The ΛCDM-NG Cosmological Model: A Possible Resolution of the Hubble Tension

We offer a cosmological model based on conventional general relativity (no speculative physics) that may resolve the Hubble tension. A reanalysis of the foundation of the Lambda-CDM model shows that general relativity alone does not specify what fraction of the mass density acts as the source term in Friedmann’s equation and what fraction acts as the source of the gravitational potential of condensed objects. This observation opens the way to alternative cosmological models within conventional general relativity, and it proves that the ΛCDM model is not the unique solution of Einstein’s equations for the usual cosmological sources of gravitation. We emphasize that the source of the gravitation potential in the ΛCDM model is the deviation δ ρ m of the mass density away from its average value, and not the total density of condensed masses as in Newtonian theory. Though not often stated, this is a modification of Newtonian gravitation within the ΛCDM model. The ΛCDM-NG model uses the freedom to move matter between source terms to restore the source of gravitational potential to its Newtonian form. There is no Hubble tension in the ΛCDM-NG model if the gravitational potential of condensed objects (stars, galaxies, and dark matter clouds) falls in a certain range, a range that does not seem unreasonable for the actual Universe. The deceleration parameter in the ΛCDM-NG model differs from that in the ΛCDM model, suggesting a test to distinguish between the two models.

At the present time the ΛCDM model is being challenged on three fronts: (1) The long standing cosmological constant problem (Weinberg 1989), (2) Initial observations by the Jame Webb Space Telescope (JWST); Advanced Deep Extragalactic Survey (JADES) (Finkelstein et al. 2022;Labbe et al. 2022;Adams et al. 2023;Rodighiero et al. 2011) suggest galaxy formation at times earlier than expected from the ΛCDM model (Boylan-Kolchin 2022); and (3) Measurements of the Hubble constant H 0 using "late universe" (redshift z ≈ 0-1) Type Ia supernovae give a value [∼ 74 km s −1 M pc −1 ] significantly different from the value [∼ 68 km s −1 M pc −1 ] obtained from observations of the "early universe" (redshift z = 1100) Cosmic Microwave Background (CMB) radiation (Planck 2018).The cosmological model proposed here makes a contributions to the last of these quandaries.
The present paper in based on the insight, explained in Sec.2, that general relativity alone does not specify how much of the mass-energy of the universe serves as source term ρ F in Friedmann's equation and how much ρ Φ is the source of gravitational potential Φ, with different splittings of the total energy among the two source terms giving a Correspondence welcomed at: richard.cook43@gmail.comarXiv:2404.13712v1[gr-qc] 21 Apr 2024 Cook different cosmological models.Section 3 identifies the sources ρ F and ρ Φ which give the ΛCDM model, and points out that the source of gravitational potential in this model is not the same as in Newtonian gravitation.The ΛCDM-NG model is introduced in Sec.4 as the cosmological model which restores the source of the gravitational potential to its Newtonian form.Sections 5-10 develop the formalism of the ΛCDM-NG model.Section 11 shows there is no Hubble tension in the ΛCDM-NG model, if the gravitational potential of the universe at the present time falls in a given range.Section 12 calculates the deceleration parameter for the ΛCDM-NG model, and points to the difference between this and that of the ΛCDM model as the basis of a possible test to distinguish between the two models.Several expansion histories, within the ΛCDM-NG model, illustrate, in Sec.13, the variety of solutions which have the observed values of cosmological parameters Ω Λ , Ω m , Ω r , H 0 , and q 0 ; and at the same time exhibit no Hubble tension in this model.A physical interpretation of the ΛCDM-NG model is given in Sec.14 in term of gravitational time dilation of the standard comoving clocks which measure cosmic time.The paper concludes in Sec.15 with some final remarks.

