A comparison theorem for cosmological lightcones

Abstract

Let (M, g) denote a cosmological spacetime describing the evolution of a universe which is isotropic and homogeneous on large scales, but highly inhomogeneous on smaller scales. We consider two past lightcones, the first, ${{\mathcal {C}}_{L}^{-}}(p, g)$, is associated with the physical observer $p\in \,M$ who describes the actual physical spacetime geometry of (M, g) at the length scale L, whereas the second, ${\mathcal {C}_{L}^{-}}(p, \hat{g})$, is associated with an idealized version of the observer p who, notwithstanding the presence of local inhomogeneities at the given scale L, wish to model (M, g) with a member $(M, \hat{g})$ of the family of Friedmann–Lemaitre–Robertson–Walker spacetimes. In such a framework, we discuss a number of mathematical results that allows a rigorous comparison between the two lightcones ${\mathcal {C}_{L}^{-}}(p, g)$ and ${\mathcal {C}_{L}^{-}}(p, \hat{g})$. In particular, we introduce a scale-dependent (L) lightcone-comparison functional, defined by a harmonic type energy, associated with a natural map between the physical ${\mathcal {C}_{L}^{-}}(p, g)$ and the FLRW reference lightcone ${\mathcal {C}_{L}^{-}}(p, \hat{g})$. This functional has a number of remarkable properties, in particular it vanishes iff, at the given length-scale, the corresponding lightcone surface sections (the celestial spheres) are isometric. We discuss in detail its variational analysis and prove the existence of a minimum that characterizes a natural scale-dependent distance functional between the two lightcones. We also indicate how it is possible to extend our results to the case when caustics develop on the physical past lightcone ${\mathcal {C}_{L}^{-}}(p, g)$. Finally, by exploiting causal diamond theory, we show how the distance functional is related (to leading order in the scale L) to spacetime scalar curvature in the causal past of the two lightcones, and briefly illustrate a number of its possible applications.

1 Introduction

For many among us, the first encounter with Boris Dubrovin has been through his classic books G$\acute{e}$ometrie Contemporaine: m$\acute{e}$thodes et applications, coauthored with S. Novikov and A. Fomenko, and published by MIR in a silk-bonded three-volume set, an edition that, notwithstanding the later expanded Springer version [9], we treasured with care. To the best of our knowledge, Boris did not work in general relativity but in his G$\acute{e}$ometrie Contemporaine there are two wonderful little chapters, just short of a total of fifty tersely written pages, which provide the fastest way to acquaint yourself with general relativity. Thinking of these elegant pages, we hope that it is not inappropriate to dedicate to Boris the present work addressing a long-standing problem in mathematical cosmology.

Let us recall that the observed universe is described by a spacetime (M, g), a four-dimensional manifold M endowed with a Lorentzian metric g, which is (statistically) isotropic and homogeneous only on sufficiently large scales, say $L\,\ge \,L_0$, where the current acceptable figure for the homogeneity scale is $L_0\ge 100h^{-1}\,Mpc$, where h is the dimensionless Hubble parameter describing the relative uncertainty of the true value of the present-epoch Hubble–Lemaitre constant $H_0\,=\,100\,h\,\,Km/s/Mpc$. At these homogeneity scales (M, g) is described with great accuracy by a member of the homogeneous and isotropic family of Friedman–Lemaitre–Robertson–Walker (FLRW) spacetimes $(M, \hat{g})$. At smaller scales, where inhomogeneities statistically dominate, we should resort to the full-fledged spacetime geometry of (M, g) in order to provide the correct dynamical description of cosmological observations. However, coming to mathematical terms with the geometrical and physical structure of (M, g) is a daunting task and typically we keep on modeling the dynamics of the universe over these inhomogeneity scales with the FLRW model $(M, \hat{g})$, thought of as providing a background around which the actual spacetime geometry (M, g) is perturbatively expanded. If we want to go beyond perturbation theory, we face the mathematically delicate problem of finding a way for comparing the past lightcone region¹${\mathcal {C}_{L}^{-}}(p, {g})$, associated with an (instantaneous) observer p sampling the inhomogeneities in (M, g) at the given length scale L, with the corresponding past lightcone region ${\mathcal {C}_{L}^{-}}(p, \hat{g})$ in the assumed FLRW background $(M, \hat{g})$. In modern high-precision cosmology this is one of the most delicate issues when modeling the observed universe. Inspired by the foundational works by Ellis et al. [10, 11], in this paper we provide a number of mathematical results that allow to compare the past lightcone regions ${\mathcal {C}_{L}^{-}}(p, {g})$ and ${\mathcal {C}_{L}^{-}}(p, \hat{g})$. In particular, we introduce a scale-dependent conformal diffeomorphism $\varphi _L$ and a corresponding lightcone comparison functional $E_{{\widehat{\Sigma }}\Sigma }[\varphi _L]$ between the physical and the FLRW reference celestial spheres $\Sigma _L\,\subset \,{\mathcal {C}_{L}^{-}}(p, {g})$ and ${\widehat{\Sigma }}_L\, \subset \, {\mathcal {C}_{L}^{-}}(p, \hat{g})$ probed, at the given length scale L, on the respective lightcones regions ${\mathcal {C}_{L}^{-}}(p, {g})$ and ${\mathcal {C}_{L}^{-}}(p, \hat{g})$. It is important to stress that the scale-dependent map $\varphi _L$, is not an abstract map, but it actually relates the physical observations on $\Sigma _L$ with those, described with a FLRW bias, on ${\widehat{\Sigma }}_L$. The functional $E_{{\widehat{\Sigma }}\Sigma }[\varphi _L]$ is defined by a harmonic map type energy and has a number of remarkable properties. In particular, it vanishes iff , at the given length-scale L, the corresponding lightcone surface sections $\Sigma _L$ and ${\widehat{\Sigma }}_L$ (which are topologically 2-spheres, as long as null-caustics are absent) are isometric. Moreover, by taking the $\inf$ of $E_{{\widehat{\Sigma }}\Sigma }[\varphi _L]$ over a suitable class of extended maps $\varphi _L$ (extension necessary in order to account also for the presence of lightcone caustics), we provide a scale-dependent distance functional, $d_L[{\widehat{\Sigma }}, \Sigma ]$, between the physical and the FLRW reference lightcones ${\mathcal {C}_{L}^{-}}(p, {g})$ and ${\mathcal {C}_{L}^{-}}(p, \hat{g})$. This distance significantly extends the lightcone theorem proved in [7]. In order to characterize the physical interpretation of $E_{{\widehat{\Sigma }}\Sigma }[\varphi _L]$ we exploit small causal diamond theory, as described in [13], to show that in the caustic-free region near the tip p of ${\mathcal {C}_{L}^{-}}(p, {g})$ and ${\mathcal {C}_{L}^{-}}(p, \hat{g})$, namely for L small enough, $d_L[{\widehat{\Sigma }}, \Sigma ]$ is perturbatively related (at first order in the scale L), to the gravitational lensing distortion on ${\mathcal {C}_{L}^{-}}(p, {g})$ and to the spacetime scalar curvatures R(g) and $R(\hat{g})$ in the interior of these lightcones. This connection between $d_L[{\widehat{\Sigma }}, \Sigma ]$ and the rich literature on causal diamond is a property that may play an important role in cosmological modeling, and the characterization of a full non-perturbative relation between the analysis presented in this paper and causal diamond theory is a relevant open problem to address². In particular, it would be interesting to explore if there is a link between the functionals $E_{{\widehat{\Sigma }}\Sigma }[\varphi _L]$ and $d_L[{\widehat{\Sigma }}, \Sigma ]$ introduced here and the comparison theorems for causal diamonds discussed in detail in the remarkable paper by Berthiere et al. [2]. Even if there is no direct relation between their approach and our motivation, geometric analysis techniques and results, the perturbative characterization mentioned above indicates that these two approaches may profitably interact. We believe more progress can be made in cosmological modeling if we join these two points of view. Finally, it must be stressed that the results presented here can be easily extended to fiducial reference metrics more general than those described by the FLRW family, e.g., to a homogeneous solution of Einstein equations. Our choice of a reference FLRW is naturally related to the actual prevalence of this family of metrics in discussing cosmological dynamics.

