# Consequences of using a smooth cosmic distance in a lumpy Universe: I

Obinna Umeh  1Institute of Cosmology & Gravitation, University of Portsmouth,
Portsmouth PO1 3FX, United Kingdom
2Department of Physics, University of the Western Cape, Cape Town 7535, South
Africa
3Department of Mathematics and Applied Mathematics, University of Cape Town,
Rondebosch 7701, South Africa<EMAIL_ADDRESS>
###### Abstract

How do we appropriately fit a model based on an idealised Friedmann-Lemaître
Robertson-Walker spacetime to observations made from a single location in a
lumpy Universe? We address this question for surveys that measure the imprints
of the baryon acoustic oscillation in galaxy distribution and the peak
apparent magnitude of the Type Ia supernova. These observables are related to
the cosmological model through the Alcock-Paczyński parameters and the
distance-redshift relation. Using the corresponding inhomogeneous spacetime
expressions of these as observed data, we perform a parameter inference
assuming that the background Friedmann-Lemaître Robertson-Walker model is the
correct model of the Universe. This process allows us to estimate the best fit
Hubble rate and the deceleration parameter. We find that the inferred Hubble
rate from the monopole of the Alcock-Paczyński parameters is in tension with
the Hubble rate determined using the distance-redshift relation. The latter
gives the best fit Hubble rate for the cosmological expansion. The constraint
on the Hubble rate from the Alcock-Paczyński parameters is contaminated by the
environment. When the environmental contribution is restricted to modes in the
Hubble flow, we find about (9-12)% discrepancy in the Hubble rate. Finally, we
comment on the insufficiency of the method of cosmography in constraining the
deceleration parameter.

##  I Introduction

The distribution of large scale structures in the Universe on ultra-large
scales appear homogeneous and isotropic [1, 2, 3]. On scales of superclusters,
there are inhomogeneities [4]. The Universe seen on scales of clusters appear
very lumpy with sources moving with discernible peculiar velocities within
clusters and clusters interacting gravitationally with each other [5, 6].
Ellis and Stoeger pointed out in [7] that fitting models based on homogeneous
and isotropic Friedmann-Lemaître Robertson-Walker (FLRW) spacetime to
observation of a lumpy Universe leads to a geometry that ignores on average
the details present on small scales. Such a description of physical reality by
the best-fit FLRW model embodies a smoothing scale that is usually not made
explicit. Ellis and Stoeger further argued that any relevant averaging
procedure that yields a best-fit FLRW model must be performed on the past
light-cone of the observer [8, 9]. This is particularly important because
averaging on the hypersurface of constant time could be gauge dependent. The
question of which of the N-possible observers in the Universe sees a
homogeneous and isotropic FLRW spacetime is still unaddressed [9, 10]. It is
important to note that observing isotropic distribution of matter on the past
light cone of an observer does not immediately imply a homogeneous
distribution of matter without a further assumption of the Copernican
principle [11, 12]. This work is about the cosmological fitting problem in the
light of the current Hubble tension.

Hubble tension refers to the discrepancies in the value of the Hubble rate,
$H_{0}$, when late/early time observations are interpreted using the FLRW
spacetime. For example, the SH0ES collaboration constrains $H_{0}$ to be
$H_{0}=73.2\pm 1.3$ ${\rm{km/sec/Mpc}}$ from the intercept of the Hubble
diagram of SNIa with the absolute luminosity calibrated using nearby Cepheid
variables [13], detached eclipsing binary system in the Large Magellanic Cloud
(LMC) [14] and distance to the NGC 4258 [15]. The Carnegie-Chicago Hubble
Program (CCHP) constrains $H_{0}$ to be $H_{0}=69.6\pm 1.9$
${\rm{km/sec/Mpc}}$ using the information contained in the Hubble diagram at
low redshift. Here, the absolute luminosity is calibrated using the Tip of the
Red Giant Branch (TRGB) stars in the Hertzsprung-Russell diagram [16]. The
calibration of the absolute luminosity using different anchors by the two
groups does not seem to be the source of the discrepancy but rather the
ability to synchronise the zero-point of the distance modulus [17]. The Baryon
Acoustic Oscillation (BAO) spectroscopic survey however, constrains a
combination of the Alcock-Paczyński parameters [18] with respect to the BAO
peak at an effective redshift different from zero. The Alcock-Paczyński
parameters are then interpreted in terms of the flat FLRW model to make a
determination of $H_{0}=68.6\pm 1.1$ ${\rm{km/sec/Mpc}}$ [19, 20, 21]. There
are a whole lot of other determinations of $H_{0}$ with values falling into
either within BAO or SH0ES categories, for details see [22].

All these efforts assume that the FLRW spacetime provides the correct
description of the Universe on all scales [19, 13, 23, 16, 21, 24], yet they
return different determinations of $H_{0}$. We study this tension by averaging
over the spacetime independent inhomogeneous generalisation of each of the
observable-redshift relations used in the determination of $H_{0}$. We make
use of the formalism developed by Kristian and Sachs [25] to derive these
equations in the limit of small redshifts. We fit FLRW model at fixed redshift
to the corresponding monopole (past-light cone average) of the generalised
observable-redshift relations. We made use of the $1+3$ covariant
decomposition formalism developed in [26, 27] to decompose the observable-
redshift relations into covariant irreducible units. In addition, we made use
of the observer past-light cone moment decomposition formalism developed in
[28, 29] to extract the monopole in a coordinate independent way. This allows
us to obtain the best-fit Hubble rate and the deceleration parameter, $q_{0}$,
using each of the observable-redshift relations by matching the monopole of
the arbitrary inhomogeneous model to that of the homogeneous and isotropic
FLRW spacetime order by order in redshift.

We find that the difference between the Hubble rate obtained from fitting the
local distance modulus for the SNIa and the Alcock-Paczyński parameters to the
FLRW model is proportional to the square of the shear tensor associated with a
geodesic observer in our local group. In a perturbed FLRW spacetime, this term
is proportional to the variance in the mass density field fluctuation smoothed
at scale $R$. We show that the smoothing scale $R$ corresponds to a scale
where the geometry of the local observer decouples from the large scale
expansion of the Universe(i.e the radius of the zero-velocity surface). We
argue that the smoothing scale must be set to this value in order to avoid the
caustics or the conjugate point at the boundary of our local group [30].
Taking the mass of our local group to be
$M_{\rm{LG}}\sim(10^{11}-10^{12})M_{\odot}$ [31, 32, 33] gives a smoothing
scale of $R\sim(0.8-1.2){\rm{Mpc}}$. The translates to about (9-12)%
discrepancy in the Hubble rate and this is the amount needed to resolve the
supernova absolute magnitude tension as well [34]. In addition, we find that
the deceleration parameter, $q_{0}$ obtained from cosmography is also impacted
by cosmic structures. We find a discrepancy in the determination of $q_{0}$.
The value we find is in agreement with the recent measurement of $q_{0}$ from
Pantheon supernova sample [35]. It is important to note that monopole of the
low redshift Taylor series expansion of the area distance differs
significantly from the full expression [36], therefore, this result may not be
trusted. We argue that it is rather an indication of a break-down of the low
redshift Taylor series expansion since the determination using the full
expression gives a much lower value [37].

The rest of the paper is organised as follows: we describe the underlying
philosophy behind the cosmological fitting problem in section II. This is
followed by a description of the generalised spacetime independent
inhomogeneous models for the area/luminosity distance in sub-section II.1. We
describe how to fit inhomogeneous models of various observables to the FLRW
model in section III. We specialised the discussion to a perturbed FLRW model
in section IV. We discuss the existence of a causal horizon and how to
identify it in section IV.1. We conclude in section V. We provide details on
how the generalised inhomogeneous observable-redshift relations were derived
in Appendix A.

Cosmology: We adopt the following values for the cosmological parameters of
the standard model [23]: the dimensionless Hubble parameter, $h=0.674$, baryon
density parameter, $\Omega_{\rm b}=0.0493$, dark matter density parameter,
$\Omega_{\rm{cdm}}=0.264$, matter density parameter, $\Omega_{\rm
m}=\Omega_{\rm{cdm}}+\Omega_{\rm b}$, spectral index, $n_{\rm s}=0.9608$, and
the amplitude of the primordial perturbation, $A_{\rm s}=2.198\times 10^{-9}$.

##  II Introduction to the cosmological fitting problem

There are two approaches for building a model of the Universe; the pragmatic
and observational approaches. The pragmatic approach makes assumptions about
the geometry of the Universe and then use observations to validate those
assumptions. For example, the standard model of cosmology is built on _a
priori_ assumption that all physical quantities measured by a comoving
observer are spatially homogeneous and isotropic (cosmological principle),
This assumption restricts a set of all possible spacetimes of the Universe to
the FLRW [38]. The current effort in cosmology is mainly directed towards
making a very precise determination of the free parameters of this model [23,
39, 40]. While the alternative observational approach uses observational data
from our past light cone such as apparent luminosities, angular diameters and
number count of sources without assuming the cosmological principle to
construct the geometry of the Universe[41, 42, 43]. The observational approach
encounters difficulties due to the absence of the initial data [44]. However,
it holds a huge promise that when all the issues are resolved, it will not
only provide us with exact geometry of the Universe, it will also allow us to
quantify homogeneity scale if anything like that exists.

The cosmological fitting approach we describe here is an intermediate approach
between these two approaches [45]. It does not assume _a priori_ that the FLRW
spacetime describes the observable Universe accurately at all times and at all
distances, rather it considers it as a fiducial model of the Universe. For
example, we assume that the fiducial model is specified by

$\displaystyle\bar{U}=\left\\{\bar{M},\bar{g}_{ab},\bar{u}^{a},\bar{\rho},\bar{N},\bar{\mu},\bar{a}_{b},\bar{\alpha}_{{}_{{\text{\tiny$\|$}}}},\bar{\alpha}_{\bot}\right\\}\,,$
(1)

where $\bar{M}$ is Riemannian manifold, $\bar{g}_{ab}$ is the metric,
$\bar{u}^{a}$ is the four velocity, $\bar{N}$ is the number count of sources,
$\bar{\mu}$ is the distance-redshift relation (distance modulus), $a_{b}$ is
the intercept of the Hubble diagram, $\bar{\alpha}_{{}_{{\text{\tiny$\|$}}}}$
and $\bar{\alpha}_{\bot}$ are the radial and orthogonal components of the
Alcock and Paczyński parameters [18]. Then, it takes some hints from the
observational approach such that it assumes that there exists a model of the
observed Universe which gives a realistic representation of the Universe
including all inhomogeneities down to some specified length scale
$|\bm{x}_{1}-\bm{x}_{2}|>R$

$\displaystyle
U=\left\\{M,g_{ab},u^{a},\rho,N,\mu,a_{b},\alpha_{{}_{{\text{\tiny$\|$}}}},\alpha_{\bot}\right\\}\,.$
(2)

The task then is to determine a best fit model of the Universe given $U$. Just
like in the observational approach, the most optimal procedure for obtaining
the best fit model is to fit the observed data or observables on the past
light cones $C^{-}(\bar{p})$ and $C^{-}(p)$ of points $\bar{p}$, $p$ in
$\bar{U}$ and $U$ respectively. Since $\bar{U}$ is homogenous and isotropic,
any point is equally likely, for the points in $U$, we smooth over angular
dependence of any observable, $X$, associated with $U$, on the past light cone
of an observer with four velocity $u^{a}$ within a sphere of constant redshift

$\displaystyle\left\langle
X(z,{{\bf\hat{n}}})\right\rangle_{\Omega}=\frac{1}{4\pi}\int\rm{d}^{2}\Omega
X(z,{{\bf\hat{n}}})=\int{\rm{d}}{{\bf\hat{n}}}\,X(z,{{\bf\hat{n}}})\,,$ (3)

where ${{\bf\hat{n}}}$ is line of sight direction. We assume that the redshift
is monotonic for points separated by distance greater than $R$. We compare the
monopole of the observable $X$ obtained by smoothing out anisotropies on the
past null cone $C^{-}(p)$ to the corresponding prediction by the fiducial
model of the same observable. The free parameters of the fiducial model, i.e
$\bar{U}$ are then adjusted to obtain the best-fit value to the angular
average of the observable in the lumpy Universe. In principle, we evaluate the
$\chi^{2}$ of any observable $X$:

$\chi^{2}_{X}=\sum_{i}\left[\frac{\left\langle
X_{i}\right\rangle_{\Omega}-\bar{X}\left(z_{i}|H^{X}_{0},q^{X}_{0}\right)}{\sigma^{X}_{i}}\right]^{2}\,,$
(4)

where $\sigma^{X}_{i}$ is the covariance, in a single parameter, it reduces to
the variance, $H_{0}$ and $q_{0}$ are the Hubble rate and the deceleration
parameter respectively. Then we find the optimum values of $H^{X}_{0}$ and
$q^{X}_{0}$ by minimising the $\chi^{2}_{X}$ with respect to $H^{X}_{0}$ and
$q^{X}_{0}$

$\displaystyle\frac{\partial\chi_{X}^{2}}{\partial
H^{X}_{0}}=0\,,\qquad\qquad{\rm{and}}\qquad\qquad\frac{\partial\chi^{2}_{X}}{\partial
q^{X}_{0}}=0\,.$ (5)

In order to build a complete picture of the best-fit model, the ideal thing we
will be to obtain the optimum values of $H_{0}$ and $q_{0}$ by minimaxing the
joint
$\chi^{2}=\chi^{2}_{\mu}+\chi^{2}_{a_{b}}+\chi^{2}_{\alpha_{{}_{{\text{\tiny$\|$}}}}}+\chi^{2}_{\alpha_{\bot}}+\cdots$
with respect to $H_{0}$ and $q_{0}$ simultaneously. We do not consider this
here because our focus is to identify the source of tension in the inferred
Hubble rate. This is best done by independently fitting the late and early
Universe observables to their corresponding FLRW space models.

