# A scale-dependent distance functional between past-lightcones in cosmology

Mauro Carfora  University of Pavia  Italian National Group of Mathematical
Physics, and INFN Pavia Section<EMAIL_ADDRESS>and  Francesca
Familiari  University of Pavia  Italian National Group of Mathematical
Physics, and INFN Pavia Section<EMAIL_ADDRESS>
###### Abstract.

We discuss a rigorous procedure for quantifying the difference between our
past lightcone and the past lightcone of the fiducial Friedmann-Lemaitre-
Robertson-Walker spacetime modeling the large scale description of
cosmological data in the standard $\Lambda\mathrm{CDM}$ scenario. This result
is made possible by exploiting the scale-dependent distance functional between
past lightcones recently introduced by us in [12]. We express this harmonic
map type functional in terms of the physical quantities that characterize the
actual measurements along our past lightcone, namely the area distance and the
lensing distortion, also addressing the very delicate problem of the presence
of lightcone caustics. This analysis works beautifully and seems to remove
several of the difficulties encountered in comparing the actual geometry of
our past lightcone with the geometry of the fiducial FLRW lightcone of choice.
We also discuss how, from the point of view of the FLRW geometry, this
distance functional may be interpreted as a scale-dependent effective field,
the pre-homogeneity field, that may be of relevance in selecting the FLRW
model that best fits the observational data.

##  1\. INTRODUCTION

_It is a pleasure to dedicate this paper to Maurizio Gasperini who has always
liked it best on the
past light cone even if the routes are tough, but in such a rugged landscape
that is to be expected_

The $\Lambda\mathrm{CDM}$ model and the Friedman-Lemaitre-Robertson-Walker
(FLRW) spacetimes provide a rather accurate physical and geometrical
representation of the universe in the present era111Characterized by the
actual temperature of the cosmic microwave background $T_{CMB}\,=\,2.725\,K$
as measured in the frame centered on us but stationary with respect to the
CMB. and over spatial scales ranging from222 The actual averaging scale
marking the statistical onset of isotropy and homogeneity is still much
debated. For the sake of the argument presented in this paper, we adopt the
rather conservative estimate of the scales over which an average isotropic
expansion is seen to emerge, namely $70-120\,h^{-1}Mpc$, and ideally extending
to a few times this scale [54]. $\approx\,100\,h^{-1}\;\mathrm{Mpc}$ to the
visual horizon of our past light cone [27], [34], [50], where $h$ is the
dimensionless parameter describing the relative uncertainty of the true value
of the present-epoch Hubble-Lemaitre constant. Within such observational
range, and on scales significantly smaller than the Hubble scale333At the
Hubble scale, the problem of _cosmic variance_ may alter the statistical
significance of the data samples we gather., we have a testable ground for
statistical isotropy in the distribution of the dark and visible matter
components on our past light cone. Homogeneity of this distribution is
difficult to test directly via astronomical surveys, but a number of
observational results [41] and in particular the kinematic Sunyaev-Zeldovich
effect [55], [18] imply that fluctuations around spatial homogeneity cannot be
too large. Thus, without resorting to an axiomatic use of the Copernican
principle, we have an observational ground for assuming that spatial
homogeneity holds, in a statistically averaged sense, over large scales. It
must be stressed that it is in a statistical sense and only over large scales
that this weak form of the cosmological principle provides observational
support for best fitting the description of spacetime geometry in terms of a
member of the FLRW family of solutions of the Einstein equations. In
particular, to whatever degree one accepts this FLRW scenario, one has to
address the fact that the role of FLRW spacetime geometry becomes delicate to
interpret when past light cone data are gathered in our cosmological
neighborhood. As we probe spatial regions in the range
$\lesssim\,100h^{-1}\;\mathrm{Mpc}$, the actual distribution of matter (dark
and visible) becomes extremely anisotropic with a high density contrast. In
particular, gravitational clustering gives rise to a complex network of
structures, characterized by the presence of a foam-like web of voids and
galaxy filaments often extending well into the $100h^{-1}\;\mathrm{Mpc}$
range. At these scales, the Einstein evolution of the FLRW geometry uncouples
from the dynamics of the matter sources and survives more as a useful
computational assumption (often assisted by Newtonian theory) rather than as a
_bona fide_ perturbative background gravitationally determined by the actual
matter distribution. FLRW is thus a very strong assumption and not a correct
representation of spacetime geometry at the pre-homogeneity scales, not even
in a statistical sense. If we want or need to go beyond FLRW perturbation
theory and enter into a fully relativistic regime, it is fair to say that we
have little mathematical control over the actual spacetime at these pre-
homogeneity scales. In particular, the transition from the large-scale FLRW to
the actual inhomogeneous and anisotropic spacetime geometry emergent at these
local scales is poorly understood in a model-independent way, and the idea
that around $100h^{-1}\;\mathrm{Mpc}$ we have a gradual and smooth transition
between these two regimes is somewhat illusive. To wit, we may have non-
perturbative correction terms due to the coupling between gravitationally
bound structures and the emergent spacetime geometry (e.g. structure
formation-induced curvature) that can be significant in cosmological modeling.
For instance, they can back-react, in a top-down causation way [52], on the
choice of the large-scale FLRW spacetime that best fits the observational
data. This complex scenario gives rise to a number of delicate and to some
extent controversial issues that are currently much debated in discussing the
existence of possible tensions between cosmological observations and the
standard $\Lambda\mathrm{CDM}$ model and in preparation to the coming era of
high-precision cosmology [11]. Some of the very delicate reasons444We wish to
thank one of the referees for pointing this out to us. motivating this tension
is that large-scale isotropy can hold for a much wider class of models, the
so-called _effective model_ [30], [31] that need not even be a solution of
Einstein’s equations. As an illustrative example, one may consider
inhomogeneous spatial sections that can be smoothed into a constant-curvature
space, _e.g._ with Ricci flow deformation techniques [8, 9, 4, 5, 6, 10].
While spatially, such slices can be identified with spatial sections of a FLRW
model, their Einstein time-evolution in general does not follow the FLRW class
of solutions, a backreaction is present [4, 5, 6]. Thus, at least in
principle, one may actually deal with an effective model with global
backreaction that can be large-scale isotropic and homogeneous, or almost so,
and it is not necessarily perturbatively away from a FLRW model. Thus,
restricting a priori the ”best-fit” to the class of FLRW models is indeed a
strong assumption. By its very nature, a discussion of this very complex
scenario should be related, as far as possible, to a model–independent _direct
observational cosmology approach_ , namely to the analysis of data determined
on our past lightcone without using any theory of gravity. Since the dark
matter and dark energy components cannot be measured yet via direct
observations, it must be stressed that a full model-independent cosmographic
approach is not actually possible [21]. Model hypotheses must be imposed for
the dark components, in particular on how they interact with observed matter.
The simplest assumptions made are that the dark matter component follows the
baryonic component, namely that: i) we know the primordial ratio of cold dark
matter (CDM) density to baryonic density; ii) they have the same 4-velocity;
iii) we know their relative concentration in matter clusters. To these, one
typically adds the working assumption that the dark energy component is
described in the form of a cosmological constant $\Lambda$, the value of which
should be known from non–cosmological physics and independently from
cosmological observations (for a thorough discussion of the implications of
these assumptions in cosmography see Chap. 8 of [21]). However, although there
are efforts to derive $\Lambda$ from non-cosmological physics, it remains a
fitting parameter of the model. The appropriate cosmographical framework was
put forward in the ’80s by G.F.R. Ellis, R. Maartens, W. Stoeger, and A.
Whitman [20] (see also [21]) by characterizing the set of cosmological
observables on the past lightcone which, together with the Einstein field
equations, allows to reconstruct the spacetime geometry in a way adapted to
the process of observation [20], [15], [16].

In this paper we address an important step in this cosmographical framework.
In particular we discuss a rigorous procedure for quantifying the difference
between our past lightcone and the reference past lightcone that, for
consistency, we associate with the fiducial large-scale FLRW spacetime. This
result is made possible by exploiting the scale-dependent (harmonic map type)
distance functional between past lightcones recently introduced by us in [12],
and which extended the light-cone theorem [14]. We express this functional in
terms of the physical quantities that characterize measurements along our past
lightcone, namely the area distance and the lensing distortion, also briefly
addressing the very delicate problem of the presence of lightcone caustics.
This analysis works beautifully and seems to remove several of the
difficulties encountered in comparing the actual geometry of our past
lightcone with the geometry of a fiducial FLRW lightcone of choice. We also
discuss how, from the point of view of the FLRW geometry, this distance
functional may be interpreted as a scale-dependent effective field that may be
of relevance in selecting the FLRW model that best fits the observative data.
In this connection and in line with the introductory remarks above its
worthwhile to stress that our choice of a reference FLRW spacetime is strictly
related to the prevalence of this family of metrics in discussing the
$\Lambda\mathrm{CDM}$ model. The results presented here can be easily extended
to more general reference metrics. It is also important to make clear that in
this paper we are not addressing the extremely delicate averaging problem on
the past lightcone, a problem to which Maurizio Gasperini has significantly
contributed with the seminal paper [24], and that has seen importat recent
progress in [7]… _but the past lighcone routes are still tough and the
landscape rugged …_ .

##  2\. The past light cone and the celestial sphere

Throughout this paper $(M,g)$ denotes a cosmological spacetime where $g$ is a
Lorentzian metric, and where $M$ is a smooth $4$-dimensional manifold which
for our purposes we can assume diffeomorphic to $\mathbb{R}^{4}$ (or to
$V^{3}\times\mathbb{R}$, for some smooth compact or complete $3$–manifold
$V^{3}$). In local coordinates $\\{x^{i}\\}_{i=1}^{4}$, we write
$g=g_{ik}dx^{i}\otimes dx^{k}$, where the metric components
$g_{ik}\,:=\,g(\partial_{i},\partial_{k})$ in the coordinate basis
$\\{\partial_{i}:=\partial/\partial x^{i}\\}_{i=1}^{4}$ have the Lorentzian
signature $(+,+,+,-)$, and the Einstein summation convention is in effect555If
not otherwise stated we adopt geometrical units, $c\,=\,1\,=\,G$.. We assume
that $(M,g)$ is associated with the evolution of a universe which is
(statistically) isotropic and homogeneous on sufficiently large scales
$L\,>\,L_{0}$ where, according to the introductory remarks, we indicatively
assume $L_{0}\,\cong\,100h^{-1}\;\mathrm{Mpc}$, and let local inhomogeneities
dominate for $L\,<\,L_{0}$. The matter content in $(M,g)$ is
phenomenologically described by a (multi-component) energy-momentum tensor
$T\,=\,T_{ik}\,dx^{i}\otimes dx^{k}$, (typically in the form of a perfect
fluid, dust, and radiation). If not otherwise stated, the explicit expression
of $T$ is not needed for our analysis. We assume that in $(M,g)$ the motion of
the matter components characterize a _phenomenological Hubble flow_ that
generates a family of preferred world-lines parametrized by proper time $\tau$

(1) $\displaystyle\gamma_{s}\,:\,\mathbb{R}_{>0}\,$
$\displaystyle\longrightarrow$ $\displaystyle\,(M,g)$ $\displaystyle\tau\,$
$\displaystyle\longmapsto$ $\displaystyle\,\gamma_{s}(\tau)\;,$

and labeled by suitable comoving (Lagrangian) coordinates $s$ adapted to the
flow. We denote by $\dot{\gamma}_{s}\,:=\,\frac{d\gamma_{s}(\tau)}{d\tau}$,
$g(\dot{\gamma}_{s},\dot{\gamma}_{s})\,=\,-1$, the corresponding $4$-velocity
field. For simplicity, we assume that at the present era these worldlines are
geodesics, i.e. $\nabla_{\dot{\gamma}_{s}}\,\dot{\gamma}_{s}\,=\,0$. This
phenomenological Hubble flow is strongly affected by the peculiar motion of
the astrophysical sources and by the complex spacetime geometry that dominates
on the pre-homogeneity scales. In particular, it exhibits a complex pattern of
fluctuations with respect to the linear _FLRW Hubble flow_ that sets in,
relatively to the standard of rest provided by the cosmic microwave background
(CMB), when we probe the homogeneity scales,
$L\,\gtrsim\,100h^{-1}\;\mathrm{Mpc}$. Again, we stress that the transitional
region between the phenomenological Hubble flow and the statistical onset of
the large-scale FLRW linear Hubble flow is quite uncertain and still actively
debated [54]. If we adopt the weak form of the cosmological principle
described in the introduction, $(M,g,\gamma_{s})$ can be identified with the
phenomenological background spacetime or Phenomenological Background Solution
(PBS)[39] associated with the actual cosmological data gathered from our past
lightcone observations. In the same vein, we define Phenomenological Observers
the collection of observers $\\{\gamma_{s}\\}$ comoving with the
phenomenological Hubble flow (1). Since in our analysis we fix our attention
on a given observer, we drop the subscript $s$ in (1), and describe a finite
portion of the observer’s world-line with the timelike geodesic segment
$\tau\,\longmapsto\,\gamma(\tau)$, $-\delta<\tau<\delta$, for some $\delta>0$,
where $p\,:=\,\gamma(\tau=0)$ is the selected event corresponding to which the
cosmological data are gathered. To organize and describe these data in the
local rest frame of the observer $p\,:=\,\gamma(\tau=0)$, let
$\left(T_{p}M,\,g_{p},\,\\{E_{(i)}\\}\right)$ be the tangent space to $M$ at
$p$ endowed with a $g$-orthonormal frame $\\{E_{(i)}\\}_{i=1,\ldots,4}$,
$g_{p}\left(E_{(i)},E_{(k)}\right)=\eta_{ik}$, where $\eta_{ik}$ is the
Minkowski metric, and where we identify $E_{(4)}$ with the observer
$4$-velocity $\dot{\gamma}(\tau)|_{\tau=0}$,  _i.e._
$E_{(4)}\,:=\,\dot{\gamma}(\tau)|_{\tau=0}$. Thus, if we denote by
$\\{\breve{E}^{\,(i)}\\}_{i=1,\ldots,4}$, the $1$-forms basis dual to
$\\{E_{(i)}\\}_{i=1,\ldots,4}$, we write

(2) $\displaystyle
g_{p}\,=\,\eta_{ik}\,\breve{E}^{\,(i)}\,\otimes\,\breve{E}^{\,(k)}\,.$

Since we have the distinguished choice
$E_{(4)}\,:=\,\dot{\gamma}(\tau)|_{\tau=0}$ for the timelike basis vector
$E_{(4)}$, we can also introduce in $\left(T_{p}M,\,\\{E_{(i)}\\}\right)$ a
reference positive definite metric $g^{(\delta)}_{p}$ associated with the
frame $\\{E_{(i)}\\}_{i=1,\ldots,4}$ by setting

(3) $\displaystyle
g^{(\delta)}_{p}\,:=\,\delta_{ik}\,\breve{E}^{\,(i)}\,\otimes\,\breve{E}^{\,(k)}\,,$

where $\delta_{ik}$ denote the components of the standard Euclidean metric. As
discussed in detail by Chen and LeFloch [13], this reference metric comes in
handy in the characterization of the functional Lipschitz and Banach space
norms of tensor fields defined on the past lightcone666The indefinite
character of a Lorentzian metric makes it unsuitable for defining integral
norms of tensor fields, and for such a purpose one is forced to introduce a
reference positive definite metric. In particular, by exploiting the Nash
embedding theorem, one typically uses the Euclidean metric and the associated
definitions of the functional space of choice, say a Sobolev space of tensor
fields. Different choices of reference metrics, as long as they are of
controlled geometry, induce equivalent Banach space norms. In our case, we can
exploit the natural choice provided by (3) by using normal coordinates and
identifying $(T_{p}M,\,\\{E_{(i)}\\},\,g_{p}^{\delta})$ with the Euclidean
space $(\mathbb{R}^{4},\,g_{p}^{\delta})$..

###  2.1. The celestial sphere

Let

(4)
$C^{-}\left(T_{p}M,\,\\{E_{(i)}\\}\right)\,:=\,\left\\{X\,=\,\mathbb{X}^{i}E_{(i)}\,\not=\,0\,\in\,T_{p}M\,\,|\,g_{p}(X,X)\,=\,0,\,\,\mathbb{X}^{4}+r=0\right\\}\;,$
(5)
$\overline{C^{-}\left(T_{p}M,\,\\{E_{(i)}\\}\right)}\,:=\,\left\\{X\,=\,\mathbb{X}^{i}E_{(i)}\,\not=\,0\,\in\,T_{p}M\,\,|\,g_{p}(X,X)\,\leq\,0,\,\,\mathbb{X}^{4}+r\,\leq\,0\right\\}\;,$

respectively denote the set of past-directed null vectors and the set of past-
directed causal vectors in $(T_{p}M,\,\\{E_{(i)}\\})$, where

(6) $r:=(\sum_{a=1}^{3}(\mathbb{X}^{a})^{2})^{1/2}\,,$

is the radial coordinate in the hyperplane
$\mathbb{X}^{4}\,=\,0\,\subset\,T_{p}M$ parametrizing the one-parameter family
of $2$-spheres

(7) $\mathbb{S}^{2}_{r}(T_{p}M)\,:=\,\\{X\in
C^{-}\left(T_{p}M,\\{E_{(i)}\\}\right)\,|\,\,\mathbb{X}^{4}\,=\,-\,r,\,\,\,\sum_{a=1}^{3}(\mathbb{X}^{a})^{2}=r^{2},\,\,r\in\,\mathbb{R}_{>0}\\}\,,$

that foliates $C^{-}\left(T_{p}M,\\{E_{(i)}\\}\right)/\\{p\\}$. The sphere
$\mathbb{S}^{2}_{r}(T_{p}M)$ can be thought of as providing a representation
of the sky directions, at a given value of $r$, in the rest space
$\left(T_{p}M,\\{E_{(i)}\\}\right)$ of the (instantaneous) observer
$(p,\dot{\gamma}(0))$. In particular, the 2-sphere
$\left.\mathbb{S}^{2}_{r}(T_{p}M)\right|_{r=1}$ or, equivalently, its
projection on the hyperplane $\mathbb{X}^{4}\,=\,0$ in $T_{p}M$,

(8)
$\mathbb{S}^{2}\left(T_{p}M\right)\,:=\,\left\\{X\,=\,\mathbb{X}^{i}E_{(i)}\,\not=\,0\,\in\,T_{p}M\,\,|\,\,\mathbb{X}^{4}=0,\,\,\sum_{a=1}^{3}(\mathbb{X}^{a})^{2}=1\right\\}\;,$

can be used to parametrize the (spatial) past directions of sight constituting
the field of vision of the observer $(p,\,\dot{\gamma}(0))$. In the sense
described by R. Penrose [46], this is a representation of the abstract sphere
$\mathcal{S}^{-}(p)$ of past null directions parameterizing the past-directed
null geodesics through $p$. Explicitly, let

$\displaystyle{n}(\theta,\phi)\,$ $\displaystyle:=$
$\displaystyle\,\sum_{a=1}^{3}\,{n}^{a}(\theta,\phi)\,{E}_{(a)}$
$\displaystyle=$
$\displaystyle\,\cos\phi\sin\theta\,{E}_{(1)}\,+\,\sin\phi\sin\theta\,{E}_{(2)}\,+\,cos\,\theta\,{E}_{(3)}\,,\,\,\,\,0\leq\theta\leq\pi,\,\,0\leq\phi<2\pi\,,$

denote the spatial direction in $T_{p}M$ associated with the point
$(\theta,\phi)\,\in\,\mathbb{S}^{2}\left(T_{p}M\right)$, (by abusing notation,
we often write ${n}(\theta,\phi)\,\in\,\mathbb{S}^{2}\left(T_{p}M\right)$).
Any such spatial direction characterizes a corresponding past-directed null
vector $\ell(\theta,\phi)\,\in\,\left(T_{p}M,\\{E_{(i)}\\}\right)$,

(10)
$\ell(\theta,\phi)\,=\,\left(n(\theta,\,\phi),\,-\,\dot{\gamma}(\tau)|_{\tau=0}\right)\,=\,\sum_{a=1}^{3}\,n^{a}(\theta,\phi)E_{(a)}\,-\,E_{(4)}\,,$

normalized according to

(11)
$g_{p}\left(\ell(\theta,\phi),\dot{\gamma}(\tau)|_{\tau=0}\right)\,=\,g_{p}\left(\ell(\theta,\phi),E_{(4)}\right)\,=\,1\,.$

The corresponding past-directed null rays

(12) $\mathbb{R}_{\geq
0}\,\ni\,r\,\longmapsto\,r\,\ell({n}(\theta,\phi))\,,\,\,\,\,\,\,\,\,(\theta,\phi)\,\in\,\mathbb{S}^{2}\left(T_{p}M\right)\,,$

generate $C^{-}\left(T_{p}M,\\{E_{(i)}\\}\right)$. Note that in such a
kinematical setup for the instantaneous rest space
$\left(T_{p}M,\,\\{E_{(a)}\\}\right)$ of the observer $(p,\,\dot{\gamma}(0))$,
a photon reaching $p$ from the past-directed null direction
$\ell(\theta,\phi)$, is characterized by the (future-pointing) wave vector