2 Cosmological observers and observational coordinates along the past lightcones

Throughout this paper (M, g) denotes a cosmological spacetime where g is a Lorentzian metric, and where M is a smooth four-dimensional manifold which for our purposes we can assume diffeomorphic to $\mathbb {R}^4$. In local coordinates $\{x^i\}_{i=1}^4$, we write $g=g_{ik}\text {d}x^i\otimes \text {d}x^k$, 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 effect. We denote by $\nabla _{(g)}$ (or $\nabla$ if there is no danger of confusion) the Levi–Civita connection of g, and let $\mathcal {R}m(g)=\mathcal {R}^{i}_{klm}\,\partial _i\otimes \text {d}x^k\otimes \text {d}x^l\otimes \text {d}x^m$, $\mathcal {R}ic(g)=\mathcal {R}_{ab}\,\text {d}x^a\otimes \text {d}x^b$ and $\mathcal {R}(g)$ be the corresponding Riemann, Ricci and scalar curvature operators, respectively. We assume that (M, g) is associated with the evolution of a universe which is (statistically) isotropic and homogeneous on sufficiently large scales, whereas local inhomogeneities dominate over smaller scales. The mass–energy content in (M, g) is phenomenologically described by an energy-momentum tensor T the explicit expression of which is not needed in our analysis, we only assume that its matter components characterize a Hubble flow that generates a family of preferred worldlines parametrized by proper time $\tau$

$$\begin{aligned} \gamma _s\,:\,\mathbb {R}_{>0}\,\longrightarrow & {} \,(M, g)\nonumber \\ \tau \,\longmapsto & {} \,\gamma _s(\tau )\;, \end{aligned}$$

(1)

and labeled by suitable comoving (Lagrangian) coordinates s. We set $c\,=\,1$, and denote by $\dot{\gamma }_s\,:=\,\frac{\text {d}\gamma _s(\tau )}{\text {d}\tau }$, with  $g(\dot{\gamma }_s, \dot{\gamma }_s)\,=\,-1$, the corresponding 4-velocity field. For simplicity, we assume that the worldlines (1) are geodesics, i.e., $\nabla _{\dot{\gamma }_s}\,\dot{\gamma }_s\,=\,0$. This is the spacetime within which we can frame the actual cosmological data gathered from our past lightcone observations. If we adopt the weak form of the cosmological principle, $(M, g, \gamma _s)$ can be identified with the phenomenological background spacetime or phenomenological background solution (PBS), according to the notation introduced in [21]. In the same vein, we define phenomenological observers the collection of observers $\{\gamma _s\}$ comoving with the Hubble flow.

2.1 The phenomenological lightcone metric

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 worldline with the timelike geodesic segment $\tau \,\longmapsto \,\gamma (\tau )$, $-\delta<\tau <\delta$,   for some $\delta >0$,   where $p\,:=\,\gamma (\tau =0)$ is the selected observational event. To set up the appropriate coordinates along $\gamma (\tau )$, 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 $E_{(4)}\,:=\,\dot{\gamma }(\tau )|_{\tau =0}$. Notice that by parallel transport, this basis can be propagated along $\gamma (\tau )$. Let us introduce the set of past-directed null vectors and the set of past-directed causal vectors in $(T_pM, g_p)$ according to

$$\begin{aligned} C^{-}\left( T_pM, g_p \right) \,:= & {} \,\left\{ X\,=\,\mathbb {X}^iE_{(i)}\,\not =\,0\,\in \,T_pM\,\,|\,\,\mathbb {X}^4+r=0 \right\} \;, \end{aligned}$$