NON-UNIQUENESS OF THE LAMBDA-CDM MODEL
From the time of Friedmann onward, cosmological models, including the ΛCDM model, have taken the Friedmann-Lemaître-Robertson-Walker (FLRW) line element as background geometry, and have assumed a source term in Friedmann's equation equal to the space-averaged density of all mass-energy in the universe (Friedmann 1922;Lemaître 1931;Eddington 1930;Weinberg 2008).With the mean value ρ(t) of the inhomogeneous density ρ(x, t) of the universe taken as the source term in Friedmann's equation, there remains only the deviation δρ = ρ(x, t) − ρ(t) of ρ(x, t) away from its average value ρ(t) as the source of gravitational potential Φ(x, t).In this section, and the next two, we show that this very plausible approach to cosmological theory has two consequences: (1) it obscures the existence of alternative cosmological models within conventional general relativity, and with the same sources of gravitation as the ΛCDM model; and (2) it necessarily involves a modification of Newtonian gravitation where Newtonian theory is expected to be valid in general relativity.
The non-uniqueness of the ΛCDM model is easily shown if we refrain from specifying the source term in Friedmann's equation a priori, and instead consider, from the beginning, a line element of sufficient generality to describe both the large-scale expansion of the universe, as described by the scale factor a(t), and the smaller scale perturbations of the metric by the inhomogeneous density ρ(x, t).It is usual in structure formation calculations to write such a line element as (often with different sign conventions and labels Φ and Ψ switched).This is the flat space LFRW line element perturbed by scalar functions Φ(x, t) and Ψ(x, t) (gauge invariant Bardeen potentials (Bardeen 1980)) in what is called the conformal Newtonian gauge.If we are willing to ignore certain small effects due to neutrino anisotropic stresses during the early radiation-dominated era, in order to simplify the following argument, then Ψ and Φ are equal, and line element (1) becomes (Dodelson 2021) where Φ is recognized as the Newtonian gravitational potential.This then is the simplest line element with sufficient detail to make the following argument.It contains the scale factor a(t) describing the large-scale dynamics of the universe and the gravitational potential Φ(x, t) describing smaller scale gravitational interactions.Line element (2) is usually applied in situations where Φ/c 2 is small (Φ/c 2 << 1) and Φ/c 2 constitutes a weak perturbation to the unperturbed line element (3) We, however, because we are interested in late times when Φ/c 2 is not necessarily small, shall treat Eq.( 2) as an exact line element without the restriction that Φ/c 2 be small.The exact Einstein equation for metric (2), with energy-momentum-tensor component T 0 0 = −ρc 2 , reads.
where (∇Φ) i = ∂Φ/∂x i , ∇ 2 Φ = δ ij ∂ 2 Φ/∂x i ∂x j , and we have written the density as the sum of a perfectly uniform dark energy, ρ Λ = constant, homogeneous radiation, ρ r (t) = ρ r (t 0 )/a 4 (t), and an inhomogeneous matter density, ρ m (x, t), describing the combined densities of dark matter and baryonic matter (stars, galaxies, and dark matter clouds).We learn from Eq.( 5) the simple, but important, fact that general relativity does not specify how the various densities on the right in this equation are partitioned into source terms for the various field terms on the left side of the equation.Any reasonable association of densities with field terms gives a solution of Einstein equation (4).
To be clear, we split the total density ρ into a source term ρ F for what will be the Friedmann equation, and a source term ρ Φ for what will be the equation for potential Φ.We argue that, if the source term ρ Φ is null, the potential Φ will also vanish.Using these values in (5), we obtain the familiar Friedmann equation and subtracting this from Eq.( 5), we have the equation for potential Φ, As a check on this result, we evaluate Eq.( 8) to first order in the dimensionless potential Φ/c 2 , and obtain the linearized potential equation familiar from structure formation theory except that the source δρ m is replaced by the density ρ Φ (Baumann 2022).
The sum of ρ F and ρ Φ is, of course, the total density, It is important to note that, according to this equation, whatever density ρ Φ is chosen as the source term for the gravitational potential, Eq.( 8), this density is unavailable to influence the evolution of the scale factor a(t) as part of the source term in Friedmann's equation ( 7).The all-important point of this section is that different partitions of the total density into source terms ρ F and ρ Φ for Eqs.( 7) and ( 8) give different cosmological models.One partition gives the ΛCDM model and a different partition gives the ΛCDM-NG model.We emphasize that the splitting of the total density into ρ Φ and ρ F is a choice made outside of general relativity.It is an independent assumption, or hypothesis, requiring its own justification.

