Luminosity Distance-redshift Relation Research Articles

Remarks on overestimating the effects of inhomogeneities on the Hubble constant

The Hubble constant is one of the most important parameters in cosmology. Discrepancies in values of the Hubble constant estimated from various measurements, the so-called Hubble tension, are a serious problem. In this paper, we study the effects of small-scale inhomogeneities of structure formation on the measurement of the Hubble constant using the luminosity distance-redshift relation. By adopting the adhesion model in Newtonian cosmology as the model of structure formation, we investigate whether or not the effects of inhomogeneities can be sufficiently large to affect the current observations of the Hubble constant. We show that inappropriate treatment of the effects of inhomogeneities can cause a large deviation of the measured value of the Hubble constant from the background value, whose magnitude is comparable with the Hubble tension. Our main message is the importance of adopting an appropriate model of structure formation to investigate the effects of inhomogeneities. We also add discussion on the spatial averaging approach used to estimate the measured Hubble constant in the inhomogeneous universe.

Gravitational wave luminosity distance in viscous cosmological models

We study the so-called Gravitational Wave luminosity distance-redshift relation d L GW(z) during cosmological eras driven by non-perfect fluids. In particular, we show that the presence of a shear viscosity in the energy momentum tensor turns out to be the most relevant effect. Within this scenario, a constant shear viscosity imprints the gravitational wave propagation through a friction term δ(z) with a uniquely given redshift dependence. This peculiar evolution predicts a specific shape for the ratio d L GW/d L EM which tends to a constant value when the sources are at z ≳ 1, whereas scales linearly with the shear viscosity at lower redshifts, regardless of the value of Ω m0. According to our final discussion, the predicted redshift dependence δ(z) provided by a shear viscosity could be tested by upcoming surveys of multi-messenger sources against analogous scenarios provided by some widely studied theories of modified gravity.

Cosmic variance of H0 in light of forthcoming high-redshift surveys

Forthcoming surveys will extend the understanding of cosmological large scale structures up to unprecedented redshift. According to this perspective, we present a fully relativistic framework to evaluate the impact of stochastic inhomogeneities on the determination of the Hubble constant. To this aim, we work within linear perturbation theory and relate the fluctuations of the luminosity distance-redshift relation, in the Cosmic Concordance model, to the intrinsic uncertainty associated to the measurement of $H_0$ from high-redshift surveys ($0.15\le z\le3.85$). We first present the detailed derivation of the luminosity distance-redshift relation 2-point correlation function and then provide analytical results for all the involved relativistic effects, such as peculiar velocity, lensing, time delay and (integrated) Sachs-Wolfe, and their angular spectra. Hence, we apply our analytical results to the study of high-redshift Hubble diagram, according to what has been recently claimed in literature. Following the specific of Euclid Deep Survey and LSST, we conclude that the cosmic variance associated with the measurement of the Hubble constant is at most of 0.1 %. Our work extends the analysis already done in literature for closer sources, where only peculiar velocity has been taken into account. We then conclude that deep surveys will provide an estimation of the $H_0$ which will be more precise than the one obtained from local sources, at least in regard of the intrinsic uncertainty related to a stochastic distribution of inhomogeneities.

Cosmography by orthogonalized logarithmic polynomials

Cosmography is a powerful tool for investigating the Universe kinematic and then for reconstructing the dynamics in a model-independent way. However, recent new measurements of supernovae Ia and quasars have populated the Hubble diagram up to high redshifts (z ∼ 7.5) and the application of the traditional cosmographic approach has become less straightforward due to the large redshifts implied. Here we investigate this issue through an expansion of the luminosity distance–redshift relation in terms of orthogonal logarithmic polynomials. In particular, we point out the advantages of a new procedure called orthogonalization, and we show that such an expansion provides a very good fit in the whole z = 0 ÷ 7.5 range to both real and mock data obtained assuming various cosmological models. Moreover, although the cosmographic series is tested well beyond its convergence radius, the parameters obtained expanding the luminosity distance–redshift relation for the Lambda cold dark matter (ΛCDM) model are broadly consistent with the results from a fit of mock data obtained with the same cosmological model. This provides a method for testing the reliability of a cosmographic function to study cosmological models at high redshifts, and it demonstrates that the logarithmic polynomial series can be used to test the consistency of the ΛCDM model with the current Hubble diagram of quasars and supernovae Ia. We confirm a strong tension (at > 4σ) between the concordance cosmological model and the Hubble diagram at z > 1.5. This tension is dominated by the contribution of quasars at z > 2 and also starts to be present in the few supernovae Ia observed at z > 1.

