, stability, mass–radius ratio, and surface redshift, are described through mathematical calculations as well as graphical plots. It is found that obtained mass–radius ratios of the compact star candidates like 4U 1820–30, PSR J 1614–2230, Vela X-1, and Cen X-3 are very much consistent with the observed data by Gangopadhyay et al. (Mon. Not. R. Astron. Soc. 431, 3216 (2013)). So our proposed model would be useful in the investigation of the possible clustering of dark energy.

Growth of perturbations in dark energy parametrization scenarios

In this paper, we study the evolution of dark matter perturbations in the linear regime by considering the possibility of dark energy perturbations. To do this, two popular parameterizations, CPL and BA with same number of free parameters and different redshift dependency have been considered. We integrate the full relativistic equations to obtain the growth of matter fluctuations for both clustering and smooth versions of CPL and BA dark energy. The growth rate is larger (smaller) than the $\Lambda$CDM in the smooth cases when $w<-1$ ($w>-1$) but the dark energy clustering gives a larger (smaller) growth index when $w>-1$ ($w<-1$). We measure the relative difference of the growth rate with respect to concordance $\Lambda$CDM and study how it changes depending on the free parameters. Furthermore, it is found that the difference of growth rates between smooth CPL and BA is negligible, less than $0.5\%$, while for clustering case, the difference is considerable and might be as large as 2$\%$. Eventually, using the latest geometrical and growth rate observational data, we perform an overall likelihood analysis and show that both smooth and clustering cases of CPL and BA parameterizations are consistent with observations. In particular, we find the dark energy FoM $\sim70$ for the BA and $\sim30$ for the CPL which indicates BA model constraints relatively better than CPL one.

Spherical collapse models with clustered dark energy

We investigate the clustering effect of dark energy (DE) in the formation of galaxy clusters using the spherical collapse model. Assuming a fully clustered DE component, the spherical overdense region is treated as an isolated system which conserves the energy separately for both matter and DE inside the spherical region. Then, by introducing a parameter $r$ to characterize the degree of DE clustering, which is defined by the nonlinear density contrast ratio of matter to DE at turnaround in the recollapsing process, i.e. $r\equiv \nld_{\de,\ta}/\nld_{\m,\ta}$, we are able to uniquely determine the spherical collapsing process and hence obtain the virialized overdensity $\Dvir$ through a proper virialization scheme. Estimation of the virialized overdensities from current observation on galaxy clusters suggests that $0.5 < r < 0.8$ at $1\sigma$ level for the clustered DE with $w < -0.9$. Also, we compare our method to the linear perturbation theory that deals with the growth of DE perturbation at early times. While both results are consistent with each other, our method is practically simple and it shows that the collapse process is rather independent of initial DE perturbation and its evolution at early times.