(13) $k(\theta,\phi)\,:=\,-\,{\nu}\,\ell(\theta,\phi)\,\in\,T_{p}M\,,$

where $\nu\,=\,-\,g_{p}\left(k,\,E_{(4)}\right)$ is the photon frequency as
measured by the instantaneous observer ${\gamma}(\tau)|_{\tau=0}$. The
spherical surface $\mathbb{S}^{2}\left(T_{p}M\right)$ endowed with the
standard round metric

(14)
$\widetilde{h}(\mathbb{S}^{2})\,=\,d\theta^{2}\,+\,\sin^{2}\theta\,d\phi^{2}\,,$

and the associated area form
$d\mu_{\mathbb{S}^{2}}\,=\,\sqrt{\det(\widetilde{h}(\mathbb{S}^{2}))}\,d\theta
d\phi\,=\,\sin\theta\,d\theta d\phi$, defines [46] the _celestial sphere_

(15)
$\mathbb{C\,S}(p)\,:=\,\left(\mathbb{S}^{2}\left(T_{p}M\right),\,\widetilde{h}(\mathbb{S}^{2})\right)\,$

providing, in the instantaneous rest space
$\left(T_{p}M,\\{E_{(i)}\\}\right)$, the geometrical representation of the set
of all directions towards which the observer can look at astrophysical sources
from her instantaneous location in $(M,g)$. In this connection,
$d\mu_{\mathbb{S}^{2}}$ can be interpreted as the element of solid angle
subtended on the celestial sphere $\mathbb{C\,S}(p)$ by the observed
astrophysical sources. It is also useful to keep track of the radial
coordinate777To avoid any misunderstanding we stress that $r$ is not a
distance parameter on the past light cone with vertex in $p\in(M,g)$. $r$ as a
possible parametrization of the past-directed null geodesics, and introduce a
celestial sphere that provides also this information according to

(16)
$\mathbb{C\,S}_{r}(p)\,:=\,\left(\mathbb{S}^{2}_{r}\left(T_{p}M\right),\,r^{2}\widetilde{h}(\mathbb{S}^{2})\right)\,.$

Lacking a better name, we shall refer to $\mathbb{C\,S}_{r}(p)$ as the
_celestial sphere at radius $r$_ in $\left(T_{p}M,\\{E_{(i)}\\}\right)$. The
celestial sphere $\mathbb{C\,S}(p)$ plays a basic role in what follows since
it provides the logbook where astrophysical data are recorded.
Let $m_{(\alpha)}(\theta,\phi)\in\,T_{p}M$, with $\alpha\,=\,2,3$, denote two
spatial $g_{p}$-orthonormal vectors spanning the tangent space
$T_{(\theta,\phi)}\mathbb{S}^{2}\left(T_{p}M\right)$ to
$\mathbb{S}^{2}\left(T_{p}M\right)$ at the point $(\theta,\phi)$, _i.e._ ,

(17)
$g_{p}\left(m_{(\alpha)},n\right)\,=\,0\,=\,g_{p}\left(m_{(\alpha)},E_{(4)}\right),\,\,g_{p}\left(m_{(\alpha)},m_{(\beta)}\right)\,=\,\delta_{\alpha\beta}\,.$

The tetrad

(18) $\left(n,m_{(2)},m_{(3)},\ell(n)\right)$

provides a basis for $T_{p}M$ (the Sachs basis), and the pair
$\left(T_{(\theta,\phi)}\mathbb{S}^{2}\left(T_{p}M\right),\;m_{(\alpha)}(\theta,\phi)\right)$
defines the _screen plane_ $T_{n}\mathbb{C\,S}(p)$ associated with the
direction of sight ${n}(\theta,\phi)\in\,\mathbb{C\,S}(p)$ in the celestial
sphere $\mathbb{C\,S}(p)$, _i.e._

(19)
$T_{n}\mathbb{C\,S}(p)\,:=\,\left(T_{(\theta,\phi)}\mathbb{S}^{2}\left(T_{p}M\right),\;m_{(\alpha)}(\theta,\phi)\right)\,.$

In the instantaneous rest space of the observer, the screen
$T_{(\theta,\phi)}\mathbb{C\,S}(p)$ is the (spatial) $2$-plane on which the
apparent image of the astrophysical source, pointed by the direction
$n\in\mathbb{C\,S}(p)$, is by convention displayed.

###  2.2. Sky sections and observational coordinates on the past light cone

We transfer the above kinematical setup from $T_{p}M$ to $(M,g)$ by using the
exponential map based at $p$,

(20) $\displaystyle\exp_{p}\,:\,W_{p}\,\subseteq\,T_{p}M\,$
$\displaystyle\longrightarrow$ $\displaystyle\;\;\;\;M$ $\displaystyle
X\;\;\;\;\;\;$ $\displaystyle\longmapsto$
$\displaystyle\;\;\;\;exp_{p}\,(X)\,:=\,\lambda_{X}(1)\,,$

where $\lambda_{X}\,:\,I_{W}\,\longrightarrow\,(M,g)$, for some maximal
interval $I_{W}\subseteq\mathbb{R}_{\geq 0}$, is the past-directed causal
geodesic emanating from the point $p$ with initial tangent vector
$\dot{\lambda}_{X}(0)\,=\,X\in W_{p}$, and where $W_{p}\,\subseteq\,T_{p}M$ is
the maximal domain of $\exp_{p}$. Thus, the past lightcone
$\mathscr{C}^{-}(p,g)\,\in\,(M,g)$ with the vertex at $p$, i.e. the set of all
events $q\in(M,g)$ that can be reached from $p$ along the past-pointing null
geodesics $r\,\longmapsto\,\exp_{p}(r\ell(n(\theta,\phi)))$, $r\in\,I_{W}$,
$(\theta,\phi)\in\mathbb{C\,S}(p)$, can be represented as

(21) $\mathscr{C}^{-}(p,g)\,:=\,\exp_{p}\left[W_{p}\cap
C^{-}\left(T_{p}M,g_{p}\right)\right]\,,$

and the portion of $\mathscr{C}^{-}(p,g)$ accessible to observations for a
given value $r_{0}\,\in\,I_{W}$ of the affine parameter $r$ is given by

$\mathscr{C}^{-}(p,g;\,r_{0})\,:=\,\left\\{\left.q\,\in\,M\,\right|\,\,q\,=\,\exp_{p}(r\ell(n(\theta,\phi))),\,\,0\leq
r<r_{0},\,\,(\theta,\phi)\in\mathbb{C\,S}(p)\right\\}\,.$

The exponential map representation, on the celestial spheres
$\mathbb{C\,S}(p)$ and $\mathbb{C\,S}_{r}(p)$, provides a natural setup for a
description of observational data gathered from $\mathscr{C}^{-}(p,g)$. It
emphasizes the basic role of past-directed null geodesics and provides the
framework for interpreting the physical data in the local rest frame of the
observer at $p$. In particular, it allows us to represent on
$\mathbb{C\,S}(p)$ and $\mathbb{C\,S}_{r}(p)$ the actual geometry of the
observed sky at a given length scale. This role is quite effective in a
neighborhood of $p$, where we can introduce normal coordinates associated with
$\exp_{p}$, but it is delicate to handle in regions where $\exp_{p}$ is not a
diffeomorphism of $W_{p}\cap C^{-}\left(T_{p}M,g_{p}\right)$ onto its image.
To set notation, our strategy is to start with the standard description [20],
[21] of observational coordinates on $\mathscr{C}^{-}(p,g)$ associated with
the usual assumption that the exponential map is a diffeomorphism888From an
observational point of view, this is the geometrical set-up proper of the
_weak lensing regime_ describing the alteration, due to the effect of gravity,
of the apparent shape and brightness of astrophysical sources. in a
sufficiently small neighborhood of $p$, and then we move to the more general,
low regularity, Lipschitz case. In this connection, it is worthwhile to stress
that the standard normal coordinates description is strictly associated with
the assumption that the metric of $(M,g)$ is sufficiently regular, with
components $g_{ij}(x^{\ell})$ which are at least twice continuously
differentiable, _i.e._
$g_{ij}(x^{\ell})\,\in\,C^{k}(\mathbb{R}^{4},\mathbb{R})$, for $k\geq 2$.
Under this hypothesis, there is a star-shaped neighborhood $N_{0}(g)$ of $0$
in $W_{p}\subseteq T_{p}M$ and a corresponding geodesically convex
neighborhood of $p$, $U_{p}\subseteq\,(M,g)$, restricted to which
$\exp_{p}\,:\,N_{0}\,\subseteq\,T_{p}M\,\longrightarrow\;U_{p}\,\subseteq\,M$
is a diffeomorphism. In such $U_{p}$ we can introduce geodesic normal
coordinates $(x^{i})$ according to

(22) $\displaystyle
x^{i}\,:=\,\mathbb{X}^{i}\,\circ\,\exp_{p}^{-1}\,:\,M\cap\,U_{p}\,$
$\displaystyle\longrightarrow$ $\displaystyle\,\mathbb{R}^{4}$ $\displaystyle
q\,\,\,\,\,\,\,$ $\displaystyle\longmapsto$
$\displaystyle\,x^{i}(q)\,:=\,\mathbb{X}^{i}\left(\exp_{p}^{-1}(q)\right)$

where $\mathbb{X}^{i}\left(\exp_{p}^{-1}(q)\right)$ are the components, in the
$g$-orthonormal frame $\\{E_{(i)}\\}$, (or with respect to the corresponding
basis (18)), of the vector $\exp_{p}^{-1}(q)\,\in\,W_{p}\subseteq T_{p}M$.
Thus, in $\mathscr{C}^{-}(p,g)\cap\,U_{p}$ we can write,

(23) $\displaystyle\exp_{p}\
\,:\,C^{-}\left(T_{p}M,\,\\{E_{(i)}\\}\right)\cap\,N_{0}(g)\,$
$\displaystyle\longrightarrow$
$\displaystyle\,\mathscr{C}^{-}(p,g)\,\cap\,U_{p}$ $\displaystyle
r\ell(n(\theta,\phi))\,=\,r\left(n^{a}(\theta,\phi)E_{(a)}\,-\,E_{(4)}\right)\,\,\,\,\,\,\,\,\,\,$
$\displaystyle\longmapsto$ $\displaystyle\,\exp_{p}(r\ell(n))\,=\,q$
$\displaystyle\Longrightarrow\,q$ $\displaystyle\longmapsto$
$\displaystyle\,\\{x^{i}(q)\,:=\,\exp_{p}^{-1}(q)\,=\,\left(r\,n^{a}(\theta,\phi),\,-\,r\right)\\}\,\;.$

According to (21) and to the Gauss lemma applied to
$\exp_{p}\,:\,C^{-}\left(T_{p}M,\,\\{E_{(i)}\\}\right)\cap\,N_{0}(g)\,\longrightarrow\,\mathscr{C}^{-}(p,g)\,\cap\,U_{p}$,
the past ligh cone region $\mathscr{C}^{-}(p,g)\,\cap\,U_{p}\setminus\\{p\\}$
is foliated by the $r$-dependent family of 2–dimensional surfaces
$\Sigma(p,r)$, the _cosmological sky sections_ , defined by

(24)
$\Sigma(p,r)\,:=\,\exp_{p}\left[\mathbb{C\,S}_{r}(p)\right]\,=\,\left\\{\left.\exp_{p}\left(r\,\ell({n}(\theta,\phi))\right)\,\right|\,\,(\theta,\phi)\,\in\,\mathbb{C\,S}(p)\right\\}\,,$

and $g$-orthogonal to all null geodesics originating at $p$, _i.e._

(25)
$\left.g\left(d_{(r,\theta,\phi)}\exp_{p}(\ell(r,\underline{n})),\,d_{(r,\theta,\phi)}\exp_{p}(\underline{v})\right)\right|_{\exp_{p}(\ell(r,\underline{n}))}\,=\,0\,.$

Here $d_{(r,\theta,\phi)}\exp_{p}(...)$ denotes the tangent mapping associated
to $\exp_{p}$ evaluated at the point
$(\theta,\phi)\,\in\,\mathbb{S}^{2}_{r}(p)$, and
$\underline{v}\,\in\,T_{\theta,\phi}\,\mathbb{S}^{2}_{r}(p)$ is the generic
vector tangent to $\mathbb{S}^{2}_{r}(p)$. In
$\mathscr{C}^{-}(p,g)\,\cap\,U_{p}\setminus\\{p\\}$, each surface
$\Sigma(p,r)$ is topologically a 2-sphere endowed with the $r$-dependent two-
dimensional Riemannian metric

(26)
$g|_{\Sigma(p,r)}\,:=\,\iota_{r}^{*}\,\left.g\right|_{\mathscr{C}^{-}(p,g)}$

induced by the inclusion
$\iota_{r}:\Sigma(p,r)\,\hookrightarrow\,\mathscr{C}^{-}(p,g)$ of
$\Sigma(p,r)$ into $\mathscr{C}^{-}(p,g)\,\cap\,U_{p}\setminus\\{p\\}$. We can
pull back this metric to the celestial sphere
$\mathbb{C\,S}_{r}(p):=\,\left(\mathbb{S}^{2}_{r}\left(T_{p}M\right),\,r^{2}\widetilde{h}(\mathbb{S}^{2})\right)$
by using the exponential map according to

(27)
$h(r,\theta,\phi)\,:=\,\left.\left(\exp_{p}^{*}\,g|_{\Sigma(p,r)}\right)_{\alpha\beta}\,dx^{\alpha}dx^{\beta}\right|_{r},\,\,\,\,\alpha,\,\beta\,=\,2,3,\,\,\,\,\,\,x^{2}:=\theta,\,x^{3}:=\phi\,.$

This metric can be profitably compared with the pre-existing round metric
$r^{2}\widetilde{h}(\mathbb{S}^{2})$ on $\mathbb{C\,S}_{r}(p)$ (see (14) and
(16)). To this end, let $r\,n(\theta,\phi)\,\in\,\mathbb{C\,S}_{r}(p)$ be the
direction of sight pointing, in the celestial sphere $\mathbb{C\,S}_{r}(p)$,
to the (extended) astrophysical source located around the point
$q\,\in\,\Sigma(p,r)$. If
$r\ell(n(\theta,\phi))\,=\,r\left(n^{a}(\theta,\phi)E_{(a)}\,-\,E_{(4)}\right)$
is the corresponding null direction in
$C^{-}\left(T_{p}M,\,\\{E_{(i)}\\}\right)$, then according to (23) we have
$\exp_{p}(r\ell(n))\,=\,q$ and, via the exponential map along the past-
directed null geodesic reaching the observer located at $p$ from the
astrophysical source located at $q$, we can pull-back the area element of
$\left(\Sigma(p,r),\,g|_{\Sigma(p,r)}\right)$ on the celestial sphere
$\mathbb{C\,S}_{r}(p)$ of the observer at $p$. We have

(28)
$d\mu_{h(r)}(p,n(\theta,\phi),r)\,:=\,\exp_{p}^{*}d\mu_{g|_{\Sigma(p,r)}}\,\circ\,\exp_{p}(r\ell(n))\,=\,\sqrt{\det(h(r,\theta,\phi))}\,d\theta
d\phi\,.$

This defines the area element associated with the metric (27), and can be
interpreted [21] as the cross-sectional area element at the source location as
seen by the observer at $p$. Since the round measure
$d\mu_{\mathbb{S}^{2}_{r}}=r^{2}\,d\mu_{\mathbb{S}^{2}}=r^{2}\,\sin\theta\,d\theta\,d\varphi$
and the actual physical measure $d\mu_{h(r)}$ are both defined over the
celestial sphere $\mathbb{C\,S}_{r}(p)\in\,T_{p}M$, we can introduce the
relative density of $d\mu_{h(r)}$ with respect to the Euclidean solid angle
measure $d\mu_{\mathbb{S}^{2}}$, _viz._ the function $D(r,\theta,\phi)$
defined by the relation

(29) $d\mu_{h(r)}\,=\,D^{2}(r,\theta,\phi)\,d\mu_{\mathbb{S}^{2}}\,,$

or equivalently,
$\sqrt{\det(h(r,\theta,\phi))}\,=\,D^{2}(r,\theta,\phi)\,\sqrt{\det(\widetilde{h}(\mathbb{S}^{2}))}$.
The function $D(r,\theta,\phi)$ is _the observer area distance_ [20], [21],
[33]. By definition, it provides the ratio of an object’s cross sectional area
to its (apparent) angular size as seen on the celestial sphere
$\mathbb{S}^{2}(p)\,\subset\,T_{p}M$. Roughly speaking, it converts the
angular separations as seen in the images of an astrophysical source, gathered
by the observer at $p$, into proper separations at the source. In general,
$D(r)\,:=\,\left.D(r,\theta,\phi)\right|_{\theta,\phi=const.}$ cannot be used
as an affine parameter along the past-directed null geodesic
$r\,\mapsto\,\exp_{p}(k(r,\underline{n}))$ since it is not a monotonic
function of $r$, (for instance in FLRW models, monotonicity fails around
$z\,\sim\,1$). However, if we have an accurate knowledge of the brightness and
of the spectrum of the astrophysical source seen at the past light cone
location $q\,:=\,\exp_{p}(\ell(r,\underline{n}))\,\in\,\mathscr{C}^{-}(p,g)$,
then $D(r,\theta,\phi)$ is, at least in principle, a measurable quantity (see
paragraph 4.3 of [20] and 7.4.3 of [21] for a discussion of this
point999Beware that in [20], the observer area distance $D^{2}(r,\theta,\phi)$
is denoted by $r$, whereas our $r$ corresponds to their $y$.). As stressed
above, we can also compare the physical metric (27),
$h(r,\theta,\phi)\,:=\,\left.\left(\exp_{p}^{*}\,g|_{\Sigma(p,r)}\right)_{\alpha\beta}\,dx^{\alpha}dx^{\beta}\right|_{r}$,
with the round metric $r^{2}\widetilde{h}(\mathbb{S}^{2})$ pre-existing on the
celestial sphere $\mathbb{C\,S}_{r}(p)$, and introduce [20], [21] the set of
functions $\mathcal{L}_{\alpha\beta}(r,\theta,\phi)$,
$\alpha,\,\beta\,=\,2,3$, implicitly defined by representing (27) in the
distorted polar form

(30)
$\left.h_{\alpha\beta}\right|_{\mathbb{S}^{2}_{r}}\,=\,D^{2}(r,\theta,\phi)\left(\widetilde{h}_{\alpha\beta}(\mathbb{S}^{2})\,+\,\mathcal{L}_{\alpha\beta}\right)\;.$

We normalize this representation by imposing [20] that, in the limit
$r\,\searrow\,0$, the distortion,
$\mathcal{L}_{\alpha\beta}(r,\theta,\phi)=\frac{h_{\alpha\beta}(r,\theta,\phi)}{D^{2}(r,\theta\,\phi}-\widetilde{h}_{\alpha\beta}(\mathbb{S}^{2})$,
of the normalized metric $h(r)/D^{2}(r)$ with respect to the round metric
$\widetilde{h}(\mathbb{S}^{2})$ goes to zero uniformly, i.e.,

(31) $\left.\lim_{r\searrow
0}\right|_{x^{4}=0}\,\frac{h_{\alpha\beta}(r,\theta,\phi)\,dx^{\alpha}dx^{\beta}}{D^{2}(r,\theta,\phi)}\,=\,d\theta^{2}\,+\,\sin^{2}\theta\,d\phi^{2}\;.$

From the relation
$D^{-\,2}\,h_{\alpha\beta}\,=\,\widetilde{h}_{\alpha\beta}(\mathbb{S}^{2})\,+\,\mathcal{L}_{\alpha\beta}$
we also compute

(32)
$D^{-\,2}\,\widetilde{h}^{\mu\beta}\,h_{\alpha\beta}=\,\delta_{\alpha}^{\mu}\,+\,\mathcal{L}_{\alpha}^{\mu}\,\Longrightarrow\,\det\left(\delta_{\alpha}^{\mu}\,+\,\mathcal{L}_{\alpha}^{\mu}\right)\,=\,1\,,$

where, for rising indexes, we used the inverse round metric
$\widetilde{h}^{\mu\beta}(\mathbb{S}^{2})$ to write
$\mathcal{L}_{\alpha}^{\mu}:=\,\widetilde{h}^{\mu\beta}(\mathbb{S}^{2})\,\mathcal{L}_{\alpha\beta}$,
and where we have exploited the relation
$\det\left({\widetilde{h}^{\mu\beta}\,h_{\alpha\beta}}\right)\,=\,D^{4}$,
direct consequence of
${\det(h)}\,=\,D^{4}\,\det(\widetilde{h}(\mathbb{S}^{2}))$ (see (29)). Since