THE LAMBDA-CDM MODEL
The ΛCDM model is based on the assumption that the source term in Friedmann's equation is the space-averaged density of all mass-energy in the universe, where ρm =< ρ m (x, t) >.Then, according to Eq.( 10), the source of the gravitational potential Φ is necessarily which is the deviation of ρ m (x, t) away from its average value (Fluctuations of radiation also contribute to the source of gravitational potential, but for simplicity of presentation we consider only the dominant matter fluctuations).One version of the basic equations of the ΛCDM model (neglecting neutrino anisotropic stresses) is written as Cook and These equations do not capture all details of the ΛCDM model, but are sufficient for the following arguments.
The ΛCDM model explains a multitude of observed phenomena and, despite a couple of "tensions" in recent decades, nothing has yet falsified the model.But this author is struck by one feature of the model which seems a bit odd, and it is this oddness of the source term (12) which motivates construction of the ΛCDM-NG model.

The Source of the Gravitation Potential in the ΛCDM Model
Though not often emphasized, the equation for the potential in the ΛCDM model, Eq.( 15), is a modification of Newtonian gravitation, even in limits where Newtonian theory is expected to apply in general relativity.On the scale of galaxies and galaxy clusters, the terms containing (1/a)(da/dt) = H ∼ 2 × 10 −18 s −1 in Eq.( 15) are negligible, and the term ∇ 2 Φ/a 2 is actually the Newtonian Laplacian acting on Φ written in terms of comoving coordinates.So, over small regions, Eq.( 15) is the Newtonian Poisson equation, except that δρ m appears as source term instead of the full density of condensed objects as in Newtonian theory.The source term δρ m has a number of curious properties which we list below: • We are accustomed to thinking of the source term in the Newtonian field equation ∇ 2 Φ = 4πGρ Φ as the density of active gravitational mass.But the space average of the density ρ Φ = δρ m in Eq.( 12) is zero.So the active gravitational mass density ρ Φ (x, t) of the ΛCDM model is as much negative as it is positive.That is to say, there are regions where ρ Φ (x, t) is negative and these regions have repulsive gravitational fields.In addition, the total active gravitational mass is zero in this model!
• If we move from cosmology to a different page of the general relativity textbook, we learn that the total mass of an isolated, gravitationally bound system (including the gravitational binding energy) may be determined using flux integral methods, if we know the weak asymptotic gravitational field far from the system (Misner et al. 1973).Such methods are used to establish the masses of black holes and neutron stars.But, if there exist a uniform negative gravitational mass density −ρ m throughout space, as in Eq.( 12), such methods will not work.
The flux integral over a spherical surface, say, will necessarily go to zero as the radius tends to infinity, because the total active gravitational mass measured in this way is zero for the ΛCDM model.
• Finally, it is odd that the source of the gravitational potential at point P should depend on masses at a great distance from P .This is indeed the case for the ΛCDM model because the source term at P , namely Eq.( 12), depends on the average density ρm which in turn depends on masses at all distances from P .
None of these comments rises to the status of a serious objection to the ΛCDM model.And, as a practical matter, the subtraction of ρm in the source term ρ N is of no concern for the ΛCDM model because ρm is very small (of the order of the critical density ρ cr = 9.5 × 10 −27 kg m −3 ) and is negligible compared to typical densities of stars, galaxies, and dark-matter clouds.But these thoughts do motivate one to ask: "Is there a reasonable cosmological model which does not require a modification of Newtonian gravitation?"The affermative answer to this question is the ΛCDM-NG model; a topic to which we now turn.

