The $\Lambda$CDM model provides a good fit to a large span of cosmological data but harbors areas of phenomenology. With the improvement of the number and the accuracy of observations, discrepancies among key cosmological parameters of the model have emerged. The most statistically significant tension is the $4-6\sigma$ disagreement between predictions of the Hubble constant $H_0$ by early time probes with $\Lambda$CDM model, and a number of late time, model-independent determinations of $H_0$ from local measurements of distances and redshifts. The high precision and consistency of the data at both ends present strong challenges to the possible solution space and demand a hypothesis with enough rigor to explain multiple observations--whether these invoke new physics, unexpected large-scale structures or multiple, unrelated errors. We present a thorough review of the problem, including a discussion of recent Hubble constant estimates and a summary of the proposed theoretical solutions. Some of the models presented are formally successful, improving the fit to the data in light of their additional degrees of freedom, restoring agreement within $1-2\sigma$ between {\it Planck} 2018, using CMB power spectra data, BAO, Pantheon SN data, and R20, the latest SH0ES Team measurement of the Hubble constant ($H_0 = 73.2 \pm 1.3{\rm\,km\,s^{-1}\,Mpc^{-1}}$ at 68\% confidence level). Reduced tension might not simply come from a change in $H_0$ but also from an increase in its uncertainty due to degeneracy with additional physics, pointing to the need for additional probes. While no specific proposal makes a strong case for being highly likely or far better than all others, solutions involving early or dynamical dark energy, neutrino interactions, interacting cosmologies, primordial magnetic fields, and modified gravity provide the best options until a better alternative comes along.[Abridged]

https://iopscience.iop.org/article/10.1088/1361-6382/ac086d/pdf

In the realm of the Hubble tension - a review of solutions

The strong discrepancy between local and early-time (inverse distance ladder) estimates of the Hubble constant H0 could be pointing towards new physics beyond the concordance ΛCDM model. Several attempts to address this tension through new physics rely on extended cosmological models, featuring extra free parameters beyond the six ΛCDM parameters. However, marginalizing over additional parameters has the effect of broadening the uncertainties on the inferred parameters (including H0), and it is often the case that within these models the H0 tension is addressed due to larger uncertainties rather than a genuine shift in the central value of H0. In this paper I consider an alternative viewpoint: what happens if a physical theory is able to fix the extra parameters to a specific set of nonstandard values? In this case, the degrees of freedom of the model are reduced with respect to the standard case where the extra parameters are free to vary. Focusing on the dark energy equation of state w and the effective number of relativistic species Neff, I find that physical theories able to fix w≈-1.3 or Neff≈3.95 would lead to an estimate of H0 from cosmic microwave background, baryon acoustic oscillation, and type Ia supernovae data in perfect agreement with the local distance ladder estimate, without broadening the uncertainty on the former. These two nonstandard models are, from a model-selection perspective, strongly disfavored with respect to the baseline ΛCDM model. However, models that predict Neff≈3.45 would be able to bring the tension down to 1.5σ while only being weakly disfavored with respect to ΛCDM, whereas models that predict w≈-1.1 would be able to bring the tension down to 2σ (at the cost of the preference for ΛCDM being definite). Finally, I estimate dimensionless multipliers relating variations in H0 to variations in w and Neff, which can be used to swiftly repeat the analysis of this paper in light of future more precise local distance ladder estimates of H0, should the tension persist. As a caveat, these results were obtained from the 2015 Planck data release, but these findings would be qualitatively largely unaffected were I to use more recent data.

New physics in light of the $H_0$ tension: an alternative view

Cosmology Intertwined: A Review of the Particle Physics, Astrophysics, and Cosmology Associated with the Cosmological Tensions and Anomalies

We investigate the observational consequences of a novel class of stable interacting dark energy (IDE) models, featuring interactions between dark matter (DM) and dark energy (DE). In the first par ...

https://iopscience.iop.org/article/10.1088/1475-7516/2018/09/019/pdf

Tale of stable interacting dark energy, observational signatures, and the H0 tension