(2)

$$\begin{aligned} \overline{C^{-}}\left( T_pM, g_p \right) \,:= & {} \,\left\{ X\,=\,\mathbb {X}^iE_{(i)}\,\not =\,0\,\in \,T_pM\,\,|\,\,\mathbb {X}^4+r\,\le \,0 \right\} \;, \end{aligned}$$

(3)

where $r:=(\sum _{a=1}^3(\mathbb {X}^a)^2)^{1/2}$. We use these sets of vectors in order to introduce observational coordinates in (a region of) the causal past of p, ${J}^{-}(p, g)$, by exploiting the exponential mapping based at p,

$$\begin{aligned} \exp _p\,:\,\overline{C^{-}}\left( T_pM, g_p \right) \,\longrightarrow & {} \;\;\;\;M \end{aligned}$$

(4)

$$\begin{aligned} X\;\;\;\;\;\;\longmapsto & {} \;\;\;\;\mathrm {exp}_p\,(X)\,:=\,\lambda _X(1) \end{aligned}$$

(5)

where $\lambda _X\,:\,[0, \infty )\,\longrightarrow \,(M, g)$ is the past-directed causal geodesic emanating from the point p with initial tangent vector $\dot{\gamma }_X(0)\,=\, X\in \overline{C^{-}}\left( T_pM, g_p \right)$. If we assume that the metric is sufficiently regular³, then there is a neighborhood $N_0(g)$ of 0 in $T_pM$ and a geodesically convex neighborhood of p, $U_p \subset \,(M, g)$, defined by all points $q\in M$ which are within the domain of injectivity of $\mathrm {exp}_p$, where we can introduce geodesic normal coordinates $(X^i)$ according to

$$\begin{aligned} X^i\,:=\,\mathbb {X}^i\,\circ \,\exp _p^{-1}\,:\,M\cap \,U_p\,\longrightarrow & {} \,\mathbb {R}^4\nonumber \\ q\,\,\,\,\,\,\,\longmapsto & {} \,X^i(q)\,:=\, \mathbb {X}^i\left( \exp _p^{-1}(q)\right) \end{aligned}$$

(6)

and where $\mathbb {X}^i\left( \exp _p^{-1}(q)\right)$ are the components, in the g-orthonormal frame $\{E_{(i)}\}$, of the vector $\exp _p^{-1}(q)\,\in \,T_pM$. In particular, if we consider the past lightcone $\mathcal {C}^{-}(p,g)$ with vertex at p, then away from the past null cut locus of p, i.e., away from the set of lightcone caustics, normal coordinates can be used to parametrize the past light cone region $\mathcal {C}^{-}(p,g)\cap \,U_p$,

$$\begin{aligned} \exp _p\,:\,C^{-}\left( T_pM,\,g_p\right) \cap \,N_0(g)\,\longrightarrow & {} \,\mathcal {C}^{-}(p,g)\,\cap \,U_p\nonumber \\ X\,=\,\mathbb {X}^i E_{(i)}\,\,\,\,\,\,\,\,\,\,\longmapsto & {} \,\exp _p(\mathbb {X}^iE_{(i)})\,=\,q\, \Rightarrow \,\{X^i(q) \}\,. \end{aligned}$$

(7)

Similarly, by restricting $\exp _p$ to $\overline{C^{-}}\left( T_pM,\,g_p\right) \cap \,N_0(g)$ we can parametrize with normal coordinates the region ${J}^{-}(p,g)\,\cap \,U_p$ within the causal past ${J}^{-}(p,g)$ of p. In particular, we can foliate ${J}^{-}(p,g)\,\cap \,U_p$ with the family of past lightcones $\mathcal {C}^{-}(\gamma (\tau ),g)$ associated with the events $\gamma (\tau )\cap \,U_p$,   $-\delta \,<\,\tau \,\le \,0$, along the observer past-directed worldline. We can specialize the normal coordinates so introduced by setting

$$\begin{aligned} x^1\,:=\,r\,:=\,\sqrt{\sum _{a=1}^3(X^a)^2},\,\,\,\, x^2\,:=\,\theta \left( {X^a}/{r}\right) ,\,\,\,\, x^3\,:=\,\varphi \left( {X^a}/{r}\right) ,\,\,\,\, x^4\,:=\,\tau \,=\,X^4\,+\,r\,, \end{aligned}$$

(8)

where $\theta \left( {X^a}/{r}\right)$,   $\varphi \left( {X^a}/{r}\right)$,  $a=1,2,3$, denote the standard angular coordinates of the direction $\left( {X^a}/{r}\right)$ on the unit 2-sphere $\mathbb {S}^2$ in $T_pM$ and where, according to (2), $x^4= 0$ corresponds to the light cone region $\mathcal {C}^{-}(p,g)\cap \,U_p$. Notice that at the vertex $p\,=\,\gamma (\tau =0)$, the coordinate function $x^4$ is not differentiable (but it is continuous).

Remark 1

Under the stated hypotheses, and as long as we stay away from the vertex p and from its null cut locus, we have that the past lightcone region $\mathcal {C}^{-}(p, g)\,\cap \,U_p\setminus \{p\}$ is topologically foliated by the r-dependent family of two-dimensional surfaces $\Sigma (p, r)$, the celestial spheres at scale r, reached by the set of past directed null geodesics as the affine parameter r varies, i.e.,

$$\begin{aligned} \Sigma (p, r)\,:=\,\left\{ \left. \exp _p\left( r\, \underline{n}\right) \,\right| \,\, \underline{n}:=(\theta ,\,\varphi ) \,\in \,\mathbb {S}^2\,\subset \,T_pM\right\} \;. \end{aligned}$$

(9)

Each $\Sigma (p, r)$ is topologically a 2-sphere endowed with the r-dependent family of two-dimensional Riemannian metrics

$$\begin{aligned} h(r)\,:=\,\left. \left( \exp _p^*\,g|_{\Sigma (p, r)}\right) _{\alpha \beta }\,\text {d}x^\alpha \text {d}x^\beta \right| _{r} \end{aligned}$$

