††thanks: Observatorna str., 3, 04058, Kyiv, Ukraine

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# POSSIBLE MODIFICATION TO STANDARD COSMOLOGICAL MODEL TO RESOLVE
TENSION WITH HUBBLE CONSTANT VALUES

S.L. Parnovsky  Astronomical Observatory of Taras Shevchenko Kyiv National
University

###### Abstract

The tensions between the values of Hubble constant obtained from the early and
the late Universe data pose a significant challenge to modern cosmology.
Possible modifications of the flat homogeneous isotropic cosmological
$\Lambda$CDM model are considered, in which the Universe contains dark energy,
cold baryonic matter and dark matter. They are based on general relativity and
satisfy two requirements: (1) the value of the Hubble constant, calculated
from the value of the Hubble parameter at the recombination by the formulas of
flat $\Lambda$CDM model, should be equal to 92% of the one based on low-
reshift observations; (2) deviations from the $\Lambda$CDM model should not
lead to effects that contradict astronomical observations and estimations
obtained thereof. The analysis showed that there are few opportunities for
choice. Either we should consider DM with negative pressure
$-\rho_{dm}c^{2}\ll p_{dm}<0$, which weakly affects the evolution of the
Universe and the observed manifestations of DM, or we should admit the
mechanism of generation of new matter, for example, by a decay of DE.

cosmology, Hubble constant tension, dark matter, dark energy.

##  1 Introduction

A great interest among cosmologists was caused by tensions between the values
of the Hubble constant obtained from the observations of the early and the
late Universe indicated recently in [1]. I simply note that the estimation
$H_{0}=67.4$ km s-1 Mpc-1 obtained from observations in the recombination era
account for about 92% of the average of the estimations based on observations
of not very distant objects $H_{0}=73.3$ km s-1 Mpc-1. The corresponding
difference is at the level of $4\sigma-6\sigma$, which, according to [1],
should be classified as something from a discrepancy or a problem to a crisis.
They are robust to exclusion of any single method, team or source.

It is known that the expansion rate of the Universe is characterized by the
time-dependant Hubble parameter $H$. Its current value is called the Hubble
constant and is denoted by $H_{0}$. The value of the Hubble constant is
calculated from the Hubble parameter measured in some era. This requires
knowledge of both the characteristics of this era, usually its redshift $z$,
and the cosmological model to relate these values.

Estimations of the Hubble constant obtained by different methods are given in
[1]. Most measurements of the Hubble parameter occur at distances, which are
small by cosmological standards. They have small redshifts and these
measurements relate to the late Universe. However, a few measurements relate
to the early Universe, more precisely to the recombination era (redshift
$z=z_{r}\approx 1100$).

First of all, these are CMB data from Planck satellite [2] and data from Dark
Energy Survey Year 1 clustering combined with data on weak lensing, baryon
acoustic oscillations and Big Bang nucleosynthesis [3].

Naturally, the differences could be explained by measurement errors, including
errors in observational data, their processing, or interpretation, or by an
influence of some poorly accounted factors. That would mean cosmology is not
yet an exact science as it pretends to be. For the purpose of this work I
choose to ignore possible issues with observational data and consider them to
be correct. This article demonstrates that the contradiction can be possibly
eliminated by a modification of the standard $\Lambda$CDM cosmological model.
It is clear that deviations from the $\Lambda$CDM model can not be large,
because it is compatible with most observations.

I considered an isotropic homogeneous cosmological model in which the Universe
is filled with dark energy (DE), dark matter (DM), and cold baryonic matter.
The Universe has passed the era of recombination, so the influence of
radiation and ultrarelativistic particles can be neglected. However, this
model must be different from the $\Lambda$CDM model so that it can explain the
8% difference in the Hubble constant at $z=z_{r}$ and $z=0$.

