Non-flat Models Research Articles

Revisiting the Epoch of Cosmic Acceleration

We revisit the epoch of cosmic speed-up characterized by the redshift of transition from a decelerated to an accelerated phase. This redshift is termed the transition redshift (z t ). We use the spatially flat and non-flat variants of the most common ΛCDM and XCDM models to put constraints on the transition redshift along with the other model parameters. The data for this analysis come from the recent and updated Pantheon+ supernova (SN) data set and the Hubble parameter measurements obtained from Cosmic Chronometers. We consider both data sets with their respective covariance matrices incorporating all statistical and systematic uncertainties. We observe that using the combined data sets of H(z) and SNe, the best fit value of transition redshift lies in the range 0.61 < z t < 0.79 for all four dark energy models. Incidentally, we observe a positive curvature for the non-flat models, correlations between several model parameters and a strong degeneracy between the curvature and the equation of state parameter.

Current data are consistent with flat spatial hypersurfaces in the ΛCDM cosmological model but favor more lensing than the model predicts

We study the performance of three pairs of tilted $\Lambda$CDM cosmological models, two pairs allowing for non-flat spatial hypersurfaces with CMB temperature and polarization power spectrum data (P18) from Planck, P18 lensing (lensing), and non-CMB data (non-CMB). For the six models, we measure cosmological parameters and study whether or not pairs of the data sets are mutually consistent in these models. Half of these models allow the lensing consistency parameter $A_L$ to be an additional free parameter, while the other three have $A_L = 1$. The tilted spatially-flat models assume the usual primordial spatial inhomogeneity power spectrum. The tilted non-flat models assume either the primordial power spectrum used in the Planck group analyses [Planck $P(q)$] or a recently computed power spectrum [new $P(q)$]. In the tilted non-flat models with $A_L=1$ we find differences between P18 data and non-CMB data cosmological parameter constraints, which are large enough to rule out the Planck $P(q)$ model at 3$\sigma$ but not the new $P(q)$ model. While both P18 data and non-CMB data separately favor a closed geometry when P18+non-CMB data are jointly analyzed the evidence in favor of non-flat hypersurfaces subsides. Differences between P18 data and non-CMB data cosmological constraints subside when $A_L$ is allowed to vary. From the most restrictive P18+lensing+non-CMB data combination we get almost model-independent constraints and find that the $A_L>1$ option is preferred over the $\Omega_k<0$ one, with the $A_L$ parameter, for all models, being larger than unity by $\sim 2.5\sigma$. According to the deviance information criterion, in the P18+lensing+non-CMB analysis, the varying $A_L$ option is on the verge of being strongly favored over the $A_L=1$ one, which could indicate a problem for the standard tilted flat $\Lambda$CDM model (Abridged abstract).

Non-flat and non-extensive thermodynamic effects of Møller tetradic theory of gravitation on cosmology

We derive non-flat cosmological models for two cases (i.e., dust and radiation) in the context of Møller’s tetradic theory (MTT) of gravitation using the tetrad that creates the non-flat Friedmann–Robertson–Walker (FRW) metric. These two models are affected by the free dimensional parameter, λ, that characterized MTT, which approaches zero in the flat case for both models. Using standard definitions of thermodynamics, we calculate the radius horizon, Hawking temperature, and entropy of our non-flat models in the framework of cosmology and show the effect of λ on open and closed universes. We then use the first law of thermodynamics to construct non-flat cosmological models via the non-extensive thermodynamic approach. The resulting models are affected by λ and the extensive parameter, δ, which quantifies the effect of non-extensive thermodynamics. When we set, λ=0 and δ=1, we return to Einstein’s general relativity models. We study the evolution of our models in the presence of collisionless non-relativistic matter and describe precise forms of the dark energy density and equation-of-state parameter constraining the non-extensive thermodynamic parameter. We show that insertion of the non-extensive thermodynamic parameter affects the non-flat FRW universe in a manner that noticeably differs from that observed under normal thermodynamics. We also show that the deceleration of the open universe behaves as dark energy in a future epoch, i.e., when the redshift approaches −1, i.e., z≈−1.