Combinations of Standard Pings and Standard Candles: An Effective and Hubble Constant-free Probe of Dark Energy Evolution

Abstract On the basis of the theoretical predication that a small fraction of fast radio bursts (FRBs) might be associated with gravitational waves (GWs), Wei et al. proposed FRB/GW association systems, i.e., the joint measurement of dispersion measures (DM) of FRBs and luminosity distances D L of GWs (the combination DM · D L), as an effective complementary cosmic probe, which has the advantage of being independent of the Hubble constant H 0. However, it is still unclear what the progenitor of FRBs is and whether a GW signal should be associated with them. In this paper, we propose that, with the fundamental assumption that luminosity distance is a monotonously increasing function of redshift, this DM · D L method could be further extended to the combination of DM measurements of localized FRBs and independent D L measurements of type Ia supernovae (SNe Ia) at similar redshifts. In practice, this extended combination, i.e., DM · D L as a function of redshift from independent DM and D L measurements, may be more applicable. In addition, we investigated the constraining power of this extended combination on dark energy evolution. It was obtained that, in the frame of the Chevalier–Polarski–Linder parameterization, the power of constraints on the dark energy equation of state from only ∼500 DM · D L combinations is even stronger than the one from the currently available 1048 Pantheon SNe Ia. More interestingly, we also found that, compared with the popular luminosity distance–redshift relation alone, the combination DM · D L, as a function of redshift, is more effective in constraining the parameter characterizing the evolution of dark energy. This benefit, along with rapid progress in the upcoming FRB observations, can be of great importance for studying the nature of dark energy.

The Local Perspective on the Hubble Tension: Local Structure Does Not Impact Measurement of the Hubble Constant

Abstract We use the largest sample to date of spectroscopic supernova (SN) Ia distances and redshifts to look for evidence in the Hubble diagram of large-scale outflows caused by local voids suggested to exist at z < 0.15. Our sample combines data from the Pantheon sample with the Foundation survey, and the most recent release of light curves from the Carnegie Supernova Project, to create a sample of 1295 SNe over a redshift range of 0.01 < z < 2.26. We make use of an inhomogeneous and isotropic Lemaitre–Tolman–Bondi metric to model a void in the SN Ia distance–redshift relation. We conclude that the SN luminosity distance–redshift relation is inconsistent at the 4–5σ confidence level with large local underdensities ( , where the density contrast δ = Δρ/ρ) proposed in some galaxy count studies, and find no evidence of a change in the Hubble constant corresponding to a void with a sharp edge in the redshift range 0.023 < z < 0.15. With an empirical precision of , we conclude that the distance ladder measurement is not affected by local density contrasts, in agreement with a cosmic variance of , predicted from simulations of large-scale structure. Given that uncertainty in the distance ladder value is , this does not affect the Hubble tension. We derive a 5σ constraint on local density contrasts on scales larger than of . The presence of local structure does not appear to impede the possibility of measuring the Hubble constant to 1% precision.

The KBC Void: Consistency with Supernovae Type Ia and the Kinematic SZ Effect in a ΛLTB Model

Abstract There is substantial and growing observational evidence from the normalized luminosity density in the near-infrared that the local universe is underdense on scales of several hundred megaparsecs. We test whether our parameterization of the observational data of such a “void” is compatible with the latest supernovae type Ia data and with constraints from line-of-sight peculiar-velocity motions of galaxy clusters with respect to the cosmic microwave background rest-frame, known as the linear kinematic Sunyaev–Zel’dovich (kSZ) effect. Our study is based on the large local void (LLV) radial profile observed by Keenan, Barger, and Cowie (KBC) and a theoretical void description based on the Lemaître–Tolman–Bondi model with a nonzero cosmological constant (ΛLTB). We find consistency with the measured luminosity distance–redshift relation on radial scales relevant to the KBC LLV through a comparison with 217 low-redshift supernovae type Ia over the redshift range . We assess the implications of the KBC LLV in light of the tension between “local” and “cosmic” measurements of the Hubble constant, H 0. We find that when the existence of the KBC LLV is fully accounted for, this tension is reduced from to . We find that previous linear kSZ constraints, as well as new ones from the South Pole Telescope and the Atacama Cosmology Telescope, are fully compatible with the existence of the KBC LLV.