This intermediate approach offers tremendous advantages over the other two
approaches, such as;

  * •

Clear description of the relevant physics: It makes apparent the geometrical
and physical interpretation of the FLRW models we use because it provides a
clear link between the highly symmetric fiducial model and more realistic
descriptions of the lumpy Universe we observe.

  * •

Delineation of scales: It helps to establish the appropriate scale of validity
of the best-fit model of the Universe. That is it helps to address statement
that the Universe can be regarded as almost FLRW Universe if averaged out over
a specified length scale.

  * •

Guidance in the presence of tension: It does not only enable one to determine
the best-fit FLRW Universe model, it allows to predict the presence of tension
in the model parameters. For example, under this framework tensions appear in
the model parameters when the fiducial model is fit to data from length scales
or time in the evolution of the Universe where the background FLRW spacetime
is not applicable(i.e nonlinear scales). We discuss this point in greater
detail in the subsequent sections.

  * •

Wider application: It is possible to use it repeatedly; that is, to consider
which of the lumpy Universe models $U$ and another lumpiness model, say
$U^{\prime}$ give an even better description of the real Universe than $U$,
i.e which one of them describes the inhomogeneities at an even more detail.

###  II.1 Inhomogeneous distances in a realistic model of the Universe

We work in the geometric optics limit i.e we assume that the wavelength of
photon is small compared to the radius of curvature of the Universe. In this
limit, null geodesics describes photon propagation and a tangent vector to the
null geodesic is given by $k^{a}={\rm{d}{x}^{a}}/{\rm{d}\lambda}$ and it
satisfies

${k}_{a}{k}^{a}=0\,\qquad{\rm{and}}\qquad{k}^{b}{\nabla}_{b}{k}^{a}=0\,.$ (6)

where $\lambda$ is the affine parameter. When initialised at the observer
position, it increases monotonically [46]. $\nabla$ is the covariant
derivative of the physical spacetime of the Universe. With respect to a set of
fundamental comoving observers with the average 4-velocity $u^{a}$, we can
decompose ${k}^{a}$ into parallel and orthogonal components:

$\displaystyle{k}^{a}=(-{u}_{b}{k}^{b})\left({u}^{a}-{n}^{a}\right)={E}\left({u}^{a}-{n}^{a}\right),$
(7)

where $n^{a}$ is the Line of Sight (LoS) spatial direction vector with a
normalization $n_{a}n^{a}=1,$ $E=-{u}_{b}{k}^{b}$ is the photon energy
measured by $u^{a}$. $n^{a}$ is orthogonal to $u^{a}$; $n_{a}u^{a}=0$. Note
that our choice of $n^{a}$ is opposite to the direction of photon propagation.
$k^{a}$ is pointing along an incoming light ray on the past light cone of the
observer. The apparent magnitude of any source, $m$, is related to the
observed flux density $F_{d_{L}}$ measurement at a given luminosity distance
$d_{L}$ in a given spectral filter according to

$\displaystyle m=-2.5\log_{10}\left[F_{d_{L}}\right]\,.$ (8)

The absolute magnitude, $M$, of the same source is defined as the apparent
magnitude measured at a distance $D_{F}$

$\displaystyle M=-2.5\log_{10}\left[{F_{D_{F}}}\right]\,,$ (9)

where $F_{D_{F}}$ is called the reference flux or the zero-point of the filter
[47, 48, 49]. The observed flux density at both distances obey the inverse
square law with the luminosity distance, $d_{L}$:
${F_{d_{L}}}/{F_{D_{F}}}=\left[{D_{F}}/{d_{L}}\right]^{2}.$ The distance
modulus is defined as the difference between $m$ and $M$

$\displaystyle
m-M=-2.5\log\left[\frac{F_{d_{L}}}{F_{D_{F}}}\right]=5\log\left[\frac{d_{L}}{D_{F}}\right]=5\log\left[\frac{d_{L}}{[\rm{pc}]}\right]-5\,.$
(10)

In the last equality, we set $D_{F}=10\,{\rm{pc}}$ for historical reason. In
this case, it means that the absolute magnitude is the apparent magnitude if
the telescope is placed at a distance of 10 pc. In cosmology however, $D_{F}$
is set to $D_{F}=1\,{\rm{Mpc}}$ leading to

$\displaystyle{\mu}(z,{{\bf\hat{n}}})$ $\displaystyle=$ $\displaystyle
m(z,{{\bf\hat{n}}})-M=5\log\left[\frac{d_{L}}{[\rm{Mpc}]}\right]+25\,,$ (11)

where ${d}_{L}$ is in the units of ${\rm{Mpc}}$. This implies that for
cosmological purposes, the consistently calibrated cosmic distance ladder
would have a reference flux determined at $1{\rm{Mpc}}$. The generalised
coordinate independent expressions for the area and the luminosity distance at
low redshift are (see appendix A for details of the derivation) [25, 50]

$\displaystyle d_{A}(z,{{\bf\hat{n}}})$ $\displaystyle=$
$\displaystyle\frac{z}{[K^{c}K^{d}\nabla_{c}u_{d}]_{o}}-\frac{1}{2}\left[\frac{K^{c}K^{d}K^{e}\nabla_{e}\nabla_{d}u_{c}}{\left(K^{c}K^{d}\nabla_{c}u_{d}\right)^{3}}\right]_{o}z^{2}+\mathcal{O}(z)^{3}\,,$
(12) $\displaystyle d_{L}(z,{{\bf\hat{n}}})$ $\displaystyle=$
$\displaystyle\frac{z}{[K^{c}K^{d}\nabla_{d}u_{c}]_{o}}\left\\{1+\frac{1}{2}\left[4-\frac{K^{c}K^{d}K^{e}\nabla_{e}\nabla_{d}u_{c}}{\left(K^{c}K^{d}\nabla_{c}u_{d}\right)^{2}}\right]_{o}z+\mathcal{O}(z^{2})\right\\},$
(13)

where $K^{a}$ is a normalized null vector: $K^{a}=u^{a}-n^{a}.$ The associated
null geodesic tangent vector is given by $k^{a}=(1+z)K^{a}$. In general,
$d_{L}$ is related to $d_{A}$ via distance duality relation:
$d_{L}=(1+z)^{2}d_{A}$ [51, 52, 53]. We neglect the effect of the heliocentric
peculiar velocity. Its contribution will not substantially change the the
determination of the Hubble rate, expecially via the local distance ladder
[54]. The redshift in equations (12) and (13) corresponds to the cosmological
redshifts.

On the FLRW background space, equation (12) and (13) reduce to

$\displaystyle\bar{d}_{A}(z)$ $\displaystyle=$
$\displaystyle\frac{z}{H_{0}}\left[1-\frac{1}{2}\left(3+q_{0}\right)z+\mathcal{O}(z)^{2}\right]\,,$
(14) $\displaystyle\bar{d}_{L}(z)$ $\displaystyle=$ $\displaystyle{\;z\over
H_{0}}\left[1+{1\over 2}\left[1-q_{0}\right]{z}+\mathcal{O}(z)^{2}\right]\,.$
(15)

The magnitude-redshift relation is given by equation (11) with the luminosity
distance given by equation (15). We have truncated the Taylor series expansion
at second order in redshift expansion because our focus is on how the cosmic
structures impact the measurement of the Hubble rate, $H_{0}$ and the
deceleration parameters, $q_{0}$. It is certainly debatable whether these
series expansions on arbitrary spacetime have much relation with the
magnitude-redshift relation in the observable Universe. In particular, is it
analytic, especially in the region where shell crossing occurs? These doubts
are valid, but, this is how we address them:

  * •

Differentiability at $z=0$: We showed in [36] that $d_{A}$ expanded up to
second order in standard cosmological perturbation theory on an FLRW
background spacetime is differentiable at $z=0$. The linearly perturbed FLRW
equations can be re-written in terms of the equations of the exact
inhomogeneous Szekeres models [55]. This shows that the generalised
inhomogeneous model is also differentiable at $z=0$.

  * •

Shell crossing singularity: It is likely that in certain directions, the
Taylor series expansion of observables will be multivalued as a light beam
passes through collapsing regions, this is a valid concern. We show in section
IV.1, that collapsing region around the observer can be isolated from the
expanding spacetime by setting the smoothing scale at the zero-velocity
surface [36]. Similar restriction was deployed recently in an attempt to
estimate equation (13) from a general relativistic N-body simulation [56].

  * •

Convergences of the series expansion: Convergence of series expansion is a
problem in general, in our case, it is well-known that for $z<0.1$, equations
(14) and (15) converge at second order in redshift expansion. It is likely
that in the presence of structures, higher-order redshift correction will be
needed to achieve convergence at the same redshift. We find some hints of this
in [36] but a more detailed study is needed since next to leading order terms
were neglected in the analysis. Currently, we are essentially interested in
what happens in the limit where $z\ll 1$ and we have no reason to believe that
second-order series expansion in redshift will not be enough to determine the
Hubble rate.

###  II.2 Covariant and multipole moment decomposition of observables

We are primarily interested in the multipole moment decomposition of the
observable-redshift relations so that the monopole or the all-sky average can
be compared to the corresponding expression based on the background FLRW
spacetime. To accomplish this, we decompose $K^{a}K^{b}\nabla_{a}u_{b}$ and
$K^{a}K^{b}K^{c}\nabla_{a}\nabla_{b}u_{c}$ into irreducible physical
quantities that live on the hypersurface orthogonal to $u^{a}$ using in 1+3
covariant decomposition formalism [26, 57, 52, 58, 27, 59]. The irreducible
decomposition of the spacetime covariant derivative of $u^{a}$ for geodesic
observers is given by [27, 59]

$\displaystyle\nabla_{b}u_{a}=\frac{1}{3}\Theta h_{ab}+\sigma_{ab}\,,$ (16)

where $\Theta$ denotes the expansion ($\Theta>0$)/contraction $(\Theta<0)$ of
the nearby geodesics associated with $u^{a}$, $\sigma_{ab}$ is the shear
tensor, it describes the rate of change of the deformation of spacetime in the
neighbourhood of the observer, $h_{ab}=g_{ab}+u_{a}u_{b}$ is the metric on the
hypersurface orthogonal to the time-like $u^{a}$ ($u^{a}u_{a}=-1$) and
$g_{ab}$ is the physical spacetime metric. The vorticity vanishes for
irrotational fluids or geodesic observer. There is a $1$-to-$1$ mapping
between all symmetric trace-free tensors of rank $\ell$ and the spherical
harmonics of order $\ell$ [28, 60, 61, 62]. We can see this by decomposing the
LoS direction vector $n^{a}$ at the observer position on an orthonormal tetrad
basis [61, 63]

$n^{a}(\theta,\phi)=\left(0,\sin\theta\sin\phi,\sin\theta\cos\phi,\cos\theta\right)\,.$
(17)

In the standard spherical harmonics decomposition formalism, any function $f$
on the sky may be expanded in spherical harmonics, $Y^{\ell
m}({{\bf\hat{n}}})$ according to

$\displaystyle f({{\bf\hat{n}}})=\sum_{\ell=0}^{L}\sum_{m=-\ell}^{\ell}F^{\ell
m}Y^{\ell
m}({{\bf\hat{n}}})=\sum_{\ell=0}^{\infty}F_{A_{\ell}}n^{A_{\ell}}=F+F_{a}n^{a}+F_{ab}n^{a}n^{b}+F_{abc}n^{a}n^{b}n^{c}\,.$
(18)

where $F^{\ell m}$ is the spherical harmonic coefficients.
$A_{\ell}=a_{1}a_{2}...a_{\ell}$ is a compound index notation that denotes
number of indices, $F_{A_{\ell}}$ is the moments, it is symmetric, trace-free
and orthogonal to $u^{a}$, $F_{A_{\ell}}=F_{(A_{\ell})}\,,$
$~{}F_{A_{\ell}ab}h^{ab}=0\,,$ and $F_{A_{\ell}a}u^{a}=0\,$ respectively.
Here, $``(\cdots)"$ brackets in the subscript denote the symmetrisation
overall indices and the angle brackets “$\langle\cdots\rangle$” denotes PSTF
part. In the second equality, we show an equivalent way of performing the
spherical harmonic expansion in Projected Symmetric Trace-Free (PSTF) tensors
in $n^{a}$basis [28, 60, 61, 62]. The PSTF part of any index-tensor is given
by [28, 64, 65, 66, 67],

$\displaystyle{F_{\langle
A_{\ell}\rangle}}=\sum_{n=0}^{[\ell/2]}\mathcal{B}^{\ell
n}h_{(a_{1}a_{2}}....h_{a_{2n-1}a_{2n}}F_{a_{2n+1}...a_{\ell})}\,,\quad{\rm{with}}\quad\mathcal{B}_{\ell
n}={\frac{{(-1)^{n}\ell!(2\ell-2n-1)!!}}{{(\ell-2n)!(2\ell-1)!!(2n)!!}}}\,.$
(19)

$[\ell/2]$ means the largest integer part less than or equal to $\ell/2$,
$\ell!=\ell(\ell-1)(\ell-2)(\ell-3)...(1)$, and
$\ell!!=\ell(\ell-2)(\ell-4)(\ell-6)...(2~{}~{}or~{}~{}1)$. The moment and
covariant decomposition of $K^{a}K^{b}\nabla_{a}u_{b}$ and
$K^{a}K^{b}K^{c}\nabla_{a}\nabla_{b}u_{c}$ within General Relativity for dust
is given by [9, 68, 69]

$\displaystyle K^{a}K^{b}\nabla_{a}u_{b}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\Theta+\sigma_{ab}n^{a}n^{b}\,,$ (20) $\displaystyle
K^{a}K^{b}K^{c}\nabla_{a}\nabla_{b}u_{c}$ $\displaystyle=$
$\displaystyle\mathcal{O}+\mathcal{O}_{a}n^{a}+\mathcal{O}_{ab}n^{a}n^{b}+\mathcal{O}_{abc}n^{a}n^{b}n^{c}$
(21)

where $\mathcal{O}$, $\mathcal{O}_{a}$, $\mathcal{O}_{ab}$ and
$\mathcal{O}_{abc}$ are the monopole, dipole, quadrupole and octupole moments
of $K^{a}K^{b}K^{c}\nabla_{a}\nabla_{b}u_{c}$

$\displaystyle\mathcal{O}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\rho+\frac{1}{3}\Theta^{2}-\frac{1}{3}\Lambda+\sigma_{ab}\sigma^{ab}\,,$
(22) $\displaystyle\mathcal{O}_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{3}{\tilde{\rm{D}}}_{a}\Theta+\frac{2}{5}{\tilde{\rm{D}}}_{b}{\sigma}_{a}^{b}\,,$
(23) $\displaystyle\mathcal{O}_{ab}$ $\displaystyle=$ $\displaystyle
E_{ab}+2\Theta\sigma_{ab}+3{\sigma^{c}}_{a}\sigma_{bc}\,,$ (24)
$\displaystyle\mathcal{O}_{abc}$ $\displaystyle=$
$\displaystyle{\tilde{\rm{D}}}_{a}\sigma_{bc}\,.$ (25)

Here $\rho$ is the matter density, $\Lambda$ is the cosmological constant,
$E_{ab}$ is the Electric part of the Weyl tensor and ${\tilde{\rm{D}}}_{a}$ is
the spatial derivative on the hypersurface. $E_{ab}$ describes how nearby
geodesics tear apart from the equation. It is crucial in the discussion that
follows. An extension to modified gravity theories and general matter fields
should be straightforward.

