Generalized Dark Energy Model Research Articles

Growth factor parametrization

The growth rate of matter perturbations can be used to distinguish between different general relativity dark energy models and also between general relativity dark energy and modified gravity models in explaining cosmic acceleration. We propose as an approximation to the growth factor [Formula: see text] the analytical expression [Formula: see text] (with [Formula: see text]) when the radiation [Formula: see text] is included. The growth factor is analyzed for the dark energy models within general relativity. The growth index [Formula: see text] takes distinct values for each model. We find that the new analytical form [Formula: see text] approximates [Formula: see text] better than [Formula: see text].

Quasilinear observables in dark energy cosmologies

What are the fundamental limitations of reconstructing the properties of dark energy, given cosmological observations in the quasilinear regime in a range of redshifts, to be as precise as required? The aim of this paper is to address this question by constructing model-independent observables, whilst completely ignoring practical problems of real-world observations. Non-Gaussianities already present in the initial conditions are not directly accessible from observations, because of a perfect degeneracy with the non-Gaussianities arising from the nonlinear matter evolution in generalized dark energy models. By imposing a specific set of evolution equations that should cover a range of dark energy cosmologies, we however find a constraint equation for the linear structure growth rate $f_1$ expressed in terms of model-independent observables. Entire classes of dark energy models which do not satisfy this constraint equation could be ruled out, and for models satisfying it we could reconstruct e.g. the nonlocal bias parameters $b_1$ and $b_2$.

The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample

We present cosmological results from the final galaxy clustering data set of the Baryon Oscillation Spectroscopic Survey, part of the Sloan Digital Sky Survey III. Our combined galaxy sample comprises 1.2 million massive galaxies over an effective area of 9329 deg^2 and volume of 18.7 Gpc^3, divided into three partially overlapping redshift slices centred at effective redshifts 0.38, 0.51, and 0.61. We measure the angular diameter distance DM and Hubble parameter H from the baryon acoustic oscillation (BAO) method after applying reconstruction to reduce non-linear effects on the BAO feature. Using the anisotropic clustering of the pre-reconstruction density field, we measure the product DM*H from the Alcock-Paczynski (AP) effect and the growth of structure, quantified by f{\sigma}8(z), from redshift-space distortions (RSD). We combine measurements presented in seven companion papers into a set of consensus values and likelihoods, obtaining constraints that are tighter and more robust than those from any one method. Combined with Planck 2015 cosmic microwave background measurements, our distance scale measurements simultaneously imply curvature {\Omega}_K =0.0003+/-0.0026 and a dark energy equation of state parameter w = -1.01+/-0.06, in strong affirmation of the spatially flat cold dark matter model with a cosmological constant ({\Lambda}CDM). Our RSD measurements of f{\sigma}_8, at 6 per cent precision, are similarly consistent with this model. When combined with supernova Ia data, we find H0 = 67.3+/-1.0 km/s/Mpc even for our most general dark energy model, in tension with some direct measurements. Adding extra relativistic species as a degree of freedom loosens the constraint only slightly, to H0 = 67.8+/-1.2 km/s/Mpc. Assuming flat {\Lambda}CDM we find {\Omega}_m = 0.310+/-0.005 and H0 = 67.6+/-0.5 km/s/Mpc, and we find a 95% upper limit of 0.16 eV/c^2 on the neutrino mass sum.

Gravitational wave source counts at high redshift and in models with extra dimensions

Gravitational wave (GW) source counts have been recently shown to be able to test how gravitational radiation propagates with the distance from the source. Here, we extend this formalism to cosmological scales, i.e. the high redshift regime, and we discuss the complications of applying this methodology to high redshift sources. We also allow for models with compactified extra dimensions like in the Kaluza-Klein model. Furthermore, we also consider the case of intermediate redshifts, i.e. 0 < z ≲ 1, where we show it is possible to find an analytical approximation for the source counts dN/d(S/N). This can be done in terms of cosmological parameters, such as the matter density Ωm,0 of the cosmological constant model or the cosmographic parameters for a general dark energy model. Our analysis is as general as possible, but it depends on two important factors: a source model for the black hole binary mergers and the GW source to galaxy bias. This methodology also allows us to obtain the higher order corrections of the source counts in terms of the signal-to-noise S/N. We then forecast the sensitivity of future observations in constraining GW physics but also the underlying cosmology by simulating sources distributed over a finite range of signal-to-noise with a number of sources ranging from 10 to 500 sources as expected from future detectors. We find that with 500 events it will be possible to provide constraints on the matter density parameter at present Ωm,0 on the order of a few percent and with the precision growing fast with the number of events. In the case of extra dimensions we find that depending on the degeneracies of the model, with 500 events it may be possible to provide stringent limits on the existence of the extra dimensions if the aforementioned degeneracies can be broken.

