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

Cosmological $N$-body simulations are now being performed using Newtonian gravity on scales larger than the Hubble radius. It is well known that a uniformly expanding, homogeneous ball of dust in Newtonian gravity satisfies the same equations as arise in relativistic Friedmann-Lema\^{\i}tre-Robinson-Walker cosmology, and it also is known that a correspondence between Newtonian and relativistic dust cosmologies continues to hold in linearized perturbation theory in the marginally bound/spatially flat case. Nevertheless, it is far from obvious that Newtonian gravity can provide a good global description of an inhomogeneous cosmology when there is significant nonlinear dynamical behavior at small scales. We investigate this issue in the light of a perturbative framework that we have recently developed [S. R. Green and R. M. Wald, Phys. Rev. D 83, 084020 (2011).], which allows for such nonlinearity at small scales. We propose a relatively straightforward dictionary---which is exact at the linearized level---that maps Newtonian dust cosmologies into general relativistic dust cosmologies, and we use our ``ordering scheme'' to determine the degree to which the resulting metric and matter distribution solve Einstein's equation. We find that, within our ordering scheme, Einstein's equation fails to hold at ``order 1'' at small scales and at ``order $ϵ$'' at large scales. We then find the additional corrections to the metric and matter distribution needed to satisfy Einstein's equation to these orders. While these corrections are of some interest in their own right, our main purpose in calculating them is that their smallness should provide a criterion for the validity of the original dictionary (as well as simplified versions of this dictionary). We expect that, in realistic Newtonian cosmologies, these additional corrections will be very small; if so, this should provide strong justification for the use of Newtonian simulations to describe relativistic cosmologies, even on scales larger than the Hubble radius.

/pdf/newtonian-and-relativistic-cosmologies-5f3t3a1l5k.pdf

Newtonian and Relativistic Cosmologies

Funding Information: IB is supported by an Alexander von Humboldt Foundation postdoctoral research fellowship.

The KBC void and Hubble tension contradict ΛCDM on a Gpc scale − Milgromian dynamics as a possible solution

We study if the discrepancy between the local and cosmological measurements of the Hubble constant ${H}_{0}$ can be affected by the choice of the cosmic microwave background (CMB) monopole temperature ${T}_{0}$. The latter is customarily fixed to the Far-Infrared Absolute Spectrophotometer (FIRAS) best-fit value in CMB data analyses. Although this value was confirmed by several independent experiments, it is interesting to see how much parameter constraints depend on this prior. We first provide a detailed pedagogical description of the ${T}_{0}$ effects on cosmological observables. We show that the recombination history and transfer functions do not depend on ${T}_{0}$, provided they are parametrized by the energy scale rather than redshift and at fixed dark matter and baryon densities per CMB photon. Thus, ${T}_{0}$ is only a property of the observer, quantifying the amount of expansion between key cosmological events and today. As a consequence, the sole effect of ${T}_{0}$ on small-scale primary CMB anisotropies is through the angular diameter distance to the epoch of last scattering, resulting in a near-perfect degeneracy between ${T}_{0}$ and ${H}_{0}$. This geometric degeneracy is partially lifted by the late-time integrated Sachs-Wolfe effect and CMB lensing. Still, Planck data alone are consistent with a broad region in the ${H}_{0}\ensuremath{-}{T}_{0}$ plane, implying that removing the FIRAS prior on ${T}_{0}$ can make Planck and SH0ES less discrepant, without introducing new physics beyond $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$. One may break the degeneracy by combining Planck with SH0ES, yielding an independent measurement of ${T}_{0}$, but the latter happens to be in $3\ensuremath{\sigma}$ tension with FIRAS. The agreement with FIRAS is restored anyway when combining Planck with the baryon acoustic oscillation data instead of SH0ES. Thus, the tension between SH0ES and cosmological measurements of ${H}_{0}$ persists even if we discard the FIRAS ${T}_{0}$ prior.

H 0 tension or T 0 tension

The physical interpretation of perturbations of homogeneous, isotropic cosmological models in the early Universe, when the perturbation is larger than the particle horizon, is clarified by defining a complete set of gauge-invariant variables. The linearized perturbation equations written in these variables are simpler than the usual versions, and easily accommodate an arbitrary background equation of state, entropy perturbations, and anisotropic pressure perturbations. Particular attention is paid to how a scalar (density) perturbation might be generated by stress perturbations at very early times, when the non-gauge-invariant perturbation in the density itself is ill-defined. The amplitude of the fractional energy density perturbation at the particle horizon cannot be larger, in order of magnitude, than the maximum ratio of the stress perturbation to the background energy density at any earlier time, unless the perturbation is inherent in the initial singularity.