##  III The cosmological fitting problem

In this section, we consider how surveys that measure the imprints of the
Baryon Acoustic Oscillation (BAO) in galaxy distribution and the apparent
magnitude of the type Ia supernova are compared to a cosmological model
through the Alcock-Paczyński parameters, the distance-redshift relation. Our
key focus will be to estimate how inhomogeneity affects the determination of
the Hubble rate and the deceleration parameter.

###  III.1 Cosmological fitting problem: Alcock-Paczyński parameters

The BAO scale constitutes an important link between the physics, before and
around the drag and present epoch. Similar to the Cosmic Microwave Background
radiation, CMB, the plasma physics of acoustic density waves before decoupling
imprints a characteristic scale in the matter distribution. The characteristic
scale, i.e BAO scale appears as an excess in the probability of finding
galaxies separated by some distance in the distribution of sources seen today
on the sky. Since a fiducial model for the separation is required for
calculating the separation between galaxies, Alcock and Paczyński [18]
introduced a clever way to recover the true separation. For a spherical
symmetric distribution of sources, the two-point correlation function,
$\xi_{g}$, is given by [70]

$\displaystyle\xi_{g}(r_{12_{{\text{\tiny$\|$}}}},r_{12\bot})=\frac{1}{\alpha_{{}_{{\text{\tiny$\|$}}}}\alpha_{\bot}^{2}}\xi_{g}(\alpha_{{}_{{\text{\tiny$\|$}}}}r^{\rm{fid}}_{12_{{\text{\tiny$\|$}}}},\alpha_{\bot}{\bm{r}}^{\rm{fid}}_{12\bot})\,,$
(26)

where $r_{12}$ and $r^{\rm{fid}}_{12}$ are the true and fiducial separation
between two galaxies respectively. Within the cosmological standard model
treatment, the difference between the fiducial model and true model are
parameterised by

$\displaystyle\bar{\alpha}_{{}_{{\text{\tiny$\|$}}}}=\frac{\bar{H}^{\rm{fid}}}{\bar{H}}\,,\qquad\qquad\bar{\alpha}_{\bot}=\frac{\bar{d}_{A}}{\bar{d}^{\rm{fid}}_{A}}\,.$
(27)

Here $\bar{H}^{\rm{fid}}$ and $\bar{d}^{\rm{fid}}_{A}$ are the fiducial Hubble
rate and area distance respectively. $\bar{H}$ and $\bar{d}_{A}$ are then
adjusted to obtain the best-fit to the observed data. BAO is sensitive to the
Hubble rate through the distortions ($\alpha_{{}_{{\text{\tiny$\|$}}}}$ and
$\alpha_{\bot}$) at the survey mean redshift $z$. The determination of the
value of the Hubble rate today is made assuming a model. The measurement of
the monopole of equation (26) constrains
$\alpha=\alpha_{\bot}^{{2}/{3}}\alpha{{}_{{\text{\tiny$\|$}}}}^{{1}/{3}}$ [19,
71], while the quadrupole moment is most sensitive to the Alcock-Paczyński
ratio $\epsilon=\alpha_{\bot}/\alpha_{{}_{{\text{\tiny$\|$}}}}$ [70]. Using
the void-galaxy cross-correlation from the BOSS survey at the mean redshift of
$z=0.57$, [70] finds a strong constraint on
$\alpha_{\bot}/\alpha{{}_{{\text{\tiny$\|$}}}}=1.016\pm 0.011$ using the CMASS
galaxy sample of the BOSS DR12 data release. This constraint when interpreted
in within the $\Lambda$CDM model(i.e equation (27) with the sound horizon,
$r_{s}$, set to the Planck value) is consistent with the Planck
collaboration’s determination of $H_{0}$ from the analysis of the anisotropies
in the cosmic microwave background (CMB) [23]:

$\displaystyle H_{0}=67.4\pm 0.5\,\,{\rm{km/sec/Mpc}}\,.$ (28)

In a lumpy Universe, the structure of equation (26) will change slightly,
however, it holds for spherical symmetric distribution of sources. We will
discuss modifications away from this limit elsewhere. For an almost FLRW
spacetime for which we are interested here, equation (27) generalises as
follows:

$\displaystyle\alpha_{{}_{{\text{\tiny$\|$}}}}$ $\displaystyle=$
$\displaystyle\frac{\partial r_{{}_{{\text{\tiny$\|$}}}}}{\partial
r^{\rm{fid}}_{{}_{{\text{\tiny$\|$}}}}}=\frac{\partial
r_{{}_{{\text{\tiny$\|$}}}}}{\partial z}\frac{\partial z}{\partial
r^{\rm{fid}}_{{}_{{\text{\tiny$\|$}}}}}\simeq\frac{\rm{d}\lambda}{\rm{d}z}\frac{\rm{d}z}{\rm{d}\lambda^{\rm{fid}}}\,,$
(29) $\displaystyle=$
$\displaystyle\frac{{\bar{H}}^{\rm{fid}}}{{H}}\left[1+\frac{\sigma_{ab}}{{H}}{n}^{a}n^{b}\right]^{-1}\,,$
(30) $\displaystyle=$
$\displaystyle\frac{\bar{H}^{\rm{fid}}}{{H}}\bigg{[}1-\frac{\sigma_{ab}}{{H}}n^{a}n^{b}+\frac{\sigma_{ab}}{{H}}\frac{\sigma_{cd}}{{H}}n^{a}n^{b}n^{c}n^{d}+\mathcal{O}(\sigma^{4})\bigg{]}\,,$
(31)

where, $H=\Theta/3$, in the last equality, This is the Hubble rate associated
with the expansion of the nearby congruence in a lumpy Universe. Moreover, we
assumed that the real Universe is almost statistical homogeneous and
isotropic, i.e ${\sigma_{ab}}n^{a}n^{b}/{{H}}\ll 1$. To move from the first
line of equation (29) to the second, we made use of the propagation equation
for the redshift [72]

$\displaystyle\frac{\rm{d}z}{\rm{d}\lambda}$ $\displaystyle=$
$\displaystyle-(1+z)^{2}\left[{H}+\sigma_{ab}n^{a}n^{b}\right]\,,\qquad\qquad\frac{\rm{d}z}{\rm{d}\lambda^{\rm{fid}}}=-(1+z)^{2}\bar{H}^{\rm{fid}}\,.$
(32)

Note that on the fiducial FLRW spacetime, the shear vanishes $\sigma_{ab}=0$.
At low redshift, the generalised counter-part to $\bar{\alpha}_{\bot}$
involves replacing the background area distance with equation (12).

To obtain the best-fit model from the radial component of the separation in
the limit $z\to 0$, we compare $\bar{\alpha}_{{}_{{\text{\tiny$\|$}}}}$ to
all-sky average of $\alpha_{{}_{{\text{\tiny$\|$}}}}$

$\displaystyle\bar{\alpha}{{}_{{\text{\tiny$\|$}}}}=\left\langle\alpha_{{}_{{\text{\tiny$\|$}}}}\right\rangle_{\Omega}$
$\displaystyle=$
$\displaystyle\frac{\bar{H}^{\rm{fid}}}{{H}}\bigg{[}1+\left\langle\frac{\sigma_{ab}}{{H}}\frac{\sigma_{cd}}{{H}}n^{a}n^{b}n^{c}n^{d}\right\rangle_{\Omega}+\mathcal{O}(\sigma^{4})\bigg{]}\,,$
(33) $\displaystyle=$
$\displaystyle\frac{\bar{H}^{\rm{fid}}}{{H}}\bigg{[}1+\frac{2}{15}\frac{\sigma_{ab}\sigma^{ab}}{H^{2}}+\mathcal{O}(\sigma^{4})\bigg{]}\,.$
(34)

To perform the all-sky average we decompose $\sigma_{(ab}\sigma_{cd)}$ into
irreducible unit and then extract the monopole using

$\displaystyle
n^{a}n^{b}n^{c}n^{d}\sigma_{(ab}\sigma_{cd)}=n^{a}n^{b}n^{c}n^{d}\sigma_{\langle
ab}\sigma_{cd\rangle}+\frac{4}{7}n^{a}n^{b}\sigma^{c}_{\langle
a}\sigma_{b\rangle c}+\frac{2}{15}\sigma_{ab}\sigma^{ab}.$ (35)

This can also be done by performing the angular integration in equation (33)
right away [29]. Comparing equation (27) to equation (34), we obtain the
effective Hubble rate

$\displaystyle
H^{\alpha{{}_{{\text{\tiny$\|$}}}}}={H}\bigg{[}1-\frac{2}{15}\frac{\sigma_{ab}\sigma^{ab}}{H^{2}}+\mathcal{O}(\sigma^{4})\bigg{]}\,.$
(36)

The Hubble rate inferred from the constraint on
$\alpha_{{}_{{\text{\tiny$\|$}}}}$ is biased by
$-{2}\sigma_{ab}\sigma^{ab}/{15H^{2}}$. For the orthogonal component,
$\alpha_{\bot}$, we focus on $d_{A}$, since the fiducial model is the same, it
reduces to comparing equation (14) to the all-sky average of equation (12) at
a fixed redshift. At the leading order in redshift, the effective Hubble rate
from fitting the monopole of area distance in a lumpy Universe to the FLRW
spacetime is given by

$\displaystyle\frac{1}{H_{0}^{{d}_{A}}}$ $\displaystyle=$
$\displaystyle\left\langle\frac{1}{{H}_{0}}\left(1+\frac{\sigma_{ab}}{{H}_{0}}n^{a}n^{b}\right)^{-1}\bigg{|}_{z=z_{o}}\right\rangle_{\Omega}=\frac{1}{{H}_{0}}\bigg{[}1-\frac{\sigma_{\langle
ab\rangle}}{{H}_{0}}n^{a}n^{b}+\frac{\sigma_{(ab}}{{H}_{0}}\frac{\sigma_{cd)}}{{H}_{0}}n^{a}n^{b}n^{c}n^{d}+\mathcal{O}(\sigma^{4})\bigg{]}\,,$
(37)

where $\Theta_{o}=3{H}_{0}$ is the monopole component of
$K^{a}K^{b}\nabla_{a}u_{b}$. Again, we expanded
$\left(K^{a}K^{b}\nabla_{a}u_{b}\right)^{-1}$ up to second order in
$\sigma_{ab}n^{a}n^{b}/H_{0}\ll 1$. Using equation (35) again and noting that
$\sigma^{a}{}_{a}=0$, we find

$\displaystyle\frac{1}{H_{0}^{d_{A}}}\equiv\frac{1}{{H}_{0}}\left[1+\frac{2}{15}\frac{\sigma_{ab}\sigma^{ab}}{{H}^{2}}\bigg{|}_{z_{o}}+\mathcal{O}(\sigma^{4})\right]\,.$
(38)

The Hubble rate inferred from the area distance by fitting the area distance
information from the lumpy Universe to an FLRW background spacetime is given
by

$\displaystyle{H_{0}^{d_{A}}}={H}_{0}\left[1-\frac{2}{15}\frac{\sigma_{ab}\sigma^{ab}}{{H}^{2}}\bigg{|}_{z_{o}}+\mathcal{O}(\tilde{\sigma}^{4})\right]\,.$
(39)

Furthermore, the evolution of $\sigma_{ab}$ is sourced by the electric part of
the Weyl tensor, ${E}_{ab}$ [59, 27]

$\displaystyle\frac{{\rm{D}}{\sigma}_{ab}}{\rm{D}\tau}$ $\displaystyle=$
$\displaystyle-\frac{2}{3}\Theta{\sigma}_{ab}-{\sigma}^{c}{}_{\langle
a}{\sigma}_{b\rangle c}-{E}_{ab}\,.$ (40)

Again, ${E}_{ab}$ contains information about the tidal forces due to gravity,
it represents how nearby geodesics tear apart from each other [59]. In an
almost FLRW spacetime (i.e small perturbation on top of an FLRW spacetime),
${\sigma}_{ij}$ is related to $E_{ij}$ in comoving spacetime

$\displaystyle{\sigma}_{ij}(\eta,{\bm{x}})$ $\displaystyle\sim$
$\displaystyle-\int_{\eta_{\rm{ini}}}^{\eta}{d\eta^{\prime}}{E}_{ij}(\eta^{\prime},{\bm{x}}^{\prime})\,.$
(41)

In this form, it is straightforward to see that $\sigma_{ij}$ describes the
cumulative effect of the local geometry deformation from the initial seed time
to the time of observation. It also helps to understand why $\sigma_{ij}$
characterises the cosmic web more transparently than $E_{ij}$ [73, 74].