(33)
$\,\det\left(\delta_{\alpha}^{\mu}\,+\,\mathcal{L}_{\alpha}^{\mu}\right)\,=\,1\,+\,\mathrm{tr}_{\mathbb{S}^{2}}\left(\mathcal{L}_{\alpha}^{\mu}\right)\,+\,\det\left(\mathcal{L}_{\alpha}^{\mu}\right)\,,$

from relation (32) it follows that

(34)
$\mathrm{tr}_{\mathbb{S}^{2}}\left(\mathcal{L}_{\alpha}^{\mu}\right)\,+\,\det\left(\mathcal{L}_{\alpha}^{\mu}\right)\,=\,0\,,$

which implies that $\mathcal{L}_{\alpha}^{\mu}$ cannot be trace-free. Roughly
speaking, $\mathcal{L}_{\alpha\beta}(r)$ can be interpreted as the image
distortion of the sources on $\left(\Sigma(p,r),h(r)\right)$ as seen by the
observer at $p$ on her celestial sphere. It can in principle be directly
observed and it can be related to the gravitational lensing shear [20], (see
also chap. 8 of [21]). Explicitly, let us compute the deformation tensor
$\Theta_{\alpha\beta}$ defined by the rate of variation of the metric tensor
$h(r)$ as $r$ varies. Dropping the angular dependence for notational ease, we
get

$\displaystyle\Theta_{\alpha\beta}\,:=\,\frac{d}{dr}\,h_{\alpha\beta}(r)\,$
$\displaystyle=$
$\displaystyle\,\frac{d}{dr}\,\left[D^{2}(r)\left(\widetilde{h}_{\alpha\beta}(\mathbb{S}^{2})\,+\,\mathcal{L}_{\alpha\beta}(r)\right)\right]$
$\displaystyle=$ $\displaystyle\,2h_{\alpha\beta}(r)\,\frac{d}{dr}\,\ln
D(r)\,+\,D^{2}(r)\,\frac{d}{dr}\,\mathcal{L}_{\alpha\beta}(r)\,,$

where we exploited $d\widetilde{h}_{\alpha\beta}(\mathbb{S}^{2})/dr\,=\,0$ and
rewrote $D(r)dD(r)/dr$ as $D^{2}(r)d\ln\,D(r)/dr$. Similarly, from the
defining relation
$\sqrt{\det(h(r,\theta,\phi))}\,=\,D^{2}(r,\theta,\phi)\,\sqrt{\det(\widetilde{h}(\mathbb{S}^{2}))}$,
(see (29)), we compute

(36) $\displaystyle\frac{d}{dr}\,\sqrt{\det(h(r))}\,$ $\displaystyle=$
$\displaystyle\,\frac{d}{dr}\,\left(D^{2}(r)\,\sqrt{\det(\widetilde{h}(\mathbb{S}^{2}))}\right)\,=\,2\,\sqrt{\det(h(r))}\,\frac{d}{dr}\,\ln
D(r)$ $\displaystyle\Rightarrow\,\,\frac{d}{dr}\,\ln\sqrt{\det(h(r))}\,$
$\displaystyle=$ $\displaystyle\,2\,\frac{d}{dr}\,\ln D(r)\,.$

Inserting this relation in (2.2) we obtain

(37)
$\Theta_{\alpha\beta}\,:=\,h_{\alpha\beta}(r)\,\frac{d}{dr}\,\ln\sqrt{\det(h(r))}\,+\,D^{2}(r)\,\frac{d}{dr}\,\mathcal{L}_{\alpha\beta}(r)\,.$

The shear $\widetilde{\sigma}_{\alpha\beta}$ is the trace-free part of this
expression,
$\widetilde{\sigma}_{\alpha\beta}:=\Theta_{\alpha\beta}-\frac{1}{2}h_{\alpha\beta}\,h^{\mu\nu}\Theta_{\mu\nu}$.
Since

(38)
$\frac{1}{2}h_{\alpha\beta}\,h^{\mu\nu}\Theta_{\mu\nu}\,=\,\frac{1}{2}h_{\alpha\beta}\,h^{\mu\nu}\frac{d}{dr}h_{\mu\nu}\,=\,h_{\alpha\beta}\,\frac{d}{dr}\,\ln\sqrt{\det(h(r))}\,,$

we eventually get

(39)
$\widetilde{\sigma}_{\alpha\beta}\,=\,D^{2}(r)\,\frac{d\mathcal{L}_{\alpha\beta}(r)}{dr}\;,$

as might have been expected. Note that, in contrast to
$\mathcal{L}_{\alpha\beta}$, $\widetilde{\sigma}_{\alpha\beta}$ is trace-free
(but with respect to the physical metric $h_{\alpha\beta}$). Now, let us
introduce the other basic player of our narrative.

##  3\. The background FLRW past light cone.

As already pointed out, the standard $\Lambda$CDM model is built on the
assumption that over scales $L\,>\,100\,h^{-1}\,\mathrm{Mpc}$, the
phenomenological background spacetime $(M,g,\gamma_{s})$ follows on average
the dynamics of a FLRW model with a (linear) Hubble expansion law. It is also
assumed that below the scale of statistical homogeneity, deviations from this
average scenario can be described by FLRW perturbation theory. Since there is
no smooth transition between the large-scale FLRW Hubble flow and the
phenomenological Hubble flow, this latter assumption rests on quite delicate
ground. For instance, the field of peculiar velocities
$\\{\dot{\gamma}_{s}(\tau)\\}$ of the phenomenological observers
$\left\\{\tau\longrightarrow\gamma_{s}(\tau)\right\\}$ shows a significant
statistical variance [53] with respect to the average FLRW Hubble flow and the
standard of rest provided by the cosmic microwave background (CMB). This
remark has an important effect on the relation between the celestial sphere
$\mathbb{C\,S}_{r}(p)$ of the phenomenological observer
$(p,\,\dot{\gamma}(0))$ and the corresponding celestial sphere
$\widehat{\mathbb{C\,S}}_{\widehat{r}}(p)$ of the idealized FLRW observer
$(p,\,\widehat{\dot{\gamma}}(0))$. They cannot be identified and must be
connected by a Lorentz boost that takes into account the origin of this
statistical variance. The actual scenario is significantly constrained by the
coupling of the matter inhomogeneities with a spacetime geometry that is no
longer Friedmannian. As a consequence, the peculiar velocity field of the
phenomenological observer may have a rather complex origin, and its variance
with respect to the FLRW average expansion may become a variable of relevance
in cosmography. This scenario naturally calls into play a delicate comparison
between the geometry of $\mathcal{C}^{-}(p,g)$ and the geometry of the
associated FLRW past light cone that sets in at scales
$L\,>\,100\,h^{-1}\,\mathrm{Mpc}$. For this purpose, along with the physical
metric $g$, we consider on the spacetime manifold $M$ a reference FLRW metric
$\hat{g}$ and the associated family of global Friedmannian observers
$\hat{\tau}\longmapsto\hat{\gamma}_{s}(\hat{\tau})$. Strictly speaking, the
FLRW model $(M,\hat{g},\hat{\gamma}_{s}(\hat{\tau}))$ should be used only over
the scales $L\,>\,L_{0}\simeq\,100\,h^{-1}\,\mathrm{Mpc}$. We need to consider
it also over the inhomogeneity scales $L\,<\,L_{0}$ where it plays the role of
the geometrical background used to interpret the data according to the
standard perturbative FLRW point of view recalled above. In such an extended
role, the chosen FLRW is the Global Background Solution (GBS according to
[39]) we need to check against the physical metric $g$ representing the
phenomenological background solution. In this section, we set up the
kinematical aspects for such a comparison. First some standard verbiage for
introducing the FLRW model $(M,\hat{g},\hat{\gamma}_{s}(\hat{\tau}))$. In
terms of the radial, and angular FRLW coordinates
$y^{\alpha}\,:=\,\left(\hat{r},\hat{\theta},\hat{\varphi}\right)$, and of the
proper time of the comoving fundamental observers $y^{4}\,:=\,\hat{\tau}$, we
set

$\displaystyle\widehat{g}\,$ $\displaystyle:=$
$\displaystyle\,-d\hat{\tau}^{2}\,+\,a^{2}(\hat{\tau})\,\left[d\hat{r}^{2}\,+\,f^{2}(\hat{r})\,\left(d\hat{\theta}^{2}\,+\,\sin^{2}\hat{\theta}\,d\hat{\varphi}^{2}\right)\right]\,,\;\;\;\;\;\widehat{\dot{\gamma}}^{h}\,=\,\delta^{h}_{4},$
$\displaystyle f(\hat{r})\,$ $\displaystyle:=$
$\displaystyle\,\begin{cases}&\sin\,\hat{r},\;\;\;k=+1\\\
&\hat{r},\;\;\;\;\;\;\;\;\;k=0\\\ &\sinh\,\hat{r},\;\;k=-1\,,\end{cases}$

where $a(\hat{\tau})$ is the time-dependent scale factor, $k$ is the
normalized dimensionless spatial curvature constant, and
$\widehat{\dot{\gamma}}^{h}$ are the components of the $4$-velocity
$\widehat{\dot{\gamma}}$ of the fundamental FLRW observers. According to the
above remarks, the geodesics ${\tau}\longmapsto{\gamma}({\tau})$, and
$\hat{\tau}\longmapsto\hat{\gamma}(\hat{\tau})$,
$-\delta<\,\tau,\,\hat{\tau}<\delta$, associated with the corresponding Hubble
flow in $(M,g,\gamma)$ and $(M,\hat{g},\hat{\gamma})$, are assumed to be
distinct, but in line with the scale-dependent cosmographic approach adopted
here we assume that they share a common observational event $p\,\in\,M$. We
denote by $\widehat{\mathscr{C}}^{-}(p,\hat{g})$ the associated FLRW past
light cone, and normalize the proper times $\tau$ and $\hat{\tau}$ along
${\gamma}({\tau})$ and $\hat{\gamma}(\hat{\tau})$ so that at
$\tau\,=\,0\,=\,\hat{\tau}$ we have $\gamma(0)\,=\,p\,=\,\hat{\gamma}(0)$. As
stressed, the two instantaneous observers $(p,\dot{\gamma}(0))$ and
$(p,\widehat{\dot{\gamma}}(0))$ have different 4-velocities,
$\dot{\gamma}(0)\not=\widehat{\dot{\gamma}}(0)$, and their respective
celestial spheres, $\mathbb{C\,S}(p)$ and $\widehat{C\,\mathbb{S}}^{2}(p)$ are
quite distinct. They are related by a Lorentz trasformation describing the
aberration of the sky mapping of one instantaneous observer with respect to
the other. This mapping will play a basic role in our analysis, and to provide
an explicit description of its properties, we start by adapting to the FLRW
instantaneous observer
$(p,\widehat{\dot{\gamma}}(0))\,\in\,(M,\hat{g},\hat{\gamma})$ the setup
characterizing the celestial spheres $\mathbb{C\,S}(p)$ and
$\mathbb{C\,S}_{r}(p)$ of the instantaneous observer
$(p,\dot{\gamma}(0))\,\in\,(M,g,\gamma)$.

###  3.1. The FLRW celestial sphere and the associated sky sections

Let
$\left(\widehat{T}_{p}M,\,\widehat{g}_{p},\,\\{\widehat{E}_{(i)}\\}\right)$ be
the tangent space to $(M,\hat{g},\hat{\gamma})$ at $p$ endowed with a
$\widehat{g}$-orthonormal frame $\\{\widehat{E}_{(i)}\\}_{i=1,\ldots,4}$,
$\widehat{g}_{p}\left(\widehat{E}_{(i)},\widehat{E}_{(k)}\right)=\eta_{ik}$,
where $\eta_{ik}$ is the Minkowski metric, and where we identify
$\widehat{E}_{(4)}$ with the FLRW-observer’s $4$-velocity
$\widehat{\dot{\gamma}}(\tau)|_{\tau=0}$,  _i.e._
$\widehat{E}_{(4)}\,:=\,\widehat{\dot{\gamma}}(\tau)|_{\tau=0}$. For ease of
notation, we shall often use the shorthand $\widehat{T}_{p}M$ when referring
to the tangent space to $(M,\hat{g},\hat{\gamma})$ at $p$. Let

(41)
$C^{-}\left(\widehat{T}_{p}M,\,\\{\widehat{E}_{(i)}\\}\right)\,:=\,\left\\{Y\,=\,\mathbb{Y}^{i}\widehat{E}_{(i)}\,\not=\,0\,\in\,\widehat{T}_{p}M\,\,|\,\widehat{g}_{p}(Y,Y)\,=\,0,\,\,\mathbb{Y}^{4}+\widehat{r}=0\right\\}\;,$
(42)
$\overline{C^{-}\left(\widehat{T}_{p}M,\,\\{\widehat{E}_{(i)}\\}\right)}\,:=\,\left\\{Y\,=\,\mathbb{Y}^{i}\widehat{E}_{(i)}\,\not=\,0\,\in\,\widehat{T}_{p}M\,\,|\,\widehat{g}_{p}(Y,Y)\,\leq\,0,\,\,\mathbb{Y}^{4}+\widehat{r}\,\leq\,0\right\\}\;,$

respectively denote the set of past-directed null vectors and the set of past-
directed causal vectors in $(\widehat{T}_{p}M,\,\\{\widehat{E}_{(i)}\\})$,
where $\widehat{r}:=(\sum_{a=1}^{3}(\mathbb{Y}^{a})^{2})^{1/2}$ is the radial
coordinate (see (6)) in the hyperplane
$\mathbb{Y}^{4}\,=\,0\,\subset\,\widehat{T}_{p}M$ parametrizing the one-
parameter family of $2$-spheres

(43) $\mathbb{S}^{2}_{\widehat{r}}(\widehat{T}_{p}M)\,:=\,\\{Y\in
C^{-}\left(\widehat{T}_{p}M,\\{\widehat{E}_{(i)}\\}\right)\,|\,\,\mathbb{Y}^{4}\,=\,-\,\widehat{r},\,\,\,\sum_{a=1}^{3}(\mathbb{Y}^{a})^{2}=\widehat{r}^{2},\,\,\widehat{r}\in\,\mathbb{R}_{>0}\\}\,,$

that foliate
$C^{-}\left(\widehat{T}_{p}M,\\{\widehat{E}_{(i)}\\}\right)/\\{p\\}$. The
$2$-spheres $\mathbb{S}^{2}_{\widehat{r}}(\widehat{T}_{p}M)$, endowed with the
round metric

(44)
$\widehat{\widetilde{h}}(\mathbb{S}^{2})\,=\,\widehat{\widetilde{h}}_{\alpha\beta}(\mathbb{S}^{2})dy^{\alpha}dy^{\beta}\,=\,d\widehat{\theta}^{2}\,+\,\sin^{2}\widehat{\theta}\,d\widehat{\phi}^{2}\,,\,\,\,\,0\leq\widehat{\theta}\leq\pi,\,\,0\leq\widehat{\phi}<2\pi\,$

can be thought of as providing a representation of the sky, at a given value
of the radial coordinate $\widehat{r}$, in the instantaneous rest space
$\left(\widehat{T}_{p}M,\\{\widehat{E}_{(i)}\\}\right)$ of the FLRW observer.
In analogy with the characterization (8) of the celestial sphere
$\mathbb{C\,S}(p)$, we use the projection of
$\left.\mathbb{S}^{2}_{\widehat{r}}(\widehat{T}_{p}M)\right|_{\widehat{r}=1}$
on the hyperplane $\mathbb{Y}^{4}\,=\,0$ in $\widehat{T}_{p}M$, to define the
FLRW _celestial sphere_

(45)
$\widehat{\mathbb{C\,S}}(p)\,\left(\left.\mathbb{S}^{2}_{\widehat{r}}(\widehat{T}_{p}M)\right|_{\widehat{r}=1},\,\widehat{\widetilde{h}}(\mathbb{S}^{2})(p)\right)\,:=\,\left\\{Y\,=\,\mathbb{Y}^{i}E_{(i)}\,\not=\,0\,\in\,\widehat{T}_{p}M\,\,|\,\,\mathbb{Y}^{4}=0,\,\,\sum_{a=1}^{3}(\mathbb{Y}^{a})^{2}=1\right\\}\;,$

parametrizing the directions of sight

(46)
$\widehat{n}(\widehat{\theta},\widehat{\phi})\,:=\,(\cos\widehat{\phi}\sin\widehat{\theta},\,\sin\widehat{\phi}\sin\widehat{\theta},\,cos\,\widehat{\theta})\,,\,\,\,\,0\leq\widehat{\theta}\leq\pi,\,\,0\leq\widehat{\phi}<2\pi\,$

in the instantaneous rest space
$\left(\widehat{T}_{p}M,\\{\widehat{E}_{(i)}\\}\right)$ of the FLRW observer.
In full analogy with (16), we define the FLRW _celestial sphere at radius
$\widehat{r}$_ in $\left(\widehat{T}_{p}M,\\{\widehat{E}_{(i)}\\}\right)$
according to

(47)
$\widehat{\mathbb{C\,S}}_{\widehat{r}}(p)\,:=\,\left(\mathbb{S}^{2}_{\widehat{r}}\left(\widehat{T}_{p}M\right),\,\widehat{r}^{2}\widehat{\widetilde{h}}(\mathbb{S}^{2}(p))\right)\,.$

With a straightforward adaptation to the FLRW geometry of the definitions
(10), (18), and (19), we also introduce in $\widehat{T}_{p}M$ the tetrad

(48)
$\left(\widehat{n},\widehat{m}_{(2)},\widehat{m}_{(3)},\widehat{\ell}(\widehat{n})\right)$

and associate with the pair
$\left(\widehat{T}_{(\widehat{\theta},\widehat{\phi})}\mathbb{S}^{2}\left(\widehat{T}_{p}M\right),\;\widehat{m}_{(\alpha)}(\widehat{\theta},\widehat{\phi})\right)$
the _screen plane_ $T_{\widehat{n}}\widehat{\mathbb{C\,S}}(p)$ associated with
the direction of sight $\widehat{n}(\widehat{\theta},\widehat{\phi})$ in the
FLRW celestial sphere $\widehat{\mathbb{C\,S}}(p)$,

(49)
$T_{\widehat{n}}\widehat{\mathbb{C\,S}}(p)\,:=\,\left(T_{(\widehat{\theta},\widehat{\phi})}\mathbb{S}^{2}\left(\widehat{T}_{p}M\right),\;\widehat{m}_{(\alpha)}(\widehat{\theta},\widehat{\phi})\right)\,.$

Together with the observational normal coordinates $\\{X^{i}\\}$ in
$(M,g,\gamma)$, describing the local geometry on the past lightcone
$\mathscr{C}^{-}(p,g)\,\cap\,U_{p}$, we introduce corresponding (normal)
coordinates $\\{Y^{k}\\}$ on the past light cone
$\widehat{\mathscr{C}}^{-}(p,\hat{g})$ in the reference FLRW spacetime
$(M,\hat{g},\hat{\gamma})$. To begin with, let $\widehat{\exp}_{p}$ denote the
exponential mapping based at the event $p=\hat{\gamma}(0)$, _i.e._

(50)
$\displaystyle\widehat{\exp}_{p}\,:\,\widehat{W}_{p}\,\subseteq\,\widehat{T}_{p}M\,$
$\displaystyle\longrightarrow$ $\displaystyle\;\;\;\;(M,\hat{g}),$
$\displaystyle\mathbb{Y}\;\;\;\;\;\;$ $\displaystyle\longmapsto$
$\displaystyle\;\;\;\;exp_{p}\,(\mathbb{Y})\,:=\,\lambda_{\mathbb{Y}}(1)\,,$

where $\widehat{W}_{p}$ is the maximal domain of $\widehat{\exp}_{p}$. To keep
on with the notation set by (21) and (2.2), we characterize the past lightcone
$\widehat{\mathscr{C}}^{-}(p,\hat{g})\,\in\,(M,\widehat{g})$, with vertex at
$p$, according to

(51)
$\widehat{\mathscr{C}}^{-}(p,\hat{g})\,:=\,\widehat{\exp}_{p}\left[\widehat{W}_{p}\cap
C^{-}\left(\widehat{T}_{p}M,\widehat{g}_{p}\right)\right]\,,$

and we denote by

$\widehat{\mathscr{C}}^{-}(p,\widehat{g};\,\widehat{r}_{0})\,:=\,\left\\{\left.q\,\in\,M\,\right|\,\,q\,=\,\widehat{\exp}_{p}(\widehat{r}\widehat{\ell}(\widehat{n}(\widehat{\theta},\widehat{\phi}))),\,\,0\leq\widehat{r}<\widehat{r}_{0},\,\,(\widehat{\theta},\widehat{\phi})\in\widehat{\mathbb{C\,S}}(p)\right\\}\,,$

the portion of $\widehat{\mathscr{C}}^{-}(p,\hat{g})$ accessible to
observations for a given value $\widehat{r}_{0}$ of the radial parameter
$\widehat{r}$. That said, if $\hat{U}_{p}\subset\,(M,\hat{g})$ denotes the
region of injectivity of $\widehat{\exp}_{p}$, then normal coordinates are
defined by