THE LAMBDA-CDM-NG MODEL
The ΛCDM-NG model assumes a relativistic theory of gravitation should reduce to Newtonian theory in the limit where the latter is expected to apply.Indeed, Einstein made this very assumption when choosing the coupling constant between matter and gravitational field, so that his theory would reduce to Newtonian theory in the weak-field static limit (Einstein 1915).But, as we have seen, the ΛCDM model does not have the Newtonian limit, because its source for the gravitational potential is δρ m , and not the full mass density of condensed masses.The ΛCDM-NG model returns the source of gravitational potential to its full Newtonian value using the freedom to move mass between source terms established in Sec.2 (the NG in the moniker ΛCDM-NG remind us of the source term from Newtonian Gravitation).According to Eq.( 10), this step necessitates a decrease in the source term of Friedmann's equation.
The ΛCDM-NG model is determined by specifying the source terms ρ F ( t) and ρ Φ (x, t) for field equations ( 7) and ( 8).If F( t) is the fraction of matter which has condensed out of the near-uniform background density, then the average density of condensed objects is ρΦ ( t) = F( t)ρ m ( t), ( 16) and this is the space-averaged source of gravitational potential in the ΛCDM-NG model.Models of gravitational clustering, such as the collapse of a spherical slight overdensity (Baumann 2022), suggest that the background density deviates very little from uniformity as structure formation progresses, and so, we expect the background density of matter to decrease with time as [1 − F( t)]ρ m ( t), and for this to be the contribution to ρ F from matter, ρ Then Eq.( 10) determines the full source for the gravitational potential, Density ρ Φ (x, t) is the density of active gravitational mass, which we insist must be a positive quantity in the ΛCDM-NG model.For now we shall work with Eqs.( 17) and ( 18), but ρ Φ (x, t) and F( t) are not yet fully defined.In section 9 we give more precise definitions of these quantities in terms of the Fourier components of δρ m ; definitions which garentee ρ Φ (x, t) ≥ 0.
A word on notation.In the section after next, it is shown that the coordinates t and xi we are using for the ΛCDM-NG model (those with a tilde " ˜") are not the cosmic time t and comoving coordinates x i used by astronomers to report their observations; the symbols t and x i (or x) are reserved for this purpose.In addition, we use the tilde to denote quantities that are not directly observed or measured.For example, the scale factor a(t) in the ΛCDM model is directly observable, because a = 1/(1 + z) and the redshift z is observable.But in the ΛCDM-NG model, as we shall see, the scale factor ã is not directly observable, hence the notation.
Using standard formulas ρ Λ =constant, ρ r ( t) = ρ r ( t0 )/ã 4 , and ρm ( t) = ρm ( t0 )/ã 3 in Eqs.( 17) and ( 18), and the results in Eqs.( 7) and ( 8), we obtain the basic equations of the ΛCDM-NG model: where, as usual, Ω x = ρ x /ρ cr , ρ cr = 3H 0 /8πG, and is the observed Hubble constant, and now ( ∇Φ The monotonically increasing condensed fraction F( t) appearing in these equations is essentially zero in the very early universe, is expected to start growing when the radiation era transitions into the matter era, grows in earnest after decoupling, and finishes at some final value F 0 = F( t0 ) at the present time t0 .It is often more convenient, as in Eq.( 20), to write F[ã( t)] as a function of the scale factor ã( t), in which case F 0 = F(ã 0 ).

THE GRAVITATIONAL POTENTIAL OF THE UNIVERSE
Solution of the rather cumbersome equation ( 21) for potential Φ is simplified by initially ignoring the inhomogeneous part δρ m of the source term, and writing this equation for the position-independent potential Φ( t) produced by the average density ρΦ = F( t)ρ m ( t0 )/ã 3 ( t) alone, namely

Cook
Solving this equation for d( Φ/c 2 )/d t and expanding H using Eq.( 20), we have where and This is our working equation for Φ/c 2 ; a quantity we call the gravitational potential of the universe.
It is then straightforward, using Eq.( 21), to write an equation for the first order perturbation δΦ to potential Φ produced by the inhomogeneous part δρ m of the source density in Eq.( 21); an equation important for structure formation in the ΛCDM-NG model.However, the principle purpose of this paper is to show how the potential Φ( t) could resolve the Hubble tension, and we shall not consider δΦ further at this time.