For the deceleration parameter today, we compare equation (14) to the monopole
of equation (12) at the second order in redshift

$\displaystyle\frac{\left(3+q^{d_{A}}_{0}\right)}{H^{d_{A}}_{0}}$
$\displaystyle=$
$\displaystyle\left\langle\frac{K^{a}K^{b}K^{c}\nabla_{a}\nabla_{b}u_{c}}{(K^{a}K^{b}\nabla_{a}u_{b})^{3}}\right\rangle_{\Omega}\,.$
(42)

Decomposing the denominator in terms of the expansion and shear tensor and
expressing the result into irreducible unit, we found

$\displaystyle{\left(K^{a}K^{b}\nabla_{a}u_{b}|_{0}\right)^{3}}$
$\displaystyle=$
$\displaystyle\frac{1}{27}\Theta^{3}+\frac{1}{3}\Theta^{2}\sigma_{ab}+\sigma_{ab}\sigma_{cd}\Theta
n^{a}n^{b}n^{c}n^{d}+\mathcal{O}(\sigma^{3})\,,$ (43) $\displaystyle=$
$\displaystyle\mathcal{H}^{(3)}\,+\mathcal{H}^{(3)}\,_{ab}n^{a}n^{b}+\mathcal{H}^{(3)}\,_{abcd}n^{a}n^{b}n^{c}n^{d}+\mathcal{O}(\sigma^{3})\,.$
(44)

In the second equality, we have introduced the first few multipole moments as

$\displaystyle\mathcal{H}^{(3)}\,$ $\displaystyle=$ $\displaystyle
H^{3}+\frac{2}{15}{\Theta\sigma_{ab}\sigma^{ab}}=H^{3}\left(1+\frac{6}{15}\frac{\sigma_{ab}\sigma^{ab}}{H^{2}}\right)\,,$
(45) $\displaystyle\mathcal{H}^{(3)}\,_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{3}{\Theta^{2}\sigma_{ab}}+\frac{4}{7}{\sigma_{\langle
a}^{c}\sigma_{b\rangle c}\Theta}\,,$ (46)
$\displaystyle\mathcal{H}^{(3)}\,_{abcd}$ $\displaystyle=$
$\displaystyle{\sigma_{\langle ab}\sigma_{cd\rangle}\Theta}\,.$ (47)

Here $\mathcal{H}^{(3)}\,$ is the monopole, $\mathcal{H}^{(3)}\,_{ab}$, and
$\mathcal{H}^{(3)}\,_{abcd}$ are quadrupole and the Hexadecapole moments of
$\left(K^{a}K^{b}\nabla_{a}u_{b}|_{0}\right)^{3}$ respectively. Using equation
(21) and (44) we find the best-fit deceleration parameter is given by

$\displaystyle q^{d_{A}}_{0}$ $\displaystyle=$
$\displaystyle-3+\frac{H^{d_{A}}_{0}}{\mathcal{H}^{(3)}\,}\bigg{[}\mathcal{O}\left(1+\frac{2}{15}\frac{\mathcal{H}^{(3)}\,_{ab}\mathcal{H}^{(3)}\,{}^{ab}}{\mathcal{H}^{(3)}\,{}^{2}}\right)-\frac{2}{15}{\frac{\mathcal{H}^{(3)}\,_{ab}}{\mathcal{H}^{(3)}\,}\mathcal{O}^{ab}}\bigg{]}\bigg{|}_{z=z_{0}}\,,$
(48) $\displaystyle\approx$
$\displaystyle\frac{1}{H^{2}}\bigg{\\{}\frac{1}{6}\rho-\frac{1}{3}\Lambda+\frac{3}{5}\sigma_{ab}\sigma^{ab}+\frac{2}{3}\bigg{(}\frac{1}{6}\rho-\frac{1}{3}\Lambda\bigg{)}\frac{\sigma^{ab}\sigma_{ab}}{H^{2}}-\frac{2}{5}\frac{\sigma^{ab}E_{ab}}{H}\bigg{\\}}\bigg{|}_{z=z_{0}}\,.$
(49)

Given that the distance duality relation or Etherington reciprocity theorem
holds, the Hubble and deceleration parameters obtained from the monopole of
luminosity distance must correspond to the Hubble rate given in equations (39)
and (49) respectively [75]. We checked and find that this holds, thus for the
Hubble rate ${H_{0}^{d_{L}}}={H_{0}^{d_{A}}}$ and the deceleration parameter
$q^{d_{L}}_{0}=q^{d_{A}}_{0}$.

###  III.2 Cosmological fitting problem: Distance modulus

The Carnegie-Chicago Hubble Program, CCHP, uses the distance modulus with the
luminosity distance given by the flat FLRW spacetime (equation (15)) to
estimate $H_{0}$ and $q_{0}$ [16].

$\displaystyle\mu(z,H_{0},q_{0})$ $\displaystyle=$ $\displaystyle m-M\,,$ (50)
$\displaystyle=$ $\displaystyle
5\log_{10}d_{L}+25=5\log_{10}\left[\frac{z}{H_{0}}\left(1+\frac{1}{2}(1-q_{0})z+\mathcal{O}(z^{2})\right)\right]+25\,.$
(51)

One unique feature of the CCHP approach is that it uses the Tip of the Red
Giant Branch stars, TRGB, to calibrate the SNIa samples. According to [76],
the best-fit $H_{0}$ depends crucially on the accuracy with which the absolute
magnitude of the SNIa is calibrated

$\displaystyle M=m-\mu_{0}^{\rm{TRGB}}\,,$ (52)

where $m$ is the apparent magnitude of the peak of the SNIa light curve for a
given subsample of SNIa co-located with the TRGB, $\mu_{0}^{\rm{TRGB}}$ is the
true calibrator distance modulus. After calibrating $M$ [16, 77], equation
(50) is used for the SNIa sample in the Hubble flow (i.e within the redshift
range ($0.03\leq z\leq 0.4$)). CCHP usually put uninformative prior(the value
of $q_{0}$ does not vary in the model estimates of distance to each of the
SNIa in the sample) on $q_{0}=-0.53$ to determine the Hubble rate to be [16]

$\displaystyle H_{0}=69.6\pm 1.9\,\,{\rm{km/sec/Mpc}}\,.$ (53)

The generalised form of the distance modulus is obtained by replacing $d_{L}$
given in (15) with the generalised form given equation (13). Then compare the
all-sky average to equation (50)

$\displaystyle\left\langle\mu\right\rangle_{\Omega}=\left\langle
m\right\rangle_{\Omega}-\left\langle M\right\rangle_{\Omega}$ $\displaystyle=$
$\displaystyle{5}\left\langle\,\log_{10}\left[{d_{L}(z,{{\bf\hat{n}}}}\right]\right\rangle_{\Omega}+25$
(54)

Comparing equation (54) to equation (50) requires that we perform the angular
integration over logarithm of $d_{L}$

$\displaystyle\left\langle\log_{10}d_{L}(z,{{\bf\hat{n}}})\right\rangle_{\Omega}$
$\displaystyle=$
$\displaystyle-\left\langle\log_{10}[K^{c}K^{d}\nabla_{d}u_{c}]_{o}\right\rangle_{\Omega}+\left\langle\log_{10}\left[\hat{d}_{L}(z,{{\bf\hat{n}}})\right]\right\rangle_{\Omega}\,,$
(55)

where we have used the quotient rule for logarithm to re-write equation (13).
We have also introduced the generalised form of the normalised luminosity
distance

$\displaystyle\hat{d}_{L}(z,{{\bf\hat{n}}})$ $\displaystyle=$ $\displaystyle
z\left\\{1+\frac{1}{2}\left[4-\frac{K^{c}K^{b}K^{a}\nabla_{a}\nabla_{b}u_{c}}{\left(K^{c}K^{b}\nabla_{c}u_{b}\right)^{2}}\right]_{0}z+\mathcal{O}(z^{2})\right\\}\,.$
(56)

In order to simplify
$\left\langle\log_{10}[K^{c}K^{d}\nabla_{d}u_{c}]_{o}\right\rangle_{\Omega}$
further, we use the irreducible decomposition of $K^{a}K^{b}\nabla_{a}u_{b}$

$\displaystyle\log_{10}\left[K^{a}K^{b}\nabla_{a}u_{b}\big{|}_{0}\right]$
$\displaystyle=$
$\displaystyle\log_{10}\left[H_{0}\left(1+\frac{\sigma_{ab}}{H_{0}}n^{a}n^{b}\right)\right]=\log_{10}H_{0}+\log_{10}\left(1+\frac{\sigma_{ab}}{H_{0}}n^{a}n^{b}\right)\,.$
(57)

For $\sigma_{ab}n^{a}n^{b}/H_{0}\ll 1$, we can expand the second term in
Taylor series and use equation (35) to obtain the PSTF part

$\displaystyle\log_{10}\left(1+\frac{\sigma_{ab}}{H_{0}}n^{a}n^{b}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{\log
10}\left[\frac{\sigma_{ab}}{H_{0}}n^{a}n^{b}-\frac{1}{2}\frac{\sigma_{ab}}{H_{0}}\frac{\sigma_{cd}}{H_{0}}n^{a}n^{b}n^{c}n^{d}+\mathcal{O}\left(\sigma\right)^{3}\right]\,,$
(58) $\displaystyle=$ $\displaystyle\frac{1}{\log
10}\bigg{[}-\frac{1}{15}\frac{\sigma_{ab}\sigma^{ab}}{H_{0}^{2}}+\left(\frac{\sigma_{ab}}{H_{0}}-\frac{2}{7}\frac{\sigma^{c}_{\langle
a}\sigma_{b\rangle c}}{H_{0}^{2}}\right)n^{\langle
a}n^{b\rangle}-\frac{1}{2}\frac{\sigma_{\langle
ab}\sigma_{cd\rangle}}{H^{2}_{0}}n^{\langle
a}n^{b}n^{c}n^{d\rangle}+\mathcal{O}\left(\sigma\right)^{3}\bigg{]}\,.$ (59)

Now it straight forward to perform the all-sky average to obtain the monopole

$\displaystyle\left\langle\log_{10}\left[K^{a}K^{b}\nabla_{a}u_{b}\big{|}_{0}\right]\right\rangle_{\Omega}$
$\displaystyle=$
$\displaystyle\log_{10}H_{0}-\log_{10}\left[1+\frac{1}{15}\frac{\sigma_{ab}\sigma^{ab}}{H_{0}^{2}}\right]\,.$
(60)

Extracting the monopole of
$\left\langle\log_{10}\left[\hat{d}_{L}(z,{{\bf\hat{n}}})\right]\right\rangle_{\Omega}$
is little more algebraically more involved. We go through it step-by-step.
Firstly, we decompose $\hat{d}_{L}(z,{{\bf\hat{n}}})$ in multipole moments

$\displaystyle\hat{d}_{L}(z,{{\bf\hat{n}}})$ $\displaystyle=$
$\displaystyle\sum_{\ell=0}^{\infty}\hat{d}_{A_{\ell}}(z)n^{\langle
A_{\ell}\rangle}=\hat{d}^{L}_{0}+\hat{d}^{L}_{a}n^{a}+\hat{d}^{L}_{ab}n^{a}n^{b}+\hat{d}^{L}_{abc}n^{a}n^{b}n^{c}+\mathcal{O}\left(\hat{d}^{L}_{A_{\ell>3}}\right)\,,$
(61)

where $\hat{d}^{L}_{0}$, $\hat{d}^{L}_{a}$ and $\hat{d}^{L}_{ab}$ are the
monopole, dipole and quadrupole moments of the Hubble rate-normalised
luminosity distance respectively. Then by factoring out the monopole of
$\hat{d}_{L}$ and requiring that $\hat{d}^{L}_{A_{\ell>1}}/\hat{d}^{L}_{0}\ll
1$ allows to expand the anisotropic part on the background of its monopole
moment

$\displaystyle\log_{10}(\hat{d}_{L}(z,{{\bf\hat{n}}})$ $\displaystyle=$
$\displaystyle\log_{10}{\hat{d}^{L}_{0}}+\frac{1}{\log
10}\bigg{[}\frac{\hat{d}^{L}_{a}}{\hat{d}^{L}_{0}}n^{a}+\frac{\hat{d}^{L}_{ab}}{\hat{d}^{L}_{0}}n^{a}n^{b}-\frac{1}{2}\left(\frac{\hat{d}^{L}_{a}}{\hat{d}^{L}_{0}}\frac{\hat{d}^{L}_{b}}{\hat{d}^{L}_{0}}n^{a}n^{b}+\frac{\hat{d}^{L}_{ab}}{\hat{d}^{L}_{0}}\frac{\hat{d}^{L}_{cd}}{\hat{d}^{L}_{0}}n^{a}n^{b}n^{c}n^{d}\right)+\mathcal{O}\left(\frac{\hat{d}^{L}_{A_{\ell}}}{\hat{d}^{L}_{0}}\right)^{3}\bigg{]}\,.$
(62)

Then taking the all sky average leads to

$\displaystyle\left\langle\log_{10}(\hat{d}_{L}(z,{{\bf\hat{n}}})\right\rangle_{\Omega}$
$\displaystyle=$
$\displaystyle\log_{10}{\hat{d}^{L}_{0}}-\frac{1}{2}\frac{1}{\log
10}\left(\frac{2}{15}\frac{\hat{d}^{L}_{ab}}{\hat{d}^{L}_{0}}\frac{\hat{d}_{L}^{ab}}{\hat{d}^{L}_{0}}+\frac{1}{3}\frac{\hat{d}^{L}_{a}}{\hat{d}^{L}_{0}}\frac{\hat{d}_{L}^{a}}{\hat{d}^{L}_{0}}\right)+\mathcal{O}\left(\frac{\hat{d}^{L}_{A_{\ell}}}{\hat{d}^{L}_{0}}\right)^{3}\,.$
(63)