(52)
$y^{i}\,:=\,\mathbb{Y}^{i}\,\circ\,\widehat{\exp}_{p}^{-1}\,:\,(M,\widehat{g})\cap\,\widehat{U}_{p}\,\longrightarrow\,\mathbb{R}\,,$

where $\mathbb{Y}^{i}$ are the components of the vectors
$\mathbb{Y}\in\,\widehat{T}_{p}M$ with respect to a $\hat{g}$-orthonormal
frame $\\{\hat{E}_{(i)}\\}_{i=1,\ldots,4}$ with
$\hat{E}_{(4)}\,:=\,\hat{\dot{\gamma}}(0)$. We can parametrize
$\widehat{\mathscr{C}}^{-}(p,\widehat{g})\,\cap\,\widehat{U}_{p}$ in terms of
the 2-dimensional FLRW sky sections

(53)
$\widehat{\Sigma}(p,\hat{r})\,:=\,\widehat{\exp}_{p}\left[\widehat{\mathbb{C\,S}}_{\widehat{r}}(p)\right]\,=\,\left\\{\left.\widehat{\exp}_{p}\left(\widehat{r}\,\widehat{\ell}(\widehat{n}(\widehat{\theta},\widehat{\phi}))\right)\,\right|\,\,(\widehat{\theta},\widehat{\phi})\,\in\,\widehat{\mathbb{C\,S}}(p)\right\\}\;,$

endowed with the metric induced by the inclusion of
$\widehat{\Sigma}(p,\hat{r})$ into $\widehat{\mathscr{C}}^{-}(p,\hat{g})$, _i.
e._

(54)
$\left.\widehat{g}\right|_{\widehat{\Sigma}(p,\hat{r})}\,:=\,\left.(\widehat{g})_{\alpha\beta}\,dy^{\alpha}dy^{\beta}\right|_{\hat{r}}\,=\,a^{2}(\widehat{\tau}(\widehat{r}))\,f^{2}\left(\widehat{r}\right)\left(d\widehat{\theta}^{2}\,+\,\sin^{2}\widehat{\theta}d\widehat{\phi}^{2}\right)\;,$

where $a(\widehat{\tau}(\widehat{r}))$ is the FLRW expansion factor
$a(\widehat{\tau})$ (see (LABEL:FLRWg)) evaluated in correspondence of the
given value of the radial coordinate $\widehat{r}\in\widehat{T}_{p}M$. We
proceed as in Subsection 2.2, and exploit the exponential map
$\widehat{\exp}_{p}$ to pull back
$\left.\widehat{g}\right|_{\widehat{\Sigma}(p,\hat{r})}$ on the celestial
sphere $\widehat{\mathbb{C\,S}}_{\widehat{r}}(p)$,

(55)
$\widehat{h}(\widehat{r},\widehat{\theta},\widehat{\phi})\,:=\,\left.\left(\widehat{\exp}_{p}^{*}\,\widehat{g}|_{\widehat{\Sigma}(p,\widehat{r})}\right)_{\alpha\beta}\,dy^{\alpha}dy^{\beta}\right|_{\widehat{r}},\,\,\,\,\alpha,\,\beta\,=\,2,3,\,\,\,\,\,\,y^{2}:=\widehat{\theta},\,y^{3}:=\widehat{\phi}\,.$

This pull-back can be explicitly computed. To wit, let
$y^{i}_{q}=(\widehat{r}_{q},\widehat{\theta}_{q},\widehat{\phi}_{q},\widehat{\tau}_{q})$
the normal coordinates of the event
$q\in\,\widehat{\mathscr{C}}^{-}(p,\hat{g})$ associated with the observation
of a given astrophysical source. The equation for the radial, past-directed,
null geodesic connecting $q$ to the observation event $p$ reduces in the FLRW
case to [19]

(56)
$d\widehat{r}\,=\,-\,\frac{d\widehat{\tau}}{a(\widehat{\tau})}\,,\,\,\,\,\widehat{\tau}(p)\,=\,0\,=\,\widehat{r}(p)\,,$

that integrates to the expression providing the (matter-comoving) radial
coordinate distance between $p$ and $q$

(57)
$\widehat{r}_{q}\,=\,\int_{0}^{\widehat{\tau}_{q}}\,\frac{d\widehat{\tau}}{a(\widehat{\tau})}\,.$

Thus, the metric (55), evaluated at $\widehat{\exp_{p}}^{-1}(q)$, can be
written in terms of $\widehat{\tau}_{q}$ as

(58)
$\widehat{h}_{q}\,:=\,\widehat{h}(\widehat{r}_{q},\widehat{\theta}_{q},\widehat{\phi}_{q})\,=\,a^{2}(\widehat{\tau}_{q})\,f^{2}\left(\widehat{r}_{q}\right)\left(d\widehat{\theta}_{q}^{2}\,+\,\sin^{2}\widehat{\theta}_{q}d\widehat{\phi}_{q}^{2}\right)\;,$

If we introduce the dimensionless FLRW cosmological redshift corresponding to
the event $q$,

(59)
$z_{q}\,:=\,z\left(\widehat{\tau}_{q}\right)\,=\,\frac{a_{0}}{a(\widehat{\tau}_{q})}\,-\,1\,,$

where $a_{0}\,:=\,a(\widehat{\tau}=0)$, then we can rewrite
$\widehat{h}(\widehat{r}_{q},\widehat{\theta}_{q},\widehat{\phi}_{q})$ as

(60)
$\widehat{h}_{q}\,=\,\frac{a^{2}_{0}}{(1\,+\,z_{q})^{2}}\,f^{2}\left(\widehat{r}_{q}\right)\left(d\widehat{\theta}_{q}^{2}\,+\,\sin^{2}\widehat{\theta}_{q}d\widehat{\phi}_{q}^{2}\right)\;.$

Note that the area element associated with the metric $\widehat{h}_{q}$,

(61)
$d\mu_{\widehat{h}_{q}}\,=\,\frac{a^{2}_{0}}{(1\,+\,z_{q})^{2}}\,f^{2}\left(\widehat{r}_{q}\right)\,d\mu_{\mathbb{S}^{2}}\,$

characterizes the FLRW _observer area distance_ (see (29)) of the event
$q\in\,\widehat{\mathscr{C}}^{-}(p,\hat{g})$ according to

(62)
$\widehat{D}(\widehat{r}_{q})\,=\,\frac{a_{0}}{1\,+\,z_{q}}\,f\left(\widehat{r}_{q}\right)\,.$

##  4\. Comparing the celestial spheres $\mathbb{C\,S}(p)$ and
$\widehat{\mathbb{C\,S}}(p)$

As stressed in the previous Section, the celestial sphere $\mathbb{C\,S}(p)$
of the phenomenological observer $(p,\dot{\gamma}(0))$, and the celestial
sphere $\widehat{\mathbb{C\,S}}(p)$ of the FLRW ideal observer
$(p,\widehat{\dot{\gamma}}(0))$ cannot be directly identified as they stand.
The velocity fields $\dot{\gamma}(0)$ and $\widehat{\dot{\gamma}}(0)$ are
distinct and to compensate for the induced aberration, the celestial spheres
$\mathbb{C\,S}(p)$ and $\widehat{\mathbb{C\,S}}(p)$ can be identified only up
to Lorentz boosts. In the standard FLRW view, this is the familiar global
boost taking care of the kinematical dipole component in the CMB spectrum due
to our peculiar motion with respect to the standard of rest provided by the
CMB. However, in a cosmographic setting and presence of a complex pattern of
local inhomogeneities coupled with a non-FLRW spacetime geometry over scales
$\lesssim\,100h^{-1}\;\mathrm{Mpc}$, the peculiar motion of the
phenomenological observer has a dynamical origin, driven by the gravitational
interaction and not just by a kinematical velocity effect. Even if we factor
out the effect of coherent bulk flows due to the non-linear local
gravitational dynamics, and average the rate of expansion over spherical
shells at increasing distances from $(p,\,\dot{\gamma}(0))$, the variance in
the peculiar velocity of $(p,\,\dot{\gamma}(0))$ with respect to the average
rate of expansion is significant [54]. These remarks imply that the Lorentz
boosts connecting $\mathbb{C\,S}(p)$ and $\widehat{\mathbb{C\,S}}(p)$ acquire
a dynamical meaning that plays a basic role in what follows. As a first step,
we describe the Lorentz boost in the idealized pure kinematical situation
where we need to compensate for a well-defined velocity field of the celestial
sphere $\mathbb{C\,S}(p)$ with respect to the celestial sphere
$\widehat{\mathbb{C\,S}}(p)$ taken as providing a well-defined standard of
rest. As a second step, we move to the more general setting required in the
pre-homogeneity region where we sample scales
$\lesssim\,100h^{-1}\;\mathrm{Mpc}$. In this latter case, a pure kinematical
Lorentz boost will not suffice, the large fluctuations in the sources
distribution require a suitable localization of the Lorentz boosts to compare
the data on $\mathbb{C\,S}(p)$ with those on $\widehat{\mathbb{C\,S}}(p)$.

###  4.1. The kinematical setting

To describe a kinematical Lorentz boost acting between
$\widehat{\mathbb{C\,S}}(p)$ and $\mathbb{C\,S}(p)$, we find it convenient to
use in this section the well-known correspondence between the restricted
Lorentz group and the six-dimensional projective special linear group
$\mathrm{PSL}(2,\mathbb{C})$ describing the automorphisms of the Riemann
sphere $\mathbb{S}^{2}\,\simeq\,\mathbb{C}\,\cup\,\\{\infty\\}$. More
expressively, $\mathrm{PSL}(2,\mathbb{C})$ can be viewed as the group of the
conformal transformations of the celestial spheres that correspond to the
restricted Lorentz transformations connecting $\mathbb{C\,S}(p)$ to
$\widehat{\mathbb{C\,S}}(p)$. In oder to set notation, let us recall that the
elements of $\mathrm{PSL}(2,\mathbb{C})$ can be identified with the set of the
Möbius transformations of the Riemann sphere
$\mathbb{S}^{2}\,\simeq\,\mathbb{C}\,\cup\,\\{\infty\\}$, _i.e._ the
fractional linear transformations of the form

(63) $\displaystyle\zeta\,:\,\mathbb{C}\,\cup\,\\{\infty\\}\,$
$\displaystyle\longrightarrow$ $\displaystyle\,\mathbb{C}\,\cup\,\\{\infty\\}$
$\displaystyle w\,$ $\displaystyle\longmapsto$
$\displaystyle\,\zeta(w)\,:=\,\frac{aw+b}{cw+d}\,,\,\,\,\,\,a,b,c,d\,\in\,\mathbb{C}\,,\,\,\,ad\,-\,bc\,\not=\,0\,,$

where, to avoid a notational conflict with the redshift parameter $z$, we have
labeled the complex coordinate in $\mathbb{C}\,\cup\,\\{\infty\\}$ with $w$
rather than with the standard $z$. Let
$Y\,=\,\widehat{n}(\widehat{\theta},\widehat{\phi})$ denote a point on the
celestial sphere $\widehat{\mathbb{C\,S}}(p)$, and let $\widehat{w}$ denote
its stereographic projection101010From the north pole
$\theta\,=\,0\in\,\widehat{\mathbb{C\,S}}(p)$. on the Riemann sphere
$\mathbb{C}\cup\,\\{\infty\\}$, _i.e._ ,

(64)
$\displaystyle\mathcal{P}_{\mathbb{S}^{2}}\,:\,\widehat{\mathbb{C\,S}}(p)\,\longrightarrow\,\mathbb{C}\cup\\{\infty\\}$
$\displaystyle\mathbb{Y}^{\alpha}\,\longmapsto\,\mathcal{P}_{\mathbb{S}^{2}}(\mathbb{Y}^{\alpha})\,=\,\widehat{w}\,:=\,\frac{\mathbb{Y}^{1}+i\,\mathbb{Y}^{2}}{1-\mathbb{Y}^{3}}\,=\,\frac{\cos\widehat{\phi}\sin\widehat{\theta}\,+\,i\,\sin\widehat{\phi}\sin\widehat{\theta}}{1\,-\,\cos\widehat{\theta}}\,,\
$

with $0<\theta\leq\pi,\,\,0\leq\phi<2\pi$. It is worthwhile to stress once
more that the celestial spheres $\widehat{\mathbb{C\,S}}(p)$ and
${\mathbb{C\,S}}(p)$ play the role of a mapping frame, a celestial globe where
astrophysical positions are registered, and where the Lorentz boost
$\widehat{\mathbb{C\,S}}(p)\,\longrightarrow\,{\mathbb{C\,S}}(p)$ must be
interpreted actively as affecting only the recorded astrophysical data. In
other words, the Lorentz boost affects the null directions in
$\widehat{\mathbb{C\,S}}(p)$, mapping them in the corresponding directions in
${\mathbb{C\,S}}(p)$. To quote a few illustrative examples [46] of the
$\mathrm{PSL}(2,\mathbb{C})$ transformations associated to the Lorentz group
action between the celestial spheres $\widehat{\mathbb{C\,S}}(p)$ and
${\mathbb{C\,S}}(p)$, let $v$ denote the modulus of the relative $3$-velocity
of the FLRW ideal observer $(p,\,\widehat{\dot{\gamma}}(0))$ with respect to
the phenomenological observer $(p,\,\dot{\gamma}(0))$, (where ${E}^{4}$ is
identified with the observer’s 4-velocity ${\dot{\gamma}}(0)$). If the map
between $\widehat{\mathbb{C\,S}}(p)$ and ${\mathbb{C\,S}}(p)$ is a pure
Lorentz boost in a common direction, say $E^{3}$, then the associated
$\mathrm{PSL}(2,\mathbb{C})$ transformation is provided by

(65)
$\displaystyle\mathrm{PSL}(2,\mathbb{C})\times\widehat{\mathbb{C\,S}}(p)\,$
$\displaystyle\longrightarrow$ $\displaystyle\,{\mathbb{C\,S}}(p)$
$\displaystyle\left(\zeta_{\,boost},\,\widehat{w}\right)\,$
$\displaystyle\longmapsto$
$\displaystyle\,\zeta(\widehat{w})\,=\,w\,=\,\sqrt{\frac{1\,+\,v}{1\,-\,v}}\,\widehat{w}\,,$

where $\sqrt{\frac{1\,+\,v}{1\,-\,v}}$ is the relativistic Doppler factor and
$w$ is the point in the Riemann sphere corresponding, under stereographic
projection, to the direction $n(\theta,\phi)\in{\mathbb{C\,S}}(p)$. Similarly,
if $\widehat{\mathbb{C\,S}}(p)$ and ${\mathbb{C\,S}}(p)$ differ by a pure
rotation through an angle $\alpha$ about the $E^{3}$ direction, then the
associated $\mathrm{PSL}(2,\mathbb{C})$ transformation is given by

(66)
$\displaystyle\mathrm{PSL}(2,\mathbb{C})\times\widehat{\mathbb{C\,S}}(p)\,$
$\displaystyle\longrightarrow$ $\displaystyle\,{\mathbb{C\,S}}(p)$ (67)
$\displaystyle\left(\zeta_{\,rot},\,\widehat{w}\right)\,$
$\displaystyle\longmapsto$
$\displaystyle\,\zeta(\widehat{w})\,=\,w\,=\,e^{i\,\alpha}\,\widehat{w}\,.$

By composing them, _e. g._ by considering a rotation through an angle $\alpha$
about the $E^{3}$ direction, followed by a boost with rapidity
$\beta\,:=\,\log\,\sqrt{\frac{1\,+\,v}{1\,-\,v}}$ along the $E^{3}$ axis, we
get

(68)
$\displaystyle\mathrm{PSL}(2,\mathbb{C})\times\widehat{\mathbb{C\,S}}(p)\,$
$\displaystyle\longrightarrow$ $\displaystyle\,\mathbb{C\,S}(p)$
$\displaystyle\left(\zeta,\,\widehat{w}\right)\,$ $\displaystyle\longmapsto$
$\displaystyle\,\zeta(\widehat{w})\,=\,w\,=\,\sqrt{\frac{1\,+\,v}{1\,-\,v}}\,e^{i\,\alpha}\,\widehat{w}\,,$

describing the general fractional linear transformation mapping
$\widehat{\mathbb{C\,S}}(p)$ and ${\mathbb{C\,S}}(p)$. From the physical point
of view, this corresponds to the composition of the adjustment of the relative
orientation of the spatial bases $\\{{E}_{(\alpha)}\\}$ with respect to
$\\{\widehat{{E}}_{(\alpha)}\\}$, $\alpha=1,2,3$, followed by a Lorentz boost
adjusting for the relative velocity of $(p,\dot{\gamma}(0))$ with respect to
$(p,\widehat{\dot{\gamma}}(0))$. Since the spatial directions
$n(\theta,\phi)\in\,{\mathbb{C\,S}}(p)$ and
$\widehat{n}(\widehat{\theta},\widehat{\phi})\in\,\widehat{\mathbb{C\,S}}(p)$
characterize corresponding past-directed null vectors
$\ell(\theta,\phi)\,\in\,\left(T_{p}M,\\{E_{(i)}\\}\right)$ and
$\widehat{\ell}(\widehat{\theta},\widehat{\phi})\,\in\,\left(\widehat{T}_{p}M,\\{\widehat{E}_{(i)}\\}\right)$
(see (10) and (48)), we can associate with the spatial directions
$\\{{{E}}_{(\alpha)}\\}$ and $\\{\widehat{{E}}_{(\alpha)}\\}$ the respective
null directions

$\displaystyle\ell_{(\alpha)}\,$ $\displaystyle=$
$\displaystyle\,E_{(\alpha)}\,-\,E_{(4)}\,=\,E_{(\alpha)}\,-\,\dot{\gamma}(0)\,,$
$\displaystyle\widehat{\ell}_{(\alpha)}\,$ $\displaystyle=$
$\displaystyle\,\widehat{E}_{(\alpha)}\,-\,\widehat{E}_{(4)}\,=\,\widehat{E}_{(\alpha)}\,-\,\widehat{\dot{\gamma}}(0)\,.$

###  4.2. The pre-homogeneity setting

From the above remarks, it follows that the Lorentz mapping from
$\widehat{\mathbb{C\,S}}(p)$ to ${\mathbb{C\,S}}(p)$ is fully determined if we
specify the three distinct null directions on the FLRW celestial sphere
$\widehat{\mathbb{C\,S}}(p)$ that are the images, under the
$\mathrm{PSL}(2,\mathbb{C})$-transformation, of three chosen distinct sources
on ${\mathbb{C\,S}}(p)$. The selection of these three distinct sources of
choice and of the corresponding null directions on ${\mathbb{C\,S}}(p)$ will
depend on the scale $L$ we are probing in our cosmological observations. This
is a particularly delicate matter when looking at the pre-homogeneity scales
$L\,\lesssim\,100\,h^{-1}\,\mathrm{Mpc}$, where astrophysical sources are
characterized by a complex distribution of peculiar velocities with respect to
the assumed Hubble flow. To keep track of this scale dependence, let us
consider the celestial spheres ${\mathbb{C\,S}}_{r}(p)$ and
$\widehat{\mathbb{C\,S}}_{\widehat{r}}(p)$ defined by (16) and (47),
respectively. For $L\,>\,0$, let $\widehat{r}(L)$ be the value of
$\widehat{r}$ such that the FLRW sky section (53)

(70)
$\widehat{\Sigma}(p,\hat{r}(L))\,:=\,\widehat{\exp}_{p}\left[\widehat{\mathbb{C\,S}}_{\widehat{r}(L)}(p)\right]\,=\,\left\\{\left.\widehat{\exp}_{p}\left(\widehat{r}(L)\,\widehat{\ell}(\widehat{n}(\widehat{\theta},\widehat{\phi}))\right)\,\right|\,\,(\widehat{\theta},\widehat{\phi})\,\in\,\widehat{\mathbb{C\,S}}(p)\right\\}\;,$

probes the length scale $L$. Similarly, we let ${r}(L)$ denote the value of
${r}$ such that the physical sky section (71)

(71)
$\Sigma(p,r(L))\,:=\,\exp_{p}\left[\mathbb{C\,S}_{r(L)}(p)\right]\,=\,\left\\{\left.\exp_{p}\left(r(L)\,\ell({n}(\theta,\phi))\right)\,\right|\,\,(\theta,\phi)\,\in\,\mathbb{C\,S}(p)\right\\}\,,$

probes the length scale $L$. Since the FLRW area distance (62),

(72)
$\widehat{D}(\widehat{r})\,=\,\frac{a_{0}}{1\,+\,z}\,f\left(\widehat{r}\right)\,,$

is isotropic and may be directly expressed in terms of $z$, we may well use
the redshift parameter $z$ as the reference $L$. Given $z$, we denote by
$L(z)$ the corresponding length-scale of choice. As long as
$\widehat{D}(\widehat{r})$ is an increasing function, we can identify $L(z)$
with the area distance $\widehat{D}(\widehat{r})$, but in general, we leave
the selection of the most appropriate $L(z)$ to the nature of the
cosmographical observations one wants to perform. Given
$\zeta\in\mathrm{PSL}(2,\mathbb{C})$ and a value of the redshift $z$, we have
a corresponding relation between the ”radial” variables $\widehat{r}(L(z))$
and $r(L(z))$ in (70) and (71). We can take advantage of this relation to
simplify the notation for the celestial spheres and the associated sky
sections according to

(73)
$\widehat{\mathbb{C\,S}}_{z}(p)\,:=\,\widehat{\mathbb{C\,S}}_{\widehat{r}(L(z))}(p)\,\Longrightarrow\,\widehat{\Sigma}_{z}\,:=\,\widehat{\Sigma}(p,\hat{r}(L(z)))\,:=\,\widehat{\exp}_{p}\left[\widehat{\mathbb{C\,S}_{z}}(p)\right]\,,$

and

(74)
$\mathbb{C\,S}_{z}(p)\,:=\,\mathbb{C\,S}_{r(L(z))}(p)\,\Longrightarrow\,\Sigma_{z}\,:=\,\Sigma(p,r(L(z)))\,:=\,\exp_{p}\left[\mathbb{C\,S}_{z}(p)\right]\,,$

a notation that, if not otherwise stated, we adopt henceforth. Since in the
pre-homogeneity region $L(z)\,\lesssim\,100\,h^{-1}\,\mathrm{Mpc}$, the large
variance in peculiar velocities of the astrophysical sources implies a great
variability in the selection of the three reference null directions that fix
the $\mathrm{PSL}(2,\mathbb{C})$ action, we localize this action according to
the following construction.