Non-singular non-flat universes

The quest to understand better the nature of the initial cosmological singularity is with us since the discovery of the expanding universe. Here, we propose several non-flat models, among them the standard cosmological scenario with a critical cosmological constant, the Einstein–Cartan cosmology, the Milne–McCrea universe with quantum corrections and a non-flat universe with bulk viscosity. Within these models, we probe into the initial singularity by using different techniques. Several nonsingular universes emerge, one of the possibilities being a static non-expanding and stable Einstein universe.

Using SPT polarization, Planck 2015, and non-CMB data to constrain tilted spatially-flat and untilted nonflat ΛCDM , XCDM, and ϕCDM dark energy inflation cosmologies

We use six tilted spatially-flat and untilted non-flat dark energy cosmological models in analyses of South Pole Telescope polarization (SPTpol) cosmic microwave background (CMB) data, alone and in combination with Planck 2015 CMB data and non-CMB data, namely, the Pantheon Type Ia supernovae apparent magnitudes, a collection of baryon acoustic oscillation data points, Hubble parameter measurements, and growth rates. Although the cosmological models that best-fit the Planck CMB and non-CMB data do not provide good fits to the SPTpol data, with the $\chi^2$'s exceeding the expected value, given the uncertainties, in each model the cosmological parameter constraints from the SPTpol data and from the Planck CMB and non-CMB data are largely mutually consistent. When the smaller angular scale SPTpol data are used jointly with either the Planck data alone or with the Planck CMB and the non-CMB data to constrain untilted non-flat models, spatially-closed models remain favored over their corresponding flat limits. When used in conjunction with Planck data, non-CMB data (baryon acoustic oscillation measurements in particular, from six experiments) have significantly more constraining power than the SPTpol data.

First study of reionization in tilted flat and untilted non-flat dynamical dark energy inflation models

ABSTRACTWe examine the effects of dark energy dynamics and spatial curvature on cosmic reionization by studying reionization in tilted spatially flat and untilted non-flat XCDM and ϕCDM dynamical dark energy inflation models that best fit the Planck 2015 cosmic microwave background (CMB) anisotropy and a large compilation of non-CMB data. We carry out a detailed statistical study, based on a principal component analysis and a Markov chain Monte Carlo analysis of a compilation of lower-redshift reionization data to estimate the uncertainties in the cosmological model reionization histories. We find that, irrespective of the nature of dark energy, there are significant differences between the reionization histories of the spatially flat and non-flat models. Although both the flat and non-flat models can accurately match the low-redshift (z ≲ 6) reionization observations, there is a clear discrepancy between high-redshift (z &amp;gt; 7) Lyman-α emitter data and the predictions from non-flat models. This is solely due to the fact that the non-flat models have a significantly larger electron scattering optical depth, τel, compared to the flat models, which requires an extended and much earlier reionization scenario supported by more high-redshift ionizing sources in the non-flat models. Non-flat models also require strong redshift evolution in the photon escape fraction, which can become unrealistically high (≳1) at some redshifts. However, τel is about 0.9-σ lower in the tilted flat ΛCDM model when the new Planck 2018 data are used and this reduction will partially alleviate the tension between the non-flat model predictions and the data.

General theory of cosmological perturbations in open and closed universes from the Horndeski action

Our Universe is nearly spatially flat, but this does not mean that it is exactly spatially flat. In this paper we derive general quadratic actions for cosmological perturbations in nonflat models from the Horndeski theory. This allows us to study how the spatial curvature influences the behavior of cosmological perturbations in the early Universe described by some general scalar-tensor theory. We show that a tiny spatial curvature at the onset of inflation is unlikely to yield large [or $\mathcal{O}(1)$] effects on the primordial spectra even if one modifies gravity. We also argue that nonsingular cosmological solutions in the Horndeski theory are unstable in spatially open cases as well as in flat cases.