We neglect the contribution of the second term since their contribution is
second order in redshift $\mathcal{O}(z^{2})$

$\frac{\hat{d}^{L}_{ab}}{\hat{d}^{L}_{0}}\frac{\hat{d}_{L}^{ab}}{\hat{d}^{L}_{0}}=\frac{\hat{d}^{L}_{a}}{\hat{d}^{L}_{0}}\frac{\hat{d}_{L}^{a}}{\hat{d}^{L}_{0}}\approx
0\,.$ (64)

We can now make use of the product rule for logarithms to re-write
$\left\langle\log_{10}\hat{d}_{L}(z,{{\bf\hat{n}}})\right\rangle_{\Omega}$ as

$\displaystyle\left\langle\log_{10}\hat{d}_{L}(z,{{\bf\hat{n}}})\right\rangle_{\Omega}$
$\displaystyle=$
$\displaystyle\log_{10}cz+\log_{10}\bigg{\\{}1+\frac{1}{2}\left[4-\left\langle\frac{K^{c}K^{b}K^{a}\nabla_{a}\nabla_{b}u_{c}}{\left(K^{c}K^{b}\nabla_{c}u_{b}\right)^{2}}\right\rangle_{\Omega}z\right]+\mathcal{O}(z^{2})\bigg{\\}}$
(65)

We simplify this further by first decomposing the denominator into irreducible
parts

$\displaystyle\left(K^{a}K^{b}\nabla_{a}u_{b}|_{0}\right)^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{9}\Theta^{2}+\frac{2}{3}n^{a}n^{b}\sigma_{ab}\Theta+n^{a}n^{b}n^{c}n^{d}\sigma_{ab}\sigma_{cd}\,,$
(66) $\displaystyle=$ $\displaystyle\mathcal{H}^{(2)}\,+n^{\langle
a}n^{b\rangle}\mathcal{H}^{(2)}\,_{\langle ab\rangle}+n^{\langle
a}n^{b}n^{c}n^{d\rangle}\mathcal{H}^{(2)}\,_{\langle abcd\rangle}\,,$ (67)

where $\mathcal{H}^{(2)}\,$ is the monopole, $\mathcal{H}^{(2)}\,_{\langle
ab\rangle}$ is the normalised quadrupole and $\mathcal{H}^{(2)}\,_{\langle
abcd\rangle}$ is the hexadecapole

$\displaystyle\mathcal{H}^{(2)}\,$ $\displaystyle=$
$\displaystyle\frac{1}{9}\Theta^{2}+\frac{2}{15}\sigma_{ab}\sigma^{ab}=H^{2}\left(1+\frac{2}{15}\frac{\sigma_{ab}\sigma^{ab}}{H^{2}}\right)\,,$
(68) $\displaystyle\mathcal{H}^{(2)}\,_{\langle ab\rangle}$ $\displaystyle=$
$\displaystyle\frac{4}{7}\sigma_{\langle a}^{c}\sigma_{b\rangle
c}+\frac{2}{3}\Theta\sigma_{\langle ab\rangle}\,,$ (69)
$\displaystyle\mathcal{H}^{(2)}\,_{\langle abcd\rangle}$ $\displaystyle=$
$\displaystyle\sigma_{\langle ab}\sigma_{cd\rangle}\,.$ (70)

Therefore, the monopole of the argument of the second term in equation (65)
becomes

$\displaystyle\left\langle\frac{K^{c}K^{b}K^{a}\nabla_{a}\nabla_{b}u_{c}}{\left(K^{c}K^{b}\nabla_{c}u_{b}\right)^{2}}\bigg{|}_{0}\right\rangle_{\Omega}$
$\displaystyle=$
$\displaystyle\frac{1}{\mathcal{H}^{(2)}\,}\bigg{[}\mathcal{O}\left(1+\frac{2}{15}\frac{\mathcal{H}^{(2)}\,_{ab}\mathcal{H}^{(2)}\,{}^{ab}}{\mathcal{H}^{(2)}\,{}^{2}}\right)-\frac{2}{15}{\frac{\mathcal{H}^{(2)}\,_{ab}}{\mathcal{H}^{(2)}\,}\mathcal{O}^{ab}}\bigg{]}\,,$
(71) $\displaystyle=$ $\displaystyle
3+\frac{1}{H^{2}}\bigg{\\{}\frac{1}{6}\rho-\frac{1}{3}\Lambda+\frac{3}{5}\sigma_{ab}\sigma^{ab}+\frac{6}{15}\left[\frac{1}{6}\rho-\frac{1}{3}\Lambda\right]\left[\frac{\sigma_{\langle
ab\rangle}\sigma^{\langle
ab\rangle}}{H^{2}}\right]-\frac{4}{15}\frac{\sigma^{\langle
ab\rangle}E_{\langle ab\rangle}}{H}\bigg{\\}}\,.$ (72)

Finally, we find that the monopole of the logarithm of the Hubble rate
normalised luminosity distance is given by

$\displaystyle\left\langle\log_{10}\hat{d}_{L}(z,{{\bf\hat{n}}})\right\rangle_{\Omega}$
$\displaystyle=$
$\displaystyle\log_{10}\bigg{\\{}z\bigg{[}1+\frac{1}{2}\bigg{[}1-\frac{1}{H^{2}}\bigg{(}\frac{1}{6}\rho-\frac{1}{3}\Lambda+\frac{3}{5}\sigma_{ab}\sigma^{ab}+\frac{6}{15}\left[\frac{1}{6}\rho-\frac{1}{3}\Lambda\right]\frac{\sigma_{\langle
ab\rangle}\sigma^{\langle ab\rangle}}{H^{2}}$
$\displaystyle-\frac{4}{15}\frac{\sigma^{\langle ab\rangle}E_{\langle
ab\rangle}}{H}\bigg{)}\bigg{]}z+\mathcal{O}(z^{2})\bigg{\\}}\,.$

Putting everything back to equation (54), we find that the monopole of the
distance modulus is given by

$\displaystyle\left\langle\mu\right\rangle_{\Omega}$ $\displaystyle=$
$\displaystyle\left\langle m\right\rangle_{\Omega}-\left\langle
M\right\rangle_{\Omega}^{R}=25+5\log_{10}\bigg{\\{}\frac{1}{{H}^{\mu}_{0}}\bigg{[}z\bigg{(}1+\frac{1}{2}\big{(}1-{q}^{\mu}\big{)}z+\mathcal{O}(z^{2})\bigg{)}\bigg{]}\bigg{\\}}\,,$
(74)

where the following terms were defined by comparing to the equivalent FLRW
expression given in equation (50) and (51)

$\displaystyle\left\langle M\right\rangle_{\Omega}^{R}$ $\displaystyle=$
$\displaystyle\left\langle
M\right\rangle_{\Omega}+5\log_{10}\left[1+\frac{1}{15}\frac{\sigma_{ab}\sigma^{ab}}{H^{2}}\bigg{|}_{z=0}\right]\,,$
(75) $\displaystyle{H}^{\mu}_{0}$ $\displaystyle=$ $\displaystyle H_{0}\,,$
(76) $\displaystyle{q}^{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{H^{2}}\bigg{(}\frac{1}{6}\rho-\frac{1}{3}\Lambda+\frac{3}{5}\sigma_{ab}\sigma^{ab}+\frac{2}{3}\left[\frac{1}{6}\rho-\frac{1}{3}\Lambda\right]\frac{\sigma_{\langle
ab\rangle}\sigma^{\langle ab\rangle}}{H^{2}}-\frac{4}{15}\frac{\sigma^{\langle
ab\rangle}E_{\langle ab\rangle}}{H}\bigg{)}\bigg{|}_{z=0}\,.$ (77)

This indicates that the best-fit absolute magnitude, $\left\langle
M\right\rangle_{\Omega}^{R}$ of the SNIa includes the effect of tidal
deformation of the local geometry. Stated differently, the measurement of the
Hubble rate via the local distance ladder accounts for the impact of tidal
deformation on the luminosity distance. The Hubble rate determined using the
local distance ladder with a properly calibrated absolute magnitude
corresponds to the global volume expansion.

###  III.3 Cosmological fitting problem: intercept of the Hubble diagram

On the other hand, the SH0ES collaboration uses the Cepheid variable stars to
calibrate the absolute magnitude of the SNIa. The estimate of the Hubble rate
is obtained from the constraint on the intercept of the magnitude-redshift
relation with the luminosity distance given by the flat FLRW spacetime.
Starting from equation (11), they re-write the distance modulus as follows
[13]

$\displaystyle m$ $\displaystyle=$ $\displaystyle
M+25+5\log_{10}\bar{d}_{L}=-5a_{b}+5\log_{10}\hat{d}_{L}(z)\,,$ (78)

where $a_{b}$ is the intercept of the of the Hubble diagram and $\hat{d}_{L}$
is the Hubble parameter normalised luminosity distance:

$\displaystyle 5a_{b}$ $\displaystyle=$
$\displaystyle-\left(M_{b}+25-5\log_{10}H_{0}\right)\,,$ (79)
$\displaystyle\hat{d}_{L}(z)$ $\displaystyle=$
$\displaystyle{H_{0}}d_{L}(z)=z\left[1+\frac{1}{2}(1-q_{0})z+\mathcal{O}(z^{2})\right]\,.$
(80)

The Hubble rate is then determined from equation (79)

$\displaystyle\log_{10}H_{0}=\frac{M_{b}+25+5a_{b}}{5}\,,$ (81)

where $M_{b}$ is called standardizable absolute luminosity in [13] but it does
the same job as equation (52) in CCHP

$\displaystyle M_{b}=m_{b,\rm{SNIa}}-\mu_{0,\rm{ceph}}$ (82)

where, $\mu_{0,\rm{ceph}}$, is an independent distance modulus to cepheids and
$m_{b,\rm{SNIa}}$, is the apparent magnitude of a sub-sample of nearby SNIa
that live in the same host as the cepheids. The SH0ES collaboration has since
made use of different geometrical distance estimates such as the NGC 4258
obtained by modelling of the water masers in the nucleus of the galaxy orbit
about its supermassive black hole [15]; Large Magellanic Cloud using eclipsing
binary systems composed of late-type stars [14]; Milky Way Cepheids using
parallax methods [78, 79] to calibrate $M_{b}$. Even though the SH0ES
collaboration limits type Ia supernova sample to $z=0.023$ due to systematics
associated with disentangling the peculiar velocities of the sources from the
coherent Hubble flow, they have accurate information on the intercept

$\displaystyle a_{b}$ $\displaystyle=$
$\displaystyle\log_{10}\bigg{\\{}cz\left[1+\frac{1}{2}(1-q_{0})z+\mathcal{O}(z^{3})\right]\bigg{\\}}-0.2M_{b}\,,$
(83) $\displaystyle\approx$ $\displaystyle\log_{10}cz-0.2M_{b}\,,$ (84)

Similarly, the SH0ES collaboration puts uninformative prior on
$\bar{q}_{0}=-0.55$ [80] to determine $a_{b}=0.71273\pm 0.00176$. With the
constraint on $M_{b}$, they found that the Hubble rate is given by [81, 82]

$\displaystyle H_{0}=73.1\pm 1.4\,\,{\rm{kms}^{-1}}{\rm{Mpc}}^{-1}\,.$ (85)

Note that the quoted error here are the error associated with the
determination of $a_{b}$ and $M_{b}$ added in quadrature [13].

We will now show how the approach taken by the SH0ES collaboration may be
generalised to apply to arbitrary inhomogeneous models. The key ingredient in
the generalised expression is the luminosity distance given in equation (13).
Starting from equation (11) and following similar steps that starts from
equation (78) in the FLRW limit

$\displaystyle\left\langle m\right\rangle_{\Omega}$ $\displaystyle=$
$\displaystyle\left\langle
M\right\rangle_{\Omega}-{5}\left\langle\log_{10}[K^{c}K^{d}\nabla_{d}u_{c}]_{o}\right\rangle_{\Omega}+5\left\langle\log_{10}\left[\hat{d}_{L}(z,{{\bf\hat{n}}})\right]\right\rangle_{\Omega}+25\,,$
(86) $\displaystyle=$ $\displaystyle-5\left\langle
a_{b}\right\rangle_{\Omega}+5\left\langle\log_{10}\hat{d}_{L}(z,{{\bf\hat{n}}})\right\rangle_{\Omega}\,,$
(87)

where $\hat{d}_{L}(z,{{\bf\hat{n}}})$ is given in equation (56) and the
generalised intercept is given by

$\displaystyle\left\langle
a_{b}({{\bf\hat{n}}})\right\rangle_{\Omega}=-\frac{1}{5}\left(\left\langle
M\right\rangle_{\Omega}+25-5\left\langle\log_{10}\left[K^{a}K^{b}\nabla_{a}u_{b}\big{|}_{0}\right]\right\rangle_{\Omega}\right)\,.$
(88)

Recall that the decomposition of $\left\langle
K^{a}K^{b}\nabla_{a}u_{b}\right\rangle_{\Omega}$ is discussed between equation
(57) to (60). Putting this back into equation (88), gives

$\displaystyle\left\langle a_{b}({{\bf\hat{n}}})\right\rangle_{\Omega}$
$\displaystyle=$ $\displaystyle-\frac{1}{5}\left(\left\langle
M\right\rangle_{\Omega}^{R}+25-5\log_{10}H_{0}\right)\,,$ (89)

where we have introduced a renormalised absolute magnitude defined in equation
(75). Similarly, the Hubble rate depends on the monopole of the intercept and
renormalised absolute luminosity according to

$\displaystyle\log_{10}H_{0}=\frac{\left\langle
M\right\rangle_{\Omega}^{R}+25+5\left\langle
a_{b}\right\rangle_{\Omega}}{5}\,.$ (90)