Nonlinear evolution of the baryon acoustic oscillation scale in alternative theories of gravity

The scale of Baryon Acoustic Oscillations (BAO) imprinted in the matter power spectrum provides an almost-perfect standard ruler: it only suffers sub-percent deviations from fixed comoving length due to non-linear effects. We study the BAO shift in the large Horndeski class of gravitational theories and compute its magnitude in momentum space using second-order perturbation theory and a peak-background split. The standard prediction is affected by the modified linear growth, as well as by non-linear gravitational effects that alter the mode-coupling kernel. For covariant Galileon models, we find a $14-45\%$ enhancement of the BAO shift with respect to standard gravity and a distinct time evolution depending on the parameters. Despite the larger values, the shift remains well below the forecasted precision of next-generation galaxy surveys. Models that produce significant BAO shift would cause large redshift-space distortions or affect the bispectrum considerably. Our computation therefore validates the use of the BAO scale as a comoving standard ruler for tests of general dark energy models.

The luminosity distance–redshift relation up to second order in the Poisson gauge with anisotropic stress

We present the generalization of previously published results, about the perturbed redshift and the luminosity–redshift relation up to second order in perturbation theory, for the case of the Poisson gauge and in the presence of anisotropic stress. The results are therefore valid for general dark energy models and (most) modified gravity models. We use an innovative approach based on the recently proposed ‘geodesic light-cone’ gauge. We then compare our findings with other results, which recently appeared in the literature, for the particular case of vanishing anisotropic stress. Arriving at a common accepted expression for the nonlinear and relativistic corrections to the redshift and distance–redshift relation is of fundamental importance in view of future cosmological surveys. Thanks to these surveys the Universe will be further probed with high precision and at very different scales, where nonlinear and relativistic effects can play a key role.

Galaxy number counts to second order and their bispectrum

We determine the number counts to second order in cosmological perturbation theory in the Poisson gauge and allowing for anisotropic stress. The calculation is performed using an innovative approach based on the recently proposed ``geodesic light-cone" gauge. This allows us to determine the number counts in a purely geometric way, without using Einstein's equation. The result is valid for general dark energy models and (most) modified gravity models. We then evaluate numerically some relevant contributions to the number counts bispectrum. In particular we consider the terms involving the density, redshift space distortion and lensing.

Lagrangian perturbation theory: exact one-loop power spectrum in general dark energy models

Recently, we find that the correction for the EdS assumption on the one-loop matter power spectrum for general dark energy models using the standard perturbation theory is not negligible \cite{LPB}. Thus, we investigate the same problem by obtaining the exact displacement vector and kernels up to the third order for the general dark energy models in the Lagrangian perturbation theory (LPT). Using these exact solutions, we investigate the present one-loop matter power spectrum in the $\Lambda$CDM model with $\Omega_{m0} = 0.25 (0.3)$ to obtain about 0.2 (0.18) \% error correction compared to that obtained from the EdS assumption for $k = 0.1 \rm{h\, Mpc}^{-1}$ mode. If we consider the total matter power spectrum, the correction is only 0.05 (0.03) \% for the same mode. It means that EdS assumption is a good approximation for $\Lambda$CDM model in LPT theory. However, one can use this method for general models where EdS assumption is improper.

