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Testing gravity and the concordance model of cosmology, $\Lambda$CDM, at large scales is a key goal of this decade's largest galaxy surveys. Here we present a comparative study of dark matter power spectrum predictions from different numerical codes in the context of three popular theories of gravity that induce scale-independent modifications to the linear growth of structure: nDGP, Cubic Galileon and K-mouflage. In particular, we compare the predictions from full $N$-body simulations, two $N$-body codes with approximate time integration schemes, a parametrised modified $N$-body implementation and the analytic halo model reaction approach. We find the modification to the $\Lambda$CDM spectrum is in $2\%$ agreement for $z\leq1$ and $k\leq 1~h/{\rm Mpc}$ over all gravitational models and codes, in accordance with many previous studies, indicating these modelling approaches are robust enough to be used in forthcoming survey analyses under appropriate scale cuts. We further make public the new code implementations presented, specifically the halo model reaction K-mouflage implementation and the relativistic Cubic Galileon implementation., Comment: 20 pages, 4 figures, 4 tables

A new degravitation mechanism within the framework of scalar tensor gravity is proposed. The mechanism eliminates all constant contributions from the potential to the Friedmann equation, leaving only the kinematic and the dynamic terms of the potential to drive cosmic acceleration. We explore a scenario involving a density-triggered phase transition in the late-time universe, and argue that the resulting effective energy density and equation of state parameter can explain late-time cosmology when extrapolated to a region of the parameter space., Comment: 5 pages, 2 figures

For most of cosmic history, the evolution of our Universe has been governed by the physics of a 'dark sector', consisting of dark matter and dark energy, whose properties are only understood in a schematic way. The influence of these constituents is mediated exclusively by the force of gravity, meaning that insight into their nature must be gleaned from gravitational phenomena. The advent of gravitational-wave astronomy has revolutionised the field of black hole astrophysics, and opens a new window of discovery for cosmological sources. Relevant examples include topological defects, such as domain walls or cosmic strings, which are remnants of a phase transition. Here we present the first simulations of cosmic structure formation in which the dynamics of the dark sector introduces domain walls as a source of stochastic gravitational waves in the late Universe. We study in detail how the spectrum of gravitational waves is affected by the properties of the model, and extrapolate the results to scales relevant to the recent evidence for a stochastic gravitational wave background. Our relativistic implementation of the field dynamics paves the way for optimal use of the next generation of gravitational experiments to unravel the dark sector., Comment: 10 pages, 3 figures. Supplementary material and videos may be found https://oyvindchristiansen.com/projects/symmetron_GWs. The code is available https://github.com/oyvach/AsGRD

The symmetron is a dark energy and dark matter candidate that forms topological defects in the late-time universe and holds promise to resolve some of the cosmological tensions. We perform high resolution simulations of the dynamical and non-linear (a)symmetron using the recently developed relativistic N-body code asevolution. By extensively testing the temporal and spatial convergence of domain decompositioning and domain wall stability, we find criteria and physical intuition for the convergence. We apply the resolution criteria to run five high resolution, $1280^3$ grids and 500 Mpc/h boxsize, simulations of the (a)symmetron and consider the behaviour of the scalar field and the domain walls in each scenario. We find the effect on the matter power spectra, the halo mass function and observables computed over the past lightcone of an observer such as the integrated Sachs-Wolfe and non-linear Rees-Sciama effect (ISW-RS) and the lensing, compared to LCDM. We show local oscillations of the fifth force strength and the formation of planar structures in the density field. The dynamics of the field is visualised in animations with high resolution in time. The simulation code is made publicly available., Comment: 28 pages, 27 figures. Supplementary material and videos may be found https://oyvindchristiansen.com/projects/symmetron_resolve. The code is available https://github.com/oyvach/asevolution

The large-scale structure of the universe is distributed in a cosmic web. Studying the distribution and clustering of dark matter particles and halos may open up a new horizon for studying the physics of the dark universe. In this work, we investigate the nearest neighbour statistics and spherical contact function in cosmological models with massive neutrinos. For this task, we use the relativistic N-body code, gevolution and study particle snapshots at three different redshifts. In each snapshot, we find the halos and evaluate the letter functions for them. We show that a generic behaviour can be found in the nearest neighbour, $G(r)$, and spherical contact functions, $F(r)$, which makes these statistics promising tools to constrain the total neutrino mass., Comment: 9 pages, 8 figures, 1 table, 2 appendices