THE OBSERVED SCALE FACTOR
In a simple line element of the form t is the proper time reading on synchronized comoving clocks known as cosmic time, and the comoving space coordinates x i are proper distances measured in the coordinate directions at the present time t 0 .Such interpretations are critical for relating the formalism to observations.When the gravitational potential Φ is added to Eq.( 27) to obtain a line element of the form the time coordinate t is no longer the cosmic time t used by astronomers to report their results, and the space coordinates xi are not, at the present time, proper distances in the coordinate directions.In the ΛCDM model, where Φ/c 2 << 1, this distinction is unimportant, and t and xi can be treated as if these were the physical variables t and x i .But in the ΛCDM-NG model, where Φ/c 2 can be larger, such an approximation may be inappropriate.
For line element (28), the increment of proper time on a comoving clock (where and Note that t( t) is a strictly increasing function of t which has inverse t(t).
Similarly, the increments of proper distance in the coordinate directions, at the present time t0 , when ã( t0 ) = 1, are Equations ( 30) and ( 32) constitute a coordinate transformation to physical coordinates t and x i , which recasts line element (28) into the form where is the observed scale factor, because the coordinates t and x i used in line element (33) are now the measured spacetime coordinates used by astronomers.Without this connection to measured quantities, the equations of cosmology would be empty formalism.
From the observed scale factor, one calculates the observed Hubble parameter as using cosmic time t; and the observed deceleration parameter as 7. THE "FRIEDMANN EQUATION" In the ΛCDM-NG model the "Friedmann equation," reads where the quantity on the right is less than that in the corresponding ΛCDM model, Eq.( 14), because [1 − F(ã)] is less than one.With Ω Λ + Ω m + Ω r = 1 and ã( t0 ) = 1 at the present time t0 , Eq.( 37) would seem to be inconsistent, because the right side of this equation is less than the Hubble constant.But this is not an inconsistency, because, as we have seen, ã( t) is not the observed scale factor.
From the results of the preceding section we conclude that the role of Eq.( 37) in the ΛCDM-NG is different than in the ΛCDM model, in that it does not directly determine the observed Hubble parameter.We shall refer to ã( t) as the prior scale factor, and we call the prior Hubble parameter to distinguish this from the observed Hubble parameter H obs (t) which we discuss next.

THE OBSERVED HUBBLE PARAMETER AND HUBBLE CONSTANT
We are now in a position to evaluate the observed Hubble parameter.Taking the derivative of a obs (t), Eq.( 34), with respect to cosmic time t, and dividing by a obs (t), we have the observed Hubble parameter of Eq.( 35), Then using Eqs.( 20), (29), and (24), Eq.( 39) becomes It appears that the density we had subtracted from the source term in Friedman's equation ( 20) is restored to the Friedmann equation (40) when working with observed quantities.But this is not exactly correct because the ã in Eq.( 40) is not the observed scale factor.Using Eq.( 34) to write Eq.( 40) in terms of the observed scale factor, we obtain where A space average of this result, with < δρ m >= 0, gives and using ρΦ = F( t)ρ m , we have Eqs.( 50) and ( 52) are our definitions of ρ Φ (x, t) and F(t), and Eqs.( 49) and ( 51) ensure that ρ Φ ≥ 0.
Although the condensation fraction in Eq.( 52) is clearly related to the power spectrum of density fluctuations P(k), it is not derivable from the latter, and, at the present time, the form of F(t) for our universe is not known with any confidence.However, as shown in Sec.13, there is quite a variety of condensation histories for the accepted values of density parameters Ω Λ , Ω m , and Ω r , which give the observed values for H 0 and q 0 and, at the same time, resolve the Hubble tension within the ΛCDM-NG model.