I consider three possible modifications to it. In the first, the dark energy
(DE) with an arbitrary equation of state is used instead of the cosmological
constant $\Lambda$. In the second, the dark matter (DM) is not pressureless,
although its pressure is small in comparison with the energy density. An
example of such matter is the so-called warm dark matter (WDM). A hypothetic
WDM was introduced earlier in astrophysics to solve some problems associated
with clustering on subgalactic scales and formation of halos [4]. WDM is often
mentioned in explanations of a monochromatic signal around 3.5 keV in the
spectrum of X-ray emissions from galaxy clusters like Perseus and Centaurus
observed by XMM-Newton [5, 6]. The current most popular candidates for WDM
particles are sterile neutrinos [7], gravitinos, non-thermally produced WIMPs
and other particles beyond the Standard Model.

The third modification is associated with the possibility of the transition or
decay of DE into matter or vice versa. The question is being investigated as
to whether any of these modifications can explain the Hubble constant tension
without contradicting other astronomical observations. Next step will be to
consider in more detail options that meet this criterion.

##  2 The choice of a cosmological model can affect the values of the Hubble
constant

I consider the flat relativistic isotropic homogeneous cosmological model with
a scale factor $a(z)$ at the interval of redshift $z=a_{0}/a-1$ from the
recombination era ($z=z_{r}$) to the present epoch ($z=0$). I assume that the
Universe consists of dark energy (DE), cold baryonic matter, and dark matter
(DM). An influence of radiation and ultrarelativistic particles can be
neglected. The energy-momentum tensor of each component has a diagonal form
$\mathbf{T}=\mathrm{diag}(\rho c^{2},p,p,p)$ in comoving frame. Note that I
define the pressure of DM and DE just as a component of the corresponding
energy-momentum tensor.

The Hubble parameter change law is described by the first Friedmann equation
[8]

$\begin{array}[]{l}H^{2}=\frac{8\pi G}{3}\rho=\frac{8\pi
G}{3}(\rho_{m}+\rho_{de})\\\
=H_{0}^{2}\left(\frac{\rho_{m}}{\rho_{m0}}\Omega_{m0}+\frac{\rho_{de}}{\rho_{de0}}\Omega_{de0}\right).\end{array}$
(1)

Here $\rho$ and $\Omega$ are the density and the density parameter, the
subscripts m and de denote matter and dark energy, the subscript 0 denotes the
current value of the corresponding quantity, and $G$ is the gravitational
constant. I assume that the Universe contains only matter, both dark and
baryonic, and DE. This formula can also be obtained in the framework of non-
relativistic cosmology [9].

The Hubble constant value is calculated from the Hubble parameter obtained by
processing observational data by eq. (1) for the $\Lambda$CDM model. It
assumes that the dark energy is the pure cosmological constant with constant
density and both baryonic and cold dark matter are pressureless. For this
model I use the standard subscript $\Lambda$ instead of de. Using the
dependences of $\rho_{\Lambda}$=const and $\rho_{m}=\rho_{m0}(1+z)^{3}$ one
get

$H^{2}=H_{0}^{2}\left(\Omega_{\Lambda 0}+(1+z)^{3}\Omega_{m0}\right).$ (2)

According to the Planck satellite observations, the parameters of cosmological
constant and matter density in the modern era are $\Omega_{\Lambda 0}=0.68\pm
0.02$ and $\Omega_{m0}=0.32\pm 0.02$ [2], and their sum is fixed to 1 in the
flat model. Note that these quantities are of the same order.

If the real cosmological model differed from the $\Lambda$CDM one, but we used
(2) instead of (1), we would get not the value $H_{0}$, but rather the product
$A(z)H_{0}$

$\begin{array}[]{l}H=H_{0}\sqrt{\frac{\rho_{m}}{\rho_{m0}}\Omega_{m0}+\frac{\rho_{de}}{\rho_{de0}}\Omega_{de0}}\\\
=A(z)H_{0}\sqrt{\Omega_{\Lambda 0}+(1+z)^{3}\Omega_{m0}}.\end{array}$ (3)

with the factor

$A(z)=\sqrt{\frac{\frac{\rho_{m}}{\rho_{m0}}\Omega_{m0}+\frac{\rho_{de}}{\rho_{de0}}\Omega_{de0}}{\Omega_{\Lambda
0}+(1+z)^{3}\Omega_{m0}}}.$ (4)

If the $\Lambda$CDM model is correct we have $A(z)=1$. If it is not correct
then $A(z)$ is almost equal to 1 for the late Universe but could differ from 1
for the early one. Let’s try to explain the discrepancies in the values of the
Hubble constant using Eq. (4). To explain the results of the article [1] we
need to provide $A(z_{r})\approx 0.92$. It is clear that this means to go
beyond the $\Lambda$CDM-model. We try to look for such modification which
could provide the condition for $A(z_{r})$ (the condition $A(0)=1$ is done
automatically). An additional requirement is small deviations from the model,
especially during the period of existence and evolution of galaxies and stars,
i.e. from the reionization era or so-called ‘‘cosmic dawn’’, $z<z_{g}\approx
11$.