$\left\langle a_{b}\right\rangle_{\Omega}$ is obtained independently by
fitting to the intercept of the distance modulus of the SNIa peak magnitudes.
The intercept of the distance modulus constrains ${\left\langle
M\right\rangle_{\Omega}^{R}}+5\left\langle a_{b}\right\rangle_{\Omega}$, from
where we find

$\displaystyle\left\langle a_{b}({{\bf\hat{n}}})\right\rangle_{\Omega}$
$\displaystyle=$
$\displaystyle\left\langle\log_{10}\hat{d}_{L}(z,{{\bf\hat{n}}})\right\rangle_{\Omega}-0.2\left\langle
M\right\rangle_{\Omega}^{R}\,,$ (91) $\displaystyle\approx$
$\displaystyle\log_{10}\bigg{\\{}cz\left[1+\frac{1}{2}\left[4-\left\langle\frac{K^{c}K^{b}K^{a}\nabla_{a}\nabla_{b}u_{c}}{\left(K^{c}K^{b}\nabla_{c}u_{b}\right)^{2}}\right\rangle_{\Omega}\right]_{0}z+\mathcal{O}(z^{2})\right]\bigg{\\}}-0.2\left\langle
M\right\rangle_{\Omega}^{R}\,,$ (92) $\displaystyle=$
$\displaystyle\log_{10}\bigg{\\{}cz\bigg{[}1+\frac{1}{2}\bigg{[}1-\frac{1}{H^{2}}\bigg{(}\frac{1}{6}\rho-\frac{1}{3}\Lambda+\frac{3}{5}\sigma_{ab}\sigma^{ab}+\frac{6}{15}\left[\frac{1}{6}\rho-\frac{1}{3}\Lambda\right]\frac{\sigma_{\langle
ab\rangle}\sigma^{\langle ab\rangle}}{H^{2}}$
$\displaystyle-\frac{4}{15}\frac{\sigma^{\langle ab\rangle}E_{\langle
ab\rangle}}{H}\bigg{)}\bigg{]}z+\mathcal{O}(z^{2})\bigg{\\}}-0.2\left\langle
M\right\rangle_{\Omega}^{R}\,,$ $\displaystyle\approx$
$\displaystyle\log_{10}cz-0.2\left\langle M\right\rangle_{\Omega}^{R}\,.$ (94)

where we made use of equation (65) in the second equality, in the third
equality, we used equation (III.2) and the very low redshift approximation is
enough [13].

Finally, we remark that the Hubble rate and the deceleration parameter
obtained from fitting to the monopole of the Alcock-Paczyński parameters for
the BAO signal in the two-point correlation function agree:

$\displaystyle{H_{0}^{d_{L}}}$ $\displaystyle=$
$\displaystyle{H_{0}^{d_{A}}}=H^{\alpha{{}_{{\text{\tiny$\|$}}}}}_{0}\,,$ (95)
$\displaystyle q^{d_{L}}_{0}$ $\displaystyle=$ $\displaystyle
q^{d_{A}}_{0}\,.$ (96)

But they differ from the local cosmic distance ladder measurement
determination

$\displaystyle H^{\mu_{I}}_{0}$ $\displaystyle=$
$\displaystyle{H}^{\mu}_{0}\,,$ (97) $\displaystyle q_{0}^{\mu_{I}}$
$\displaystyle=$ $\displaystyle q_{0}^{\mu}\,.$ (98)

The Hubble rate from local cosmic distance measurement corresponds to volume
expansion without any contamination due to the tidal deformation of the local
spacetime geometry. The Hubble rate and the deceleration parameter determined
from the local cosmic ladder and Alcock-Paczyński parameters of the BAO signal
differ according to

$\displaystyle{H_{0}^{d_{A}}}-H^{\mu}_{0}$ $\displaystyle\approx$
$\displaystyle-\frac{2}{15}\frac{\sigma_{ab}\sigma^{ab}}{{H}}\bigg{|}_{z=0}+\mathcal{O}(\sigma^{3})\,,$
(99) $\displaystyle q^{d_{A}}_{0}-q_{0}^{\mu}$ $\displaystyle\approx$
$\displaystyle-\frac{2}{15}\frac{\sigma^{\langle ab\rangle}E_{\langle
ab\rangle}}{H^{2}}\bigg{|}_{z=0}+\mathcal{O}(\sigma^{3}).$ (100)

It is possible to calibrate the SNIa using the BAO distance information
(proper distance travelled by the acoustic waves from the initial time to the
last scattering surface) instead of the nearby distance anchors [83]. This
approach usually involves the use of a model which does not include the tidal
fields we discussed here in the calibration process [84], thus, we can
associate the absolute magnitude of the SNIa determined through this process
to $\left\langle M\right\rangle_{\Omega}$. On the other hand, the local cosmic
distance ladder approach does not assume any model for distance(apart from the
parallax formula for distance, flux inverse square law, period-luminosity
relation for cepheids) during the calibration process, hence, we can associate
the absolute magnitude it determines to $\left\langle
M\right\rangle_{\Omega}^{R}$. The difference is given by

$\displaystyle\left\langle M\right\rangle_{\Omega}^{R}-\left\langle
M\right\rangle_{\Omega}$ $\displaystyle\approx$ $\displaystyle\frac{1}{3\log
10}\frac{\sigma_{ab}\sigma^{ab}}{{H}^{2}}\bigg{|}_{z=0}+\mathcal{O}(\sigma^{3})\,.$
(101)

##  IV The perturbed FLRW spacetime

The results we have derived so far by fitting the FLRW model to an
inhomogeneous model is general, they apply to any inhomogeneous cosmological
model provided that anisotropies are small when compared to the monopole. In
this section, we specialise to a Universe where the inhomogeneities could be
described as small perturbations on top of a flat FLRW background spacetime.
We work in conformal Newtonian gauge (our result is gauge-invariant):

$\displaystyle\rm{d}s^{2}$ $\displaystyle=$
$\displaystyle\bar{a}^{2}\big{[}-\big{[}1+2\Phi\big{]}\rm{d}\eta^{2}+\big{(}\big{[}1-2\Phi\big{]}\delta_{ij}\big{]}\,,$
(102)

where $\delta_{ij}$ is the metric of the Minkowski spacetime, $\bar{a}$ is the
scale factor of the expanding background FLRW spacetime and $\Phi$ is the
Newtonian gravitational potential. The components of the time-like four
velocity in perturbation theory are given by

$\displaystyle u^{0}$ $\displaystyle=$ $\displaystyle
1-\Phi+\frac{3}{2}\Phi^{2}-\frac{1}{2}\Phi^{\text{\tiny\color[rgb]{0,1,0}{({{2}})}}}+\frac{1}{2}{\partial}_{i}v{\partial}^{i}v\,,$
(103) $\displaystyle u^{i}$ $\displaystyle=$
$\displaystyle{\partial}^{i}v+\frac{1}{2}v^{i}{}^{\text{\tiny\color[rgb]{0,1,0}{({{2}})}}}+\frac{1}{2}{\partial}^{i}v^{\text{\tiny\color[rgb]{0,1,0}{({{2}})}}}\,.$
(104)

where $v^{i}$ is the peculiar velocity, $\partial_{i}$ is the spatial
derivative on the Minkowski spacetime in cartesian coordinates, and $v$ is the
peculiar velocity potential. The indices with the small English alphabets
denote the spatial component of the spacetime. The green superscripts denote
terms evaluated at second order in perturbation theory.

Using equations (102), (103) and (104) at leading order in perturbation
theory, the shear tensor, $\sigma^{ab}$ becomes

$\displaystyle{{\sigma}}_{ij}(\eta,{\bm{x}})=\bar{a}\partial_{\langle
i}\partial_{j\rangle}v(\eta,{\bm{x}})\,.$ (105)

The ensemble average of the shear scalar is given

$\displaystyle\frac{\sigma_{ab}\sigma^{ab}}{{H}^{2}}$ $\displaystyle=$
$\displaystyle\frac{2}{3}f^{2}(z)\sigma^{2}_{R}(z)\,,$ (106)

where $f$ is the rate of growth of structures. We made use of the continuity
to express the velocity potential and the in terms of the matter density
contrast $\delta_{m}$

$\displaystyle v({\bm{k}},\eta)$ $\displaystyle=$
$\displaystyle\frac{\mathcal{H}\,}{k^{2}}f(\eta)\delta_{m}({\bm{k}},\eta)\,.$
(107)

In addition, we introduced the variance in matter density field

$\displaystyle\sigma^{2}_{R}$ $\displaystyle=$
$\displaystyle\int_{0}^{k_{\rm{UV}}}\frac{{\rm{d}}k}{2\pi^{2}}\,\left[k^{2}P_{\rm{m}}(k)\right]=\int_{0}^{\infty}\frac{{\rm{d}}k}{2\pi^{2}}\,\left[(kW(kR))^{2}P_{\rm{m}}(k)\right]\,.$
(108)

Here, $P_{m}$ is the power spectrum of the matter density field. We have
introduced a top-hat window function, $W$, instead of a UV dependent momentum
integral. The scale, $R$, will be fixed shortly. For the different
determinations of the Hubble rate, we focused on the dominant terms only,
perturbing the monopole of various definitions of the Hubble rate give

$\displaystyle H_{0}^{\mu}$ $\displaystyle=$ $\displaystyle
H_{0}^{\mu_{I}}\simeq\bar{H}_{0}\,,$ (109) $\displaystyle H_{0}^{d_{A}}$
$\displaystyle=$
$\displaystyle{H_{0}^{d_{L}}}=H^{\alpha{{}_{{\text{\tiny$\|$}}}}}_{0}\simeq\bar{H}_{o}\left[1-\frac{4}{45}f^{2}(z)\sigma^{2}_{R}(z)\bigg{|}_{0}\right]\,.$
(110)

Note that $\Theta$ is well described by the FLRW background spacetime to an
accuracy better than $0.1\%$ [85], thus we neglect its perturbations and
approximate $H_{0}^{\mu}$ and $H_{0}^{\mu_{I}}$ with the FLRW background
prediction, $\bar{H}_{0}$. The relationship between the rest of Hubble rate
determinations and $H_{0}^{\mu_{I}}$ is given

$\displaystyle\frac{H^{\alpha{{}_{{\text{\tiny$\|$}}}}}_{0}}{H_{0}^{\mu_{I}}}=1-\frac{4}{45}f^{2}(z)\sigma^{2}_{R}(z)\bigg{|}_{0}\,.$
(111)

Therefore, the BAO determination of the Hubble rate differs from the local
measurements by a factor of $1-{4}f^{2}(0)\sigma^{2}_{R}/{45}$. The electric
part of the Weyl tensor, $E_{ab}$, at leading order in perturbation theory is
given by

$\displaystyle{E}_{ij}(\eta,{\bm{x}})$ $\displaystyle=$
$\displaystyle\partial_{\langle i}\partial_{j\rangle}\Phi(\eta,{\bm{x}})\,.$
(112)

We use the Poisson equation to express $\Phi$ in terms $\delta_{m}$

$\displaystyle\Phi({\bm{k}},\eta)$ $\displaystyle=$
$\displaystyle-\frac{3}{2}\Omega_{m}(z)\left(\frac{\mathcal{H}\,}{k}\right)^{2}\delta_{m}({\bm{k}},\eta)\,,$
(113)

where $\Omega_{m}$ is the matter-energy density parameter. Using equations
(107) and (113) we find that

$\displaystyle\frac{\sigma^{ab}E_{ab}}{{H}^{3}}$ $\displaystyle=$
$\displaystyle-\Omega_{m}f(z)\sigma^{2}_{R}(z)\,.$ (114)

Expanding the first two terms in
$q_{0}=\left(\frac{1}{6}\rho-\frac{1}{3}\Lambda+\sigma^{ab}\sigma_{ab}\right)/{H}^{2}_{0}$
up to second order in perturbation theory gives

$\displaystyle\frac{1}{H^{2}_{0}}\left(\frac{1}{6}\frac{1}{\rho}-\frac{1}{3}\Lambda\right)\bigg{|}_{z_{o}}$
$\displaystyle\approx$
$\displaystyle\bar{q}_{0}-\frac{1}{3H_{0}}\Omega_{m}\delta_{m}\partial^{2}v+\frac{1}{3H^{2}_{0}}\bar{q}_{0}(\partial^{2}v)^{2}\,,$
(115) $\displaystyle\approx$
$\displaystyle\bar{q}_{0}+\frac{1}{3}\Omega_{m}f(0)\sigma_{R}^{2}+\frac{1}{3}\bar{q}_{0}f^{2}(0)\sigma_{R}^{2}\,,$
(116)

The product of $q_{0}$ and shear scalar in perturbation theory is given by

$\displaystyle\frac{1}{H^{2}_{0}}\left(\frac{1}{6}\frac{1}{\rho}-\frac{1}{3}\Lambda\right)\bigg{|}_{z_{o}}\frac{\sigma_{ab}\sigma^{ab}}{H^{2}}\bigg{|}_{z_{o}}=\frac{2}{3}\bar{q}_{0}f^{2}(0)\sigma^{2}_{R}\,,$
(117)

where $\bar{q}_{0}$ is the deceleration parameter on the background FLRW
spacetime with dust and Cosmological constant

$\displaystyle\bar{q}_{0}$ $\displaystyle=$
$\displaystyle\frac{\Omega_{m}}{2}-\Omega_{\Lambda}=-1+\frac{3}{2}\Omega_{m}\,,$
(118)

where $\Omega_{\Lambda}$ is the energy density due to the Cosmological
constant. In the second equality, we made use of the Friedmann equation
$\Omega_{m}+\Omega_{\Lambda}=1\,,$ to express $\bar{q}_{0}$ in terms of
$\Omega_{m}$ only. The matter density and the cosmological constant today are
defined in terms of these parameters as
$\bar{\rho}_{0}=3\mathcal{H}\,^{2}_{0}\Omega_{m0}$ and
$\Lambda=3\mathcal{H}\,^{2}_{0}\Omega_{\Lambda 0}$. Using equation (116),
$q_{0}$ perturbed up to second order is given by

$\displaystyle\frac{1}{{H}^{2}_{0}}\left(\frac{1}{6}\rho-\frac{1}{3}\Lambda+\sigma^{ab}\sigma_{ab}\right)\approx\bar{q}_{0}+\frac{1}{3}\Omega_{m}f(0)\sigma_{R}^{2}+\frac{1}{3}\bar{q}_{0}f^{2}(0)\sigma_{R}^{2}+\frac{2}{3}(f^{2}(0))\sigma^{2}_{R}(z)$
(119)

Finally, the deceleration parameter for both cases becomes

$\displaystyle q_{0}^{d_{A}}=q_{0}^{d_{L}}$ $\displaystyle\simeq$
$\displaystyle\bar{q}_{0}+\frac{11}{15}\Omega_{m}f(0)\sigma_{R}^{2}+\left(\frac{7}{9}\bar{q}_{0}+\frac{2}{5}\right)f^{2}(0)\sigma^{2}_{R}\,,$
(120) $\displaystyle q^{\mu}_{0}=q^{\mu_{I}}_{0}$ $\displaystyle\simeq$
$\displaystyle\bar{q}_{0}+\frac{3}{5}\Omega_{m}f(0)\sigma_{R}^{2}+\left(\frac{7}{9}\bar{q}_{0}+\frac{2}{5}\right)f^{2}(0)\sigma^{2}_{R}\,.$
(121)

The exact magnitude of the difference between the predications of the
effective Hubble rate and deceleration parameter from different ways of
fitting an FLRW model to a lumpy Universe depends on $\sigma^{2}_{R}$. We
discuss how we handle the dependence on $\sigma^{2}_{R}$ in section IV.1.