Gauge-Invariant Cosmological Perturbations

We present the gauge-invariant formalism of cosmological weak lensing, accounting for all the relativistic effects due to the scalar, vector, and tensor perturbations at the linear order. While the light propagation is fully described by the geodesic equation, the relation of the photon wavevector to the physical quantities requires the specification of the frames, where they are defined. By constructing the local tetrad bases at the observer and the source positions, we clarify the relation of the weak lensing observables such as the convergence, the shear, and the rotation to the physical size and shape defined in the source rest-frame and the observed angle and redshift measured in the observer rest-frame. Compared to the standard lensing formalism, additional relativistic effects contribute to all the lensing observables. We explicitly verify the gauge-invariance of the lensing observables and compare our results to previous work. In particular, we demonstrate that even in the presence of the vector and tensor perturbations, the physical rotation of the lensing observables vanishes at the linear order, while the tetrad basis rotates along the light propagation compared to a FRW coordinate. Though the latter is often used as a probe of primordial gravitational waves, the rotation of the tetrad basis is indeed not a physical observable. We further clarify its relation to the E-B decomposition in weak lensing. Our formalism provides a transparent and comprehensive perspective of cosmological weak lensing.

Gauge-Invariant Formalism of Cosmological Weak Lensing

The background photon temperature $\overline{T}$ is one of the fundamental cosmological parameters, and it is often set equal to the precise measurement ${⟨T⟩}^{\mathrm{obs}}$ of the comic microwave background (CMB) temperature by the COBE Far Infrared Absolute Spectrometer (FIRAS). However, even in future CMB experiments, $\overline{T}$ will remain unknown due to the unknown monopole contribution ${\mathrm{\ensuremath{\Theta}}}_{0}$ at our position to the observed (angle-averaged) temperature ${⟨T⟩}^{\mathrm{obs}}$. Using the Fisher formalism, we find that the standard analysis with $\overline{T}\ensuremath{\equiv}{⟨T⟩}^{\mathrm{obs}}$ underestimates the error bars on cosmological parameters by $1%\ensuremath{\sim}2%$ of the present errors, and the best-fit parameters obtained in the analysis are biased by $\ensuremath{\sim}1%$ of their standard deviation. These systematic errors are negligible for the Planck data analysis, providing a justification to the standard practice. However, with $\overline{T}\ensuremath{\equiv}{⟨T⟩}^{\mathrm{obs}}$, these systematic errors will always be present and irreducible, and future cosmological surveys might misinterpret the measurements.

/pdf/background-photon-temperature-t-a-new-cosmological-parameter-3zqwtdf8bv.pdf

Background photon temperature T ̄: A new cosmological Parameter?

This is the first in a series of papers devoted to fully general-relativistic $N$-body simulations applied to late-time cosmology. The purpose of this paper is to present the combination of a numerical relativity scheme, discretization method and time-integration algorithm that provides satisfyingly stable evolution. More precisely, we show that it is able to pass a robustness test and to follow scalar linear modes around an expanding homogeneous and isotropic space-time. Most importantly, it is able to evolve typical cosmological initial conditions on comoving scales down to tenths of megaparsecs with controlled constraint and energy-momentum conservation violations all the way down to the regime of strong inhomogeneity.

General Relativistic Cosmological N-body Simulations I: time integration

Gauge-Invariant Formalism of Cosmological Weak Lensing.

Analytical computations in relativistic cosmology can be split into two sets: time evolution relating the initial conditions to the observer's light-cone and light propagation to obtain observables. Cosmological perturbation theory in the Friedmann–Lemaitre–Robertson–Walker (FLRW) coordinates constitutes an efficient tool for the former task, but the latter is dramatically simpler in light-cone-adapted coordinates that trivialize the light rays toward the observer world-line. Here we point out that time evolution and observable reconstruction can be combined into a single computation that relates directly initial conditions to observables. This is possible if one works uniquely in such light-cone coordinates, thus completely bypassing the FLRW 'middle-man' coordinates. We first present in detail these light-cone coordinates, extending and generalizing the presently available material in the literature, and construct a particularly convenient subset for cosmological perturbation theory. We then express the Einstein and energy–momentum conservation equations in these coordinates at the fully non-linear level. This is achieved through a careful 2 + 1 + 1 decomposition which leads to relatively compact expressions and provides good control over the geometrical interpretation of the involved quantities. Finally, we consider cosmological perturbation theory to linear order, paying attention to the remaining gauge symmetries and consistently obtaining gauge-invariant equations. Moreover, we show that it is possible to implement statistical homogeneity on stochastic fluctuations, despite the fact that the coordinate system privileges the observer world-line.

https://iopscience.iop.org/article/10.1088/1361-6382/abd681/pdf

Ermis Mitsou

Papers

Gauge-Invariant Formalism of Cosmological Weak Lensing

Background photon temperature T ̄: A new cosmological Parameter?

General Relativistic Cosmological N-body Simulations I: time integration

Gauge-Invariant Formalism of Cosmological Weak Lensing.

Cutting out the cosmological middle man: general relativity in the light-cone coordinates