A couple of new papers have suggested very subtle modifications of General Relativity as conventionally applied, which is the paradigmatic core theory of gravity, that are worth noting. Both are driven by the flaws of the ΛCDM "Standard Model of Cosmology" in confronting the observational evidence in an elegant and satisfying manner.

One is more limited. Another, by Alexander Sobolev, is striking and bold.

Honorable mention goes to a third paper entitled "Limiting curvature models of gravity", by Valeri P. Frolov, arXiv:2111.14318 (November 29, 2021), exploring the previously proposed idea that the singularities of General Relativity could be "cured" by imposing a maximum space-time curvature called LCG for limited curvature gravity.

**General Relativistic Entropic Acceleration (GREA)**

General Relativistic Entropic Acceleration hovers between being a modification of General Relativity and a novel way of operationalizing it that produces different predictions than the standard approach. Functionally, the main achievements of this approach are dispensing with the cosmological constant and resolving the Hubble anomaly, if it turns out that it is real and not merely a measurement error artifact. The introduction spells out what the authors are trying to do:

Our understanding of the expanding universe is anchored in the geometric description provided by Einstein’s theory of General Relativity (GR). On the one hand, its approximate symmetries, i.e., homogeneity and isotropy at large scales, determine its background space-time to be described by a Friedmann-Lemaître-Robertson-Walker (FLRW) metric. On the other hand, its matter content is responsible for the dynamics of the scale factor, which tracks the growth of length-scales in the geometric expansion, as described by the Friedmann equations.The currently accepted realization of FLRW cosmology is given by the Λ – Cold Dark Matter (ΛCDM) model. According to it, baryonic matter and radiation make up only a small portion of the present content of the universe. Instead, its expansion is dominated by two components which lack a fully satisfactory microscopic description. First, a cosmological constant, usually denoted by Λ, which is added to Einstein’s field equations to account for the observed late-time accelerated expansion of the universe. Second, cold (low temperature) dark (without electromagnetic interactions) matter, which was required originally to explain anomalies in the galactic rotation curves but is nowadays consistent with many other early- and late-time cosmological observables.Even though ΛCDM seems to be the best fit to observations, the existence of a cosmological constant has been challenged on theoretical grounds. Consequently, a plethora of alternatives haven been explored, which fall systematically into two groups. First, modified gravity (MG) theories attempt to deliver new dynamics at large, cosmological, scales, while leaving invariant smaller scales at which GR has been thoroughly probed. Second, dark energy (DE) models propose the addition of exotic matter, such as quintessence.Furthermore, in the last years there have been observational challenges to ΛCDM. Early- and late-time measurements of the present-value of the Hubble parameter (H0) seem to be inconsistent. This H0 tension signals a possible failure of the ΛCDM to describe our universe. However, no available alternative MG or DE seems to be able to resolve the tension between high and low redshift probes, while providing a fit to cosmological observations that is competitive with ΛCDM. Moreover, there have been recent model-independent analyses, using machine learning approaches, that suggest that there maybe hints of deviations from ΛCDM at high redshifts.Recently, a first-principles explanation of cosmic acceleration has been proposed by two of us. This is the General Relativistic Entropic Acceleration (GREA) theory. It is not based on MG or DE. Rather, it is based on the covariant formulation of non-equilibrium formulation of thermodynamics.Entropy production during irreversible processes necessarily has an impact on Einstein field equations. This suggests the idea that entropy production or, equivalently, information coarse graining, gravitates. As such, it affects the space-time geometry.In FLRW cosmology, irreversible processes inevitably contribute with an acceleration term to the Friedmann equations. In GREA, it is the sustained growth of the entropy associated with the cosmic horizon in open inflation scenarios that explains current cosmic acceleration.The goal of this paper is to test the full viability of the GREA theory at the background level and compare it with the ΛCDM, against available cosmological data.

The paper and its abstract are as follows:

Recently, a covariant formulation of non-equilibrium phenomena in the context of General Relativity was proposed in order to explain from first principles the observed accelerated expansion of the Universe, without the need for a cosmological constant, leading to the GREA theory.

Here, we confront the GREA theory against the latest cosmological data, including type Ia supernovae, baryon acoustic oscillations, the cosmic microwave background (CMB) radiation, Hubble rate data from the cosmic chronometers and the recent H(0) measurements.

We perform Markov Chain Monte Carlo analyses and a Bayesian model comparison, by estimating the evidence via thermodynamic integration, and find that when all the aforementioned data are included, but no prior on H(0), the difference in the log-evidence is ∼ −9 in favor of GREA, thus resulting in overwhelming support for the latter over the cosmological constant and cold dark matter model (ΛCDM).

When we also include priors on H(0), either from Cepheids or the Tip of the Red Giant Branch measurements, then due to the tensions with CMB data the GREA theory is found to be statistically equivalent with ΛCDM.