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 ($c_s$) associated with clustering DE. We compare the CNN results with those from a Random Forest (RF) algorithm trained on power spectra. Varying the dark energy equation of state parameter $w_{\rm{DE}}$ 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 $40\%$ 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 $w_{\rm DE}$ for three different cases: $w_{\rm DE} \in \{-0.7, -0.8, -0.9\}$. 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 $w_{\rm{DE}}\approx -1$. 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 $20\%$ enhancement in $c_s^2$ detection accuracy in some cases., Comment: 12 pages, 6 figures, 6 tables; data available at https://doi.org/10.5281/zenodo.8220732

We present asevolution, a cosmological N-body code developed based on gevolution, which consistently solves for the (a)symmetron scalar field and metric potentials within the weak-field approximation. In asevolution, the scalar field is dynamic and can form non-linear structures. A cubic term is added in the symmetron potential to make the symmetry-broken vacuum expectation values different, which is motivated by observational tensions in the late-time universe. To study the effects of the scalar field dynamics, we also implement a constraint solver making use of the quasi-static approximation, and provide options for evaluating the background evolution, including using the full energy density averaged over the simulation box within the Friedmann equation. The asevolution code is validated by comparison with the Newtonian N-body code ISIS that makes use of the quasi-static approximation. There is found a very small effect of including relativistic and weak-field corrections in our small test simulations; it is seen that for small masses, the field is dynamic and can not be accurately solved for using the quasi-static approximation; and we observe the formation of unstable domain walls and demonstrate a useful way to identify them within the code. A first consideration indicates that the domain walls are more unstable in the asymmetron scenario., Comment: 26 pages, 13 figures

Cosmological models beyond $\Lambda$CDM, like 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 $\Lambda$CDM 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-$\Lambda$CDM 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 like 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., Comment: 5 pages, 5 figures; data available at https://zenodo.org/record/7832261

Nonlinear partial differential equations appear in many domains of physics, and we study here a typical equation which one finds in effective field theories (EFT) originated from cosmological studies. In particular, we are interested in the equation $\partial_t^2 u(x,t) = \alpha (\partial_x u(x,t))^2 +\beta \partial_x^2 u(x,t)$ in $1+1$ dimensions. It has been known for quite some time that solutions to this equation diverge in finite time, when $\alpha >0$. We study the nature of this divergence as a function of the parameters $\alpha>0 $ and $\beta\ge0$. The divergence does not disappear even when $\beta $ is very large contrary to what one might believe (note that since we consider fixed initial data, $\alpha$ and $\beta$ cannot be scaled away). But it will take longer to appear as $\beta$ increases when $\alpha$ is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to $3+1$ dimensions., Comment: 19 pages, 5 figures; v2: minor revision, matches accepted version

In this paper, we study the effective field theory (EFT) of dark energy for the $k$-essence model beyond linear order. Using particle-mesh $N$-body simulations that consistently solve the dark energy 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 new 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 $c_s$, a linear treatment is not adequate., Comment: 24 pages, 10 figures

We discuss an implementation of a deep learning framework to gain insight into dark matter (DM) structure formation. We investigate the impact of initial velocity and density field information on the construction of halo mass function (HMF) through cosmological $N$-body simulations. We train a Convolutional Neural Network (CNN) on the initial snapshot of a DM-only simulation to predict the HMF that individual particles fall into at $z=0$, in the halo mass range of $10.5< \log(M / M_{\odot})<14$. Our results show a negligible improvement from including the velocity in addition to the density information when considering simulations based on ($\Lambda$CDM) with the amplitude of initial scalar perturbations $A_s = 2\times10^{-9}$. To investigate the effect of the initial velocity field in constructing the halo mass function, we increase the initial power spectrum such that we see the effect of velocities in larger halos visible to the resolution of our simulations. The CNN model trained on the simulation snapshots with large $A_s$ shows a considerable improvement in the HMF prediction when adding the velocity field information. Eventually, for the simulation with $A_s = 8 \times 10^{-8}$, the model trained with only density information shows at least $80\%$ increase in the mean squared error relative to the model with both velocity and density information, which indicates the failure of the density-only model to predict the halo mass function in this case. Our work shows the interpretability and ability of CNNs to read higher-order information from simple images, making them an excellent tool for cosmological studies., Comment: 12 pages, 12 figures, 1 table