Planck 2015 Constraints on the Non-flat XCDM Inflation Model

Abstract We examine the Planck 2015 cosmic microwave background (CMB) anisotropy data by using a physically consistent energy density inhomogeneity power spectrum generated by quantum-mechanical fluctuations during an early epoch of inflation in the non-flat XCDM model. Here dark energy is parameterized using a fluid with a negative equation of state parameter but with the speed of fluid acoustic inhomogeneities set to the speed of light. We find that the Planck 2015 data in conjunction with baryon acoustic oscillation distance measurements are reasonably well fit by a closed-XCDM model in which spatial curvature contributes a percent of the current cosmological energy density budget. In this model, the measured non-relativistic matter density parameter and Hubble constant are in good agreement with values determined using most other data. Depending on cosmological parameter values, the closed-XCDM model has reduced power, relative to the tilted, spatially flat ΛCDM case, and can partially alleviate the low multipole CMB temperature anisotropy deficit and can help partially reconcile the CMB anisotropy and weak lensing σ 8 constraints, at the expense of somewhat worsening the fit to higher multipole CMB temperature anisotropy data. However, the closed-XCDM inflation model does not seem to improve the agreement much, if at all, compared to the closed ΛCDM inflation case, even though it has one additional free parameter.

Planck 2015 Constraints on the Non-flat ΛCDM Inflation Model

Abstract We study Planck 2015 cosmic microwave background (CMB) anisotropy data using the energy density inhomogeneity power spectrum generated by quantum fluctuations during an early epoch of inflation in the non-flat ΛCDM model. Unlike earlier analyses of non-flat models, which assumed an inconsistent power-law power spectrum of energy density inhomogeneities, we find that the Planck 2015 data alone, and also in conjunction with baryon acoustic oscillation measurements, are reasonably well fit by a closed ΛCDM model in which spatial curvature contributes a few percent of the current cosmological energy density budget. In this model, the measured Hubble constant and nonrelativistic matter density parameter are in good agreement with values determined using most other data. Depending on parameter values, the closed ΛCDM model has reduced power, relative to the tilted, spatially flat ΛCDM case, and can partially alleviate the low multipole CMB temperature anisotropy deficit and can help partially reconcile the CMB anisotropy and weak lensing σ 8 constraints, at the expense of somewhat worsening the fit to higher multipole CMB temperature anisotropy data. Our results are interesting but tentative; a more thorough analysis is needed to properly gauge their significance.

First study of reionization in the Planck 2015 normalized closed ΛCDM inflation model

We study reionization in two non-flat $\Lambda$CDM inflation models that best fit the Planck 2015 cosmic microwave background anisotropy observations, ignoring or in conjunction with baryon acoustic oscillation distance measurements. We implement a principal component analysis (PCA) to estimate the uncertainties in the reionization history from a joint quasar-CMB dataset. A thorough Markov Chain Monte Carlo analysis is done over the parameter space of PCA modes for both non-flat $\Lambda$CDM inflation models as well as the original Planck 2016 tilted, spatially-flat $\Lambda$CDM inflation model. Although both flat and non-flat models can closely match the low-redshift ($z\lesssim6$) observations, we notice a possible tension between high-redshift ($z\sim8$) Lyman-$\alpha$ emitter data and the non-flat models. This is solely due to the fact that the closed models have a relatively higher reionization optical depth compared to the flat one, which in turn demands more high-redshift ionizing sources and favors an extended reionization starting as early as $z\approx14$. We conclude that as opposed to flat-cosmology, for the non-flat cosmology models (i) the escape fraction needs steep redshift evolution and even unrealistically high values at some redshifts and (ii) most of the physical parameters require to have non-monotonic redshift evolution, especially apparent when Lyman-$\alpha$ emitter data is included in the analysis.