##  3 DE cannot solve the problem

Let’s start with the letter $\Lambda$ in the name of the model and consider
dark energy with variable density instead of the cosmological constant. The
CDM part remains the same. We get $A(z_{r})<1$ if
$\rho_{de}(z_{r})<\rho_{de0}$, but we can not obtain $A(z_{r})=0.92$. Indeed,

$A(z)>\sqrt{\frac{(1+z)^{3}\Omega_{m0}}{\Omega_{\Lambda
0}+(1+z)^{3}\Omega_{m0}}},$ (5)

but this value is less than 1 by no more than $10^{-8}$ at $z\approx 1100$.
This is caused by the value of the $(1+z)^{3}$ factor at large $z$. It’s a
dead end.

##  4 DM with non-zero pressure

Now let’s try to abandon the letters CDM in the model title and leave only the
letters DM. This is not about the effects of electromagnetic radiation or
ultrarelativistic particles and neutrinos. They exist, but their share is so
small that even at $z\sim 1100$ they do not provide the desired value of
$A(z)$. So temporally forget about them and suppose that matter is composed of
ordinary matter and dark matter. We know enough about baryonic matter to be
sure that it can be considered as pressureless one. Its density
$\rho_{b}=\rho_{b0}(1+z)^{3}$.

But we actually know just a bit about the DM. Suppose that it has some
pressure $p_{dm}=w(z)\rho_{dm}c^{2}$ which can affect the evolution of its
density $\rho=\rho_{0}F(z)(1+z)^{3}$. The first law of thermodynamics gives
for matter and DE (see § 2.7.4 in [9])

$\frac{d\rho}{\rho+pc^{-2}}=-\frac{dV}{V}=-3\frac{da}{a}.$ (6)

Here $V\propto a^{3}$ is a volume of part of space expanding with the
Universe, $a=a_{0}/(1+z)$ is the scale factor. This equation and corresponding
EoS specifies the dependencies of densities of all components on $z$. I assume
that each of the components, i.e. DE, baryonic and dark matter expands
adiabatically. This means, in particular, that none of them could decay or
transform into another. It is not difficult to find the relation

$F(z)=\exp\left(3\int_{0}^{z}\frac{w(\xi)d\xi}{\xi+1}\right).$ (7)

If the density of matter $\rho_{m}$ is equal to the sum of the densities of
baryonic $\rho_{b}$ and dark $\rho_{dm}$ matter, then from (4) we obtain the
expression for $A(z)$ at large $z$ when it is possible to neglect the terms
with subscripts de and $\Lambda$

$A(z)=\sqrt{\frac{F(z)\Omega_{dm0}+\Omega_{b0}}{\Omega_{m0}}}.$ (8)

I used the value $\Omega_{b0}=0.16\Omega_{m0}$ based on Planck data [2] to get
the rough estimation $F(z_{r})=0.82$ from the condition $A(z_{r})=0.92$. From
(7) it is seen that this is impossible for $w(z)\geq 0$. That is, the DM must
have a negative pressure on at least some time interval after recombination.
So it cannot be called WDM.

###  4.1 The case of the simplest EoS

At first I consider the DM with the simplest and most popular equation of
state (EoS)

$p_{dm}=w\rho_{dm}c^{2},$ (9)

where the subscript dm means dark matter, $\rho_{dm}$ and $p_{dm}$ are DM
density and pressure, $c$ is the speed of light and $w=\mathrm{const}$. I
calculate the parameter $w$ which could explain the difference in the
estimated values of the Hubble constant obtained from high- and low-redshift
data. From (9) and (6) we obtain the expression $F(z)=(1+z)^{3w}$. One get the
rough estimation $w\approx-0.009$ from $F(z_{r})=0.82$.