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.

Accelerating expansion or inhomogeneity? A comparison of theΛCDMand Lemaître-Tolman models

It is shown how certain observations interpreted in the background of the Friedmann model with $\mathrm{\ensuremath{\Lambda}}<0=k$ (the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model) can be reinterpreted using the $\mathrm{\ensuremath{\Lambda}}=0$ Lema\^{\i}tre-Tolman (L-T) model so as to do away with the ``dark energy.'' The purpose of this paper is to clarify the underlying geometrical relations by doing the calculations as much as possible analytically or by very simple numerical programs. In the first part of the paper (fictitious) observations of the distribution of expansion velocity along the past light cone of the observer are considered. It is shown that the whole past light cone of the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ observer can be reproduced in the L-T model with $\mathrm{\ensuremath{\Lambda}}=0=E$. This is a geometric exercise that has the advantage of being free of numerical complications. In the second part, the luminosity distance-redshift relation of the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model is duplicated using the L-T model with $\ensuremath{-}k=2E/{r}^{2}=\text{constant}\text{ }>0$. The value of $k$ and the function ${t}_{B}(r)$ are determined by the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ parameters. General properties of this L-T model are described. Difficulties of carrying the numerical calculations through the apparent horizon are presented in detail and mostly solved. The second model is a counterexample to the general belief that an L-T model mimicking $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ must contain a void around the center---it has a peak of density at $R=0$.

Constraints from Type Ia supernovae on the Λ-CDM model in Randers-Finsler space

Gravitational field equations in Randers-Finsler space of approximate Berwald type are investigated. A modified Friedmann equation and a new luminosity distance-redshift relation is proposed. A best-fit to the Type Ia supernovae (SNe) observations yields that the ΩΛ in the Λ-CDM model is suppressed to almost zero. This fact indicates that the astronomical observations on the Type Ia SNe can be described well without invoking any form of dark energy. The best-fit age of the universe is given. It is in agreement with the age of our galaxy.

Measuring a Cosmological Distance-Redshift Relationship Using Only Gravitational Wave Observations of Binary Neutron Star Coalescences

Detection of gravitational waves from the inspiral phase of binary neutron star coalescence will allow us to measure the effects of the tidal coupling in such systems. Tidal effects provide additional contributions to the phase evolution of the gravitational wave signal that break a degeneracy between the system's mass parameters and redshift and thereby allow the simultaneous measurement of both the effective distance and the redshift for individual sources. Using the population of O(10(3)-10(7)) detectable binary neutron star systems predicted for 3rd generation gravitational wave detectors, the luminosity distance-redshift relation can be probed independently of the cosmological distance ladder and independently of electromagnetic observations. We conclude that for a range of representative neutron star equations of state the redshift of such systems can be determined to an accuracy of 8%-40% for z<1 and 9%-65% for 1<z<4.

Late time solutions for inhomogeneousΛCDMcosmology, their characterization, and observation

Assuming homogeneous isotropic Lambda-CDM cosmology allows Lambda, spatial curvature and dark matter density to be inferred from large scale structure observations such as supernovae. The purpose of this paper is to extend this to allow observations to measure or constrain inhomogeneity and anisotropy. We obtain the general inhomogeneous anisotropic Lambda-CDM solution which is locally asymptotic to an expanding de Sitter solution as a late time expansion using Starobinsky's method (analogous to the `holographic renormalization' technique in AdS/CFT) together with a resummation of the series. The dark matter is modeled as perfect dust fluid. The terms in the expansion systematically describe inhomogeneous and anisotropic deformations of an expanding FLRW solution, and are given as a spatial derivative expansion in terms of data characterizing the solution - a 3-metric and a perturbation of that 3-metric. Leading terms describe inhomogeneity and anisotropy on the scale set by the cosmological constant, approximately the horizon scale today. Higher terms in the expansion describe shorter scale variations. We compute the luminosity distance-redshift relation and argue that comparison with current and future observation would allow a partial reconstruction of the characterizing data. We also comment on smoothing these solutions noting that geometric flows (such as Ricci flow) applied to the characterizing data provide a canonical averaging method.