Constraining the dark energy models with H(z) data: An approach independent of H0

We study the performance of the latest $H(z)$ data in constraining the cosmological parameters of different cosmological models, including that of Chevalier-Polarski-Linder $w_{0}w_{1}$ parametrization. First, we introduce a statistical procedure in which the chi-square estimator is not affected by the value of the Hubble constant. As a result, we find that the $H(z)$ data do not rule out the possibility of either non-flat models or dynamical dark energy cosmological models. However, we verify that the time varying equation of state parameter $w(z)$ is not constrained by the current expansion data. Combining the $H(z)$ and the Type Ia supernova data we find that the $H(z)$/SNIa overall statistical analysis provides a substantial improvement of the cosmological constraints with respect to those of the $H(z)$ analysis. Moreover, the $w_{0}-w_{1}$ parameter space provided by the $H(z)$/SNIa joint analysis is in a very good agreement with that of Planck 2015, which confirms that the present analysis with the $H(z)$ and SNIa probes correctly reveals the expansion of the Universe as found by the team of Planck. Finally, we generate sets of Monte Carlo realizations in order to quantify the ability of the $H(z)$ data to provide strong constraints on the dark energy model parameters. The Monte Carlo approach shows significant improvement of the constraints, when increasing the sample to 100 $H(z)$ measurements. Such a goal can be achieved in the future, especially in the light of the next generation of surveys.

HUBBLE PARAMETER MEASUREMENT CONSTRAINTS ON THE REDSHIFT OF THE DECELERATION–ACCELERATION TRANSITION, DYNAMICAL DARK ENERGY, AND SPACE CURVATURE

ABSTRACT We compile an updated list of 38 measurements of the Hubble parameter H(z) between redshifts 0.07 ≤ z ≤ 2.36 and use them to place constraints on model parameters of constant and time-varying dark energy cosmological models, both spatially flat and curved. We use five models to measure the redshift of the cosmological deceleration–acceleration transition, z da, from these H(z) data. Within the error bars, the measured z da are insensitive to the model used, depending only on the value assumed for the Hubble constant H 0. The weighted mean of our measurements is z da = 0.72 ± 0.05 (0.84 ± 0.03) for H 0 = 68 ± 2.8 (73.24 ± 1.74) km s−1 Mpc−1 and should provide a reasonably model-independent estimate of this cosmological parameter. The H(z) data are consistent with the standard spatially flat ΛCDM cosmological model but do not rule out nonflat models or dynamical dark energy models.

Reconstruction of f ( T ) $f(T)$ -gravity in the absence of matter

We derive an exact $f(T)$ gravity in the absence of ordinary matter in Friedmann-Robertson-Walker (FRW) universe, where $T$ is the teleparallel torsion scalar. We show that vanishing of the energy-momentum tensor $\mathcal{T}^{\mu \nu}$ of matter does not imply vanishing of the teleparallel torsion scalar, in contrast to general relativity, where the Ricci scalar vanishes. The theory provides an exponential (inflationary) scale factor independent of the choice of the sectional curvature. In addition, the obtained $f(T)$ acts just like cosmological constant in the flat space model. Nevertheless, it is dynamical in non-flat models. In particular, the open universe provides a decaying pattern of the $f(T)$ contributing directly to solve the fine-tuning problem of the cosmological constant. The equation of state (EoS) of the torsion vacuum fluid has been studied in positive and negative Hubble regimes. We study the case when the torsion is made of a scalar field (tlaplon) which acts as torsion potential. This treatment enables to induce a tlaplon field sensitive to the symmetry of the spacetime in addition to the reconstruction of its effective potential from the $f(T)$ theory. The theory provides six different versions of inflationary models. The real solutions are mainly quadratic, the complex solutions, remarkably, provide Higgs-like potential.