###  IV.1 Causal horizon and the expanding regions of spacetime

In cosmology, the well-known causal limits are determined by the dynamics on
the null cone, for example, the particle horizon, which indicates the maximum
distance light from particles could have travelled to the observer in the age
of the Universe. Ellis and Stoeger argued in [86] that there exists a causal
horizon that is determined not by the dynamics on the light cone but by the
dynamics of our time-like geodesic. It is given by the boundary of the
comoving region that has contributed most significantly to the dynamics of our
local environment. The dominant interaction within this region are not
mediated by massless particles (i.e the vector and tensor perturbations on an
FLRW background spacetime have negligible impact on the dynamics within this
region), instead, they are mediated by massive particles which travel at very
low speeds relative to the cosmic rest frame. It is these differences in speed
that cause our local environment to decouple from the Hubble expansion because
it cannot keep up the cosmic rest frame [86, 87].

Figure 1:  The horizontal line in the left panel shows the zero-
velocity(expansion) surface: $\Theta(0,R{{\bf\hat{n}}})=0$, i.e the transition
radius where the spacetime go from expansion to contraction. The exact
position of $R$ depends on the halo mass. The three curves; thick, dashed and
dash-dotted lines correspond to the following halo masses
$M=[10^{11},10^{12},10^{13}]M_{\odot}$ respectively. On the right panel, we
compute the gradient of the halo density only for the same halo masses shown
in the left panel. The splashback radius for a given halo mass is the position
of the least gradient. The splashback radius is always less than the radius of
the zero-velocity surface. In both cases, we fixed the halo concentration at
5.

We determine the causal horizon as the comoving radius where divergence of our
4-velocity vanish. This is also called the zero-velocity surface [88]. At the
leading order in cosmological perturbation theory, this is given by

$\displaystyle\Theta\simeq
3H_{0}+\partial_{i}\partial^{i}v\big{|}_{z=0}=3{H}_{0}+\frac{c}{r}\frac{\rm{d}\ln\rho}{\rm{d}\ln
r}\,.$ (122)

In the second equality, we made use of the continuity equation:
$\delta^{\prime}_{m}=-\partial_{i}\partial^{i}v$ with
$\delta_{m}\equiv{\delta\rho}/{\bar{\rho}}=({\rho-\bar{\rho}})/{\bar{\rho}}\,$
and use the chain rule to express the conformal time derivative of
$\delta_{m}$ in terms of radial derivative of $\rho$

$\displaystyle\delta^{\prime}_{m}=\frac{\rm{d}\delta_{m}}{\rm{d}\eta}=\frac{1}{\bar{\rho}}\frac{\partial
r}{\partial\eta}\frac{\partial\rho}{\partial
r}\approx-\frac{1}{\bar{\rho}}\frac{\partial\rho}{\partial
r}\approx-\frac{c}{r}\frac{\rm{d}\ln\rho}{\rm{d}\ln r}\,.$ (123)

The radius of the zero-velocity surface $\Theta(0,{r})=0$ is given by

$\displaystyle R_{0}=-\frac{c}{3H_{0}}\frac{\rm{d}\ln\rho}{\rm{d}\ln r}.$
(124)

The spacetime in the region $r<R_{0}$ are not expanding and it defines the
local sphere of influence with respect to the observer. We estimate equation
(122) using a dark matter halo model with the Einasto [89] and NFW profiles
[90] with the outer profile given by the mean matter density [91]. The result
is shown in figure 1. $R_{0}$ is obtained from figure 1 as the value of $r$
where the curves intersect the $\Theta(0,r)=0$ horizontal line. We find that
$R_{0}$ is dependent on the mass of the host halo. The larger the halo mass,
the higher the causal horizon. One might ask, how is $R_{0}$ related to the
virial radius, $r_{\Delta_{c}}$($\Delta_{c}$ is over-density constant). The
mass contained with $r_{\Delta_{c}}$ (virial mass) is the mass of a
gravitationally bound astrophysical system, assuming the virial theorem holds.
Recent studies have shown that $r_{\Delta_{c}}$ does not correspond to the
physical boundary of a gravitationally bound astrophysical system. The reason
is that the mass contained within $r_{\Delta_{c}}$ is subject to pseudo-
evolution [92, 93]. And there are physical processes that are known to
redistribute sub-structures formed during collapse from small to large radii
greater than $r_{\Delta_{c}}$ [94, 95, 96]. For these reasons, the splashback
radius, $r_{\rm{sp}}$, was introduced as a radius that includes all matter
that orbits a halo [97, 92]. The splashback radius of haloes of different
masses is shown in the right panel of figure 1. We find that $R_{0}$ is
different from the splashback radius of the host halo. In fact from figure 1,
we find that $R_{0}$ is always greater than the $r_{\rm{sp}}$ at any of the
given halo mass we considered. There are observational constraints for the
radius of the zero-velocity surface for our local group
$R_{0}\sim(0.95-1.05){\rm{Mpc}}$ [98, 31, 88] and this is the value we use for
the rest of the analysis.

###  IV.2 The cosmological tensions: Hubble rate, absolute magnitude and the
deceleration parameter

Now that we have determined the minimum length scale participating in the
cosmic expansion, it is straightforward to calculate the Hubble discrepant
(i.e equation (111)). The minimum length scale corresponds to the distance
where the reference flux or the absolute magnitudes of sources in the Hubble
flow are calibrated. This makes sense because cosmic evolutionary effect does
not contribute to the absolute magnitude. Setting the smoothing scale to the
radius of the zero-velocity surface $R=R_{0}$. The full result is shown in
figure 2. We find about $\sim(9-12)\%$ difference in the determination of the
Hubble rate between the local measurements using the cosmic distance ladder
and the BAO determinations.

Figure 2:  Left panel: We show the fractional difference between the Hubble
rates determined using the local measurement of the peak apparent magnitude of
SNIa and the interpretation of the Alcock-Paczyński parameters in terms of the
FLRW spacetime. Right panel: We show the difference between the renormalised
absolute magnitude due to the impact of the tidal deformations on the distance
to the anchor and the intrinsic absolute magnitude. We made use of the linear
theory matter power spectrum to evaluate these.

Although, the cosmological interpretation of the peak magnitude of the SNIa by
the SH0ES and CCHP groups was done using a smooth luminosity distance, Both
groups differ on the calibration of the supernova absolute magnitude. The
SH0ES build a local cosmic distance ladder using distance measurements to a
set of anchors to infer the absolute magnitude of the SNIa. Other than using
the standard formula for the parallax for the milky-way cepheids or the
inverse square law for flux for the detached eclipsing binaries, or Leavitt’s
law for cepheids [99], etc, the process of calibrating the absolute magnitude
of the SNIa is cosmological model-independent. Since these set of anchors live
nearby, the calibration of the local cosmic distance ladder includes the
impact of the tidal deformations on the local spacetime that we discuss here
[36]. We showed how the impact of the tidal deformation results in a
renormalisation of the absolute magnitude in subsection III.2. Calibrating the
SNIa absolute magnitude using the proper distance acoustic waves could have
travelled from the origin of the Universe to the surface of the last
scattering as an anchor, requires that we assume a particular model based on
the FLRW spacetime. This model does not include the impact of the tidal field
on distance measurement [83].

The supernova absolute magnitude tension refers to the disparity in the
determination of the supernova absolute magnitude between the inverse distance
ladder method with the sound horizon scale at the surface of the last
scattering as an anchor and local cosmic distance ladder with the anchors in
the nearby Universe [84, 100]. Using the inverse distance method on Pantheon
supernova peak magnitudes, [34] found the absolute magnitude to be

$\displaystyle M^{\rm{P18}}=-19.387\pm 0.021\,\,{\rm{mag}}\,.$ (125)

For this same supernova sample, but using the SH0ES Cepheid photometry for the
geometric distance estimates, [34] finds the absolute magnitude to be

$\displaystyle M^{\rm{E21}}=-19.214\pm 0.037\,\,{\rm{mag}}\,.$ (126)

The difference between equation (125) and (126) is given by

$\displaystyle M^{\rm{E21}}-M^{\rm{P18}}=0.173\pm 0.04\,\,{\rm{mag}}\,.$ (127)

We can relate equation (125) to $\left\langle M\right\rangle_{\Omega}$ and
equation (126) to $\left\langle M\right\rangle_{\Omega}^{R}$ so that the
difference between them is given by equation (75)

$\displaystyle\left\langle M\right\rangle_{\Omega}^{R}-\left\langle
M\right\rangle_{\Omega}$ $\displaystyle=$ $\displaystyle\frac{2}{9\log
10}f^{2}(0)\sigma^{2}_{R}\,.$ (128)

We show the result of calculating equation (128) in the right panel of figure
2. With $R=D=1{\rm{Mpc}}$, we find a diffrenece of about $0.12~{}{\rm{mag}}$

In addition to the Hubble rate and the absolute magnitude, we also derived the
expression for the deceleration parameter by fitting the FLRW model to the
monopole of an inhomogeneous model of the area distance, distance modulus,
etc. In this case, we find that both the local measurements and the BAO
measurements (Alcock-Paczyński parameters) give deceleration parameter that
differ substantially from the prediction of an FLRW model

$\displaystyle q_{0}^{d_{A}/\mu}-\bar{q}_{0}=\Delta q^{d_{A}/\mu}_{0}\,,$
(129)

where $\Delta q^{d_{A}/\mu}_{0}$ depends on the scalar invariant of a
combination of shear tensor and the electric part of the Weyl tensor, i.e.
${\sigma^{\langle ab\rangle}E_{\langle ab\rangle}}/{H^{2}}\big{|}_{z=0}$ and
$\sigma_{ab}\sigma^{ab}/H^{2}\big{|}_{z=0}$. The corresponding cosmological
perturbation limit of both expressions are given in equations (120) and (121).
Again the local cosmic ladder and the Alcock-Paczyński parameters
determinations of the deceleration parameter differ only in the coefficient of
this term ${\sigma^{\langle ab\rangle}E_{\langle
ab\rangle}}/{H^{2}}\big{|}_{z=0}$(see equation (100)). ${\sigma^{\langle
ab\rangle}E_{\langle ab\rangle}}/{H^{2}}\big{|}_{z=0}$ is related to the
spatial curvature of the local domain.

Figure 3:  We show the fractional difference between the deceleration
parameters derived in equations (120)(thick lines) and (121) (dashed lines)
and the FLRW predictions.

We show in figure 3, the fractional difference between the deceleration
parameters derived in equations (120) and (121) and the FLRW predictions. We
found over a 100% difference between the deceleration parameter from the local
measurement interpreted based on an FLRW model and Planck prediction. This
finding is consistent with the recent measurements of the deceleration
parameter using the Pantheon supernova samples and the cosmographical
expression for the luminosity distance in FLRW limit [35, 100]. However, it
differs from the measurement of the deceleration parameter by the SH0ES
collaboration [37]. Note that SH0ES collaboration made use of the full
expression for the luminosity distance based on the background FLRW spacetime
and not its cosmographical approximation [80]. Given that [37] made use of the
data set used by [35, 100] but supplemented with another data set which is
composed of high redshift SNIa, It is most likely that the reason for this
disparity is related to the breakdown of Taylor series expansion in the
presence of structures as reported in [36].

The Hubble tension is usually described as a discrepancy between the early and
late-time Universe determination of the Hubble rate [22]. We have shown that
it could be a consequence of how the distance measurements in the
neighbourhood of the observer are interpreted within the $\Lambda$CDM model.
The key factor is that the local spacetime around the observer is tidally
deformed, the background FLRW spacetime does not capture this effect. We
showed that the model of area/luminosity distance that includes the effects of
inhomogeneities through the perturbation of the FLRW spacetime would be able
to explain the Hubble tension without any need of invoking any exotic evolving
dark energy [101, 102] (see [103, 104] for a list of all possible models
within this framework), frame-dependent dark energy [105] (this approach
explains the SH0ES result but gives a large value of the Hubble constant from
the BAO analysis), quantum measurement uncertainties [106], evolving
gravitational constant [107], modification of gravity [108, 109]. One unique
thing about all these approaches except the frame dependent dark energy which
breaks 4D diffeomorphism invariance but retains 3D coordinate invariance is
that they assume that the FLRW background spacetime is valid on all scales and
at all times. This is contrary to the findings in [36] which showed that the
FLRW background spacetime in the presence of structures break down at about 1
Mpc. This scale is consistent with the scale where caustics or conjugate
points appears [110]. Appearance of the conjugate points is a unique indicator
for the breakdown of the coordinate system [30].