  * •

We assume that there is a finite collection of points
$\\{y_{(I)}\\}\,\in\,\widehat{\mathbb{C\,S}}_{z}(p)$ and a corresponding
collection of open disks $\\{\widehat{B}(y_{(I)},\delta)\\}$ of radius
$\delta$, centered at the points $\\{y_{(I)}\\}$, and defined by

(75)
$\widehat{B}(y_{(I)},\delta)\,:=\,\\{y^{\prime}\in\widehat{\mathbb{C\,S}}_{z}(p)\,|\,d_{\mathbb{S}^{2}}(y^{\prime},y_{(I)})\,\leq\,\delta\\}\subset\widehat{\mathbb{C\,S}}_{z}(p)\,$

where $d_{\mathbb{S}^{2}}(y^{\prime},y_{(I)})$ denotes the distance in the
round unit metric on $\mathbb{S}^{2}$. We also assume that any such
$\widehat{B}(y_{(I)},\delta)$ contains the images of three reference
astrophysical sources of choice, call them $A_{(I,\,k)}$, $k=1,2,3,$, with
celestial coordinates in $\widehat{\mathbb{C\,S}}_{z}(p)$ given by
$y_{(I,\,k)}\,=:\,\widehat{n}_{(I,\,k)}(\widehat{\theta},\widehat{\phi})$.

  * •

We adopt a similar partition on the celestial sphere $\mathbb{C\,S}_{z}(p)$,
to the effect that associated with each disk $\widehat{B}(y_{(I)},\delta)$
there is, in $\mathbb{C\,S}_{z}(p)$, a corresponding metric disk

(76)
$B(x(y_{(I)}),\,\delta)\,=\,\\{x^{\prime}\in\mathbb{C\,S}_{z}(p)\,\,|\,d_{\mathbb{S}^{2}}(x^{\prime},x(y_{(I)}))\leq\delta\\}\,\subset\,\mathbb{C\,S}_{z}(p)\,.$

We require that the images $A_{(I,\,k)}$ of the three reference astrophysical
sources of choice, that in $\widehat{B}(y_{(I)},\delta)$ have celestial
coordinates $y_{(I,\,k)}$, are represented in $B(x(y_{(I)}),\,\delta)$ by
three distinct points with celestial coordinates
$x_{(I,\,k)}\,=:\,{n}_{(I,\,k)}({\theta},{\phi})$.

  * •

We further assume that the past null directions
$\widehat{\ell}_{(I,\,k)}\,=\,\widehat{n}_{(I,\,k)}(\widehat{\theta},\widehat{\phi})\,-\,\widehat{\dot{\gamma}}(0)$,
associated with the location of the reference sources $A_{(I,\,k)}$ in the
portion of the celestial sphere
$\widehat{B}(y_{(I)},\delta)\cap\widehat{\mathbb{C\,S}}_{z}(p)$, are related
to the corresponding null directions
${\ell}_{(I,\,k)}\,=\,{n}_{(I,\,k)}({\theta},{\phi})\,-\,{\dot{\gamma}}(0)$,
locating the sources $A_{(I,\,k)}$ in
${B}(x(y_{(I)}),\delta)\cap{\mathbb{C\,S}}_{z}(p)$, by the
$\mathrm{PSL}(2,\mathbb{C})$ map

(77)
$\displaystyle\zeta_{(I)}\,:\,\widehat{B}(y_{(I)},\delta)\cap\widehat{\mathbb{C\,S}}_{z}(p)\,$
$\displaystyle\longrightarrow$
$\displaystyle\,{B}(x(y_{(I)}),\delta)\cap\mathbb{C\,S}_{z}(p)$
$\displaystyle\widehat{w}\,$ $\displaystyle\longmapsto$
$\displaystyle\,\zeta_{I}(\widehat{w})\,=\,w\,=\,\sqrt{\frac{1\,+\,v}{1\,-\,v}}\,e^{i\,\alpha(A_{(I,\,k)})}\,\widehat{w}\,,$

where $\sqrt{\frac{1\,+\,v}{1\,-\,v}}\,e^{i\,\alpha(A_{(I,\,k)})}$ is the
composition of the Lorentz boost ($v$ being the relative 3-velocity of
$\dot{\gamma}(0)$ with respect to $\widehat{\dot{\gamma}}(0)$) and of the
spatial rotation that, at the given scale $L(z)$, allow us to align the
portion of the celestial sphere $\mathbb{C\,S}_{z}(p)$ described by
${B}(x(y_{(I)}),\delta)$ with the portion of the FLRW celestial sphere
$\widehat{\mathbb{C\,S}}_{z}(p)$ described by $\widehat{B}(y_{(I)},\delta)$.

  * •

Finally, we require that the finite collections of _celestial coordinate bins_
$\\{\widehat{B}(y_{(I)},\delta)\\}$ and
$\left\\{{B}(x(y_{(I)}),\delta)\right\\}$ cover the respective celestial
spheres $\widehat{\mathbb{C\,S}}_{z}(p)$ and $\mathbb{C\,S}_{z}(p)$.

It is worthwhile to stress that the collections of bins
$\\{\widehat{B}(y_{(I)},\delta)\\}$ and
$\left\\{{B}(x(y_{(I)}),\delta)\right\\}$ can be chosen in many distinct ways,
according to the cosmographic observations one wishes to carry out (we use
disks for mathematical convenience). Whatever choice of the above type we
make, we can extend the localized $\mathrm{PSL}(2,\mathbb{C})$ maps (77) by
using a smooth partition of unity $\left\\{\chi_{(I)}\right\\}$ subordinated
to the finite covering $\\{\widehat{B}(y_{(I)},\delta)\\}$ of
$\widehat{\mathbb{C\,S}}_{z}(p)$, _i.e._ a collection of smooth functions
$\chi_{(I)}\,:\,\widehat{B}(y_{(I)},\delta)\longrightarrow\,[0,1]$ whose
support is such that
$\mathrm{supp}\,\chi_{(I)}\subseteq\,\widehat{B}(y_{(I)},\delta)$ and such
that $\sum_{y\in\,\widehat{\mathbb{C\,S}}_{z}(p)}\,\chi_{(I)}(y)\,=\,1$. We
define the _localized $\mathrm{PSL}(2,\mathbb{C})$ map_ connecting, at scale
$L(z)$, the celestial spheres $\widehat{\mathbb{C\,S}}_{z}(p)$ and
${\mathbb{C\,S}}_{z}(p)$, decorated with the respective coordinate bins
$\\{\widehat{B}(y_{(I)}\\}$ and $\\{{B}(x(y_{(I)}),\delta)\\}$, according to

(78) $\displaystyle\zeta_{(z)}\,:\,\widehat{\mathbb{C\,S}}_{z}(p)\,$
$\displaystyle\longrightarrow$ $\displaystyle\,\mathbb{C\,S}_{z}(p)$
$\displaystyle\widehat{w}\,$ $\displaystyle\longmapsto$
$\displaystyle\,\zeta_{(z)}(\widehat{w})\,:=\,\sum_{y\in\,\widehat{\mathbb{C\,S}}_{z}(p)}\,\chi_{(I)}(y)\,\zeta_{(I)}(w)\,,$

where $\zeta_{(I)}(w)$ is provided by (77). Note that, when necessary, this
localized $\mathrm{PSL}(2,\mathbb{C})$ map can be further generalized by
completing it in the Sobolev space of maps which together with their
derivatives are square-summable over $\widehat{\mathbb{C\,S}}_{z}(p)$. This
completion requires some care which we do not enter here (see [12] for
details), and it is needed when discussing the distance between the FLRW and
the cosmographic lightcones.

It is worthwhile to stress that in the pre-homogeneity region
$L(z)\,\lesssim\,100\,h^{-1}\,\mathrm{Mpc}$, the large variance in peculiar
velocities of the astrophysical sources implies a great variability in the
selection of the three reference null directions that fix the local
$\mathrm{PSL}(2,\mathbb{C})$ action characterizing the map $\zeta_{(z)}$. This
implies that $\zeta_{(z)}$ may vary considerably with $L(z)$. Recall that the
role of the celestial spheres ${\mathbb{C\,S}}_{z}(p)$ and
$\widehat{\mathbb{C\,S}}_{z}(p)$ is simply that of representing past null
directions at the observational event $p\,\in\,M$, directions that
respectively point to the astrophysical sources on the sky section
${\Sigma}_{z}$, as seen by $(p,\dot{\gamma}(0))$, and on
$\widehat{\Sigma}_{z}$, as seen according to $(p,\widehat{\dot{\gamma}}(0))$.
These data are transferred from these sky sections to the respective celestial
spheres through null geodesics, thus we can associate with the localized
$\mathrm{PSL}(2,\mathbb{C})$ action the map between the sky sections
$\widehat{\Sigma}_{z}$ and ${\Sigma}_{z}$ given by

(79) $\displaystyle\psi_{(z)}\,:\,\widehat{\Sigma}_{z}\,$
$\displaystyle\longrightarrow$ $\displaystyle\,\Sigma_{z}$ $\displaystyle q\,$
$\displaystyle\longmapsto$
$\displaystyle\,\psi_{(z)}(q)\,:=\,\exp_{p}\,\circ\,\zeta_{(z)}\,\circ\,\widehat{\exp}_{p}^{\,-1}(q)\,,$

for any point $q\,\in\,\widehat{\Sigma}_{z}$.

##  5\. The comparison between the screen planes
$T_{\widehat{n}}\widehat{C\,S}_{z}(p)$ and $T_{n}{C\,S}_{z}(p)$

The localized $\mathrm{PSL}(2,\mathbb{C})$ map $\zeta_{(z)}$ induces a
corresponding map between the _screen plane_
$T_{\widehat{n}}\widehat{\mathbb{C\,S}}(p)_{z}$ associated with the direction
of sight $\widehat{n}(\widehat{\theta},\widehat{\phi})$ in the FLRW celestial
sphere $\widehat{\mathbb{C\,S}}_{z}(p)$ (see (49)), and the _screen plane_
$T_{n}\mathbb{C\,S}_{z}(p)$ associated with the direction of sight
${n}(\theta,\phi)\,=\,\zeta_{(z)}\left(\widehat{n}(\widehat{\theta},\widehat{\phi})\right)$
in the celestial sphere $\mathbb{C\,S}_{z}(p)$ (see (19)). The geometry of
this correspondence is quite sophisticated since it is strictly related to
harmonic map theory and it will be described here in some detail. To begin
with, we denote by $T\widehat{\mathbb{C\,S}}_{z}$ and by $T\mathbb{C\,S}_{z}$
the _screen bundles_ associated with the screen planes on
$\widehat{\mathbb{C\,S}}_{z}(p)$ and $\mathbb{C\,S}_{z}(p)$, respectively.
These are just two copies of the usual tangent bundle $T\mathbb{S}^{2}$ of the
2-sphere. If there is no danger of confusion, we use both notations in what
follows. Under such notational assumptions, we can associate with the map
(78),

(80)
$\zeta_{(z)}\,:\,\widehat{\mathbb{C\,S}}_{z}(p)\,\longrightarrow\,\mathbb{C\,S}_{z}(p)\,,$

the pull–back bundle $\zeta_{(z)}^{-1}\,T\mathbb{C\,S}_{z}$ whose sections
$\mathrm{v}\equiv\zeta_{(z)}^{-1}V:=V\circ\zeta_{(z)}$, $V\in
C^{\infty}(\mathbb{C\,S}_{z}(p),T\mathbb{C\,S}_{z})$, are the vector fields
over $\mathbb{C\,S}_{z}(p)$ covering the map $\zeta_{(z)}$. In physical terms,
the vectors $v$ are the tangent vector on the celestial sphere
$\mathbb{C\,S}_{z}(p)$ that describe the (active) effect of the combination of
rotation and Lorentz boost induced by $\zeta_{(z)}$ on the null direction
$\widehat{\ell}(\widehat{n})$. More expressively, let us remark that for a
given direction of sight
$\zeta_{(z)}(\widehat{n})\,=\,n(\theta,\phi)\,\in\,\mathbb{C\,S}_{z}(p)$, the
vectors $V\in T_{n}\mathbb{C\,S}_{z}(p)$ can be used to describe the
geometrical characteristics of the astrophysical images on the screen
$T_{n}\mathbb{C\,S}_{z}(p)$, for instance, the apparent diameters of the
source. Thus, the vectors
$\mathrm{v}\equiv\zeta_{(z)}^{-1}V:=V\circ\zeta_{(z)}$, sections of the
pull–back bundle $\zeta_{(z)}^{-1}\,T\mathbb{C\,S}_{z}$, can be interpreted as
transferring the ”images” of the screens in $T\mathbb{C\,S}_{z}$ back to
$\widehat{\mathbb{C\,S}}_{z}(p)$ so as to be able to compare them with the
reference screen-shots in $T\widehat{\mathbb{C\,S}}_{z}$. In terms of the
local coordinates $y^{a}\,:=\,\left(\widehat{\theta},\widehat{\phi}\right)$,
$a=1,2$, on $\widehat{\mathbb{C\,S}}_{z}(p)$ (see (52))111111In what follows
the $(\widehat{\theta},\widehat{\phi})$, corresponding to $(y^{2},y^{2})$ in
the normal coordinates string $\\{y^{\alpha}\\}$, are relabelled as
$\\{y^{a}\\}$, with $a=1,2$; a similar relabeling is also adopted for the
normal coordinates $(\theta,\phi)$ on ${\mathbb{C\,S}}_{z}(p)$. , we can write
the section $\mathrm{v}\equiv\zeta_{(z)}^{-1}V:=V\circ\zeta_{(z)}$ as121212In
what follows we freely refer to the excellent [22], [32], and [36] for a
detailed analysis of the geometry of the computations involved in harmonic map
theory.

(81)
${\mathbb{C\,S}}_{z}(p)\,\ni\,y^{a}\,\longmapsto\,\mathrm{v}(y^{a})\,=\,\mathrm{v}^{b}(y)\,\frac{\partial}{\partial\zeta_{(z)}^{b}(y)}\,\in\,\left.\zeta_{(z)}^{-1}T\mathbb{C\,S}_{z}\right|_{y}\;,$

where $\zeta_{(z)}^{b}(y)$, $b\,=1,2$, are the coordinates of the point
(direction of sight) in $\zeta_{(z)}(y)\,\in{\mathbb{C\,S}}_{z}(p)$ given, in
terms of the $y^{a}$ by (64). In particular, if
$T^{*}\widehat{\mathbb{C\,S}}_{z}$ denotes the cotangent bundle to
$\widehat{\mathbb{C\,S}}_{z}(p)$, we can locally introduce the differential

(82) $d\zeta_{(z)}\,=\,\frac{\partial\zeta_{(z)}^{b}}{\partial
y^{a}}dy^{a}\otimes\frac{\partial}{\partial\zeta_{(z)}^{b}}\;,$

and interpret it as a section of the product bundle
$T^{*}\widehat{[}\mathbb{C\,S}]_{z}\otimes\zeta_{(z)}^{-1}\,T\mathbb{C\,S}_{z}$.
To provide a comparison between the geometrical information gathered from the
astrophysical data, let us recall that on the screens
$T\widehat{\mathbb{C\,S}}_{z}$ and $T\mathbb{C\,S}_{z}$ we have the inner
products respectively defined by the pull-back metrics (55) and (27), _i.e._

(83)
$\widehat{h}(\widehat{r}(L(z)),\widehat{\theta},\widehat{\phi})\,:=\,\left.\left(\widehat{\exp}_{p}^{*}\,\widehat{g}|_{\widehat{\Sigma}_{z}}\right)_{ab}\,dy^{a}dy^{b}\right|_{\widehat{r}(L(z))},\,\,\,\,a,\,b\,=\,1,2,\,\,\,\,\,\,y^{1}:=\widehat{\theta},\,y^{2}:=\widehat{\phi}\,.$

and

(84)
$h(r(L(z)),\theta,\phi)\,:=\,\left.\left(\exp_{p}^{*}\,g|_{\Sigma_{z}}\right)_{ab}\,dx^{a}dx^{b}\right|_{r(L(z))},\,\,\,\,a,\,b\,=\,1,2,\,\,\,\,\,\,x^{1}:=\theta,\,x^{2}:=\phi\,.$

The Riemannian metric in the pull-back screen
$\left(\zeta_{(z)}^{-1}\,T\mathbb{C\,S}_{z}\right)_{y}$ over
$y\in\widehat{\mathbb{C\,S}}_{z}(p)$ is provided by $h(\zeta_{(z)}(y))$, hence
the tensor bundle
$T^{*}\widehat{\mathbb{C\,S}}_{z}\otimes\zeta_{(z)}^{-1}\,T\mathbb{C\,S}_{z}$
over the celestial sphere $\widehat{\mathbb{C\,S}}_{z}(p)$ is endowed with the
pointwise inner product

(85)
$\langle\cdot,\cdot\rangle_{T^{*}\widehat{[}\mathbb{C\,S}]_{z}\otimes\zeta_{(z)}^{-1}\,T\mathbb{C\,S}_{z}}\,:=\,\widehat{h}^{-1}(y)\otimes
h(\zeta_{(z)}(y))(\cdot,\cdot)\;,$

where
$\widehat{h}^{-1}(y)\,:=\,\widehat{h}^{ab}(y)\,\partial_{a}\otimes\partial_{b}$
is the metric tensor in $T^{*}_{y}\widehat{\mathbb{C\,S}}_{z}$. The
corresponding Levi-Civita connection will be denoted by
$\nabla^{\langle,\rangle}$. Explicitly, if
$W\,=\,W_{a}^{b}\,dy^{a}\otimes\frac{\partial}{\partial\zeta_{(z)}^{b}}$ is a
section of
$T^{*}\widehat{\mathbb{C\,S}}_{z}\otimes\zeta_{(z)}^{-1}\,T\mathbb{C\,S}_{z}$,
the covariant derivative of $W$ in the direction $\frac{\partial}{\partial
y^{b}}$ is provided by

$\displaystyle\nabla^{\langle,\rangle}_{b}\,W\,=\,\nabla^{\langle,\rangle}_{b}\,\left(W_{a}^{c}\,dy^{a}\otimes\frac{\partial}{\partial\zeta_{(z)}^{c}}\right)$
$\displaystyle=\,\frac{\partial}{\partial
y^{b}}\,W_{a}^{c}\,dy^{a}\otimes\frac{\partial}{\partial\zeta_{(z)}^{c}}+W_{a}^{c}\left(\widehat{\nabla}_{b}\,dy^{a}\right)\otimes\frac{\partial}{\partial\zeta_{(z)}^{c}}$
$\displaystyle+\,W_{a}^{c}\,dy^{a}\otimes\,\nabla^{*}_{b}\left(\frac{\partial}{\partial\zeta_{(z)}^{c}}\right)\;,$

where $\widehat{\nabla}$ denotes the Levi–Civita connection on
$(\widehat{\mathbb{C\,S}}_{z}(p),\widehat{h})$, and $\nabla^{*}$ is the pull
back on $\zeta_{(z)}^{-1}\,T\mathbb{C\,S}_{z}$ of the Levi–Civita connection
of $({\mathbb{C\,S}}_{z},h)$. If $\widehat{\Gamma}^{a}_{bc}(\widehat{h})$ and
$\Gamma^{a}_{bc}(h)$ respectively denote the Christoffel symbols of
$(\widehat{\mathbb{C\,S}}_{z}(p),\widehat{h})$ and
$({\mathbb{C\,S}}_{z}(p),h)$, then
$\widehat{\nabla}_{b}\,dy^{a}\,=\,-\,\widehat{\Gamma}_{bc}^{a}(\widehat{h})\,dy^{c}$
and
$\nabla^{*}_{b}\left(\frac{\partial}{\partial\zeta_{(z)}^{c}}\right)\,=\,\frac{\partial\zeta_{(z)}^{i}}{\partial
y^{b}}\,\Gamma^{k}_{ci}(h)\,\frac{\partial}{\partial\zeta_{(z)}^{k}}$, and one
computes