In this letter we introduce the non-linear partial differential equation (PDE) $\partial^2_{\tau} \pi \propto (\vec\nabla \pi)^2$ showing a new type of instability. Such equations appear in the effective field theory (EFT) of dark energy for the $k$-essence model as well as in many other theories based on the EFT formalism. We demonstrate the occurrence of instability in the cosmological context using a relativistic $N$-body code, and we study it mathematically in 3+1 dimensions within spherical symmetry. We show that this term dominates for the low speed of sound limit where some important linear terms are suppressed., Comment: 5 pages, 2 figures; data available at https://zenodo.org/record/7950024

In non-linear scales, the matter density distribution is not Gaussian. Consequently, the widely used two-point correlation function is not adequate anymore to capture the matter density field's entire behaviour. Among all statistics beyond correlation functions, the spherical contact (or equivalently void function), and nearest neighbour distribution function seem promising tools to probe matter distribution in non-linear regime. In this work, we use halos from cosmological N-body simulations, galaxy groups from the volume-limited galaxy group and central galaxies from mock galaxy catalogues, to compare the spherical contact with the nearest neighbour distribution functions. We also calculate the J-function (or equivalently the first conditional correlation function), for different samples. Moreover, we consider the redshift evolution and mass-scale dependence of statistics in the simulations and dependence on the magnitude of volume-limited samples in group catalogues as well as the mock central galaxies. The shape of the spherical contact probability distribution function is nearly skew-normal, with skewness and kurtosis being approximately 0.5 and 3, respectively. On the other hand, the nearest neighbour probability distribution function is nearly log-normal, with logarithmic skewness and kurtosis being approximately 0.1 and 2.5, respectively. Accordingly, the spherical contact distribution function probes larger scales compared to the nearest neighbour distribution function, which is influenced by details of structures. We also find a linear relation between the mean and variance of the spherical contact probability distribution function in simulations and mock galaxies, which could be used as a distinguishing probe of cosmological models., Comment: 20 pages, 12 figures, 11 tables. Accepted for publication in Monthly Notices of the Royal Astronomical Society

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 $c_s$ 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 $w-c_s^2$, as the $\textit{linear}$ clustering of dark energy would change the angular power spectrum by $\sim 30\%$ at low $\ell$ 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 a few percent only. The effect of dark energy $\textit{non-linearities}$ in all the power spectra is negligible at low $\ell$, but reaches about $2\%$ and $3\%$, respectively, in the convergence and ISW-RS angular power spectra at multipoles of a few hundred when observed at redshift $\sim 0.85$. Future cosmological surveys achieving percent precision measurements will allow to probe the clustering of dark energy to a high degree of confidence., Comment: 18 pages, 15 figures, version accepted for publication in MNRAS; data available at https://doi.org/10.5281/zenodo.7950184

We present $\texttt{MG-evolution}$, an $N$-body code simulating the cosmological structure formation for parametrised modifications of gravity. It is built from the combination of parametrised linear theory with a parametrisation of the deeply nonlinear cosmological regime extrapolated from modified spherical collapse computations that cover the range of known screening mechanisms. We test $\texttt {MG-evolution}$, which runs at the speed of conventional $\Lambda$CDM simulations, against a suit of existing exact model-specific codes, encompassing linearised and chameleon $f(R)$ gravity as well as the normal branch of the Dvali-Gabadadz-Porrati braneworld model, hence covering both large-field value and large-derivative screening effects. We compare the nonlinear power spectra produced by the parametrised and model-specific approaches over the full range of scales set by the box size and resolution of our simulations, $k=(0.05-2.5)$~h/Mpc, and for two redshift slices, $z=0$ and $z=1$. We find sub-percent to one-percent level recovery of all the power spectra generated with the model-specific codes for the full range of scales. $\texttt {MG-evolution}$ can be used for generalised and accurate tests of gravity and dark energy with the increasing wealth of high-precision cosmological survey data becoming available over the next decade., Comment: 11 pages, 3 figures. Updated to match version published in Monthly Notices of the Royal Astronomical Society