(10)

obtained by using the exponential map to pull back to $\mathbb {S}^2\subset T_pM$ the two-dimensional metric $g|_{\Sigma (p, r)}$ induced on $\Sigma (p, r)$ by the embedding $\Sigma (p, r)\,\hookrightarrow \,(M, g)$. We normalize this metric by imposing that the angular variables $x^\alpha =(\theta , \varphi )$, in the limit $r\,\searrow \, 0$, reduce to the standard spherical coordinates on the unit 2-sphere $\mathbb {S}^2$, i.e.,

$$\begin{aligned} \left. \lim _{r\searrow 0}\right| _{x^4=0}\,\frac{h_{\alpha \beta }(r)\,\text {d}x^\alpha \text {d}x^\beta }{r^2}\, =\,\text {d}\Omega ^2\,:=\,\text {d}\theta ^2\,+\,\sin ^2\theta \,\text {d}\varphi ^2\;. \end{aligned}$$

(11)

For a physical interpretation [11], it is convenient to parametrize h(r) as a sky-mapping metric

$$\begin{aligned} h(r)\,=\,D^2(r)\,\left( \text {d}\Omega ^2\,+\,\mathcal {L}_{\alpha \beta }(r)\text {d}x^\alpha \text {d}x^\beta \right) \;, \end{aligned}$$

(12)

where $\text {d}\Omega ^2$ is the unit radius round metric on $\mathbb {S}^2$ (see (11)), and the coordinates $\{x^\alpha \}_{\alpha =2,3}$ provide the direction of observation (as seen at p) of the astrophysical sources on the celestial sphere $\Sigma (p, r)$.  The function D(r) is the observer area distance defined by the relation $\text {d}\mu _{h(r)} = D^2(r)\,\text {d}\mu _{\mathbb {S}^2}$ where $\text {d}\mu _{h(r)}$ is the pulled-back (via $\exp _p$) area measure of $\left( \Sigma (p, r),\,g|_{\Sigma (p, r)}\right)$, (roughly speaking, $\text {d}\mu _{h(r)}$ can be interpreted [11] as the cross-sectional area element at the source location as seen by the observer at p) and $\text {d}\mu _{\mathbb {S}^2}$ is the area element on the unit round sphere $\mathbb {S}^2\in \,T_pM$ (i.e., the element of solid angle subtended by the source at the observer location p). In the same vein, the symmetric tensor field $\mathcal {L}_{\alpha \beta }(r)$, describing the distortion of the normalized metric $h(r)/D^2(r)$ with respect to the round metric $\text {d}\Omega ^2$, 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. This term, which in general is not trace-free, involves both the gravitational lensing shear [11] and the gravitational focusing of the light rays generating the local source image magnification. By taking into account these remarks, we have the following characterization of the past lightcone metric in a neighborhood of the point p.

Lemma 2

In the geometrical coordinates introduced above, the null geodesics generators of $\mathcal {C}^{-}(p, g)\,\cap \,U_{p}$ have equation $x^4\,=\,0$, $x^\alpha \,=\, const.$, and their tangent vector is provided by $\frac{\partial }{\partial x^1}$,   with $(\exp ^*g)\left( \frac{\partial }{\partial x^1},\,\frac{\partial }{\partial x^1}\right) \,=\,0$. Since $\frac{\partial }{\partial x^1}$ is past-directed we can introduce the normalization

$$\begin{aligned} \lim _{r\searrow 0}\,(\exp ^*g)\left( \frac{\partial }{\partial x^1},\,\dot{\gamma }\right) \,=\,1\, \end{aligned}$$

(13)

and write the restriction of the spacetime metric g on $\mathcal {C}^{-}(p, g)\,\cap \,U_{p}$ according to⁴

$$\begin{aligned} \left. g\right| _{x^4=0}\,=\,g_{44}\,(\text {d}x^4)^2+2g_{14}\text {d}x^1\text {d}x^4+2g_{4\alpha }\text {d}x^4\text {d}x^\alpha +h_{\alpha \beta }\text {d}x^\alpha \text {d}x^\beta \;, \end{aligned}$$

(14)

where $\alpha , \beta =2,3$, and where the components $g_{ik}(x^i):=(\exp ^*g)(\frac{\partial }{\partial x^i},\,\frac{\partial }{\partial x^k})$, and $h_{\alpha \beta }(x^i)$ are all evaluated for $x^4\,=\,0$.

As already stressed, the coordinates $\{x^i\}$ are singular at the vertex $\gamma (\tau =0)=p$ of the cone. A detailed analysis of the limit $r\,\searrow \, 0$, besides the standard assumptions we already made, is carried out in detail in the foundational paper [10] (see paragraph 3) and in [8], (see paragraphs 4.2.1-4.2.3-4.5, the results presented there are stated for the future lightcone, but they can be easily adapted to the past lightcone).

Remark 3

Clearly the lightcone metric (14) does not hold when caustics form; however, our final result involving the characterization of a distance functional between lightcones naturally extends to the case when caustics are present.

2.2 The reference FLRW lightcone metric

Along the physical metric g, we also introduce in M the FLRW metric $\hat{g}$ and the family of global Friedmannian observers $\hat{\gamma }_s$ that, at the homogeneity scale, we can associate with the cosmological data. This is the global background solution (GBS) according to [21]. In full generality 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 $(M, \hat{g}, \hat{\gamma })$, will be distinct but, in line with the set up adopted here, we assume that they share a common observational event p. We 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 $\gamma (0)\,=\,p\,=\,\hat{\gamma }(0)$. Hence, together with the coordinates $\{x^i \}$ in $(M, g, \gamma _s)$, describing the observational metric (14) on the past lightcone $\mathcal {C}^{-}(p,g)\,\cap \,U_{p}$, we introduce corresponding (normal) coordinates $\{Y^k \}$ in the reference $(M, \hat{g}, \hat{\gamma })$. With an obvious adaptation of the analysis for (M, g), carried out in previous subsection, let $N_0(\hat{g})$ denote the domain of injectivity of the exponential mapping $\widehat{\exp }_{p}\,:\,T_{p}M\,\longrightarrow \,(M, \hat{g})$ based at the event $p=\hat{\gamma }(0)$. If $\hat{U}_{p}\subset \,(M, \hat{g})$ denotes the region of injectivity of $\widehat{\exp }_{p}$ we can consider normal coordinates

$$\begin{aligned} Y^i\,:=\,\mathbb {Y}^i\,\circ \,\widehat{\exp }_p^{-1}\,:\,(M, \hat{g})\cap \,\hat{U}_{p}\,\longrightarrow \,\mathbb {R}\,, \end{aligned}$$