It is clear that neither classical particles, nor bosons, nor fermions can
have a negative pressure. So DM in this case should have a completely
different nature than ordinary matter. On the other hand, the introduction of
such a small negative pressure has little effect on the numerous astronomical
manifestations of DM. I mean the rotation curves of galaxies, estimations of
virial masses of galaxy clusters, gravitational lensing, galaxy cluster
mergers like the Bullet Cluster (1E 0657-56) and so on (see Chap. 4 in [9]).
It could affect the results of modelling of large-scale structure formation,
however.

This pressure with $|w|\ll 1$ also weakly affects the evolution of the
Universe. I will show that the changes are insignificant taking the age of the
Universe as an example. It can be found from (1) for the flat model with
cosmological constant, cold baryonic matter, and DM with EoS (9). This age is
equal to

$T=H_{0}^{-1}\int_{0}^{1}\frac{u^{0.5+1.5w}du}{\sqrt{u^{3(1+w)}\Omega_{\Lambda
0}+u^{3w}\Omega_{b0}+\Omega_{dm0}}}.$ (10)

It is easy to calculate that at $w=-0.009$ it is 13.9 billion years or the
100.76% of the age of the Universe for the $\Lambda$CDM model, which is equal
to 13.8 billion years. As one can see, the difference from the case of the
$\Lambda$CDM model is negligible.

###  4.2 More general EoS of DM

Let us consider a more general EoS in the form

$w(z)=B(1+z)^{\alpha},\;B<0,\alpha=const.$ (11)

Consider the $w$ change during evolution. It is easy to estimate the values of
$w(z_{r})$ for (11) with different $\alpha$. At $\alpha\geq 0.1$ we get
$w(z_{r})\approx-0.066\alpha$. At $\alpha<0$ we have $w(z_{r})\approx
0.066\alpha z_{r}^{\alpha}<0.066\alpha$. Thus, for $\alpha<3$, the estimate
gives $-0.2<w(z_{r})<0$. This type of DM significantly distinguishes from DE
because of $|w(z)|\ll 1$.

In the modern era, this parameter is equal to $w(0)=B$. For $\alpha>0.2$
$B\approx-0.066\alpha$, for $\alpha<-0.2$ $B\approx 0.066\alpha$. Let‘s
estimate this parameter for ‘‘cosmic dawn’’. This corresponds to a redshift of
$z_{g}\approx 11$. At $\alpha\geq 0.2$ we get
$w(z_{g})\approx-0.066\alpha\,0.01^{\alpha}$, at $\alpha<0$ we have
$w(z_{r})\approx 0.066\alpha z_{g}^{\alpha}$. All these values are negative,
but close to zero, so the pressure has little effect on the evolution of the
universe and structures in it.

The exception is the very low values of $\alpha\ll-1$, which give a large
value of $B$ in the modern era. At $\alpha\gg 1$ this problem arises in the
recombination era. In the case $\alpha=1$ we have

$\rho_{dm}=\rho_{dm0}(1+z)^{3}\exp(3Bz).$ (12)

Setting $F(z_{r})\approx 0.82$ one can evaluate the combination

$Bz_{r}\approx-0.066.$ (13)

This is the value of $w$ at the time of recombination. At present epoch we get
$w=B\approx-0.066/1100\approx-6\cdot 10^{-5}$ and the matter can be considered
quite cold. The farthest known galaxy has $z\approx 11$. This corresponds to a
value of $w\approx-0.0066$. Thus, galaxies appeared in the era when WDM was
quite cold in the framework of the considered EoS. However, the structure
began to form during the period of warmer dark matter.

Note that all obtained estimations for (9) and (11) lie in the parameter
interval $-1/3<w<0$. Let me remind you that matter with $w>-1/3$ attracts
surrounding bodies, but when $w<-1/3$, it repels them, demonstrating
antigravity. This, in particular, is typical for the cosmological constant
with $w=-1$ and DE with $w\approx-1$. However, astronomical observations show
that DM attracts both ordinary matter and DM itself.