We explore whether nonstandard dark sector physics might be required to solve the existing cosmological tensions. The properties we consider in combination are (a) an interaction between the dark matter and dark energy components and (b) a dark energy equation of state $w$ different from that of the canonical cosmological constant $w=\ensuremath{-}1$. In principle, these two parameters are independent. In practice, to avoid early-time, superhorizon instabilities, their allowed parameter spaces are correlated. Moreover, a clear degeneracy exists between these two parameters in the case of cosmic microwave background (CMB) data. We analyze three classes of extended interacting dark energy models in light of the 2019 Planck CMB results and Cepheid-calibrated local distance ladder ${H}_{0}$ measurements of Riess et al. (R19), as well as recent baryon acoustic oscillation (BAO) and type Ia supernovae (SNeIa) distance data. We find that in quintessence coupled dark energy models, where $wg\ensuremath{-}1$, the evidence for a nonzero coupling between the two dark sectors can surpass the $5\ensuremath{\sigma}$ significance. Moreover, for both $\mathrm{Planck}+\mathrm{BAO}$ or $\mathrm{Planck}+\mathrm{SNeIa}$, we find a preference for $wg\ensuremath{-}1$ at about three standard deviations. Quintessence models are, therefore, in excellent agreement with current data when an interaction is considered. On the other hand, in phantom coupled dark energy models, there is no such preference for a nonzero dark sector coupling. All the models we consider significantly raise the value of the Hubble constant, easing the ${H}_{0}$ tension. In the interacting scenario, the disagreement between $\mathrm{Planck}+\mathrm{BAO}$ and R19 is considerably reduced from $4.3\ensuremath{\sigma}$ in the case of the $\mathrm{\ensuremath{\Lambda}}$ cold dark matter ($\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$) model to about $2.5\ensuremath{\sigma}$. The addition of low-redshift BAO and SNeIa measurements leaves, therefore, some residual tension with R19 but at a level that could be justified by a statistical fluctuation. Bayesian evidence considerations mildly disfavor both the coupled quintessence and phantom models, while mildly favoring a coupled vacuum scenario, even when late-time datasets are considered. We conclude that nonminimal dark energy cosmologies, such as coupled quintessence, phantom, or vacuum models, are still an interesting route toward softening existing cosmological tensions, even when low-redshift datasets and Bayesian evidence considerations are taken into account.

Nonminimal dark sector physics and cosmological tensions

Numerical solution of some classes of integral equations using Bernstein polynomials

The paper deals with a theoretical model for interacting dark energy. The interaction between the cold dark matter (dust) and the dark energy has been assumed to be non-gravitational in nature. Exact analytic cosmological solutions are obtained both for constant and variable equation of state for dark energy. It is found that, for very small value of the coupling parameter (in the interaction term), the model asymptotically extends up to $\Lambda$CDM, while the model can enter into the phantom domain asymptotically, if the coupling parameter is not so small. Both the solutions are then analyzed with 194 Supernovae Type Ia data. The best fit parameters are shown with 1$\sigma$ and 2$\sigma$ confidence intervals. Finally, we have discussed the cosmographic parameters for both the cases.

An analytic model for interacting dark energy and its observational constraints

In the background of a homogeneous and isotropic spacetime with zero spatial curvature, we consider interacting scenarios between two barotropic fluids, one is the pressureless dark matter (DM) and the other one is dark energy (DE), in which the equation of state (EoS) in DE is either constant or time dependent. In particular, for constant EoS in DE, we show that the evolution equations for both fluids can be analytically solved. For all these scenarios, the model parameters have been constrained using the current astronomical observations from Type Ia Supernovae, Hubble parameter measurements, and baryon acoustic oscillations distance measurements. Our analysis shows that both for constant and variable EoS in DE, a very small but nonzero interaction in the dark sector is favored while the EoS in DE can predict a slight phantom nature, i.e. the EoS in DE can cross the phantom divide line `$-1$'. On the other hand, although the models with variable EoS describe the observations better, but the Akaike Information Criterion supports models with minimal number of parameters. However, it is found that all the models are very close to the $\Lambda$CDM cosmology.

A new interacting two fluid model and its consequences

In this paper the Bernstein polynomials are used to approximate the solutions of linear Volterra integral equations. Both second and first kind integral equations, with regular, as well as weakly singular kernels, have been considered.

http://www.m-hikari.com/ams/ams-password-2008/ams-password33-36-2008/mandalAMS33-36-2008.pdf

Subhra Bhattacharya

Papers

Numerical solution of some classes of integral equations using Bernstein polynomials

An analytic model for interacting dark energy and its observational constraints

A new interacting two fluid model and its consequences

Use of Bernstein Polynomials in Numerical Solutions of Volterra Integral Equations

Numerical solution of a singular integro-differential equation