(15)

where $\mathbb {Y}^i$ are the components of the vectors $\mathbb {Y}\in \,T_pM$ with respect to a $\hat{g}$-orthonormal frame $\{\hat{E}_{(i)}\}_{i=1,\ldots ,4}$ with $\hat{E}_{(4)}\,:=\,\hat{\dot{\gamma }}(0)$. Within $\hat{U}_{p}$ we can introduce, in full analogy with (7) and (14), the coordinates $y^1\,:=\,\hat{r}\,=\,(\sum _{a=1}^3(Y^a)^2)^{1/2}$,  $y^\alpha |_{\alpha =2,3}\,=\,\left( \theta \left( {Y^a}/\hat{r}\right) ,\,\varphi \left( {Y^a}/\hat{r}\right) \right)$ and parametrize $\mathcal {C}^{-}(p,\hat{g})\,\cap \,\hat{U}_{p}$ in terms of the two-dimensional spheres

$$\begin{aligned} \hat{\Sigma }(p, \hat{r})\,:=\,\left\{ \left. \hat{\exp _p}\left( \hat{r}\, \underline{n}\right) \,\right| \,\, \underline{n}:=(\theta ,\,\varphi ) \,\in \,\mathbb {S}^2\,\subset \,T_pM\right\} \;, \end{aligned}$$

(16)

endowed with the round metric

$$\begin{aligned} \hat{h}(\hat{r})\,:=\,\left. (\hat{g})_{\alpha \beta }\text {d}y^\alpha \text {d}y^\beta \right| _{\hat{r}}\,=\, a^2(\hat{r})\,\hat{r}^2\left( \text {d}\theta ^2\,+\,\sin ^2\theta \text {d}\varphi ^2\right) \;, \end{aligned}$$

(17)

where $a(\hat{r})$ is the FLRW expansion factor corresponding to the distance $\hat{r}$. Hence, we can write the metric $\hat{g}$ on the reference FLRW past lightcone region $\mathcal {C}^{-}(p,\hat{g})\,\cap \,\hat{U}_{p}$ as

$$\begin{aligned} \left. \hat{g}\right| _{y^4=0}\,=\,\hat{g}_{44}\,(\text {d}y^4)^2+\hat{h}_{\alpha \beta }\text {d}y^\alpha \text {d}y^\beta \;. \end{aligned}$$

(18)

3 Comparing lightcones: a scale dependent comparison functional

According to our hypotheses, the spacetime $(M, g, \gamma _s)$ describes the evolution of a universe which is isotropic and homogeneous only at sufficiently large scales $L_0$. At these homogeneity scales, $(M, g, \gamma _s)$ is modeled by the FLRW spacetime $(M, \hat{g}, \hat{\gamma }_s)$. Even if at smaller scales, where inhomogeneities statistically dominate, $(M, g, \gamma _s)$ provides the bona fide spacetime describing cosmological observations, we can still use the reference $(M, \hat{g}, \hat{\gamma }_s)$ as a background FLRW model. As the observational length scale L varies from the local highly inhomogeneous regions to the homogeneity scale $L_0$, we do not assume a priori that $(M, g, \gamma _s)$ is perturbatively near to the reference FLRW spacetime $(M, \hat{g}, \hat{\gamma }_s)$. Rather, we compare $(M, g, \gamma _s)$ with $(M, \hat{g}, \hat{\gamma }_s)$ keeping track of the pointwise and global relations among the various geometric quantities involved. In particular, we will compare the lightcone region $\mathcal {C}^{-}(p, {g})\,\cap \,{U}_{p}$ with the reference FLRW lightcone region $\mathcal {C}^{-}(p,\hat{g})\,\cap \,\hat{U}_{p}$, assuming that in such a range there are no lightcone caustics. As already emphasized, this is an assumption that makes easier to illustrate some of the technical arguments presented here, in the final part of the paper we indicate how our main result, concerning the existence and the properties of the distance functional described in the introduction, holds also in the more general case when caustics are present. That said, let us consider the following scale-dependent subsets of the past light cones $\mathcal {C}^{-}(p,\,g)$ and $\mathcal {C}^{-}(p,\, \hat{g})$,

$$\begin{aligned} {\mathcal {C}_{L}^{-}}(p,g)\,:=\,\exp _p\left[ C^{-}_{L\le \,L_0}\left( T_pM, g_p \right) \right] ,\,\,\, \mathcal {C}_{L}^{-}(p,\hat{g})\, :=\,\widehat{\exp }_p\left[ C_{L\le \,L_0}^{-}\left( T_pM, \hat{g}_p \right) \right] \,, \end{aligned}$$

(19)

where

$$\begin{aligned} C_{L\le \,L_0}^{-}\left( T_pM, g_p \right) \,:= & {} \, \left\{ X\,=\,\mathbb {X}^i E_{(i)}\,\in \,(T_pM, g_p)\,|\,X^4\,+\,r\,=\,0,\;-L_0\,\le \,X^4\,\le \,0 \right\} \,,\nonumber \\ \end{aligned}$$

(20)

$$\begin{aligned} C_{L\le \,L_0}^{-}\left( T_pM, \hat{g}_p \right) \,:= & {} \, \left\{ Y\,=\,\mathbb {Y}^a \hat{E}_{(a)}\,\in \,(T_pM, \hat{g}_p)\,|\,Y^4\,+\,\hat{r}\,=\,0,\;-L_0\,\le \,Y^4\,\le \,0 \right\} \,,\nonumber \\ \end{aligned}$$