(87) $\nabla^{\langle,\rangle}_{b}\,W\,=\,\left(\frac{\partial}{\partial
y^{b}}\,W_{a}^{i}\,-\,W^{i}_{c}\widehat{\Gamma}^{c}_{ba}(\widehat{h})\,+\,W^{k}_{a}\frac{\partial\zeta_{(z)}^{j}}{\partial
y^{b}}\,\Gamma_{kj}^{i}(h)\right)\,dy^{a}\otimes\frac{\partial}{\partial\zeta_{(z)}^{i}}\;.$

These remarks on the geometry of the map (80) allow us to compare the data on
the screens $T\mathbb{C\,S}_{z}$ and $T\widehat{\mathbb{C\,S}}_{z}$. For this
purpose, the relevant quantity is the norm, evaluated with respect to the
inner product (85), of the differential (82) of the
$\mathrm{PSL}(2,\mathbb{C})$ map $\zeta_{(z)}$. Direct computation provides

$\displaystyle e(\widehat{h},\zeta_{(z)};h)\,$ $\displaystyle:=$
$\displaystyle\,\langle
d\zeta_{(z)},d\zeta_{(z)}\rangle_{T^{*}\widehat{[}\mathbb{C\,S}]_{z}\otimes\zeta_{(z)}^{-1}\,T\mathbb{C\,S}_{z}}$
$\displaystyle=$
$\displaystyle\,\widehat{h}^{ab}(y)\,\frac{\partial\zeta_{(z)}^{i}(y)}{\partial
y^{a}}\frac{\partial\zeta_{(z)}^{j}(y)}{\partial
y^{b}}\,h_{ij}(\zeta_{(z)}(y))\,=\,tr_{\widehat{h}(y)}\,(\zeta_{(z)}^{*}\,h)\,,$

where $tr_{\widehat{h}(y)}\,(\zeta_{(z)}^{*}\,h)$ denotes the trace, with
respect to the metric $\widehat{h}$ of the pull-back metric
$\zeta_{(z)}^{*}\,h$. In other words, at any point $y$,
$e(\widehat{h},\zeta_{(z)};h)(y)$ is the sum of the eigenvalues of the metric
$\zeta_{(z)}^{*}\,h$, thus providing the sum of the squares of the length
stretching generated by the (pull-back of) the physical metric
$\zeta_{(z)}^{*}\,h$ along the orthogonal directions
$(\widehat{\theta},\widehat{\phi})$. To such stretching, we can associate the
_tension field_ of the map $\zeta_{(z)}$, defined by

(89)
$\tau^{i}(\zeta_{(z)})\,:=\,\Delta_{(\widehat{h})}\,\zeta_{(z)}^{i}\,+\,\widehat{h}^{kj}\,\Gamma_{ab}^{i}(h)\,\frac{\partial\zeta_{(z)}^{a}}{\partial
y^{k}}\frac{\partial\zeta_{(z)}^{b}}{\partial y^{j}}\,.$

To provide some intuition on these geometrical quantities, we can adapt to our
case a nice heuristic remark by J. Eells and L. Lemaire described in their
classical paper on harmonic map theory [22]. Let us imagine the FLRW celestial
sphere $(\widehat{\mathbb{C\,S}}_{z}(p),\widehat{h})$ as a rubber balloon,
decorated with dots representing the astrophysical sources recorded from the
sky section $\widehat{\Sigma}_{z}$. This balloon has the geometry described by
the round metric $\widehat{h}(z,\widehat{\theta},\widehat{\phi})$ defined by
(83), explicitly (see (60))

(90)
$\widehat{h}(\widehat{r}(z),\widehat{\theta},\widehat{\phi})\,=\,\frac{a^{2}_{0}}{(1\,+\,z_{L})^{2}}\,f^{2}\left(\widehat{r}(z)\right)\left(d\widehat{\theta}^{2}\,+\,\sin^{2}\widehat{\theta}d\widehat{\phi}^{2}\right)\;,$

where $z_{L}$ is the redshift associated with the length scale $L$.
Conversely, let us imagine the physical celestial sphere
$({\mathbb{C\,S}}_{z}(p),\,{h})$ as a rigid surface with the geometry induced
by the metric $h(r(z),\theta,\phi)$ defined by (84), _i.e._ , (see (30)),

(91) $\displaystyle h\left(r(z),\theta,\phi\right)$
$\displaystyle\,=\,D^{2}(r(z),\theta,\phi)\left(d{\theta}^{2}\,+\,\sin^{2}\theta
d{\phi}^{2}\,+\,\mathcal{L}_{ab}(r(z),\theta,\phi)\,dx^{a}dx^{b}\right)\,,\;\;\;x^{1}:=\theta,\,x^{2}:=\phi\,,$

providing the geometric landscape of the astrophysical sources reaching us
along null geodesics from the physical sky section $\Sigma_{z}$. We can think
of the $PSL(2,\mathbb{C})$ map $\zeta_{(z)}$ as stretching the elastic surface
$(\widehat{\mathbb{C\,S}}_{z}(p),\widehat{h})$ on the rigid surface
$({\mathbb{C\,S}}_{z}(p),\,{h})$. The purpose of this stretching is to overlap
the images of the astrophysical sources recorded on
$(\widehat{\mathbb{C\,S}}_{z}(p),\widehat{h})$ with the images of the
corresponding sources as registered on $({\mathbb{C\,S}}_{z}(p),{h})$. In
general, this overlap is not successful without stretching the surface, and to
any point $y\,\in\,(\widehat{\mathbb{C\,S}}_{z}(p),\widehat{h})$ we can
associate a corresponding vector measuring the stretch necessary for
connecting the images of the same source on the two celestial
spheres131313This is not to be confused with the phenomenon of strong
gravitational lensing that occurs in a given celestial sphere. It is simply a
mismatch due to the comparison between the description of the same
astrophysical source on two distinct celestial spheres.
$(\widehat{\mathbb{C\,S}}_{z}(p),\widehat{h})$ and
$({\mathbb{C\,S}}_{z}(p),{h})$. To leading order, the required stretching is
provided by the tension vector $\tau^{i}(\zeta_{(z)},y)$ at $y$. Both the
Hilbert-Schmidt norm (5) and the tension vector field (89) of the map
$\zeta_{(z)}$ are basic quantities in harmonic map theory, and to understand
the strategy we will follow in comparing, at a given length scale $L$, the
FLRW past light cone $\widehat{\mathcal{C}}(p,\widehat{g})$ with the physical
observational past light cone ${\mathcal{C}}(p,{g})$ we need to look into the
harmonic map theory associated with $\zeta_{(z)}$. Let us start by associating
with $\langle
d\zeta_{(z)},d\zeta_{(z)}\rangle_{T^{*}\widehat{[}\mathbb{C\,S}]_{z}\otimes\zeta_{(z)}^{-1}\,T\mathbb{C\,S}_{z}}$
the density

(92) $e(\widehat{h},\zeta_{(z)},h)\,\,d\mu_{\widehat{h}}\,:=\,\langle
d\zeta_{(z)},d\zeta_{(z)}\rangle_{T^{*}\widehat{[}\mathbb{C\,S}]_{z}\otimes\zeta_{(z)}^{-1}\,T\mathbb{C\,S}_{z}}\,d\mu_{\widehat{h}}\,=\,tr_{\widehat{h}(y)}\,(\zeta_{(z)}^{*}\,h)\,d\mu_{\widehat{h}}\;,$

where $d\mu_{\widehat{h}}$ is the volume element defined by the metric
$\widehat{h}$ on the FLRW celestial sphere $\widehat{\mathbb{C\,S}}_{z}(p)$.
An important property of the density
$e(\widehat{h},\zeta_{(z)};h)\,\,d\mu_{\widehat{h}}$ is that it is invariant
under the two-dimensional conformal transformations

(93)
$(\widehat{\mathbb{C\,S}}_{z}(p),\,\widehat{h}_{ab})\,\longmapsto\,(\widehat{\mathbb{C\,S}}_{z}(p),\,e^{-f}\,\widehat{h}_{ab})\;,$

where $f$ is a smooth function on $\widehat{\mathbb{C\,S}}_{z}(p)$. In this
connection, it is worthwhile to recall that conformal invariance is strictly
related to the action of the Lorentz group on the celestial spheres (and it is
ultimately the rationale for the relation between Lorentz transformations and
the fractional linear transformations of $\mathrm{PSL}(2,\mathbb{C})$).
The expression
$\frac{1}{2}\,e(\widehat{h},\zeta_{(z)};h)\,\,d\mu_{\widehat{h}}$
characterizes the harmonic map energy functional associated to the map
$\zeta_{(z)}$, _viz._

(94)
$E[\widehat{h},\zeta_{(z)},\,h]\,:=\,\frac{1}{2}\,\int_{\widehat{\mathbb{C\,S}}_{z}}\,e(\widehat{h},\zeta_{(z)},h)\,\,d\mu_{\widehat{h}}\;.$

It is worthwhile to put forward a more explicit characterization of the nature
of the harmonic map functional $E[\widehat{h},\zeta_{(z)};\,h]$ by making
explicit, together with the celestial spheres $\widehat{\mathbb{C\,S}}_{z}(p)$
and ${\mathbb{C\,S}}_{z}(p)$, the role of the corresponding sky sections
$\widehat{\Sigma}_{z}$ and $\Sigma_{z}$. To this end, let us consider the map
(79) acting between the sky sections $\widehat{\Sigma}_{z}$ and
${\Sigma}_{z}$,

(95)
$\displaystyle\psi_{(z)}\,:\,(\widehat{\Sigma}_{z},\,\left.\widehat{g}\right|_{\hat{\Sigma}_{z}})\,$
$\displaystyle\longrightarrow$
$\displaystyle\,(\Sigma_{z},\,\left.{g}\right|_{{\Sigma}_{z}})$ $\displaystyle
y\,$ $\displaystyle\longmapsto$
$\displaystyle\,\psi_{(z)}(q)\,:=\,\exp_{p}\,\circ\,\zeta_{(z)}\,\circ\,\widehat{\exp}_{p}^{\,-1}(y)\,.$

The corresponding harmonic map functional is provided by

(96)
$E\left[\widehat{g}_{(z)},\,\psi_{(z)},\,{g}_{(z)}\right]\,:=\,\frac{1}{2}\,\int_{\widehat{\Sigma}_{z}}\,(\widehat{g}_{(z)})^{ab}\,\frac{\partial\psi_{(z)}^{i}(y)}{\partial
y^{a}}\frac{\partial\psi_{(z)}^{k}(y)}{\partial
y^{b}}\,(g_{(z)})_{ik}\,d\mu_{\widehat{g}_{(z)}}$

where, for notational ease, we have set
$\widehat{g}_{(z)}\,:=\,\widehat{g}_{\,\hat{\Sigma}_{z}}$ and
${g}_{(z)}\,:=\,{g}_{\,{\Sigma}_{z}}$. We can equivalently write
$E\left[\widehat{g}_{(z)},\,\psi_{(z)},\,{g}_{(z)}\right]$ in terms of pull-
backs of the relevant maps, and get the following chain of relations

$\displaystyle E\left[\widehat{g}_{(z)},\,\psi_{(z)},\,{g}_{(z)}\right]\,$
$\displaystyle=$
$\displaystyle\,\frac{1}{2}\,\int_{\widehat{\Sigma}_{z}}\,(\widehat{g}_{(z)})^{ab}\,\left(\psi_{(z)}^{*}g_{(z)}\right)_{ab}\,d\mu_{\widehat{g}_{(z)}}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2}\,\int_{\widehat{\exp_{p}}(\widehat{\mathbb{C\,S}}_{z})}\,(\widehat{g}_{(z)})^{ab}\,\left(\psi_{(z)}^{*}g_{(z)}\right)_{ab}\,d\mu_{\widehat{g}_{(z)}}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2}\,\int_{\widehat{\mathbb{C\,S}}_{z}}\,\widehat{\exp_{p}}^{*}\left[(\widehat{g}_{(z)})^{ab}\,\left(\psi_{(z)}^{*}g_{(z)}\right)_{ab}\right]\,\widehat{\exp_{p}}^{*}(d\mu_{\widehat{g}_{(z)}})$
$\displaystyle=$
$\displaystyle\,\frac{1}{2}\,\int_{\widehat{\mathbb{C\,S}}_{z}}\,\widehat{h}^{ab}\,\left(\widehat{\exp_{p}}^{*}\,\left(\psi_{(z)}^{*}g_{(z)}\right)\right)_{ab}\,d\mu_{\widehat{h}}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2}\,\int_{\widehat{\mathbb{C\,S}}_{z}}\,\widehat{h}^{ab}\,\left(\widehat{\exp_{p}}^{*}\,\left(\exp_{p}\,\circ\,\zeta_{(z)}\,\circ\,\widehat{\exp}_{p}^{\,-1}\right)^{*}g_{(z)}\right)_{ab}\,d\mu_{\widehat{h}}$
$\displaystyle=$
$\displaystyle\,\frac{1}{2}\,\int_{\widehat{\mathbb{C\,S}}_{z}}\,\widehat{h}^{ab}\,\left(\zeta_{(z)}^{*}h\right)_{ab}\,d\mu_{\widehat{h}}\,=\,E[\widehat{h},\,\zeta_{(z)},\,{h}]\,,$

from which it follows that the harmonic map energy functional associated with
the localized $\mathrm{PSL}(2,\mathbb{C})$ map $\zeta_{(z)}$ and with the map
$\psi_{(z)}$, defined by (95), can be identified. This is not surprising since
$\psi_{(z)}\,:=\,\exp_{p}\,\circ\,\zeta_{(z)}\,\circ\,\widehat{\exp}_{p}^{\,-1}$
can be seen as the representation of $\zeta_{(z)}$ on the sky sections
$\widehat{\Sigma}_{z}:=\widehat{\exp}_{p}\left(\widehat{\mathbb{C\,S}}_{z}(p)\right)$
and ${\Sigma}_{z}:={\exp}_{p}\left({\mathbb{C\,S}}_{z}(p)\right)$. From the
conformal nature of the map
$\zeta_{(z)}:\widehat{\mathbb{C\,S}}_{z}(p)\longrightarrow{\mathbb{C\,S}}_{z}(p)$,
it follows that $\psi_{(z)}$ acts as a conformal diffeomorphism between
$\hat{\Sigma}_{z}$ and ${\Sigma}_{z}$ as long as the exponential maps are
diffeomorphisms from $\widehat{\mathbb{C\,S}}_{z}(p)$ and
${\mathbb{C\,S}}_{z}(p)$ onto their respective images $\widehat{\Sigma}_{z}$
and ${\Sigma}_{z}$. Later we shall see how this result can be extended, under
suitable hypotheses, to the less regular case of Lipschitzian exponential map.
Here, we restrict our attention to the stated regularity assumptions on the
exponential maps $\widehat{\exp}_{p}$ and $\exp_{p}$. They imply that the sky
sections $\hat{\Sigma}_{z}$ and ${\Sigma}_{z}$ have the topology of a
2-sphere. Moreover, we can take advantage of the fact that
$(\widehat{\Sigma}_{z},\,\widehat{g}_{(z)})$ is a (rescaled) round sphere,
thus we can apply the Poincaré–Koebe uniformization theorem, to the effect
that there is a positive scalar function
$\Phi_{\widehat{\Sigma}\,\Sigma}\,\in\,C^{\infty}(\widehat{\Sigma}_{z},\mathbb{R}_{>0})$
such that

(98)
$\left(\psi_{(z)}^{*}g_{(z)}\right)_{ab}\,=\,\frac{\partial\psi_{(z)}^{i}(y)}{\partial
y^{a}}\frac{\partial\psi_{(z)}^{k}(y)}{\partial
y^{b}}\,(g_{(z)})_{ik}\,=\,\Phi^{2}_{\widehat{\Sigma}\,\Sigma}\,(\widehat{g}_{(z)})_{ab}\,.$

The required conformal factor
$\Phi_{\widehat{\Sigma}\,\Sigma}\,\in\,C^{\infty}(\widehat{\Sigma}_{z},\mathbb{R}_{>0})$
is the solution, (unique up to the $\mathrm{PSL}(2,\mathbb{C})$ action on
$(\widehat{\Sigma}_{z},\,\widehat{g}_{(z)})$), of the elliptic partial
differential equation on $(\widehat{\Sigma}_{z},\hat{g}_{(z)})$ defined by [2]

(99)
$-\,\Delta_{\widehat{g}_{(z)}}\ln({\Phi}_{\widehat{\Sigma}\Sigma}^{2})\,+\,R(\widehat{g}_{(z)})\,=\,R(\psi_{(z)}^{*}g_{(z)})\,{\Phi}_{\widehat{\Sigma}\Sigma}^{2}\;,$

where
$\Delta_{\widehat{g}_{(z)}}\,:=\,\widehat{g}_{(z)}^{ab}\nabla_{a}\nabla_{b}$
is the Laplace-Beltrami operator on $(\widehat{\Sigma}_{z},\hat{g}_{(z)})$,
and where we respectively denoted by $R(\widehat{g}_{(z)})$ and
$R(\psi_{(z)}^{*}g_{(z)})$ the scalar curvature of the metrics
$\widehat{g}_{(z)}$ and $\psi_{(z)}^{*}g_{(z)}$. Notice that the scalar
curvature $R(\hat{g}_{(z)})$ is associated with the metric (60) evaluated for
$\widehat{r}\,=\,\widehat{r}(L)$ and hence is given by the constant
$R(\hat{g}_{(z)})\,=\,\left[\frac{a^{2}_{0}}{(1\,+\,z)^{2}}\,f^{2}\left(\widehat{r}\right)\right]^{-1}$.
Similarly, $R(g_{(z)})$ is associated with the metric (30) evaluated for
$r\,=\,r(z)$, and as such it depends on the area distance
$D^{2}(r(z),\theta,\phi)$ and the lensing distortion $\mathcal{L}_{ab}$.

By tracing (98) with respect to $\widehat{g}_{(z)}^{ab}$, we get
$tr_{\widehat{g}_{(z)}(y)}\,\left(\psi_{(z)}^{*}g_{(z)}\right)\,=\,2\Phi^{2}_{\widehat{\Sigma}\,\Sigma}$,
and we can wite

(100)
${\Phi}_{\widehat{\Sigma}\Sigma}^{2}\,=\,\frac{1}{2}\,tr_{\widehat{g}_{(z)}(y)}\,\left(\psi_{(z)}^{*}g_{(z)}\right)\,=\,\frac{1}{2}\,\widehat{g}_{(z)}^{ab}\,\frac{\partial\psi_{(z)}^{i}(y)}{\partial
y^{a}}\frac{\partial\psi_{(z)}^{k}(y)}{\partial y^{b}}\,(g_{(z)})_{ik}\;.$

From (98) we also get
$\det\left(\psi_{(z)}^{*}g_{(z)}\right)\,=\,{\Phi}_{\widehat{\Sigma}\Sigma}^{4}\,\det(\widehat{g}_{(z)})$,
hence we can equivalently express the conformal factor
${\Phi}_{\widehat{\Sigma}\Sigma}^{2}$ as the Radon-Nikodym derivative of the
Riemannian measure $d\mu_{\psi^{*}{g}_{(z)}}\,:=\,\psi_{(z)}^{*}d\mu$ of the
pulled back metric $\psi_{(z)}^{*}g_{(z)}$ on the sky section
$\widehat{\Sigma}_{z}$, with respect to the Riemannian measure
$d\mu_{\widehat{g}_{(z)}}$ of the round metric $\widehat{g}_{(z)}$ on
$\widehat{\Sigma}_{z}$, _i.e._ ,

(101)
${\Phi}_{\widehat{\Sigma}\Sigma}^{2}\,=\,\frac{d\mu_{\psi_{(z)}^{*}{g_{(z)}}}}{d\mu_{\widehat{g}_{(z)}}}\,=\,\frac{\psi_{(z)}^{*}d\mu_{g_{(z)}}}{d\mu_{\widehat{g}_{(z)}}}\;.$

Directly from this latter relation and from
$E\left[\widehat{g}_{(z)},\,\psi_{(z)},\,{g}_{(z)}\right]\,=\,E\left[\widehat{h},\,\zeta_{(z)},\,{h}\right]$
(see (5)), we get

(102)
$E\left[\widehat{h},\,\zeta_{(z)},\,{h}\right]\,=\,\int_{\widehat{\Sigma}_{z}}\,{\Phi}_{\widehat{\Sigma}\Sigma}^{2}\,d\mu_{\hat{h}}\,,$

which expresses the harmonic map functional
$E\left[\widehat{h},\,\zeta_{(z)},\,{h}\right]$ in terms of the conformal
factor ${\Phi}_{\widehat{\Sigma}\Sigma}^{2}$. As the
$\mathrm{PSL}(2,\mathbb{C})$-localized map $\zeta_{(z)}$ varies with the scale
$L(z)$, relation (102) shows that
$E\left[\widehat{h},\,\zeta_{(z)},\,{h}\right]$ describes the
$\zeta_{(z)}$-dependent total ”energy” associated with the conformal
stretching of $(\widehat{\mathbb{C\,S}}_{z}(p),\,\widehat{h})$ over
$(\mathbb{C\,S}_{z}(p),\,h)$.