Naturally, we cannot use EoS (11) for $\alpha>0$ and huge $z$ significantly
exceeding $z_{r}$, since we come to the region with $w(z)<-1$. However, no one
considers (11) as a real EoS, but only as a kind of approximation for the
period after the recombination epoch. The evolution of the Universe before
this epoch lies outside the framework of the considered model.

##  5 Decay of DE into matter

Is it possible to provide the condition $F(z_{r})=0.82$ without invoking
negative pressure? Suppose both baryonic and dark matter is presureless.
Nevertheless, the density of matter is now $1:0.82=1.22$ times more than
follows from the equation (6). In order to explain this, one could assume that
its amount has increased from the recombination epoch. Since both types of
matter have the same laws of density decreasing with time, both the decay of
DM into ordinary matter and the transition of baryonic matter into dark matter
cannot affect the total amount of matter. However, during the transition of DE
into matter, an increase in its amount can be obtained without violating the
laws of conservation of energy.

So, one could suggest that the contents of the Universe is presureless, but
its density has been decreasing since recombination more slowly than predicted
by (6) due to the assumption that there is a source of influx of matter, dark
or baryonic, through transition DE into matter. In addition, the process must
be sufficiently intense so that from the moment of recombination the total
amount of matter has increased by approximately 20%.

This possibility is somewhat reminiscent of the now practically forgotten
theory, proposed back in 1948 [10, 11]. In it, matter was constantly born
‘‘out of nothing’’, more precisely from the mysterious C-field (C for
creation), maintaining a constant density. Now we are sure that it is
incorrect and the density of matter is constantly decreasing. However, it can
be assumed that different components of the contents of the Universe can
transform into each other, keeping its flatness. More specifically, DE can
turn into matter.

##  6 Conclusion

Hubble constant tension can be explained in different ways, from errors in
measurements, their processing and interpretation, to the manifestation of
some unknown effects. But if we try to explain it, considering the $H_{0}$
values given in [1] to be correct, and staying within the framework of
cosmological models based on general relativity, in which the Universe
contains DE, DM and cold baryonic matter, then we have few opportunities to
choose from. Either we should consider DM with negative pressure
$-\rho_{dm}c^{2}\ll p_{dm}<0$, which weakly affects the evolution of the
Universe and the observed manifestations of DM, or we should admit the
mechanism of generation of new matter, for example, by a decay of DE. None of
the evolution of the DE density can explain the differences in the estimates
given in [1].

Let me clarify that the mention of general relativity may suggest that the
conclusions of the article are based on this theory. However, to obtain them,
we needed two equations, namely (1) and (6). Each of them can be obtained
within the framework of nonrelativistic cosmology (see [9]). Equation (1) is
obtained from classical mechanics and Newtonian theory of universal
gravitation (more precisely, the limit of general relativity for small
curvature of space-time, generalizing this law), equation (6) from the first
law of thermodynamics. Therefore, it is impossible to modify the homogeneous
isotropic cosmological model by replacing general relativity with another
theory, if this does not change the classical mechanics and thermodynamics.

It seems to me that the solution to the problem is more likely associated with
the revision and refinement of the estimates of $H_{0}$ for the early and
modern Universe, including taking into account the peculiarities of processing
the initial data. A relevant analysis is presented in my later paper [12]. But
in this article I focus on the deviations from the $\Lambda$CDM model and
demonstrate that they theoretically could resolve the differences in Hubble
constant values obtained from high- and low-redshift observations with a
negligible change to the age of the Universe and other parameters. However,
they require the introduction of either a new effect, namely the possibility
of DE transition into matter, or the assumption that DM has negative pressure,
which excludes the possibility that it consists of currently known particles
and even of practically all hypothetical particles considered as candidates
for the role of DM.

This work was supported by the National Research Foundation of Ukraine under
Project No. 2020.02/0073. This paper was published as S.L. Parnovsky, Possible
Modification of the Standard Cosmological Model to Resolve a Tension with
Hubble Constant Values Ukr. J. Phys. Vol. 66 No. 9 P. 739-744 (2021). DOI:
https://doi.org/10.15407/ujpe66.9.739

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Received 09.12.20