Swiss cheese and a cheesy CMB

It has been argued that the Swiss-Cheese cosmology can mimic Dark Energy, when it comes to the observed luminosity distance-redshift relation. Besides the fact that this effect tends to disappear on average over random directions,we show in this work that based on the Rees-Sciama effect on the cosmic microwave background (CMB), the Swiss-Cheese model can be ruled out if allholes have a radius larger than about 35 Mpc. We also show that for smaller holes, the CMB is not observably affected, and that the small holes can still mimic Dark Energy, albeit in special directions, as opposed to previous conclusions in the literature.However, in this limit, the probability of looking in a special direction wherethe luminosity of supernovae is sufficiently supressed becomes very small, at leastin the case of a lattice of spherical holes considered in this paper.

Modification to the luminosity distance redshift relation in modified gravity theories

We derive an expression for the luminosity distance as a function of redshift for a flat Robertson-Walker spacetime perturbed by arbitrary scalar perturbations possibly produced by a modified gravity theory with two different scalar perturbation potentials. Measurements of the luminosity distance as function of redshift provide a constraint on a combination of the scalar potentials and so they can complement weak lensing and other measurements in trying to distinguish among the various alternative theories of gravity.

Luminosity distance in “Swiss cheese” cosmology with randomized voids. I. Single void size

Recently, there have been suggestions that the Type Ia supernova data can be explained using only general relativity and cold dark matter with no dark energy. In 'Swiss cheese' models of the Universe, the standard Friedmann-Robertson-Walker picture is modified by the introduction of mass-compensating spherical inhomogeneities, typically described by the Lemaitre-Tolman-Bondi metric. If these inhomogeneities correspond to underdense cores surrounded by mass-compensating overdense shells, then they can modify the luminosity distance-redshift relation in a way that can mimic accelerated expansion. It has been argued that this effect could be large enough to explain the supernova data without introducing dark energy or modified gravity. We show that the large apparent acceleration seen in some models can be explained in terms of standard weak field gravitational lensing together with insufficient randomization of void locations. The underdense regions focus the light less than the homogeneous background, thus dimming supernovae in a way that can mimic the effects of acceleration. With insufficient randomization of the spatial location of the voids and of the lines of sight, coherent defocusing can lead to anomalously large demagnification effects. We show that a proper randomization of the voids and lines of sight reduces the effect to the point more » that it can no longer explain the supernova data. « less

Accelerating Cosmologies with an Anisotropic Equation of State

If the dark energy equation of state is anisotropic, the expansion rate of the universe becomes direction-dependent at late times. We show that such models are not only cosmologically viable but that they could explain some of the observed anomalies in the CMB, and shed some light into the coincidence problem. The possible anisotropy can then be constrained by studying its effects on the luminosity distance-redshift relation inferred from several observations. A vector field action for dark energy is also presented as an example of such possibility.

Constraining the dark energy and smoothness parameter with supernovae

The presence of inhomogeneities modifies the cosmic distances through the gravitational lensing effect, and, indirectly, must affect the main cosmological tests. Assuming that the dark energy is a smooth component, the simplest way to account for the influence of clustering is to suppose that the average evolution of the expanding Universe is governed by the total matter-energy density whereas the focusing of light is only affected by a fraction of the total matter density quantified by the $\ensuremath{\alpha}$ Dyer-Roeder parameter. By using two different samples of SNe type Ia data, the ${\ensuremath{\Omega}}_{m}$ and $\ensuremath{\alpha}$ parameters are constrained by applying the Zeldovich-Kantowski-Dyer-Roeder luminosity distance-redshift relation for a flat ($\ensuremath{\Lambda}\mathrm{CDM}$) model. A ${\ensuremath{\chi}}^{2}$-analysis using the 115 SNe Ia data of the Astier et al. sample (2006) constrains the density parameter to be ${\ensuremath{\Omega}}_{m}={0.26}_{\ensuremath{-}0.07}^{+0.17}$ ($2\ensuremath{\sigma}$) while the $\ensuremath{\alpha}$ parameter is weakly limited (all the values $\ensuremath{\in}[0,1]$ are allowed even at $1\ensuremath{\sigma}$). However, a similar analysis based the 182 SNe Ia data of Riess et al. (2007) constrains the pair of parameters to be ${\ensuremath{\Omega}}_{m}={0.33}_{\ensuremath{-}0.07}^{+0.09}$ and $\ensuremath{\alpha}\ensuremath{\ge}0.42$ ($2\ensuremath{\sigma}$). Basically, this occurs because the Riess et al. sample extends to appreciably higher redshifts. As a general result, even considering the existence of inhomogeneities as described by the smoothness $\ensuremath{\alpha}$ parameter, the Einstein-de Sitter model is ruled out by the two samples with a high degree of statistical confidence ($11.5\ensuremath{\sigma}$ and $9.9\ensuremath{\sigma}$, respectively). The inhomogeneous Hubble-Sandage diagram discussed here highlights the necessity of the dark energy, and a transition deceleration/accelerating phase at $z\ensuremath{\sim}0.5$ is also required.

