# On the gauge invariance of the locally averaged Friedmann Universe and the
Hubble tension

Masanori Tomonaga<EMAIL_ADDRESS>Faculty of Science, Kyoto Sangyo
University, Motoyama, Kamigamo, Kita-ku, Kyoto, 603-8555 Japan  Toshifumi
Futamase<EMAIL_ADDRESS>Faculty of Science, Kyoto Sangyo University,
Motoyama, Kamigamo, Kita-ku, Kyoto, 603-8555 Japan

###### Abstract

The Hubble tension cast a blight on the standard cosmology. As a possible
attitude to the problem, the local variation of the expansion rate in an
inhomogeneous cosmology has been proposed where the spatial averaging over a
finite domain was introduced in order to construct local Friedmann spacetime.
However, it is not clear that the concept of the spatial averaging itself is
gauge invariant or not. Namely the conclusion obtained by the averaging in a
particular gauge is physically meaningful or not. In this paper we address
this question, namely the gauge invariance of the spatial averaging in
cosmology . We show that the answer is positive by studying the spatial
average in the gauge-invariant cosmological perturbation theory.

##  1 Introduction

The Hubble constant $H_{0}$ is one of the most important cosmological
parameter since it characterizes the global properties of our universe. The
standard cosmology is based on the assumption of the homogeneity and isotropy.
Thus the Hubble parameter $H_{0}$ is regarded as a constant over at least the
horizon scale which is also the prediction of the inflationary scenario.
However recent observations suggest a non negligible difference between local
and global (or recent and old) Hubble parameter.[1, 2]
There has been a large number of studies which try to resolve the
discrepancy.[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] We
regard that the difference of the local and global Hubble parameter is real
and be explained by the inhomogeneous distribution of the matter. In fact the
observation of the K-band luminosity density seems to suggest that region with
several hundred Mpc around us has low density with density contrast
$\delta_{K}\sim-0.5$ compared with the globally averaged density.[19]
Furthermore there is some indication that the voids are actually low density
by weak lensing observation. Thus it will be meaningful to pursue the
indication of the cosmological inhomogeneity.

The homogeneous and isotropic universe(here we call Friedmann Universe)
appears as the result of some kind of averaging procedure since the universe
is actually very inhomogeneous. There are various ways to averaging
inhomogeneous universe (such as the light-corn averaging that is directly
related with observational quantities). In this paper, we only consider the
scalar perturbations in the linear order and the spatial averaging. [20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34] Our purpose is not
studying the averaging itself in general inhomogeneous spacetimes, but rather
the gauge dependence of the relationship between locally averaged and globally
averaged spacetime in the linearly perturbed universe using the spatial
averaging. By adopting the spatial averaging defined below, we were able to
derive a locally averaged Friedmann universe and have obtained the following
relation between the locally average Hubble parameter and the globally
averaged Hubble parameter within the framework of the general relativistic
perturbation theory[35]

$H_{D0}=H_{0}\left(1-\frac{1}{3}\langle\delta\rangle_{Dt_{0}}\right)$ (1)

where $H_{D0}$ is the averaged Hubble parameter at the present time $t_{0}$
over a finite domain $D$, and $H_{0}$ is the global, or the horizon scale
Hubble parameter, and $\langle\varDelta\rangle_{Dt_{0}}$ is the present
density contrast average over the domain $D$. However the treatment is carried
out in the comoving synchronous and Newtonian gauge, and there is some
question if the averaging and the result are gauge invariant or not.

In order to answer the question we study the spatial averaging in the
framework of the gauge invariant cosmological perturbation theory, and find
that the averaging has a gauge invariant meaning and thus the results obtained
by the averaging has physically relevant.

##  2 3+1 decomposition of Einstein equations and the spatial averaging

We follow the previous treatment, namely 3 + 1 decomposition of Einstein
equations. We use the following convention:Greek indices $\mu,\nu,\cdots$run
from $0\sim 3$, Latin indices $i,j,\cdots$run from $1\sim 3$, and the speed of
light is unity, $c=1$.

We write the line element as follows.

$ds^{2}=-N^{2}dt^{2}+q_{ij}\left(dx^{i}+N^{i}dt\right)\left(dx^{j}+N^{j}dt\right)$
(2)

Then we obtained the $3+1$ decomposition of the Einstein equations as follows.