(21)

are the exponential map domains associated with the observational length-scale L up to the homogeneity scale $L_0$. Under the stated caustic-free assumption, both $\mathcal {C}_{L}^{-}(p,{g})$ and $\mathcal {C}_{L}^{-}(p,\hat{g})$ can be foliated in terms of the two-dimensional surfaces ${\Sigma }(p, {r})$ and $\hat{\Sigma }(p, \hat{r})$ introduced in the previous section, i.e., we can write

$$\begin{aligned} \mathcal {C}_{L}^{-}(p,g)\,=\,\bigcup _{\,0\,\le \,r\,\le \,L_0}\,{\Sigma }(p, {r}),\,\,\,\,\, \mathcal {C}_{L}^{-}(p,\hat{g})\,=\,\bigcup _{\,0\,\le \,\hat{r}\,\le \,L_0}\,\hat{\Sigma }(p, \hat{r})\;. \end{aligned}$$

On $\mathcal {C}_{L}^{-}(p,g)$ and $\mathcal {C}_{L}^{-}(p,\hat{g})$ the normal coordinates⁵$\{x^i \}$ and $\{y^a \}$, associated with the observational metric (14) and the reference metric (18), cannot be directly identified since they are defined in terms of the distinct exponential mappings $\exp _p$ and $\widehat{\exp }_p$ and, for a given initial tangent vector   $X\,\in \,C_{L\le \,L_0}^{-}\left( T_pM, {g}_p \right) \cap \,C_{L\le \,L_0}^{-}\left( T_pM, \hat{g}_p \right)$, we have

$$\begin{aligned} \exp _p(X)\,=\,q\,\not =\,\widehat{\exp }_p(X)\,=\,\hat{q}\;. \end{aligned}$$

(22)

However, q and $\hat{q}$ are in the open spacetime region defined by

$$\begin{aligned} M_p\,:=\,\exp _p\left( N_0({g}) \right) \,\cap \,\widehat{\exp _p}\left( N_0(\hat{g}) \right) \subset \,M\;, \end{aligned}$$

(23)

and since $\exp _p$ and $\widehat{\exp }_p$ are local diffeomorphisms from $N_0({g})\cap \,N_0(\hat{g})\subset \,T_pM$ into $M_p$, the map defined by

$$\begin{aligned} \psi \,:(M_p\,\cap \,\mathcal {C}_{L}^{-}(p, \hat{g}),\, \hat{g})\,\longrightarrow & {} \,(M_p\,\cap \,\mathcal {C}_{L}^{-}(p,{g}),\, {g})\nonumber \\ \hat{q}\,\longmapsto & {} \,\psi (\hat{q})\,=\,{q}\,=\,{\exp }_p\left( \widehat{\exp }_p^{- 1}(\hat{q}) \right) \end{aligned}$$

(24)

is a diffeomorphism with $\psi (p)\,=\,\mathrm {id}_M$. In particular, in terms of the coordinates $\{x^i \}$ and $\{y^a \}$ we can locally write

$$\begin{aligned} y^a(\hat{q})\,\longmapsto \,x^i({q})\,=\,\psi ^i(y^b(\hat{q})). \end{aligned}$$

(25)

In order to describe at a given length scale $0\,<\,L\,\le \,L_0$, the effect of these diffeomorphisms on the lightcone regions $\mathcal {C}_{L}^{-}(p,g)$ and $\mathcal {C}_{L}^{-}(p,\hat{g})$, let us consider the spherical surfaces

$$\begin{aligned} \left( \Sigma _L,\,h\right) \,:=\,[\Sigma (p, {r=L}),\, h],\,\,\,\,\,({\widehat{\Sigma }}_L,\,\hat{h})\,:=\, [{\widehat{\Sigma }}(p, \hat{r}=L),\,\hat{h}] \end{aligned}$$

(26)

with their respective metrics h and $\hat{h}$, and where, since the notation wants to travel light, we drop the explicit reference to the vertex p of the lightcone and where we have replaced the affine parameters r and $\hat{r}$ with the preassigned value L of the probed length scale. The surfaces $\left( \Sigma _L,\,h\right)$ and $({\widehat{\Sigma }}_L,\,\hat{h})$ characterize, at the given scale L, the celestial sphere at p as seen by the phenomenological observer and by the reference FLRW observer, respectively.

A direct application of the standard geometrical setup of harmonic map theory (see, e.g., [22]) provides the following notational lemma directly connecting our analysis to harmonic maps between surfaces.

Lemma 4

Let $\psi _L$ be the diffeomorphism $\psi$ restricted to the surfaces $({\widehat{\Sigma }}_L,\,\hat{h})$ and $(\Sigma _L,\, h)$,

$$\begin{aligned} \psi _L\,:\,({\widehat{\Sigma }}_L,\,\hat{h})\,\longrightarrow \,(\Sigma _L,\, h) \end{aligned}$$

(27)

then we can introduce the pullback bundle $\psi _L^{-1}T{\widehat{\Sigma }}_L$ whose sections $v\equiv \psi _L^{-1}V:= V\circ \psi _L$,  $V\in C^{\infty }({\widehat{\Sigma }}, T{\widehat{\Sigma }}_L)$,  are the vector fields over ${\widehat{\Sigma }}$ covering the map $\psi _L$. If $T^*{\widehat{\Sigma }}_L$ denotes the cotangent bundle to $({\widehat{\Sigma }}_L, \hat{h})$, then the differential $\text {d}\psi _L\,=\,\frac{\partial \psi ^i_L}{\partial y^{a}}\text {d}y^{a}\otimes \frac{\partial }{\partial \psi ^i}$ can be interpreted as a section of $T^*{\widehat{\Sigma }}_L\otimes \psi ^{-1}T\Sigma _L$, and its Hilbert–Schmidt norm, in the bundle metric

$$\begin{aligned} \langle \cdot ,\cdot \rangle _{T^*{\widehat{\Sigma }}_L\otimes \psi ^{-1}T\Sigma _L}\,:=\, \hat{h}^{-1}(y)\otimes h(\psi _L(y))(\cdot ,\cdot )\;, \end{aligned}$$