Non-linear structure formation and ‘apparent’ acceleration: an investigation

We present an analytically tractable non-linear model of structure formation in a Universewith only dust. The model is a Lemaître–Tolman–Bondi solution (of general relativity) andstructures are shells of different density. We show that the luminosity distance–redshiftrelation has significant corrections at low redshift when the density contrast becomesnon-linear. A minimal effect is a correction in apparent magnitudes of order . We discuss different possibilities that could further enhance this effect and mimic darkenergy.

Systematic corrections to the measured cosmological constant as a result of local inhomogeneity

We explicitly calculate the lowest order systematic inhomogeneity-induced corrections to the cosmological constant that one would infer from an analysis of the luminosities and redshifts of type Ia supernovae, assuming a homogeneous universe. The calculation entails a post-Newtonian expansion within the framework of second-order perturbation theory, wherein we consider the effects of subhorizon density perturbations in a flat, dust-dominated universe. Within this formalism, we calculate luminosity distances and redshifts along the past light cone of an observer. The luminosity distance-redshift relation is then averaged over viewing angles and ensemble averaged, assuming that density fluctuations at a given cosmic time are a homogeneous random process. The resulting relation is fit to that of a homogeneous model containing dust and a cosmological constant, in order to deduce the best-fit cosmological constant density ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}$. We find that the luminosity distance-redshift relation is indeed modified, even for large sample sizes, but only by a very small fraction, of order ${10}^{\ensuremath{-}5}$ for $z\ensuremath{\sim}0.1$. This lowest order deviation depends on the peculiar velocities of the source and the observer. However, when fitting this perturbed relation to that of a homogeneous universe, via maximizing a likelihood function, we find that the inferred cosmological constant can be surprisingly large, depending on the range of redshifts sampled. For a sample of supernovae extending from ${z}_{\mathrm{min}}=0.02$ out to a limiting redshift ${z}_{\mathrm{max}}=0.15$, we find that ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}\ensuremath{\approx}0.004$. The value of ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}$ has a large variance, and its magnitude tends to get progressively larger as the limiting redshift ${z}_{\mathrm{max}}$ gets smaller, implying that precision measurements of ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}$ from nearby supernova data will require taking this effect into account. This effect has been referred to in the past as the ``fitting problem,'' and more recently as subhorizon ``backreaction.'' We find that it is likely too small to explain the observed value ${\ensuremath{\Omega}}_{\ensuremath{\Lambda}}\ensuremath{\approx}0.7$. There have been previous claims of much larger backreaction effects. In contrast to those calculations, our work is directly related to how observers deduce cosmological parameters from astronomical data.

Luminosity distance–redshift relation for the LTB solution near the centre

Motivated by the inverse problem for the Lemaître–Tolman–Bondi dust solution in which problem the luminosity distance function DL(z) is taken as an input to select a specific model, we compute the function DL(z) of the LTB solution up to the third order of z. To perform the otherwise cumbersome computation, we introduce a new convenient form of the LTB solution, in which the solution is explicit and unified. With this form of the LTB solution we obtain the luminosity distance function with full generality. We, in particular, find that the function exactly coincides with that of a homogeneous and isotropic dust solution up to second order, if we demand that the solution be regular at the centre.