On the relation between the non-flat cosmological models and the elliptic integral of first kind

Recently we have found an analytic integration formula that describes the dependence of the luminosity distance on the redshift for the flat cosmological model with a non-zero cosmological constant. The purpose of this article is to search for a similar relation for the non-flat models. A Taylor series was used. The elliptic integral of the first kind can indeed be applied. The result shows that a combination of the numerical integration with the previous analytic formula can also be useful for the nearly flat cosmological models.

How flat is our Universe really?

Distance measurement provides no constraints on curvature independent of assumptions about the dark energy, raising the question, how flat is our Universe if we make no such assumptions? Allowing for general evolution of the dark energy equation of state with 20 free parameters that are allowed to cross the phantom divide, w(z)=−1, we show that while it is indeed possible to match the first peak in the Cosmic Microwave Background with non-flat models and arbitrary Hubble constant, H0, the full WMAP7 and supernova data alone imply −0.12<Ωk<0.01 (2σ). If we add an H0 prior, this tightens significantly to Ωk=0.002±0.009. These constitute the most conservative and model-independent constraints on curvature available today, and illustrate that the curvature-dynamics degeneracy is broken by current data, with a key role played by the Integrated Sachs Wolfe effect rather than the distance to the surface of last scattering. If one imposes a quintessence prior on the dark energy (−1⩽w(z)⩽1) then just the WMAP7 and supernova data alone force the Universe to near flatness: Ωk=0.013±0.012. Finally, allowing for curvature, we find that all datasets are consistent with a Harrison–Zelʼdovich spectral index, ns=1, at 2σ, illustrating the interplay between early and late Universe constraints.

Repeatable light paths in the shearfree normal cosmological models

Conditions for the existence of repeatable light paths (RLPs) in the shearfree normal cosmological models are investigated. It is found that in the conformally nonflat models the only RLPs are radial null geodesics (in the spherical case) and their analogues in the plane and hyperbolically symmetric cases. In the conformally flat Stephani models, there exist special spherically, plane, and hyperbolically symmetric subcases, in which all null geodesics are RLPs. They are slightly more general than the Friedmann-Lema\^{\i}tre-Robertson-Walker models of the corresponding symmetries: their curvature index function $k(t)$ and the scale factor $R(t)$ are expressed through a single function of time. In addition to that, there exist special cases of the Stephani solution in which some of the null geodesics are RLPs. All these special cases are identified.

A measure on the set of compact Friedmann–Lemaître–Robertson–Walker models

Compact, flat Friedmann–Lemaître–Robertson–Walker (FLRW) models have recently regained interest as a good fit to the observed cosmic microwave background temperature fluctuations. However, it is generally thought that a globally, exactly flat FLRW model is theoretically improbable. Here, in order to obtain a probability space on the set F of compact, comoving, 3-spatial sections of FLRW models, a physically motivated hypothesis is proposed, using the density parameter Ω as a derived rather than fundamental parameter. We assume that the processes that select the 3-manifold also select a global mass-energy and a Hubble parameter. The requirement that the local and global values of Ω are equal implies a range in Ω that consists of a single real value for any 3-manifold. Thus, the obvious measure over F is the discrete measure. Hence, if the global mass-energy and Hubble parameter are a function of 3-manifold choice among compact FLRW models, then probability spaces parametrized by Ω do not, in general, give a zero probability of a flat model. Alternatively, parametrization by a spatial size parameter, the injectivity radius rinj, suggests the Lebesgue measure. In this case, the probability space over the injectivity radius implies that flat models occur almost surely (a.s.), in the sense of probability theory, and non-flat models a.s. do not occur.