(28)

is provided by

$$\begin{aligned} \langle \text {d}\psi _L , \text {d}\psi _L \rangle _{T^*{\widehat{\Sigma }}_L\otimes \psi ^{-1}T\Sigma _L}\,=\, \hat{h}^{ab}(x)\,\frac{\partial \psi ^{i}(y)}{\partial y^{a}} \frac{\partial \psi ^{j}(y)}{\partial y^{b}}\,h_{ij}(\psi (y))=\,tr_{\hat{h}(y)}\,(\psi _L^{*}\,h)\;, \end{aligned}$$

(29)

where

$$\begin{aligned} \psi _L^* h\,\Longrightarrow \, \left( \psi _L^* h\right) _{ab}\,=\,\frac{\partial \psi ^i(y^c)}{\partial y^a}\frac{\partial \psi ^k(y^d)}{\partial y^b}\,h_{ik} \end{aligned}$$

(30)

provides the pullback of the metric h on ${\widehat{\Sigma }}_L$.

The connection between the pulled-back metric $\psi _L^* h$ and the round metric $\hat{h}$, both defined on ${\widehat{\Sigma }}_L$, is provided by the following proposition where we, respectively, denote by $R_L(\hat{h})$ and $R_L( h)$ the scalar curvature of $({\widehat{\Sigma }}_L, \hat{h})$ and $(\Sigma _L, h)$, and we let $\Delta _{\hat{h}}\,:=\,\hat{h}^{\alpha \beta }\nabla _\alpha \nabla _\beta$ be the Laplace–Beltrami operator on $({\widehat{\Sigma }}_L, \hat{h})$. Notice that the scalar curvature $R_L(\hat{h})$ is associated with the metric (17) evaluated for $\hat{r}\,=\,L$ and hence is given by the constant $R_L(\hat{h})\,=\,\frac{2}{a^2(L)\,L^2}$. In a similar way, $R_L( h)$ is associated with the metric (12) evaluated for $r\,=\,L$, and as such it depends on the area distance $D^2(L)$ and on the lensing distortion $\mathcal {L}_{\alpha \beta }(L)\text {d}x^\alpha \text {d}x^\beta$.

Proposition 5

Let ${q}_{(i)}$, $i\,=1,2,3$ three distinct points intercepted, on the observer celestial sphere $({\Sigma }_L,\,{h})$, by three past-directed null geodesics on $\mathcal {C}_{L}^{-}(p,{g})$, and let $\hat{q}_{(i)}$, $i\,=1,2,3$, three distinguished points on the reference FLRW celestial sphere $({\widehat{\Sigma }}_L,\,\hat{h})$, characterizing three corresponding past-directed null directions on $\mathcal {C}_{L}^{-}(p,\hat{g})$. If $\zeta \,\in \,\mathrm{PSL}(2, \mathbb {C})$ denotes the fractional linear transformation in the projective special linear group, describing the automorphism of $({\widehat{\Sigma }}_L,\,\hat{h})$ that brings $\{\psi ^{-1}(q_i)\}$ into $\hat{q}_{(i)}$, $i\,=1,2,3$, then there is a positive scalar function ${\Phi }_{{\widehat{\Sigma }}\Sigma }\,\in \,C^\infty (\hat{\Sigma },\,\mathbb {R})$, solution of the elliptic partial differential equation

$$\begin{aligned} -\,\Delta _{\hat{h}}\ln ({\Phi }_{{\widehat{\Sigma }}\Sigma } ^2)\,+\,R_L(\hat{h})\,=\, R_L(h)\,{\Phi }_{{\widehat{\Sigma }}\Sigma } ^2\;, \end{aligned}$$

(31)

such that $\psi _L\circ \zeta$ characterizes a conformal diffeomorphism between $({\widehat{\Sigma }}_L,\,\hat{h})$ and $(\Sigma _L,\, h)$, i.e.,

$$\begin{aligned} \left( \psi _L\circ \zeta \right) ^* h\,=\,{\Phi }_{{\widehat{\Sigma }}\Sigma } ^2\,\hat{h}\;. \end{aligned}$$

(32)

Proof

This is a direct consequence of the Poincare–Koebe uniformization theorem which implies that the 2-sphere with the pulled back metric $({\widehat{\Sigma }}_L,\,\psi _L^* h)$ can be mapped conformally, in a one-to-one way, onto the round 2-sphere $({\widehat{\Sigma }}_L,\,\hat{h})$. Recall that on the unit sphere $\mathbb {S}^2$, with its canonical round metric $\text {d}\Omega ^2$, there is a unique conformal class $[\text {d}\Omega ^2]$ and that the metric (17) on ${\widehat{\Sigma }}_L\simeq \mathbb {S}^2$, rescaled according to $\hat{h}/(a^2(\hat{r})\,\hat{r}^2)$, is isometric to $\text {d}\Omega ^2$. Hence, by the uniformization theorem, all metrics on ${\widehat{\Sigma }}_L\simeq \mathbb {S}^2$ may be pulled back by conformal diffeomorphisms to the conformal class $[\hat{h}]$ of the round metric with the chosen radius $a^2(\hat{r})\,\hat{r}^2$. Since $({\widehat{\Sigma }}_L,\, \hat{h}/(a^2(\hat{r})\,\hat{r}^2))\simeq \mathbb {S}^2$, the pullback is unique modulo the action of the conformal group of the sphere $\mathrm {Conf}(\mathbb {S}^2)$. If we denote by $\mathcal {P}_{\mathbb {S}^2}$ the stereographic projection (from the north pole (0, 0, 1) of $\mathbb {S}^2:=\{(x,y,z)\in \mathbb {R}^3\,\,|\,\,x^2+y^2+z^2=1\}$)

$$\begin{aligned} \mathcal {P}_{\mathbb {S}^2}\,:\,\mathbb {S}^2\,\subset \,\mathbb {R}^3\,\longrightarrow \,\mathbb {C}\cup \{\infty \},\,\,\,\, \mathcal {P}_{\mathbb {S}^2}(x,y,z)\,=\,\frac{x+i\,y}{1-z}\,, \end{aligned}$$