In this paper, we quantify the non-linear effects from $k$-essence dark energy through an effective parameter $\mu$ that encodes the additional contribution of a dark energy fluid or a modification of gravity to the Poisson equation. This is a first step toward quantifying non-linear effects of dark energy/modified gravity models in a more general approach. We compare our $N$-body simulation results from $k$-evolution with predictions from the linear Boltzmann code $\texttt{CLASS}$, and we show that for the $k$-essence model one can safely neglect the difference between the two potentials, $ \Phi -\Psi$, and short wave corrections appearing as higher order terms in the Poisson equation, which allows us to use single parameter $\mu$ for characterizing this model. We also show that for a large $k$-essence speed of sound the $\texttt{CLASS}$ results are sufficiently accurate, while for a low speed of sound non-linearities in matter and in the $k$-essence field are non-negligible. We propose a $\tanh$-based parameterisation for $\mu$, motivated by the results for two cases with low ($c_s^2=10^{-7}$) and high ($c_s^2=10^{-4}$) speed of sound, to include the non-linear effects based on the simulation results. This parametric form of $\mu$ can be used to improve Fisher forecasts or Newtonian $N$-body simulations for $k$-essence models., Comment: 18 pages, 8 figures. Updated to match version published in JCAP; data available at https://doi.org/10.5281/zenodo.7950666

We introduce $k$-evolution, a relativistic $N$-body code based on $\textit{gevolution}$, 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 $\texttt{CLASS}$ and $\textit{gevolution}$ 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., Comment: 38 pages, 19 figures. Final version, accepted for publication in JCAP; data available at https://doi.org/10.5281/zenodo.7953189

Cosmological N-body simulations are done on massively parallel computers. This necessitates the use of simple time integrators, and, additionally, of mesh-grid approximations of the potentials. Recently, Adamek et al. (2015); Barrera-Hinojosa et al. (2019) have developed general relativistic N-body simulations to capture relativistic effects mainly for cosmological purposes. We therefore ask whether, with the available technology, relativistic effects like perihelion advance can be detected numerically to a relevant precision. We first study the spurious perihelion shift in the Kepler problem, as a function of the integration method used, and then as a function of an additional interpolation of forces on a 2-dimensional lattice. This is done for several choices of eccentricities and semi-major axes. Using these results, we can predict which precisions and lattice constants allow for a detection of the relativistic perihelion advance in N-body simulation. We find that there are only small windows of parameters -- such as eccentricity, distance from the central object and the Schwarzschild radius -- for which the corrections can be detected in the numerics., Comment: 15 pages, 8 figures. Replaced with revised version accepted for publication in Celestial Mechanics and Dynamical Astronomy; data available at https://zenodo.org/record/7950670

Outside galaxy clusters the competition between the inwards gravitational attraction and the outwards expansion of the Universe leads to a special radius of velocity cancellation, which is called the turn-around radius. Measurements of the turn-around radius hold promises of constraining cosmological parameters, and possibly even properties of gravity. Such a measurement is, however, complicated by the fact that the surroundings of galaxy clusters are not spherical, but instead are a complicated collection of filaments, sheets and voids. In this paper we use the results of numerically simulated universes to quantify realistic error-bars of the measurement of the turn-around radius. We find that for a $\Lambda$CDM cosmology these error-bars are typically of the order of $20\%$. We numerically simulate three different implementations of dark energy models and of a scalar dark sector interaction to address whether the turn-around radius can be used to constrain non-trivial cosmologies, and we find that only rather extreme models can be distinguished from a $\Lambda$CDM universe due to the large error-bars arising from the non-trivial cluster environments., Comment: 15 pages, 8 figures. Updated to match version published in JCAP

The Cosmic Microwave Background (CMB) radiation lensing is a promising tool to study the physics of early universe. In this work we probe the imprints of deviations from isotropy and scale invariance of primordial curvature perturbation power spectrum on CMB lensing potential and convergence. Specifically, we consider a scale-dependent hemispherical asymmetry in primordial power spectrum. We show that the CMB lensing potential and convergence and also the cross-correlation of the CMB lensing and late time galaxy convergence can probe the amplitude and the scale dependence of the dipole modulation. As another example, we consider a primordial power spectrum with local feature. We show that the CMB lensing and the cross-correlation of the CMB lensing and galaxy lensing can probe the amplitude and the shape of the local feature. We show that the cross correlation of CMB lensing convergence and galaxy lensing is capable to probe the effects of local features in power spectrum on smaller scales than the CMB lensing. Finally we showed that the current data can constrain the amplitude and moment dependence of dipole asymmetry., Comment: 18 pages, 22 figures, published in JCAP, Final Version