Furthermore, we showed that the interpretation of the BAO measurement (i.e
Alcock-Paczyński parameters) using the area distance based on the background
FLRW spacetime infers a wrong $H_{0}$ because the symmetries of background
FLRW spacetime does not allow the contribution of the electric part of the
Weyl tensor or the effect of the tidal deformations around the observer. If
the Alcock-Paczyński parameters are consistently interpreted or evolved from
high redshift to low redshifts using a model of the area distance that
includes the impact of the tidal field or the area distance derived from a
perturbed FLRW spacetime as described in [36], the tensions in the Hubble
parameter would not arise.

Of course, we are not the first to claim that the effects of inhomogeneities
could resolve the Hubble tension. In fact it was argued in [111] that the
large scale outflows due to the presence of local voids could be the cause of
the Hubble tension and it could potentially can explain the late time cosmic
acceleration by dark energy [112]. The authors defined a spatial averaging
procedure following the Buchert averaging formalism [113]. They found that an
average under density of about $\langle\delta\rangle=-0.3$ is enough to
explain the discrepant Hubble rate. A more detailed study of this has shown
that this does work [114]. Secondly, the authors did not compute the average
values of what is observed rather, they calculated the average density on the
spatial hypersurface. An observer does not have access to the entire
hypersurface rather it has access only to the screen space (2D surface) due to
restriction imposed by the constant speed of light [115]. Similarly, it has
been conjectured that anisotropies in the super-horizon perturbations could
explain the Hubble tension. This approach has some fine-tuning issues to
resolve. This simplest model of this has three free parameters that have to be
chosen precisely for it to work [116].

##  V Conclusion

We have deployed the cosmological fitting approach introduced by Ellis and
Stoeger [45, 7] to study how well the FLRW spacetime fits an inhomogeneous
Universe on average. The FLRW model could be characterised by a set of free
parameters such as the Hubble rate, deceleration parameter, etc, we show how
to obtain the best-fit Hubble rate and the deceleration parameter given
generalised inhomogeneous models of the following observables: Alcock-
Paczyński parameters, magnitude-redshift relation, the intercept of the
distance modules.

We made use of the results of the low redshifts Taylor series expansion of the
generalised inhomogeneous expression of the area distance and luminosity
distance developed by Kristian and Sachs in 1966 [25]. Using the $1+3$
covariant decomposition formalism, we decomposed these expressions in terms of
irreducible observables with respect to a geodesic observer. In this limit,
the only key observables are the rate of expansion (a scalar) and the rate of
shear deformation tensor. The rate of expansion scalar describes the rate at
which nearby geodesics expands/contract with respect to a geodesic observer.
It corresponds to the Hubble rate in the FLRW limit. The shear tensor, on the
other hand, describes the rate of change of the spacetime deformation in the
neighbourhood of the observer [27]. It vanishes in the FLRW limit.

Using these tools, we derived the generalised inhomogeneous equations for the
Alcock-Paczyński parameters. The Alcock-Paczyński parameters constrain the
imprints of the Baryon Acoustic Oscillation in the galaxy distribution through
the N-point correlation function. These parameters are usually interpreted in
terms of the background FLRW model. We showed how to generalise the two
components of the Alcock-Paczyński parameters to an inhomogeneous spacetime
model. By comparing the monopole of the generalised radial component to the
corresponding FLRW counter-part, we find that the inferred Hubble rate is
biased by the impact of the tidal deformation tensor at the observer location.
The orthogonal part is proportional to the area distance, hence we compare the
monopole of the generalised area distance to the corresponding FLRW limit. At
the leading order in redshift, we find that the Hubble rate today is biased as
well by the scalar invariant of the shear tensor. At second order in redshift,
we obtain the generalised expression for the deceleration parameter which is
also biased by the scalar invariant of the shear tensor and the product of the
shear tensor and the electric part of the Weyl tensor.

Furthermore, we considered the magnitude-redshift relation. This is the key
observable from which the determination of the Hubble rate is made from the
peak magnitude of the type IA supernovae. We showed that the tidal deformation
tensor that biases the determination of the Hubble rate from the monopole of
the Alcock-Paczyński parameters, impacts the measurement of the supernova
absolute magnitude instead. The Hubble rate, in this case, is not biased, it
corresponds exactly to the rate of expansion scalar which describes the rate
of volume expansion/contraction. The deceleration parameter, however, is
biased by the same terms as in the case of the orthogonal component of the
Alcock-Paczyński parameter.

We quantified these terms within the cosmological perturbation theory assuming
Gaussian and adiabatic initial conditions. We showed that the expectation
value of these biasing terms is proportional to the smoothing scale-dependent
variance in the matter density field. We argued that the most physically
motivated smoothing scale corresponds to the comoving radius (causal horizon)
where the spacetime region goes from contracting to expanding phase. We use
the halo model to quantify this exactly. We find that it is slightly greater
than the splashback radius of the host halo. Setting the mass of our local
group to about$\sim(10^{11}-10^{12})M_{\odot}$, we find the smoothing scale in
the range of $R\sim(0.8-1.2){\rm{Mpc}}$. This leads to about$\sim(8-12)\%$
fractional difference in the determination of the Hubble rate between the
local measurements and the Alcock-Paczyński parameters. With the same range of
values for the smoothing scales, we find that it explains the supernova
absolute magnitude tension [100, 34] and the values obtained from recent
measurement of the deceleration parameter today using the Pantheon supernova
samples [35]. The interpretation of the inferred values of the deceleration
parameters should be taken with caution because cosmography in the presence of
structures at second order in redshift differs from the full expression by
more than 100% at about $z\sim 0.1$.

Finally, it is undeniable that the FLRW spacetime has played a big role in our
current understanding of the Universe. Our results show that the tidal
deformation of the observer spacetime region due to the gravitational
interaction among nearby structures cannot be neglected. This interaction is
characterised by tensors that vanish on the FLRW background spacetime. We have
shown that not taking these into account when using the FLRW model alone to
analyse cosmological observation leads to cosmological tensions. We showed
that the determination of Hubble rate from the analysis of the local distance
ladder for the SNIa peak magnitudes takes into account the impact of the tidal
deformation of the local spacetime consistently.

## Acknowledgement

I benefited immensely from discussions with Chris Clarkson, Robert Crittenden,
Pierre Fleury, Emir Gumrukcuoglu, Asta Heinesen, Mathew Hull, Antony Lewis,
Kazuya Koyama and Daniela Saadeh. The works by G.F.R. Ellis on this topic have
been very influential. OU is supported by the UK Science & Technology
Facilities Council (STFC) Consolidated Grants Grant ST/S000550/1 and the South
African Square Kilometre Array Project. The perturbation theory computations
in this paper were done with the help of tensor algebra software xPand [117]
which is based on xPert [118]. I made use of COLOSSUS (A python toolkit for
cosmology, large-scale structure, and dark matter halos) developed by Benedikt
Diemer for computations involving the dark matter halo profiles [91].

## Data Access Statement

No new data were generated or analysed in support of this research.

##  Appendix A Covariant cosmography on the past lightcone

The distance-redshift relation on arbitrary spacetime may be expanded in
Taylor series up to any order following the formalism developed by Kristain
and Sachs [25]. The formalism uses the null focusing equation to propagate the
deviation vector, $\xi^{a}$, with the tail on the central ray, $L$ and head on
the nearby geodesics from a source with 4-velocity $u^{a}_{s}$ to the observer
with 4-velocity $u^{a}_{o}$

$\frac{\rm{d}^{2}\xi^{a}}{\rm{d}\lambda^{2}}=-R^{a}{}_{cbd}k^{c}k^{d}\xi^{b}\,\,,$
(130)

where $R^{a}{}_{cbd}$ is the Riemann curvature tensor, $\xi^{a}$ lives on the
screen space and $k^{a}$ is the tangent vector to the null geodesics. The
solution to equation (130) needs to satisfy the following initial conditions
at source

$\displaystyle(\xi^{a})_{s}$ $\displaystyle=$ $\displaystyle
0,\qquad\left(\frac{\rm{d}\xi^{a}}{\rm{d}\lambda}u_{a}\right)_{s}=0,\qquad\left(\frac{\rm{d}\xi^{a}}{\rm{d}\lambda}k_{a}\right)_{s}=0,\qquad\pi\left(\frac{\rm{d}\xi^{a}}{\rm{d}\lambda}\frac{\rm{d}\xi_{a}}{\rm{d}\lambda}\right)_{s}=-(u_{a}k^{a})_{s}d\Omega\,.$
(131)

Equation (130) is valid provided there are no focal points between the
observer and the source. Expanding $(\xi_{a}\xi_{b})$ in Taylor series in
$\lambda$ up to fifth order, gives

$\displaystyle(\xi_{a}\xi_{b})_{o}$ $\displaystyle=$
$\displaystyle\left(\xi_{a}\xi_{b}\right)_{s}+\left[\frac{\rm{d}}{\rm{d}\lambda}(\xi_{a}\xi_{b})\right]_{s}\lambda_{o}+\frac{1}{2}\left[\frac{\rm{d}^{2}}{\rm{d}\lambda^{2}}(\xi_{a}\xi_{b})\right]_{s}\lambda_{o}^{2}+\mathcal{O}(\lambda_{o}^{3})\,.$
(132)

Using equation (130) and the boundary conditions (equation (131)), give

$\displaystyle(\xi_{a}\xi_{b})_{o}$ $\displaystyle=$
$\displaystyle\lambda_{o}^{2}\left(\frac{\rm{d}\xi^{c}}{\rm{d}\lambda}\frac{\rm{d}\xi_{c}}{\rm{d}\lambda}\right)_{s}\left(g_{ab}-\frac{1}{3}k^{c}k^{d}R_{acbd}\lambda_{o}^{2}+\mathcal{O}(\lambda_{o}^{3})\right)\,.$
(133)

The area element is defined $dA=\sqrt{(\xi_{a}\xi_{b})_{o}}d\Omega$, therefore

$\displaystyle dA_{o}$ $\displaystyle=$ $\displaystyle
d\Omega\lambda_{o}^{2}(u_{a}k^{a})^{2}_{s}\left(1-\frac{1}{6}k^{c}k^{j}R_{cj}\lambda_{o}^{2}+\mathcal{O}(\lambda_{2}^{3})\right)\,.$
(134)

To make contact with observation we expand the redshift in $\lambda$

$\displaystyle\left(1+z\right)=\frac{E_{s}}{E_{o}}$ $\displaystyle=$
$\displaystyle\frac{1}{(k^{a}u_{a})_{0}}\bigg{[}k^{a}u_{a}+k^{a}k^{b}\nabla_{a}u_{b}\lambda+\frac{1}{2}k^{a}k^{b}k^{c}\nabla_{a}\nabla_{b}u_{c}\lambda^{2}+\mathcal{O}(\lambda^{3})\bigg{]}\,.$
(135)

Using the fact that the area distance is the ratio of the cross-sectional area
at the observer to the cross-sectional area at the source according to

$d^{2}_{A}=\frac{dA_{0}}{d\Omega}\frac{(u_{a}k^{a})_{o}^{2}}{(u_{a}k^{a})^{2}_{s}}\,.$
(136)

Using equation (134) in equation (136) we find

$\displaystyle d^{2}_{A}$ $\displaystyle=$
$\displaystyle(u_{a}k^{a})_{0}^{2}\lambda_{0}^{2}\left\\{1-\frac{1}{6}k^{c}k^{d}R_{cd}\lambda_{0}^{2}+\mathcal{O}(\lambda_{0}^{3})\right\\}\,.$
(137)

Inverting the series is equation (137) gives the affine parameter in terms of
the area distance,

$\displaystyle\lambda_{0}$ $\displaystyle=$ $\displaystyle
d_{A}\left[1+\frac{1}{12}K^{c}K^{d}R_{cd}d^{2}_{A}+\mathcal{O}(d_{A}^{3})\right]\,,$
(138)

where we defined normalised photon tangent vector
$K^{a}:=\frac{k^{a}}{(u_{a}k^{a})_{0}}=-u^{a}+n^{a}$. Then putting equation
(138) in equation (135) gives

$\displaystyle z$ $\displaystyle=$
$\displaystyle\left.K^{a}K^{b}\nabla_{a}u_{b}\right|_{o}d_{A}+\frac{1}{2}\left.\left(K^{a}K^{b}K^{c}\nabla_{a}\nabla_{b}u_{c}\right)\right|_{o}d^{2}_{A}+\mathcal{O}(d_{A}^{3})\,.$
(139)

Inverting equation (139) gives the area distance in terms of the redshift

$\displaystyle d_{A}$ $\displaystyle=$
$\displaystyle\frac{z}{K^{c}K^{j}\nabla_{c}u_{j}|_{0}}\left\\{1-\left[\frac{1}{2}\frac{K^{c}K^{j}K^{k}\nabla_{k}\nabla_{j}u_{c}}{\left(K^{c}K^{j}\nabla_{c}u_{c}\right)^{2}}\right]_{0}z+\mathcal{O}(z^{2})\right\\}\,.$
(140)

Using the reciprocity theorem $d_{L}=d_{A}(1+z)^{2}$, we find the luminosity
distance to be

$\displaystyle d_{L}$ $\displaystyle=$
$\displaystyle\frac{z}{K^{c}K^{j}\nabla_{j}u_{c}|_{0}}\left\\{1+\frac{1}{2}\left[4-\frac{K^{c}K^{j}K^{k}\nabla_{k}\nabla_{j}u_{c}}{\left(K^{c}K^{j}\nabla_{c}u_{c}\right)^{2}}\right]_{0}z+\mathcal{O}(z^{2})\right\\}\,.$
(141)

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