A unifying description of dark energy

We review and extend a novel approach that we recently introduced, to describe general dark energy or scalar-tensor models. Our approach relies on an Arnowitt-Deser-Misner (ADM) formulation based on the hypersurfaces where the underlying scalar field is uniform. The advantage of this approach is that it can describe in the same language and in a minimal way a vast number of existing models, such as quintessence, F(R) theories, scalar tensor theories, their Horndeski extensions and beyond. It also naturally includes Horava–Lifshitz theories. As summarized in this review, our approach provides a unified treatment of the linear cosmological perturbations about a Friedmann-Lemaître-Robertson-Walker (FLRW) universe, obtained by a systematic expansion of our general action up to quadratic order. This shows that the behavior of these linear perturbations is generically characterized by five time-dependent functions. We derive the full equations of motion in the Newtonian gauge. In the Horndeski case, we obtain the equation of state for dark energy perturbations in terms of these functions. Our unifying description thus provides the simplest and most systematic way to confront theoretical models with current and future cosmological observations.

Nonlinear relativistic corrections to cosmological distances, redshift and gravitational lensing magnification: I. Key results

The next generation of telescopes will usher in an era of precision cosmology, capable of determining the cosmological model to beyond the percent level. For this to be effective, the theoretical model must be understood to at least the same level of precision. A range of subtle relativistic effects remain to be explored theoretically, and offer the potential for probing general relativity in this new regime. We present the distance–redshift relation to second order in cosmological perturbation theory for a general dark energy model. This relation determines the magnification of sources at high precision, as well as redshift space distortions in the mildly non-linear regime. We identify a range of new lensing effects, including: double-integrated and nonlinear-integrated Sachs–Wolfe contributions, transverse Doppler effects, lensing from the induced vector mode and gravitational wave backgrounds, in addition to lensing from the second-order potential. Modifications to Doppler lensing from redshift space distortions are identified. Finally, we find a new double-coupling between the density fluctuations integrated along the line of sight, and gradients in the density fluctuations coupled to transverse velocities along the line of sight. These can be large and thus offer important new probes of gravitational lensing and general relativity. This paper accompanies paper II (Umeh, Clarkson and Maartens 2014 Class. Quantum Grav. 31 205001), where a comprehensive derivation is given.

Exact third-order density perturbation and one-loop power spectrum in general dark energy models

Under the standard perturbation theory (SPT), we obtain the fully consistent third-order density fluctuation and kernels for the general dark energy models without using the Einstein–de Sitter (EdS) universe assumption for the first time. We also show that even though the temporal and spatial components of the SPT solutions cannot be separable, one can find the exact solutions to any order in general dark energy models. With these exact solutions, we obtain the less than % error correction of one-loop matter power spectrum compared to that obtained from the EdS assumption for k=0.1 hMpc−1 mode at z=0(1,1.5). Thus, the EdS assumption works very well at this scale. However, if one considers the correction for P13, the error is about 6 (9, 11)% for the same mode at z=0(1,1.5). One absorbs P13 into the linear power spectrum in the renormalized perturbation theory (RPT) and thus one should use the exact solution instead of the approximation one. The error on the resummed propagator N of RPT is about 14 (8, 6)% at z=0(1,1.5) for k=0.4 hMpc−1. For k=1 hMpc−1, the error correction of the total matter power spectrum is about 3.6 (4.6, 4.5)% at z=0(1,1.5). Upcoming observation is required to archive the sub-percent accuracy to provide the strong constraint on the dark energy and this consistent solution is prerequisite for the model comparison.

Lagrangian perturbation theory: Third-order solution for general dark energy models

We investigate the Lagrangian perturbation theory of a general dark energy models with a constant equation of state, \omega, and derive the fitting form of the fastest growing mode solutions up to the third order. These fitting forms are less than few % errors compared to the numerical calculation. We also correct the solutions of Einstein-de Sitter and the open universe with the proper initial conditions and the correct consideration for the growing mode solutions in order to compare with the general dark energy models. Even though these fitting forms are limited to the constant equation of state models, one can apply these to the time-varying \omega by interpolating between models with the constant \omega. These fitting forms can be used for the correct prediction for the two point correlation function and can give the proper predictions for the different dark energy models.