THE EARLY UNIVERSE
Here the term "early universe" is taken to mean the period from the big bang (t = 0) up to and including the time of photon decoupling or last scattering (t ls ≈ 3.7 × 10 5 yr, z ls = 1090, a ls = 9.2 × 10 −4 ).Our conclusion will be: There is essentially no difference between the ΛCDM and ΛCDM-NG models during this period.
On comparing ρΦ , Eq.( 51), with δρ m , Eq.( 48), we see that, although ρΦ is greater than δρ m (because 1 ≥ cosθ), the two quantities are of the same order of magnitude.Therefore, the density contrast δ = δρ m /ρ m in the ΛCDM model is of the same order of magnitude as the condensed fraction F = ρN /ρ m in the ΛCDM-NG model.Estimates of δ at last scattering are of order δ ∼ 10 −3 or smaller (Weinberg 1987), and a condensed fraction of this order is negligible in the prior Friedmann equation (37); which puts this equation into the same form as the Friedmann equation of the ΛCDM model ( 14).
With sources, δρ m and ρ Φ , for the gravitational potentials in the two models being of the same order of magnitude, the two potentials are also of the same magnitude.From Sachs-Wolfe theory, we know the dimensionless potential Φ/c 2 is of the same order of magnitude as the temperature fluctuations of the CMB radiation (Φ/c 2 ∼ ∆T / T ∼ 10 −5 ) (Sachs et al. 1967).Hence, the potential Φ( t(t))/c 2 at early time t in the observed scale factor (34) is negligible, and the observed scale factor a obs (t) differs from ã(t) only by the constant factor 1/ 1 − 2 Φ( t(t 0 ))/c 2 , which has no effect on the Hubble parameter.Consequently, the Hubble parameters in the two models are equal [H obs (t) = H(t)] and, in

CONDENSATION HISTORIES
We shall show, in this section, that the observed parameters H 0 and q 0 can be realized in the ΛCDM-NG model at the same time it resolves the Hubble tension, and that it does so for a variety of condensation histories F(ã), without the need for any fine tuning of parameters.Ideally, the other parameters in the ΛCDM-NG model, namely Ω m , Ω Λ , Ω r , should be obtained from a best fit of the ΛCDM-NG model to observational data, but we shall take these values from the Planck Collaboration results (Planck 2018), because these values are constrained by a variety of independent observations (Perlmutter et at. 1999;Riess 1998;Burles 1998;Cooke et al. 2018;Abbott 1918;Mantz 2014), and cannot deviate much from the Planck values.
We conclude there is a broad range of condensation histories which resolve the Hubble tension in the ΛCDM-NG model, and no fine tuning of F(ã) is required to secure this result.

PHYSICAL INTERPRETATION OF THE ΛCDM-NG MODEL
When a clock is placed near a large mass, whose gravitational potential at the clock is Φ, the time interval d t the clock would have registered in the absence of the mass is shortened (Φ is negative) to the value This is the well understood, and well tested, gravitational time dilation effect.When the average density of condensed masses ρΦ is included in the source term for the gravitational potential, it creates gravitational potential Φ, and the time d t that would have evolved on the standard comoving clocks used by astronomers is shortened to the value (65)

Figure 1 .
Figure 1.Schematic diagram of a density ρm(x) consisting of a harmonic ripple on top of a constant density.The relationships of densities ρm, δρ(x), ρΦ(x), and (1 − F )ρm are shown for this total density.

Figure 3 .
Figure 3. Three different condensation fractions F(ã) are plotted to illustrate the variety of condensation histories which resolve the Hubble tension in the ΛCDM-NG model, and have Hubble constants and deceleration constants in the observed range.One concludes that no fine tuning of the condensation fraction F(ã) is required to obtain the observed cosmological parameters in the ΛCDM-NG model.