$\displaystyle\left(K^{i}_{i}\right)^{2}-K^{i}_{j}K^{j}_{i}+~{}^{(3)}R^{i}_{i}$
$\displaystyle=16\pi G\frac{T_{00}}{N^{2}}$ (3) $\displaystyle
K^{j}_{i|j}-K^{j}_{j|i}$ $\displaystyle=8\pi G\frac{T_{0i}}{N}$ (4)
$\displaystyle\mathcal{L}_{n}K_{ij}=2K_{il}K^{l}_{j}-K^{l}_{l}K_{ij}-~{}^{(3)}R_{ij}$
$\displaystyle+N_{|ij}+8\pi G\left(T_{ij}-\frac{1}{2}q_{ij}T\right)$ (5)

where $\mathcal{L}_{n}$ is Lie derivative, $|$ denotes the three-dimensional
derivative with respect to $g_{ij}$. $~{}^{(3)}R^{i}_{j}$ is the three-
dimensional Ricci tensor and $K^{i}_{j}$ is the extrinsic curvature defined as
follows.

$K_{ij}\equiv\nabla_{i}n_{j}=\frac{1}{2N}\left(\frac{\partial q_{ij}}{\partial
t}-N_{i|j}-N_{j|i}\right)$ (6)

$n_{\mu}(\text{or}~{}n^{\mu})$ is the normal unit vector at t=const.
hypersurface, defined as

$n_{\mu}=-N\nabla_{\mu}t,\quad
n^{\mu}=N^{-1}\left[\left(\partial_{t}\right)^{\mu}-N^{\mu}\right]$ (7)

We consider the dust as our fluid and thus the energy momentum tensor is

$T^{\mu\nu}=\rho u^{\mu}u^{\nu}$ (8)

where $u^{\mu}$ is the four velocity.

We average the above equations (3),(4),(5) over sufficiently large region $D$
where sufficient number of galaxies are included and smaller than the horizon
scale $H$. The averaging of the quantity $Q$ over the region $D$ is defined as
follows.

$\langle Q\rangle_{D}\equiv\frac{1}{V_{D}}\int_{D}\sqrt{\gamma}Qd^{3}x$ (9)

with the volume defined as

$V_{D}\equiv\int_{D}\sqrt{\gamma}d^{3}x$ (10)

Since the observation of the Cosmic Microwave Background strongly suggests
that our universe is homogeneous and isotropic at least over the horizon
scale, thus it is natural to assume that the horizon scale averaging satisfies
$\langle N\rangle_{H}=1,\,\langle N^{i}\rangle_{H}=0$ and $\langle
q_{ij}\rangle_{H}=a^{2}(t)\gamma_{ij}$ where $\gamma_{ij}$ is the 3 metric of
a constant curvature space, as well as
${\langle\rho\rangle}_{H}={\bar{\rho}},\,\langle u^{0}=1\rangle_{H}=1$ and
$\langle u^{i}\rangle_{H}=0$. This give us the usual background geometry.

$ds^{2}=-dt^{2}+a^{2}\gamma_{ij}dx^{i}dx^{j}$ (11)

With this background, the extrinsic and three-dimensional curvature are given
by

$\displaystyle K^{i}_{j}=\frac{\dot{a}}{a}\delta^{i}_{j}$ (12) $\displaystyle
R^{i}_{j}=\frac{2K}{a^{2}}\delta^{i}_{j}$ (13)

where $K$ is a signature of the spatial curvature. As easily confirmed, the
above choices made that equation (4) satisfies trivially and simplify (3) and
(5) as follows.

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}+\frac{K}{a^{2}}=\frac{8\pi
G}{3}{\bar{\rho}}$ (14)
$\displaystyle\dot{{\bar{\rho}}}+3\frac{\dot{a}}{a}{\bar{\rho}}=0$ (15)

In the following we assume $K=0$ for simplicity.

##  3 Guage invariant linear perturbation theory

The universe is well described by a Friedmann universe on the horizon scale as
described above. This geometry should be regarded as the result of some kind
of averaging since the universe is very inhomogeneous in local scales.
Therefore it is natural to expect the averaged Friedmann universe may not be
the same if the averaged region is different.

Although the density contrast indicated from the K band luminosity density is
of the order of $-0.5$, the metric perturbation may not be so large when the
local scale is sufficiently smaller than the horizon scale since we have the
relation between the density contrast and the Newtonian potential as
$\Delta\Phi\sim G{\bar{\rho}}\delta\sim H^{2}_{0}\delta$. Therefore we assume
that the metric fluctuations are small and we use with linear perturbation
theory. Unfortunately this means we restrict the situation where the density
contrast is also small enough.