Simple implementation of general dark energy models

We present a formalism for the numerical implementation of general theories of dark energy, combining the computational simplicity of the equation of state for perturbations approach with the generality of the effective field theory approach. An effective fluid description is employed, based on a general action describing single-scalar field models. The formalism is developed from first principles, and constructed keeping the goal of a simple implementation into CAMB in mind. Benefits of this approach include its straightforward implementation, the generality of the underlying theory, the fact that the evolved variables are physical quantities, and that model-independent phenomenological descriptions may be straightforwardly investigated. We hope this formulation will provide a powerful tool for the comparison of theoretical models of dark energy with observational data.

Cosmographic Constraints and Cosmic Fluids

The problem of reproducing dark energy effects is reviewed here with particular interest devoted to cosmography. We summarize some of the most relevant cosmological models, based on the assumption that the corresponding barotropic equations of state evolve as the universe expands, giving rise to the accelerated expansion. We describe in detail the ΛCDM (Λ-Cold Dark Matter) and ωCDM models, considering also some specific examples, e.g., Chevallier–Polarsky–Linder, the Chaplygin gas and the Dvali–Gabadadze–Porrati cosmological model. Finally, we consider the cosmological consequences of f(R) and f(T) gravities and their impact on the framework of cosmography. Keeping these considerations in mind, we point out the model-independent procedure related to cosmography, showing how to match the series of cosmological observables to the free parameters of each model. We critically discuss the role played by cosmography, as a selection criterion to check whether a particular model passes or does not present cosmological constraints. In so doing, we find out cosmological bounds by fitting the luminosity distance expansion of the redshift, z, adopting the recent Union 2.1 dataset of supernovae, combined with the baryonic acoustic oscillation and the cosmic microwave background measurements. We perform cosmographic analyses, imposing different priors on the Hubble rate present value. In addition, we compare our results with recent PLANCK limits, showing that the ΛCDM and ωCDM models seem to be the favorite with respect to other dark energy models. However, we show that cosmographic constraints on f(R) and f(T) cannot discriminate between extensions of General Relativity and dark energy models, leading to a disadvantageous degeneracy problem.

Dark energy constraints after the new Planck data

The Planck collaboration has recently published maps of the Cosmic Microwave Background radiation with the highest precision. In the standard flat $\Lambda$CDM framework, Planck data show that the Hubble constant $H_0$ is in tension with that measured by the several direct probes on $H_0$. In this paper, we perform a global analysis from the current observational data in the general dark energy models and find that resolving this tension on $H_0$ requires the dark energy model with its equation of state (EoS) $w\neq-1$. Firstly, assuming the $w$ to be a constant, the Planck data favor $w < -1$ at about $2\,\sigma$ confidence level when combining with the supernovae "SNLS" compilation. And consequently the value derived on $H_0$, $H_0=71.3\pm2.0$ ${\rm km\,s^{-1}\,Mpc^{-1}}$ (68% C.L.), is consistent with that from direct $H_0$ probes. We then investigate the dark energy model with a time-evolving $w$, and obtain the 68% C.L. constraints $w_0=-0.81\pm0.19$ and $w_a=-1.9\pm1.1$ from the Planck data and the "SNLS" compilation. Current data still slightly favor the Quintom dark energy scenario with EoS across the cosmological constant boundary $w\equiv-1$.

DARK ENERGY FROM A POSITIVE JERK PARAMETER

In this paper, we demonstrate that the variation of acceleration, namely the jerk parameter j, could give hints for determining the dark energy equation of state (EoS). In particular, it is possible to show that a viable cosmological model is compatible with a constant jerk, here conventionally rewritten as j = 1+ϵ, with ϵ &gt; 0 representing a departure from the ΛCDM model. This suggests that the cosmological constant could be seen as a limiting case of a more general dark energy model. We use the most recent union 2.1 compilation of supernovae Ia, showing at 1σ confidence level, that j is compatible with the condition j ≥ 1. In doing so, we infer a corresponding cosmological model, viable with a negative acceleration parameter, in the observed range -1 &lt; q0 &lt; 0.