Rubén Arjona, et al., "A GREAT model comparison against the cosmological constant" arXiv:2111.13083 (November 25, 2021). Report number: IFT-UAM/CSIC-2021-136.

**Sobolev's Bold Theory**

Sobolev's just released

*magnum opus*is impressive and his modification of general relativity, discarding true general covariance, which even Einstein acknowledged was not observationally required, in favor of a subtle constraint on it, produces a stunning amount of weak field and cosmology phenomenology. His introduction sets the stage:In the light of new experimental data,GR no longer seems as unshakeable as it once did.For an explanation of the results derived within the framework of this theory, it was necessary to introduce certain hypothetical entities (the ΛCDM model) the nature of which are still unclear. “Entia non sunt multiplicanda praeter necessitatem”;it is likely that the necessity for the introduction of inflations at first, and now of dark energy and dark matter in GR (with the development of new methods of astronomical observation), are symptoms of a defect in its fundamental basis.General relativity violates the unity of the material world.In GR, the gravitational field itself does not have the properties of a material medium; its energy–momentum density is zero. This is a direct consequence of the general covariance of the gravitational field equations. Attempts to introduce a non-general covariant energy–momentum density actually mean refuting the original axiom of general covariance.In my opinion,it is the general covariance of the equations that is the source of the troubles of GR.One possible way to construct anon-generally covariant theory of gravity without violating Hilbert’s axioms(as I see it) is the introduction of an a priori constraint that restricts the choice of coordinate system. Attempts of such a kind have been made previously, for example the unimodular theory of gravity, whose origins date back to Einstein. A consequence of the introduction of this constraint is the appearance of an edge in the space–time manifold. Therefore,restrictedly covariant geometric objects are defined only on manifolds with this edge.Under such an approach, the fundamental principle of the equivalence of all reference systems compatible with the pseudo-Riemannian metric, which underlies GR, is not violated. In addition,we do not put into doubt the principle of the invariance ofHowever, this cannot be grounds for rejecting the proposed approach as contradicting the experiments verifying the strong equivalence principle for bodies of cosmic scales.matteraction relative to arbitrary transformations of coordinates. At the same time, in contrast to GR,a covariance of the gravitational equations is restricted by the constraint. Thus, a priori, only the “medium-strong principle” of equivalence is met in this case.The fact is that already in GR, within the framework of the ΛCDM model, space itself is endowed with energy. The same thing occurs when an a priori constraint is introduced. Space becomes a self-gravitating object because of the nonlinearity of the gravitational equations. One can determine the inertial and gravitational masses of such an object. The solution of the gravitational equations has enough free parameters to not only ensure the requirement of the equality of the inertial mass of the gravitational field to its gravitational mass, but also to determine inertial mass in accordance with Mach’s principle (the latter problem has not been solved in GR). From this point of view, the results of experiments should be considered as an indication thatonly such (quasi) stationary self-gravitating objects exist for which inertial mass is equal to gravitational mass.Hilbert’s axioms are formulated in a coordinate language. The gravitational field was represented by the ten componentsgμν(xλ) of the metric tensor. In addition, it was assumed that derivatives of the metrics no higher than second order could enter into the gravitational equations.There is no theorem prohibiting the existence of a constraint between the components of a metric in mathematical physics. However, the unimodular theory turned out to be unacceptable from a physical point of view, which prompted Einstein to abandon it in favor of the general covariant theory. Currently, such theories are considered as an approach to the construction of a quantum theory of gravity.Among the other possible approaches,Of course, there must be sufficiently substantial physical grounds to introduce the restrictions on the group of coordinate transformations.a restriction of general covariance has the least effect on the concepts about the world around us that are dictated by common sense.There is a deep analogy between the mathematical description of gravitational interaction in GR and the description of gauge interactions in elementary particle physics. The only way to calibrate for the latter (due to the requirement for general covariance) is by imposing the condition that the 4-divergence of the gauge fields is equal to zero. A similar condition for the gravitational field would be the requirement for an equality to zero of a 4-divergence of the connectivity consistent with the metric, simplified by a pair of indices . However, due to the fact that GR is not a gauge theory, to avoid contradictions with the initial provisions, such a condition should be considered not as a gauge, but as a constraint.

Jumping ahead to the conclusion:

The theory of gravity with a constraint, as the canonical theory, is based on the Hilbert action. Within the framework of the model proposed by the author, the fundamental differences from the standard cosmological model in a description of the evolution process of the Universe are as follows:* The constraint defines an edge with a zero-world physical anisotropic time at the restriction of the group of admissible coordinate transformations.* The gravitational field is endowed with all the properties of a material medium: energy, pressure, entropy and temperature.* It is possible to construct a space–time manifold, in which only its boundary is singular.* From the classical point of view, the process of the evolution of the Universe begins from a state with a minimum nonzero value of the scale factor and equal to zero energy.* By virtue of the definition of an energy–momentum density tensor adopted in the paper, the pressure of the gravitational field at the initial moment turns out to be negative, as a result of which the growth of the scale factor begins. At the same time, the energy density of the gravitational field also grows in proportion to the growth rate squared of the volume factor. This process has an avalanche-like character (“big bang”) and will continue until the energy density reaches its maximum value and begins to decrease due to the energy consumption for the adiabatic expansion of the Universe.* Despite the presence of a singularity at the boundary, the described classical model of the evolution of the Universe allows the construction of a canonical (or using path integrals) quantum theory of the gravitational field on its basis. The wave function of the very early Universe has been constructed.* The available experimental data on the temperature of the CMB radiation allow us to conclude that the maximum global energy density in the Universe has never exceeded 1 × 10^{50}−3 ~ (1.5 TeV)^{4}, the maximum temperature of the matter fields has not exceed 1.230 × 10^{11}K, and the relative energy density of neutrinos is currently less than 1.061 × 10^{–4}.

[Ed.The maximum energy density of the theory naïvely suggests an initial Big Bang minimum size of a sphere with a radius of about 1088 km, about the size of the dwarf planet Pluto, although this back of napkin calculation derived by dividing the total mass-energy of the universe determined in a ΛCDM cosmology by the maximum energy density in this new theory, is probably model dependent. The maximum temperature is that associated with the "quark epoch" in the "textbook" chronology of a ΛCDM universe at roughly one millisecond after the Big Bang.]

* The global energy density of the Universe is currently 94.5% composed of the energy density of the gravitational field, and all known types of matter only contribute 5.5%.* The accuracy of the available astronomical observations is still insufficient to choose between the predictions of GR and the proposed theory of gravity. However, over the past twenty years, the physical natures of dark energy, dark matter, and inflatons have not been established, and no new particles with suitable properties have been detected at the LHC. This is an essential argument for doubting their existence.* From the point of view of the theory presented here, all observable effects associated with dark energy and dark matter are only manifestations of the material essence of the gravitational field. On the one hand, in the present era of the second acceleration, the gravitational field has a negative pressure; that is, it behaves like hypothetical dark energy. On the other hand, the energy density of the gravitational field exceeds the average energy density of matter on the large-scale structure of the Universe. This energy, which is not taken into account within the framework of GR and has properties attributed to dark matter, can contribute to an increase in the speed of the observed gravitationally bound objects. In addition, the pressure of the gravitational field was negative in the very early Universe also and, as mentioned above, already within the framework of the classical approach at zero initial energy density, this leads to the Big Bang, therefore there is no need for a hypothesis about the existence of any inflatons.

The evolution of the observed Hubble constant in this theory is quite different. Another conclusion of this theory which doesn't make the conclusion section also deserves mention:

Thus, instead of the standard cosmological model (SCM), in this case, we have a continuum of cosmological models parameterized by the value of the maximum energy density ρ_{grmax}. Comparison of the data in Tables 1 and 2 shows that the results of the calculation are in good agreement, at least up to redshift of the last-scattering surface, despite a difference in the value of the maximum energy density of more than 60 orders. This circumstance excludes doubts about the possibility of an unambiguous description of the evolution of space in this range of redshift variation. It should be noted that the “last scattering” occurred less than 100 years after the beginning of the evolution process, as opposed to 373,000 years in the ΛCDM model.

[Ed.This is the end of the photon epoch and the beginning of "recombination" in the "textbook" chronology of the universe.]

The article and its citation are as follows:

The gravitational equations were derived in general relativity (GR) using the assumption of their covariance relative to arbitrary transformations of coordinates. It has been repeatedly expressed an opinion over the past century that such equality of all coordinate systems may not correspond to reality. Nevertheless, no actual verification of the necessity of this assumption has been made to date. The paper proposes a theory of gravity with a constraint, the degenerate variants of which are general relativity (GR) and the unimodular theory of gravity. This constraint is interpreted from a physical point of view as a sufficient condition for the adiabaticity of the process of the evolution of the space-time metric. The original equations of the theory of gravity with the constraint are formulated.On this basis,a unified model of the evolution of the modern, early, and very early Universe is constructed that is consistent with the observational astronomical data but does not require the hypotheses of the existence of dark energy, dark matter or inflatons.It is claimed that:physical time is anisotropic, the gravitational field is the main source of energy of the Universe, the maximum global energy density in the Universe was 64 orders of magnitude smaller the Planckian one, and the entropy density is 18 orders of magnitude higher the value predicted by GR.The value of the relative density of neutrinos at the present time and the maximum temperature of matter in the early Universe are calculated.The wave equation of the gravitational field is formulated, its solution is found, and the nonstationary wave function of the very early Universe is constructed. It is shown that the birth of the Universe was random.

Alexander P. Sobolev, "Foundations of a Theory of Gravity with a Constraint and its Canonical Quantization" 52

*Foundations of Physics*Article number: 3 arXiv:2111.14612 (open access, pre-print November 25, 2021, publication date anticipated 2022) DOI: 10.1007/s10701-021-00521-1
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