The angular-diameter distance maximum and its redshift as constraints on Λ≠ 0 Friedmann-Lemaître-Robertson-Walker models

The plethora of recent cosmologically relevant data has indicated that our Universe is very well fitted by a standard Friedmann-Lemaitre-Robertson-Walker (FLRW) model, with Ω M ≈ 0.27 and Ω Λ ≈ 0.73 - or, more generally, by nearly flat FLRW models with parameters close to these values. Additional independent cosmological information, particularly the maximum of the angular-diameter (observer area) distance and the redshift at which it occurs, would improve and confirm these results, once sufficient precise Type Ia supernovae data in the range 1.5 < z < 1.8 become available. We obtain characteristic FLRW-closed functional forms for C = C(z) and M 0 = M 0 (z), the angular-diameter distance and the density per source counted, respectively, when A ≠ 0, analogous to those we have for A = 0. More importantly, we verify that for flat FLRW models z max - as is already known but rarely recognized - the redshift of C max , the maximum of the angular-diameter distance, uniquely gives Ω Λ , the amount of vacuum energy in the universe, independent of H 0 , the Hubble parameter. For non-flat models, determination of both z max and C max gives both Ω Λ and Ω M , the amount of matter in the universe, as long as we know H 0 independently. Finally, determination of C max automatically gives a very simple observational criterion for whether or not the universe is flat - presuming that it is FLRW.

Constraining dark energy with X-ray galaxy clusters, supernovae and the cosmic microwave background

We present new constraints on the evolution of dark energy from an analysis of Cosmic Microwave Background, supernova and X-ray galaxy cluster data. Our analysis employs a minimum of priors and exploits the complementary nature of these data sets. We examine a series of dark energy models with up to three free parameters: the current dark energy equation of state w{sub 0}, the early time equation of state w{sub et} and the scale factor at transition, a{sub t}. From a combined analysis of all three data sets, assuming a constant equation of state and that the Universe is flat, we measure w{sub 0} = 1.05{sub -0.12}{sup +0.10}. Including w{sub et} as a free parameter and allowing the transition scale factor to vary over the range 0.5 < a{sub t} < 0.95 where the data sets have discriminating power, we measure w{sub 0} = -1.27{sub -0.39}{sup +0.33} and w{sub et} = -0.66{sub -0.62}{sup +0.44}. We find no significant evidence for evolution in the dark energy equation of state parameter with redshift. Marginal hints of evolution in the supernovae data become less significant when the cluster constraints are also included in the analysis. The complementary nature of the data sets leads tomore » a tight constraint on the mean matter density, {Omega}{sub m} and alleviates a number of other parameter degeneracies, including that between the scalar spectral index n{sub s}, the physical baryon density {Omega}{sub b}h{sup 2} and the optical depth {tau}. This complementary nature also allows us to examine models in which we drop the prior on the curvature. For non-flat models with a constant equation of state, we measure w{sub 0} = -1.09{sub -0.15}{sup +0.12} and obtain a tight constraint on the current dark energy density, {Omega}{sub de} = 0.70 {+-} 0.03. For dark energy models other than a cosmological constant, energy-momentum conservation requires the inclusion of spatial perturbations in the dark energy component. Our analysis includes such perturbations, assuming a sound speed c{sub s}{sup 2} = 1 in the dark energy fluid as expected for Quintessence scenarios. For our most general dark energy model, not including such perturbations would lead to spurious constraints on w{sub et} which would be tighter than those mentioned above by approximately a factor two with the current data.« less

Latest supernova data in the framework of the generalized Chaplygin gas model

We use the most recent Type Ia supernova data in order to study the dark energy–dark matter (XCDM) unification approach in the context of the generalized Chaplygin gas (GCG) model. Rather surprisingly, we find that data allow models with α > 1. We have studied how the GCG adjusts flat and non-flat models, and our results show that GCG is consistent with a flat case up to 68 per cent confidence level. Actually this holds even if one relaxes the flat prior assumption. We have also analysed what one should expect from a future experiment such as the Supernova/Acceleration Probe (SNAP). We find that there is a degeneracy between the GCG model and an XCDM model with a phantom-like dark energy component.