###  5.1. A local expression for ${\Phi}_{\widehat{\Sigma}\Sigma}^{2}$

It is worthwhile to provide a local expression for
${\Phi}_{\widehat{\Sigma}\Sigma}^{2}$ showing the explicit dependence from the
celestial coordinates $(\theta,\phi)$, the area distances
$\widehat{D}(\widehat{r}(L))$, $D(r(L),\theta,\phi)$, and the distortion
tensor $\mathcal{L}$ (see (30)). We proceed as follows. Let us consider one of
the coordinate bin $\widehat{B}(y_{(I)},\delta)$ (see (75)) in the celestial
sphere $\widehat{\mathbb{C\,S}}_{z}(p)$. For
$y=(r(z),\widehat{\theta},\widehat{\phi})\in\,\widehat{B}(y_{(I)},\delta)$ let
$q:=\widehat{\exp}_{p}(y)$ the point in the sky section $\widehat{\Sigma}_{z}$
reached, at the scale $L(z)$, along the past-directed null geodesics
associated with the observational direction
$y=(\widehat{\theta},\widehat{\phi})$. From the expression (101) of the
conformal factor ${\Phi}_{\widehat{\Sigma}\Sigma}^{2}$ in terms of the measure
$\psi_{(z)}^{*}d\mu_{g_{(z)}}$ we get, by massaging pull-backs,

$\displaystyle{\Phi}_{\widehat{\Sigma}\Sigma}^{2}\,d\mu_{\widehat{g}_{(z)}}(q)\,$
$\displaystyle=$ $\displaystyle\,\psi_{(z)}^{*}d\mu_{g_{(z)}}(q)$
$\displaystyle=$
$\displaystyle\left(\exp_{p}\,\circ\,\zeta_{(z)}\,\circ\,\widehat{\exp}_{p}^{\,-1}\right)^{*}d\mu_{g_{(z)}}$
$\displaystyle=$
$\displaystyle(\widehat{\exp}_{p}^{\,-1})^{*}(\zeta_{(z)}^{*}d\mu_{h})$
$\displaystyle\Rightarrow$
$\displaystyle\,\widehat{\exp}_{p}^{*}\left({\Phi}_{\widehat{\Sigma}\Sigma}^{2}\,d\mu_{\widehat{g}_{(z)}}(q)\right)\,=\,\zeta_{(z)}^{*}d\mu_{h}(y)$
$\displaystyle{\Phi}_{\widehat{\Sigma}\Sigma}^{2}(y)\,d\mu_{\widehat{h}}(y)\,$
$\displaystyle=$ $\displaystyle\,\zeta_{(z)}^{*}d\mu_{h}(y)\,.$

Hence, on $(\widehat{\mathbb{C\,S}}(p),\,\widehat{h})$, we need to compute the
Radon-Nicodym derivative

(104)
${\Phi}_{\widehat{\Sigma}\Sigma}^{2}(y)\,=\,\frac{\zeta_{(z)}^{*}d\mu_{h}}{d\mu_{\widehat{h}}}(y)\,.$

If we take into account the characterization
$\sqrt{\det(h(r(z),\theta,\phi))}\,=\,D^{2}(r(z),\theta,\phi)\,\sqrt{\det(\widetilde{h}(\mathbb{S}^{2}))}$
of the area distance $D^{2}(r(z),\theta,\phi)$ (see (29)), we compute

(105)
$\zeta_{(z)}^{*}d\mu_{h}(y)\,=\,\left|\mathrm{Jac}_{y}(\zeta_{(z)})\right|\,D^{2}(y)\,d\mu_{\mathbb{S}^{2}}\,,$

where $|\mathrm{Jac}_{y}(\zeta_{(z)})|$ is the Jacobian determinant associated
with the localized $\mathrm{PSL}(2,\mathbb{C})$ map $\zeta_{(z)}$, and where
$D^{2}(y)$ is a shorthand notation for the area distance
$D^{2}(\zeta_{(z)}(\widehat{r}(z),\widehat{\theta},\widehat{\phi}))$ pulled
back at $y\in(\widehat{\mathbb{C\,S}}_{z}(p),\,\widehat{h})$ by the localized
$\zeta_{(z)}$. Similarly, from (61) we compute
$d\mu_{\widehat{h}}(y)\,=\,\frac{a^{2}_{0}}{(1\,+\,z_{L})^{2}}\,f^{2}\left(\widehat{r}(L)\right)\,d\mu_{\mathbb{S}^{2}}\,$.
Thus, we can write

(106)
${\Phi}_{\widehat{\Sigma}\Sigma}^{2}(\widehat{r}(z),\widehat{\theta},\widehat{\phi})\,=\,\left|\mathrm{Jac}\left(\zeta_{(z)}(\widehat{r}(z),\widehat{\theta},\widehat{\phi})\right)\right|\,\frac{D^{2}(\zeta_{(z)}\left(\widehat{r}(z),\widehat{\theta},\widehat{\phi})\right)(1\,+\,z)^{2}}{a^{2}_{0}\,f^{2}\left(\widehat{r}(z)\right)}\,.$

In terms of the FLRW area distance

(107)
$\widehat{D}(\widehat{r}(z))\,=\,\frac{a_{0}}{1\,+\,z}\,f\left(\widehat{r}\right)\,,$

we can equivalently write (106) in the simpler form (where, to have handy the
formula for later use, we have taken the square root)

(108)
${\Phi}_{\widehat{\Sigma}\Sigma}(\widehat{r}(z),\widehat{\theta},\widehat{\phi})\,=\,\left|\mathrm{Jac}\left(\zeta_{(z)}(\widehat{r}(z),\widehat{\theta},\widehat{\phi})\right)\right|^{\frac{1}{2}}\,\frac{D\left(\zeta_{(z)}(\widehat{r}(z),\widehat{\theta},\widehat{\phi})\right)}{\widehat{D}(\widehat{r}(z))}\,.$

This clearly shows that the conformal factor ${\Phi}_{\widehat{\Sigma}\Sigma}$
is an explicit and, at least in principle, measurable quantity associated with
the local Lorentz mapping (described by the localized
$\mathrm{PSL}(2,\mathbb{C})$ map $\zeta_{(z)}$) needed for adjusting the three
reference null directions in the chosen celestial coordinates bin
$\widehat{B}(y_{(I)},\delta)$ in the celestial sphere
$\widehat{\mathbb{C\,S}}_{z}(p)$. This adjustment allows to transfer to
$\widehat{B}(y_{(I)},\delta)$ the actual area distance, namely, compute
$D(\zeta_{(z)}(\widehat{r}(z),\widehat{\theta},\widehat{\phi}))$, and compare
its distribution on the FLRW celestial sphere $\widehat{\mathbb{C\,S}}_{z}(p)$
with respect to the isotropic FLRW area distance
$\widehat{D}(\widehat{r}(z))$. The anisotropies in the angular distribution
with respect to $\widehat{D}(\widehat{r}(z))$ give rise to fluctuations in
${\Phi}_{\widehat{\Sigma}\Sigma}$. It may appear somewhat surprising that,
after all, the conformal factor does not explicitly depend also from the
distortion tensor $\mathcal{L}_{ab}$ defined by (30). This dependence is
implicit in the definition of the area distance (29) and of the coordinate
parametrization (30) characterizing $\mathcal{L}_{ab}$. These definitions give
rise to the relation (34) that, as can be easily checked, remove the explicit
$\mathcal{L}_{ab}$ dependence from ${\Phi}_{\widehat{\Sigma}\Sigma}$. As we
shall see, this fact will turn to our advantage when extending our analysis to
the more general case of fractal-like sky sections.

##  6\. The sky section comparison functional at scale $L$

The harmonic energy $E\left[\widehat{h},\,\zeta_{(z)},\,{h}\right]$, or
equivalently $E\left[\widehat{g}_{(z)},\,\psi_{(z)},\,{g}_{(z)}\right]$,
associated with the maps $\zeta_{(z)}$ and $\psi_{(z)}$, cannot be used
directly as comparison functional between the sky sections
$(\widehat{\Sigma}_{z},\,\widehat{g}_{(z)})$ and $(\Sigma_{z},\,{g}_{(z)})$.
This follows directly as a consequence of the conformal invariance (93) which
implies

(109) $\displaystyle
E\left[\widehat{g}_{(z)},\,\psi_{(z)},\,{g}_{(z)}\right]\,=\,\frac{1}{2}\,\int_{\widehat{\Sigma}_{z}}\,(\widehat{g}_{(z)})^{ab}\,\frac{\partial\psi_{(z)}^{i}(y)}{\partial
y^{a}}\frac{\partial\psi_{(z)}^{k}(y)}{\partial
y^{b}}\,(g_{(z)})_{ik}\,d\mu_{\widehat{g}_{(z)}}$
$\displaystyle=\frac{1}{2}\,\int_{\widehat{\Sigma}_{z}}\,\left[\frac{a^{2}_{0}\,f^{2}\left(\widehat{r}_{(z)}\right)}{(1\,+\,z_{L})^{2}}\right]^{-1}(\widehat{\widetilde{h}}(\mathbb{S}^{2}))^{ab}\,\frac{\partial\psi_{(z)}^{i}(y)}{\partial
y^{a}}\frac{\partial\psi_{(z)}^{k}(y)}{\partial
y^{b}}\,(g_{(z)})_{ik}\,\left[\frac{a^{2}_{0}\,f^{2}\left(\widehat{r}_{(z)}\right)}{(1\,+\,z_{L})^{2}}\right]\,d\mu_{\mathbb{S}^{2}}$
$\displaystyle=\frac{1}{2}\,\int_{\widehat{\Sigma}_{z}}\,(\widehat{\widetilde{h}}(\mathbb{S}^{2}))^{ab}\,\frac{\partial\psi_{(z)}^{i}(y)}{\partial
y^{a}}\frac{\partial\psi_{(z)}^{k}(y)}{\partial
y^{b}}\,(g_{(z)})_{ik}\,d\mu_{\mathbb{S}^{2}}\,,$

where, as usual, $\widehat{\widetilde{h}}(\mathbb{S}^{2}))$ is the round
metric on the unit 2-sphere $\mathbb{S}^{2}$. From the above relation it
follows that $E\left[\widehat{g}_{(z)},\,\psi_{(z)},\,{g}_{(z)}\right]$, (and
similarly for $E\left[\widehat{h},\,\zeta_{(z)},\,{h}\right]$), does not
depend from the area distance
$\frac{a^{2}_{0}}{(1\,+\,z)^{2}}\,f^{2}\left(\widehat{r}(z)\right)$ on the
FLRW past lightcone $\widehat{\mathcal{C}}^{-}(p)$. Thus,
$E\left[\widehat{g}_{(z)},\,\psi_{(z)},\,{g}_{(z)}\right]$ cannot be a good
candidate for the role of the functional that compares the sky sections
$(\widehat{\Sigma}_{z},\,\widehat{g}_{(z)})$ and $(\Sigma_{z},\,{g}_{(z)})$.
For this role, we introduced in [12] a functional whose structure was
suggested by the rich repertoire of functionals used in the problem of
comparing shapes of surfaces in relation to computer graphic and visualization
problems (see e.g. [35] and [28], to quote two relevant papers in a vast
literature). In particular, we were inspired by an energy functional
introduced, under the name of elastic energy, in a remarkable paper by J. Hass
and P. Koehl [29], who use it as a powerful means of comparing the shapes of
genus-zero surfaces in problems relevant to surface visualization.

In the more complex framework addressed in cosmography, we found it useful to
define the sky section comparison functional at scale $L(z)$ according to

(110)
$E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]\,:=\,\int_{\widehat{\Sigma}_{z}}({\Phi}_{\widehat{\Sigma}\Sigma}\,-\,1)^{2}\,d\mu_{\hat{g}_{(z)}}\;,$

that can be, more expressively, rewritten as (see (108))

(111)
$E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]\,:=\,\int_{\widehat{\Sigma}_{z}}\,\left[\frac{\left|\mathrm{Jac}\left(\zeta_{(z)}(\widehat{r}(z))\right)\right|^{\frac{1}{2}}\,D\left(\zeta_{(z)}(\widehat{r}(z),\widehat{\theta},\widehat{\phi})\right)\,-\,\widehat{D}(\widehat{r}(z))}{\widehat{D}(\widehat{r}(z))}\right]^{2}\,d\mu_{\hat{g}_{(z)}}\;.$

Thus, from the physical point of view,
$E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]$ describes the mean square
fluctuations of the physical area distance
$D\left(\zeta_{(z)}(\widehat{r}(z),\widehat{\theta},\widehat{\phi})\right)$
(biased by the localized $\mathrm{PSL}(2,\mathbb{C})$ mapping $\zeta_{(z)}$)
with respect to the reference FLRW isotropic area distance
$\widehat{D}(\widehat{r}(z))$.
Notice that, whereas the harmonic map energy
$E\left[\widehat{g}_{(z)},\,\psi_{(z)},\,{g}_{(z)}\right]$ is a conformal
invariant quantity, the functional $E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]$ is
not conformally invariant. Under a conformal transformation
$\hat{h}\,\longrightarrow\,e^{2f}\,\hat{h}$ we get

(112)
$\int_{\widehat{\Sigma}_{z}}\left(e^{\,-\,f}{\Phi}_{\widehat{\Sigma}\Sigma}\,-\,1\right)^{2}\,e^{2f}\,d\mu_{\hat{h}}\;.$

Since we can also write

(113)
${\Phi}_{\widehat{\Sigma}\Sigma}\,=\,\left[\frac{\psi_{(z)}^{*}d\mu_{g_{(z)}}}{d\mu_{\widehat{g}_{(z)}}}\right]^{\frac{1}{2}}\,,$

(see (101)), it is also clear from its definition that corresponding to large
linear ”stretches” in conformally mapping $\psi_{(z)}^{*}g_{(z)}$ on
$\widehat{g}_{(z)}$, $E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]$ tends to the
harmonic map energy.

In our particular framework, the functional
$E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]$ has many important properties that
make it a natural candidate for comparing, at the given length scale $L$, the
sky sections $(\widehat{\Sigma}_{z},\,\widehat{g}_{(z)})$ and
$(\Sigma_{z},\,{g}_{(z)})$ and, as the length-scale $L$ varies, the physical
lightcone region $\mathcal{C}^{-}_{L}(p,{g})$ with the FLRW reference region
$\mathcal{C}^{-}_{L}(p,\hat{g})$. These properties are discussed in detail in
[12] (see Lemma 8 and Theorem 9), here we recollect them, without presenting
their proof, in the following141414In [12], the general notation is somehow at
variance from the one adopted here, since we address the analysis of
$E_{\widehat{\Sigma}\Sigma}$ directly on the surfaces $\widehat{\Sigma}$ and
$\Sigma$. In particular, we refer to $\widehat{\Sigma}$ and $\Sigma$ as
celestial spheres rather than sky sections.

######  Theorem 1.

The functional $E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]$ is symmetric

(114)
$E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]\,=\,E_{\Sigma\widehat{\Sigma}}[\psi^{-1}_{(z)}]\;,$

where

(115)
$E_{\Sigma\widehat{\Sigma}}[\psi^{-1}_{(z)}]\,:=\,\int_{{\Sigma}_{z}}({\Phi}_{\Sigma\widehat{\Sigma}}\,-\,1)^{2}\,d\mu_{{g}_{(z)}}\;,$

is the comparison functional associated with the inverse map
$\psi_{(z)}^{-1}\,:\,{\Sigma}_{z}\,\longrightarrow\,\widehat{\Sigma}_{z}$, and
${\Phi}_{\Sigma\widehat{\Sigma}}$ is the corresponding conformal factor.
Let $(M,\,\widetilde{g})$ be another member of the FLRW family of spacetimes,
distinct from $(M,\,\hat{g})$, that we may wish to use as a control in a best-
fitting procedure for the physical spacetime $(M,g)$. Let
$(\widetilde{\Sigma}_{z},\,\widetilde{g}_{(z)})$ denote the sky section on the
past lightcone $\widetilde{\mathcal{C}}^{-}_{L_{0}}(p,\tilde{g})$, with vertex
at $p$, and let
$\widetilde{\psi}_{(z)}\,:\Sigma_{z}\,\longmapsto\,\widetilde{\Sigma}_{z}$,
and $\Phi_{\Sigma\widetilde{\Sigma}}$ respectively denote the corresponding
diffeomorphism and conformal factor. Then to the composition of maps

(116)
$\widehat{\Sigma}_{z}\,\underset{\psi_{(z)}}{\longrightarrow}\,\Sigma_{z}\,\underset{\widetilde{\psi}_{(z)}}{\longrightarrow}\,\widetilde{\Sigma}_{z}$

we can associate the triangular inequality

(117)
$E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]\,+\,E_{\Sigma\widetilde{\Sigma}}[\widetilde{\psi}_{(z)}]\,\geq\,E_{\widehat{\Sigma}\widetilde{\Sigma}}[(\widetilde{\psi}_{(z)}\circ\psi_{(z)})]\,,$

where

(118)
$E_{\widehat{\Sigma}\widetilde{\Sigma}}[(\widetilde{\psi}_{(z)}\circ\psi_{(z)})]\,:=\,\int_{\widehat{\Sigma}_{z}}({\Phi}_{\widehat{\Sigma}\widetilde{\Sigma}}\,-\,1)^{2}\,d\mu_{\widehat{g}_{(z)}}\;.$

Moreover,

(119) $E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]\,=\,0$

iff the sky sections $(\widehat{\Sigma},\,\hat{g}_{(z)})$ and
$(\Sigma,\,{g}_{(z)})$ are isometric. Finally, if we denote by
$\mathrm{W}^{1,2}_{\zeta_{(z)}}(\widehat{\mathbb{C\,S}}_{z}(p),\,\mathbb{C\,S}_{z}(p))$
the space of localized $\mathrm{PSL}(2,\mathbb{C})$\- maps $\zeta_{(z)}$ which
are of Sobolev class $\mathrm{W}^{1,2}$, (_i.e._ square summable together with
their first derivatives), then

(120)
$d_{(z)}\left[\widehat{\Sigma}_{z},\,\Sigma_{z}\right]\,:=\,\inf_{\zeta_{(z)}\in\mathrm{W}^{1,2}_{\zeta_{(z)}}(\widehat{\mathbb{C\,S}}_{z}(p),\,\mathbb{C\,S}_{z}(p))}\,E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]$

defines a scale-dependent distance between the sky sections
$(\widehat{\Sigma}_{z},\,\hat{g}_{(z)})$ and $(\Sigma_{z},\,g_{(z)})$ on the
lightcone regions $\mathcal{C}^{-}_{L}(p,\hat{g})$ and
$\mathcal{C}^{-}_{L}(p,g)$.

We need to conclude our long lightcone journey addressing the real nature of
the physical sky section $\Sigma_{z}$. This forces us to leave the comfort
zone of the assumed smoothness of the past physical lightcone
${\mathcal{C}}^{-}(p,\widehat{g})$.

##  7\. The Lipschitz geometry of the cosmological sky sections $\Sigma_{z}$

The celestial sphere description of the sky sections $\Sigma_{z}$ discussed
above is inherently vulnerable to the vagaries of the local distribution of
astrophysical sources, and the associated strong gravitational lensing
phenomena151515See [45] for a thorough analysis of the geometry of
gravitational lensing. imply that the actual past light cone
$\mathscr{C}^{-}(p,g)$ is not smooth as we have assumed161616The restrictive
nature of the smoothness assumption on the metric $g$, typically represented
by functions $g_{ab}\in C^{k}(\mathbb{R}^{4},\mathbb{R})$, $k\geq 2$, and of
the associated light cone, has been pointed out by many authors, mainly in the
context of the proof of singularity theorems and in causality theory, (see
_e.g._ [13], [17], [38],[42], [51]).. In particular, $\mathscr{C}^{-}(p,g)$
may fail to be the boundary $\partial\,\mathrm{I}^{-}(p,g)$ of the
chronological past $\mathrm{I}^{-}(p,g)$ of $p$, (the set of all events $q\in
M$ that can be connected to $p$ by a past-directed timelike curve), because
past-directed null geodesics generators of $\mathscr{C}^{-}(p,g)$,
$\lambda\,:\,[0,\delta)\,\longrightarrow\,(M,g)$, with $\lambda(0)\,=\,p$, may
leave $\partial\mathrm{I}^{-}(p,g)$ and, under the action of the local
spacetime curvature, plunge into the interior $\mathrm{I}^{-}(p,g)$. A
spacetime description of this behaviour in connection with the phenomenology
of gravitational lensing is discussed in detail in [45], with a rich
repertoire of examples of the possible singular structure that
$\mathscr{C}^{-}(p,g)$ may induce on the cosmological sky sections
$\Sigma(p,r)$. As a matter of fact, the sections $\Sigma_{z}$ may evolve into
fractal-like surfaces, and to describe them from the point of view of
geometric analysis, we need to introduce a framework tailored to the low-
regularity landscape generated by the local inhomogeneities.