Then we write the metric and the energy density in the perturbed universe as
follows:

$\displaystyle ds^{2}$
$\displaystyle=-\left(1+2A\right)dt^{2}+a^{2}\left(\delta_{ij}+2E_{,ij}+2F\delta_{ij}\right)\left(dx^{i}+B^{,i}dt\right)\left(dx^{j}+B^{,j}dt\right)$
(16) $\displaystyle\rho$ $\displaystyle=\rho_{b}\left(1+\delta\right)$ (17)

where we consider only scalar perturbation and the scalars $A,E,F$ and $B$ as
well as $\delta$ are small quantities.

In this metric, the proper time can be defined as $d\tau=Ndt$ where
$N^{2}=1+2A$ is lapse function. So the four-velocity $u^{\mu}$ is
$\frac{1}{N}\left(1,v^{i}\right)$ where $v^{i}\equiv dx^{i}/dt$.

Now consider gauge transformations such as $\bar{t}=t+\alpha$ and
$\bar{x}^{i}=x+\beta^{,i}$. Then the perturbed quantities transform as
follows.

$\displaystyle\bar{A}$ $\displaystyle=A-\dot{\alpha}$ (18)
$\displaystyle\bar{B}$ $\displaystyle=B+\frac{\alpha}{a^{2}}-\dot{\beta}$ (19)
$\displaystyle\bar{E}$ $\displaystyle=E-\dot{\beta}$ (20)
$\displaystyle\bar{F}$ $\displaystyle=F-\frac{\dot{a}}{a}\alpha$ (21)
$\displaystyle\bar{\delta}$ $\displaystyle=\delta+3\frac{\dot{a}}{a}\alpha$
(22) $\displaystyle\bar{v}^{i}$ $\displaystyle=v^{i}+\dot{\beta}^{,i}$ (23)

Using the above transformations, the following gauge invariant quantities are
defined as usual.

$\displaystyle\Phi$ $\displaystyle\equiv
A+a^{2}\dot{B}+2a\dot{a}B-a^{2}\ddot{E}-2a\dot{a}\dot{E}$ (24)
$\displaystyle\Psi$ $\displaystyle\equiv F+a\dot{a}B-a\dot{a}\dot{E}$ (25)
$\displaystyle\varDelta$
$\displaystyle\equiv\delta-3a\dot{a}B+3a\dot{a}\dot{E}$ (26) $\displaystyle
V^{i}$ $\displaystyle\equiv v^{i}+\dot{E}^{,i}$ (27)

These represent the gauge-invariant potential, density fluctuations, and
velocity, respectively. Using these quantities, we obtain gauge-invariant
linearized Einstein equations as follows:

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}-2\left(\frac{\dot{a}}{a}\right)^{2}\Phi+2\frac{\dot{a}}{a}\dot{\Psi}-\frac{2\Delta\Psi}{3a^{2}}=\frac{8\pi
G}{3}\rho_{b}\left(1+\varDelta\right)$ (28)
$\displaystyle\frac{\dot{a}}{a}\Delta\Phi-\Delta\dot{\Psi}=-4\pi
G\rho_{b}a^{2}V^{i}_{,i}$ (29) $\displaystyle\left[-3\frac{\ddot{a}}{a}\
+\ddot{\Psi}+6\frac{\dot{a}}{a}\dot{\Psi}-\frac{\dot{a}}{a}\dot{\Phi}-3\left(\frac{\dot{a}}{a}\right)^{2}\Phi-\frac{\Delta\Psi}{a^{2}}\right]\delta^{i}_{j}+\frac{1}{a^{2}}\left(\Phi+\Psi\right)^{,i}_{,j}$
$\displaystyle=4\pi G\rho_{b}\left(1+\varDelta\right)\delta^{i}_{j}$ (30)

From $i\neq j$ of the Eq.(30), we have $\Phi+\Psi=0$. Furthermore, from the
first order of perturbation in equations $(\ref{00inv})$ and $(\ref{ijinv})$,
we can lead to the equation for $\Psi$ as

$\ddot{\Psi}+4\frac{\dot{a}}{a}\dot{\Psi}=0,$ (31)

the growing solution corresponds to $\dot{\Psi}=0$. We only consider the
growing solution henceforth.

The conservation of the energy-momentum tensor
$(\nabla_{\mu}T^{\mu}_{~{}\nu}=0)$ is given in the linearized order as

$\displaystyle\dot{\varDelta}+V^{i}_{,i}=0$ (32)
$\displaystyle\dot{V}_{i}+2\frac{\dot{a}}{a}V_{i}+\frac{\Phi_{,i}}{a^{2}}=0$
(33)

From Eq.(29),(32) and $a=\left(\frac{9}{4}H_{0}^{2}\right)^{1/3}t^{2/3}$, we
obtain

$\Delta\Phi=4\pi G\rho_{b}a^{2}\varDelta$ (34)

Using eq.(32)$\sim$(34), we obtain the second-order differentiation equation
for the gauge-invariant density contrast $\varDelta$.