Constraining H0 in general dark energy models from Sunyaev-Zeldovich/X-ray technique and complementary probes

In accelerating dark energy models, the estimates of the Hubbleconstant, H0, from Sunyaev-Zel'dovich effect (SZE) and X-raysurface brightness of galaxy clusters may depend on the mattercontent (ΩM), the curvature (ΩK) and the equationof state parameter (ω). In this article, by using a sampleof 25 angular diameter distances of galaxy clusters described by theelliptical β model obtained through the SZE/X-ray technique,we constrain H0 in the framework of a general ΛCDM model(arbitrary curvature) and a flat XCDM model with a constant equationof state parameter ω = px/ρx. In order to avoid the use ofpriors in the cosmological parameters, we apply a joint analysisinvolving the baryon acoustic oscillations (BAO) and the CMBShift Parameter signature. By taking into account thestatistical and systematic errors of the SZE/X-ray technique weobtain for nonflat ΛCDM model H0 = 74+5.0−4.0 kms−1 Mpc−1(1σ) whereas for a flat universe withconstant equation of state parameter we find H0 = 72+5.5−4.0km s−1 Mpc−1(1σ). By assuming that galaxyclusters are described by a spherical β model these resultschange to H0 = 62+8.0−7.0 and H0 = 59+9.0−6.0 kms−1 Mpc−1(1σ), respectively. The results fromelliptical description are in good agreement with independentstudies from the Hubble Space Telescope key project andrecent estimates based on the Wilkinson Microwave AnisotropyProbe, thereby suggesting that the combination of these threeindependent phenomena provides an interesting method to constrainthe Hubble constant. As an extra bonus, the adoption of theelliptical description is revealed to be a quite realisticassumption. Finally, by comparing these results with a recentdetermination for a flat ΛCDM model using only the SZE/X-raytechnique and BAO, we see that the geometry has a very weakinfluence on H0 estimates for this combination of data.

Constraining the interacting dark energy models from weak gravity conjecture and recent observations

We examine the effectiveness of the weak gravity conjecture in constraining the dark energy by comparing with observations. For general dark energy models with plausible phenomenological interactions between dark sectors, we find that although the weak gravity conjecture can constrain the dark energy, the constraint is looser than that from the observations.

Properties of the exact analytic solution of the growth factor and its applications

There have been the approximate analytic solution \cite{Silveria} and several approximate analytic forms \cite{0508156,Carroll,0303112} of the growth factor $D_{g}$ for the general dark energy models with the constant values of its equation of state $\omde$ after Heath found the exact integral form of the solution of $D_{g}$ for the Universe including the cosmological constant or the curvature term. Recently, we obtained the exact analytic solutions of the growth factor for both $\omde = -1$ or $-\fr{1}{3}$ \cite{SK1} and the general dark energy models with the constant equation of state $\omde$ \cite{SK3} independently. We compare the exact analytic solution of $D_{g}$ with the other well known approximate solutions. We also prove that the analytic solutions for $\omde = -1$ or $-\fr{1}{3}$ in Ref. \cite{SK1} are the specific solutions of the exact solutions of the growth factor for general $\omde$ models in Ref. \cite{SK3} even though they look quite different. We scrutinize the issue of using the well known parameterizations of the growth index and its parameter given in Ref. \cite{WS} to obtain the growth factor for general dark energy models. We also investigate the possible extensions of the exact solution of $D_{g}$ to the time-varying $\omde$ for the comparison with observations.

Growth index with the exact analytic solution of sub-horizon scale linear perturbation for dark energy models with constant equation of state

Three decades ago Heath found the integral form of the exact analytic growing mode solution of the linear density perturbation δ on sub-horizon scales including the cosmological constant or the curvature term. Recently, we obtained the exact analytic form of this solution in our previous work [1]. Interestingly, we are able to extend this solution for general dark energy models with the constant equation of state ωde in a flat universe. This analytic solution provides the accurate and efficient tools for probing the properties of dark energy models such as the behavior of the growth factor and the growth index. We investigate the growth index and its parameter at any epoch with this exact solution for different dark energy models and find that the growth index is quite model dependent in the redshift space, 0.25⩽z⩽1.5, so observations of the structure growth around this epoch would be very interesting. Also one may be able to rule out some dark energy models by using the analysis from this exact solution. Thus, the analytic solution for the growth factor provides the very useful tools for future observations to constrain the exact values of observational quantities at any epoch related to the growth factor in the dark energy models.