###  7.1. The Lipschitz landscape

Given a past-directed null geodesic
$I_{W}\,\ni\,r\longmapsto\exp_{p}(rk(n(\theta,\phi)))$, issued from $p\in\,M$
in the direction $n(\theta,\phi)\,\in\,\mathbb{C\,S}_{z}$, we follow [37] and
define its _terminal point_ as the last–point

(121) $q(r_{*},n(\theta,\phi)):=\exp_{p}(rk(n(\theta,\phi)))$

that lies on the boundary $\partial\mathrm{I}^{-}(p,g)$ of the chronological
past of $p$. Any such terminal point $q(r_{*},n(\theta,\phi))$ is said to be:
_i)_ a _conjugate terminal point_ if the exponential map $\exp_{p}$ is
singular at $\left(r_{*},\,n(\theta,\phi)\right)$;  _ii)_ a _cut locus
terminal point_ if the exponential map $\exp_{p}$ is non–singular at
$\left(r_{*},\,n(\theta,\phi)\right)$ and there exists another null geodesic,
issued from $p$, passing through $q(r_{*},n(\theta,\phi))$, (see also [1],
[45]). We denote [37] by $\mathcal{T}^{-}(p)$ the set of all terminal points
associated with the past null geodesic flow issuing from $p$. In presence of
cut points, $\mathscr{C}^{-}(p,g)$ fails to be an embedded submanifold of
$(M,g)$. Failure to be an immersed manifold is more directly related to
conjugate points along the generators of $\mathscr{C}^{-}(p,g)$ and of the
associated conjugate locus [45]. It follows that in presence of terminal
points the mapping

(122)
$\left.\exp_{p}\right|_{\mathscr{C}^{-}(p,g)}\,:\,\mathbb{C\,S}_{z}\,\longrightarrow\,\Sigma_{z}\,:=\,\exp_{p}\,[\mathbb{C\,S}_{z}]$

is no longer one-to-one, and the cosmological sky section $\Sigma_{z}$ fails
to be a smooth surface. From the physical point of view, this is the
geometrical setting associated with the generation of multiple images of
astrophysical sources171717If the sources are not pointlike, we also have the
more complex ring patterns typical of strong gravitational lensing. in the
observer celestial sphere $\mathbb{C\,S}_{z}$. The mathematical framework for
handling such a scenario is to assume that the past null cone
$\mathscr{C}^{-}(p,g)$ has the regularity of a Lipschitz manifold,
characterized by a maximal atlas
$\mathcal{A}\,=\,\left\\{(U_{\alpha},\varphi_{\alpha})\right\\}$ such that all
transition maps between the coordinate charts $(U_{\alpha},\varphi_{\alpha})$
of $\mathscr{C}^{-}(p,g)$,

(123)
$\varphi_{\alpha\beta}:=\varphi_{\beta}\circ\varphi_{\alpha}^{-1}\,:\,\varphi_{\alpha}\left(U_{\alpha}\cap\,U_{\beta}\right)\,\longrightarrow\,\varphi_{\beta}\left(U_{\alpha}\cap\,U_{\beta}\right),\,$

are locally Lipschitz maps between domains of the Euclidean space
$(\mathbb{R}^{3},\delta)$. On $\mathscr{C}^{-}(p,g)$, the condition of being
Lipschitz can be viewed as a weakened version of the differentiability. In
particular, if $f:\,\mathscr{C}^{-}(p,g)\,\ni\,U\longrightarrow\mathbb{R}^{3}$
is a continuous map between open sets, then $f$ is Lipschitz if and only if it
admits distributional partial derivatives that are in $L^{\infty}(U)$ with
respect to the Lebesgue measure. This statement of Rademacher’s theorem [23],
[48] implies that the transition maps $\varphi_{\alpha\beta}$ on
$\mathscr{C}^{-}(p,g)$ have differentials $d\varphi_{\alpha\beta}$ that are
defined almost everywhere, and which are locally bounded an measurable on
their domains. In such a low-regularity setting the exponential map is quite
delicate to handle. However, a key result, geometrically proved by M.
Kunzinger, R. Steinbauer, M. Stojkovic [40], (based on work by B.-L. Chen and
P. LeFloch [13]), and by E. Minguzzi [42], implies that the exponential map
associated with a $C^{1,\,1}$ metric can still be defined as a local bi-
Lipschitz homeomorphism, namely a bijective map which along with its inverse
is Lipschitz continuous in a sufficiently small neighborhood of $p$. Thus, the
exponential map retains an appropriate form of regularity in the sense that
locally, for each point $p\in\,M$, there exist open star-shaped neighborhoods,
$N_{0}(p)$ of $0\in\,T_{p}M$ and $U_{p}\,\subset\,(M,g)$, such that
$\exp_{p}\,:\,N_{0}(p)\,\longrightarrow\,U_{p}$ is a bi-Lipschitz
homeomorphism [40]. In particular, each point $p\in(M,g)$ possesses a basis of
totally normal neighborhoods. It is worthwhile to stress that geodesic normal
coordinates (see (22)) can be still defined, but the transition from the
current smooth coordinate systems181818Recall that $M$ is a smooth manifold,
and that the low Lipschitz $C^{1,\,1}$ regularity is caused by the metric $g$,
and not by the differentiable structure of $M$. used around $p\in M$ to the
normal coordinates associated with $\exp_{p}$ is only continuous.

###  7.2. The fractal-like sky section $\Sigma_{z}$

We are interested in the geometry that such past light cone scenario induces
on the cosmological sky section
$\Sigma_{z}\,:=\,\exp_{p}\,[\mathbb{C\,S}_{z}]$ of $\mathcal{C}^{-}(p,g)$. As
long as $\exp_{p}$ is bi-Lipschitz, the sky sections $\Sigma_{z}$ are
topological 2-spheres, and the results above seem to suggest that after all
there is no such a strong motivation to abandon the comforts of the smooth
framework in favor of a Lipschitzian rugged landscape. However, as the length
scale $L$ varies, the development of caustics in $\mathscr{C}^{-}(p,g)$
generates cusps and crossings in the surfaces $\Sigma_{z}$, to the effect that
they are no longer homeomorphic to 2-spheres. In such a setting, the
restriction of the exponential map to the celestial sphere
$\mathbb{C\,S}_{z}$, characterizing the surface $\Sigma_{z}$, (see (71)),

(124)
$\exp_{p}\,:\,\mathbb{C\,S}_{z}\,\subset\,T_{p}M\,\longrightarrow\,\Sigma_{z}\,:=\,\exp_{p}\left[\mathbb{C\,S}_{z}\right]\,\subset\,\mathscr{C}^{-}(p,g)\,,$

is only a Lipschitz map between the metric spaces
$\left(\mathbb{C\,S}_{z},\,d_{\mathbb{S}_{r}^{2}}\right)$ and
$\left(\Sigma_{z},\,d_{g|\Sigma}\right)$, where $d_{\mathbb{S}_{r}^{2}}$ is
the standard distance function on the round 2-sphere ${\mathbb{S}^{2}_{r}}$ or
radius $r$, and $d_{g|\Sigma}$ is the distance function induced (almost
everywhere) on $\Sigma_{z}$ by the metric $g|_{\Sigma_{z}}$ defined191919In
presence of cut points the inclusion map
$\iota_{r}:\Sigma_{z}\,\hookrightarrow\,\mathscr{C}^{-}(p,g)$ of the sky
section $\Sigma_{z}$ into $\mathscr{C}^{-}(p,g)$ is Lipschitz, thus
Rademacher’s theorem allows us to define the pull-back metric
$g|_{\Sigma_{z}}\,:=\,\iota_{r}^{*}\,\left.g\right|_{\mathscr{C}^{-}(p,g)}$
only almost-everywhere by (26). In general, the sky section $\Sigma_{z}$ can
be topologically very complex since it may contain terminal points of the
exponential map $\exp_{p}$, giving rise to cusps and swallow-tail points
associated with self-intersections of $\Sigma_{z}$. Even if this may evolve in
a very complex picture of $\Sigma_{z}$, we still have quite a geometric
control over its metric structure. The Lipschitz regularity of $\exp_{p}$
implies that there is a constant $c_{r}$, depending on the parameter $r$, such
that

(125)
$d_{\Sigma(p,r)}\left(\exp_{p}(x),\exp_{p}(y)\right)\,\leq\,c_{r}\,d_{\mathbb{S}^{2}_{r}}(x,y),\,\,\,\,\forall\,x,\,y\,\in\,\mathbb{S}^{2}_{r}\,,$

and we can define the pull-back on the celestial sphere
$\mathbb{C\,S}_{z}\,\in\,T_{p}M$ of the distance function $d_{\Sigma_{z}}$
according to

(126)
$\exp_{p}^{*}d_{\Sigma_{z}}\,=\,d_{g\Sigma}\left(\exp_{p}(x),\exp_{p}(y)\right)\,,\,\,\,\,\forall\,x,\,y\,\in\,\mathbb{C\,S}_{z}\,.$

We can also pull-back the metric $g|_{g|\Sigma_{z}}$ to $\mathbb{C\,S}_{z}$.
By Rademacher’s theorem $\exp_{p}$ is differentiable almost everywhere, and

(127)
$h(\theta,\phi)\,:=\,\left(\exp_{p}^{*}\,g|_{\Sigma_{z}}\right)_{\alpha\beta}\,dx^{\alpha}dx^{\beta}\,,$

is a metric defined, almost everywhere on the celestial sphere
$\mathbb{C\,S}_{z}$, (by a slight abuse of language, we have used the same
notation as for the smooth version(27)). We can also define almost everywhere
the volume element $d\mu_{h}$ associated with the metric (127), _i.e._

(128)
$d\mu_{h}\,:=\,\exp_{p}^{*}d\mu_{g|_{\Sigma_{z}}}\,=\,\sqrt{\det(h(r(z),\theta,\phi))}\,d\theta
d\varphi\,,$

in full analogy with its smooth version (28). All this implies that with the
proviso of the almost everywhere meaning, the characterization (29) of the
angular diameter distance $D(r,\theta,\phi)$ and of the shear-inducing
distortion $L_{\alpha\beta}$ defined by (30), carry over to the bi-Lipschitz
case.

To put these geometrical remarks at work, let us stress that we cannot have
reasonable control over the very complex topological structure of the sky
section $\Sigma_{z}$ induced by a cascade of (strong) lensing events.
Moreover, the corresponding caustics and singularities at the terminal points
on $\Sigma_{z}$ provide a level of detail that is not relevant to the present
analysis. Thus, as a reasonable compromise, we assume that the exponential map
$\exp_{p}$ is bi-Lipschitz, that $\Sigma_{z}$ is topologically a 2-sphere, and
we mimic the effect of the many lensing events that may affect $\Sigma_{z}$ by
assuming that the sky section $\Sigma_{z}$ has the irregularities of a metric
surface with the fractal geometry of a 2-sphere with the locally-finite
Hausdorff 2-measure associated with (128). Under such assumptions, it can be
shown that our smooth analysis can be safely extended, (in particular, we can
still exploit the Poincaré–Koebe uniformization theorem [44]), and the results
obtained hold also in the more general setting of a Lipschitz description of
the cosmographic past lightcone $\mathcal{C}^{-}(p,g)$.

##  8\. Concluding remarks:
$d_{(z)}\left[\widehat{\Sigma}_{z},\,\Sigma_{z}\right]$ as a scale-dependent
field

According to the physical characterization (111) of
$E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]$, and the results described in Theorem
1, the distance function
$d_{(z)}\left[\widehat{\Sigma}_{z},\,\Sigma_{z}\right]$, (for simplicity, one
may work with the $E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]$ realizing the
minimum), can be interpreted as defining a $z$-dependent field on the FLRW
past light cone $\widehat{\mathcal{C}}^{-}(p,\widehat{g})$ describing the mean
square fluctuations of the anisotropies of the physical area distance
$D(\zeta_{(z)})$ with respect to the reference FLRW area distance
$\widehat{D}(\widehat{r}(z))$. These fluctuations provide information on how
much the local area element on the physical sky section $\Sigma_{z}$ differs
from the corresponding (round) area element on the reference FLRW sky section
$\Sigma_{z}$. Since for 2-dimensional surfaces the local Riemannian geometry
is fully described by the area element, the fluctuations in $D(\zeta_{(z)})$
give information on how much the geometries of the sky sections
$\widehat{\Sigma}_{z}$ and $\Sigma_{z}$ differ. When we reach the scale of
homogeneity, the physical area distance $D(\zeta_{(z)})$ becomes isotropic and
can be identified with the reference FLRW $\widehat{D}(\widehat{r}(z))$. The
localized null-directions alignment between the corresponding celestial
spheres $\mathbb{C\,S}_{z}(p)$ and $\widehat{\mathbb{C\,S}}_{z}(p)$ reduces to
a global kinematical Lorentz boost (and a rotation). Thus, corresponding to
this homogeneity scale, the distance function
$d_{(z)}\left[\widehat{\Sigma}_{z},\,\Sigma_{z}\right]$ field vanishes.

Thus, we have an interesting scenario whereby it is possible to associate with
the distance functional
$d_{(z)}\left[\widehat{\Sigma}_{z},\,\Sigma_{z}\right]$ a scale-dependent
field that describes a global effect that the reference FLRW past lightcone
$\widehat{\mathcal{C}}^{-}(p,\widehat{g})$ misses in describing the pre-
homogeneity anisotropies of the actual past lightcone
${\mathcal{C}}^{-}(p,\widehat{g})$. This _pre-homogeneity field_ is, in line
of principle, measurable since it is the mean-square variation of the physical
area distance $D(\zeta_{(z)})$. The delicate question concerns its possible
role in selecting the large-scale FLRW model that best fits the cosmological
observations on large scales. A few qualitative indications in this direction,
mainly of a perturbative nature, are discussed in [12]. The results presented
here are however more precise since they connect directly the distance
functional $d_{(z)}\left[\widehat{\Sigma}_{z},\,\Sigma_{z}\right]$ to the area
distance $D(\zeta_{(z)})$. To describe an important consequence of these
results, let us consider the light cone regions
$\mathcal{C}^{-}_{L}(p,\hat{g})$ and $\mathcal{C}^{-}_{L}(p,g)$ over a
sufficiently small length scale $L(z)$. If $\zeta_{(z)}$ and the corresponding
$\psi_{(z)}$ denote the minimizing maps characterized in Theorem 1, then we
can write [12]

$\displaystyle E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]\,$ $\displaystyle=$
$\displaystyle\,\int_{\widehat{\Sigma}_{z}}({\Phi}_{\widehat{\Sigma}\Sigma}\,-\,1)^{2}\,d\mu_{\widehat{g}_{(z)}}\,=\,\int_{\widehat{\Sigma}_{z}}{\Phi}_{\widehat{\Sigma}\Sigma}^{2}\,d\mu_{\widehat{g}_{(z)}}\,+\,\int_{\widehat{\Sigma}_{z}}\,d\mu_{\widehat{g}_{(z)}}\,-\,2\int_{\widehat{\Sigma}_{z}}{\Phi}_{\widehat{\Sigma}\Sigma}\,d\mu_{\widehat{g}_{(z)}}$
$\displaystyle=$
$\displaystyle\int_{\widehat{\Sigma}_{z}}\frac{\psi_{g_{(z)}}^{*}d\mu_{g_{(z)}}}{d\mu_{\widehat{g}_{(z)}}}\,d\mu_{\widehat{g}_{(z)}}\,+\,A\left(\widehat{\Sigma}_{z}\right)\,-\,2\int_{\widehat{\Sigma}_{z}}{\Phi}_{\widehat{\Sigma}\Sigma}\,d\mu_{\widehat{g}_{(z)}}$
$\displaystyle=$
$\displaystyle\int_{\psi_{(z)}(\widehat{\Sigma}_{z})}{d\mu_{g_{(z)}}}\,+\,A\left(\widehat{\Sigma}_{z}\right)\,-\,2\int_{\widehat{\Sigma}_{z}}{\Phi}_{\widehat{\Sigma}\Sigma}\,d\mu_{\widehat{g}_{(z)}}$
$\displaystyle=$ $\displaystyle
A({\Sigma}_{z})\,+\,A\left(\widehat{\Sigma}_{z}\right)\,-\,2\int_{\widehat{\Sigma}_{z}}{\Phi}_{\widehat{\Sigma}\Sigma}\,d\mu_{\widehat{g}_{(z)}}\;,$

where we have exploited the Radon-Nikodyn characterization of
$\widehat{\Phi}_{\widehat{\Sigma}\Sigma}^{2}$, (see (101)), the identification
$\psi_{(z)}(\widehat{\Sigma}_{z})\,=\,\Sigma_{z}$, and the relation

(130)
$\int_{\widehat{\Sigma}_{z}}\frac{\psi_{(z)}^{*}d\mu_{{g}_{(z)}}}{d\mu_{\widehat{g}_{(z)}}}\,d\mu_{\widehat{g}_{(z)}}=\int_{\widehat{\Sigma}_{z}}{\psi_{(z)}^{*}d\mu_{g_{(z)}}}=\int_{\psi(\widehat{\Sigma}_{z})}{d\mu_{g_{(z)}}}=\int_{{\Sigma}_{z}}{d\mu_{g_{(z)}}}\,=\,A({\Sigma}_{z})\;,$

where $A({\Sigma}_{z})$ and $A\left(\widehat{\Sigma}_{z}\right)$ respectively
denote the area of the sky sections $(\hat{\Sigma}_{z},\,\widehat{g}_{(z)})$
and $(\Sigma_{z},\,g_{(z)})$. Thus,

(131)
$d_{(z)}\left[\widehat{\Sigma}_{L},\,{\Sigma}_{L}\right]\,=\,E_{\widehat{\Sigma}\Sigma}[\psi_{(z)}]\,:=\,A\left(\widehat{\Sigma}_{z}\right)\,+\,A({\Sigma}_{z})\,-\,2\int_{\widehat{\Sigma}_{z}}{\Phi}_{\widehat{\Sigma}\Sigma}\,d\mu_{\widehat{g}_{(z)}}\,.$

To simplify matters, we assume that at the given length scale $L(z)$ the
corresponding region $\mathcal{C}^{-}_{L}(p,g)$ is caustic free. Let us
rewrite ${\Phi}_{\Sigma\widehat{\Sigma}}$ as

$\displaystyle{\Phi}_{\Sigma\widehat{\Sigma}}\,$ $\displaystyle=$
$\displaystyle\,\left({\Phi}_{\Sigma\widehat{\Sigma}}\,-\,1\right)\,+\,1$
$\displaystyle=$
$\displaystyle\,\frac{\left|\mathrm{Jac}\left(\zeta_{(z)}\right)\right|^{\frac{1}{2}}\,D(\zeta_{(z)})\,-\,\widehat{D}(\widehat{r}(z))}{\widehat{D}(\widehat{r}(z))}\,+\,1\,,$

where we have simplified the notation used in (108). By introducing this in
(131) we get

(133)
$d_{L}\left[\widehat{\Sigma}_{z},\,{\Sigma}_{z}\right]\,=\,A\left({\Sigma}_{z}\right)\,-\,A(\widehat{\Sigma}_{z})\,-\,2\int_{\widehat{\Sigma}_{z}}\,\left[\frac{\left|\mathrm{Jac}\left(\zeta_{(z)}\right)\right|^{\frac{1}{2}}\,D(\zeta_{(z)})\,-\,\widehat{D}(\widehat{r}(z))}{\widehat{D}(\widehat{r}(z))}\right]\,d\mu_{\widehat{g}_{(z)}}\,.$

This expression can be further specialized if we exploit the asymptotic
expressions of the area $A\left(\widehat{\Sigma}_{z}\right)$ and
$A\left({\Sigma}_{z}\right)$ of the two surfaces
$(\widehat{\Sigma}_{z},\,\widehat{g}_{(z)})$, $({\Sigma}_{z},\,{g}_{(z)})$ on
the corresponding lightcones $\mathcal{C}^{-}_{{L}}(p,\widehat{g})$ and
$\mathcal{C}^{-}_{{L}}(p,{g})$. These asymptotic expressions can be obtained
if we consider the associated causal past regions
$\mathcal{J}^{-}_{{L}}(p,\widehat{g})$ and $\mathcal{J}^{-}_{{L}}(p,{g})$
sufficiently near the (common) observation point $p$, in particular when the
length scale $L(z)$ we are probing is small with respect to the ”cosmological”
curvature scale. Under such assumption, there is a unique maximal
3-dimensional region $V_{L}^{3}(p)$, embedded in
$\mathcal{J}^{-}_{{L}}(p,{g})$, having the surface $({\Sigma}_{z},\,{h})$ as
its boundary. This surface intersects the world line $\gamma(\tau)$ of the