$\ddot{\varDelta}+2\frac{\dot{a}}{a}\dot{\varDelta}-4\pi G\rho_{b}\varDelta=0$
(35)

##  4 Spatial averaging over local domain

We now consider the local domain large enough for the averaging meaningful but
small enough compared with the horizon scale. For example the domain with
several hundred Mega parsec scale applies. Since we approximate the domain as
a homogeneous and isotropic local universe, namely we describe our local
domain by Friedman equations with appropriately defined scale factor $a_{D}$,
we expect that the (non-relativistic) density $\langle\rho\rangle_{D}$ defined
in region $D$ obeys the following equation.

$\frac{d}{dt}\langle\rho\rangle_{D}+3\frac{\dot{a}_{D}}{a_{D}}\langle\rho\rangle_{D}=0$
(36)

We regard that this equation defines the local Hubble parameter
$H_{D}=\frac{\dot{a}_{D}}{a_{D}}$.

The locally averaged Friedman universe is obtained as the following step.
First we consider the spatial average of equations (28) and (30).

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}+\frac{2\langle\Delta\Phi\rangle_{D}}{3a^{2}}=\frac{8\pi
G}{3}\rho_{b}\left(1+\langle\varDelta\rangle_{D}+2\langle\Phi\rangle_{D}\right)$
(37)
$\displaystyle\frac{\ddot{a}}{a}-\frac{\langle\Delta\Phi\rangle_{D}}{3a^{2}}=-\frac{4\pi
G}{3}\rho_{b}\left(1+\langle\varDelta\rangle_{D}+2\langle\Phi\rangle_{D}\right)$
(38)

From the above equation, we expect the right-hand side would be the density of
the local region $D$. That is, the density $\langle\rho\rangle_{D}$, defined
below, is considered satisfy the equation (36).

$\langle\rho\rangle_{D}\equiv\rho_{b}\left(1+\langle\varDelta\rangle_{D}+2\langle\Phi\rangle_{D}\right)$
(39)

Then by taking the time derivative of the averaged $00$ component Eq.(37)
combined with the averaged equation of the spatial component Eq.(38), we
obtain

$\frac{\dot{a}_{D}}{a_{D}}=\frac{\dot{a}}{a}\left(1-\frac{2a\langle\Delta\Phi\rangle_{D}}{9H_{0}^{2}}\right)=\frac{\dot{a}}{a}\left(1-\frac{1}{3}\langle\varDelta\rangle_{D}\right)$
(40)

where we have used

$\langle\Delta\Phi\rangle_{D}=4\pi
G\rho_{b}a^{2}\langle\varDelta\rangle_{D},\quad\frac{2a}{3H_{0}^{2}}=\frac{1}{4\pi
G\rho_{b}a^{2}}$ (41)

This equation also shows that the potential $\Phi$ is much smaller than the
density contrast $\varDelta$, the result is essentially same with the previous
result. The Friedman equation may be expressed as

$\left(\frac{\dot{a}_{D}}{a_{D}}\right)^{2}+\frac{10\langle\Delta\Phi\rangle_{D}}{9a_{D}^{2}}=\frac{8\pi
G}{3}\langle\rho\rangle_{D}$ (42)

##  5 Conclusion and Discussion

Motivated by the Hubble tension, there have been many studies on the possible
resolutions. One of them is the local variation of the cosmological parameters
due to inhomogeneous matter distribution. We have also studied the
inhomogeneous universe by spatial averaging and obtained an interesting
results on the relation between the local and global Hubble parameters which
might explain the Hubble tension.

However the question of the gauge invariance of the result is not fully
understood. In this paper we address this question. We employ the gauge-
invariant linear cosmological perturbation theory to show that the spatial
averaging is gauge invariant, namely the averaged Friedmann equation and the
relation between the local and global cosmological parameters take the same
form.

It is some interest to develop this treatment to the second order since the
density contrast report by the observation of the K-band luminosity density is
of the order $-0.5$. Although we gave an argument based on the order of
magnitude discussion of the cosmological Poisson equation, it is clearly not
sufficient. Another direction of this study is to consider the possible
interpretation by the inhomogeneity of the observation of m-z relation of Type
Ia supernovae and CMB Power spectrum. We hope to study this possibility in
future.

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