(33)

then we can identify $\mathrm {Conf}(\mathbb {S}^2)$ with the six-dimensional projective special linear group $\mathrm{PSL}(2, \mathbb {C})$ describing the automorphisms of $\mathbb {S}^2\simeq \mathbb {C}\cup \{\infty \}$. The elements of $\mathrm{PSL}(2, \mathbb {C})$ are the fractional linear transformations the Riemann sphere $\mathbb {S}^2\,\simeq \,\mathbb {C}\,\cup \,\{\infty \}$

$$\begin{aligned} \mathbb {C}\,\cup \,\{\infty \}\,\longrightarrow & {} \,\mathbb {C}\,\cup \,\{\infty \}\nonumber \\ z\,\longmapsto & {} \,\zeta (z)\,:=\,\frac{az+b}{cz+d}\,,\,\,\,\,\,a, b, c, d\,\in \,\mathbb {C}\,,\,\,\,ad\,-\,bc\,\not =\,0\,. \end{aligned}$$

(34)

These transformations act on the diffeomorphism (27) according to

$$\begin{aligned} \mathrm{PSL}(2, \mathbb {C})\times ({\widehat{\Sigma }}_L,\,\hat{h})\,\longrightarrow & {} \,(\Sigma _L,\, h)\nonumber \\ \left( \zeta ,\,y \right) \,\longmapsto & {} \,\psi _L(\zeta (y)) \end{aligned}$$

(35)

where, abusing notation, we have denoted by $\zeta (y)$ the action that the fractional linear transformation $\zeta (z)$ defines on the point $y\in {\widehat{\Sigma }}_L$ corresponding, via stereographic projection, to the point $z\in \mathbb {C}\,\cup \,\{\infty \}$. This action may be a potential source of a delicate problem since $\mathrm{PSL}(2,\mathbb {C})$ is non-compact and ${\Phi }_{{\widehat{\Sigma }}\Sigma }$ is evaluated on the composition $\psi _L\circ \zeta$ defined by (35). This is not problematic as long as $\zeta$ varies in the maximal compact subgroup of $\mathrm{PSL}(2,\mathbb {C})$ generated by the isometries of $({\widehat{\Sigma }}, \hat{h})$. However, if we consider a sequence $\{\zeta _k\}_{k\in \,\mathbb {N}}\in PSL(2,\mathbb {C})$ defined by larger and larger dilation (corresponding to larger and larger (local) Lorentz boosts of the surface ${\widehat{\Sigma }}$ in the reference spacetime $(M, \hat{g})$), then the composition $\psi _L\circ \zeta _k$ may generate a sequence of conformal factors $\{{\Phi ^2_{(k)}}_{{\widehat{\Sigma }}\Sigma }\}$ converging to a non-smooth function. To avoid these pathologies, we exploit the fact that a linear fractional transformation is fully determined if we fix its action on three distinct points of the sphere. In our setting, this corresponds to fixing the action on three distinct null direction in the lightcone region $\mathcal {C}_{L}^{-}(p,\hat{g})$. In physical terms this is equivalent to require that the FLRW reference observer at p has to adjust his velocity and orientation in such a way that three given astrophysical sources of choice are in three specified position on the celestial sphere $({\widehat{\Sigma }}_L, \hat{h})$ at scale L. This is a gauge fixing of the action of $\mathrm{PSL}(2,\mathbb {C})$ that corresponds in a very natural way to adjust the location of three reference observations in order to be able to compare the data on the phenomenological past lightcone $\mathcal {C}_{L}^{-}(p,{g})$ with the data on the reference past lightcone $\mathcal {C}_{L}^{-}(p,\hat{g})$. By fixing in this way the $\mathrm{PSL}(2,\mathbb {C})$ action, the pullback $\left( \psi _L\circ \zeta \right) ^* h$ on $\hat{\Sigma }_L$ of the metric h is well defined. By the Poincare-Koebe uniformization theorem, the metric $\left( \psi _L\circ \zeta \right) ^* h$ is in the same conformal class of $\hat{h}$. Let us denote by ${\Phi }^2_{{\widehat{\Sigma }}\Sigma }\,\in \,C^\infty (\hat{\Sigma },\,\mathbb {R})$ the corresponding conformal factor such that $\left( \psi _L\circ \zeta \right) ^* h\,=\,{\Phi }^2_{{\widehat{\Sigma }}\Sigma }\,\hat{h}$. If we set $e^f\,:=\,{\Phi }^2_{{\widehat{\Sigma }}\Sigma }\,$, then the properties of the scalar curvature under the conformal transformation $h\,=\,e^f\,\hat{h}$ (see, e.g., [1]) provide the relation

$$\begin{aligned} R\left( \left( \psi _L\circ \zeta \right) ^* h\right) \,=\,e^{\,-f}\,\left[ R(\hat{h})\,+\,\Delta _{\hat{h}}\,f \right] \;. \end{aligned}$$

(36)

If for notational ease we keep on writing R(h) for $R\left( \left( \psi _L\circ \zeta \right) ^* h\right) =R(h(\psi _L\circ \zeta ))$, then it follows from (36) that ${\Phi }^2_{{\widehat{\Sigma }}\Sigma }$ necessarily is a solution on $({\widehat{\Sigma }}_L, \hat{h})$ of the elliptic partial differential equation (31), solution that under the stated hypotheses always exists [1]. $\square$

According to the above result, there is a positive scalar function ${\Phi }_{{\widehat{\Sigma }}\Sigma }\,\in \,C^\infty (\hat{\Sigma },\,\mathbb {R})$ such that $\psi _L\circ \zeta$ characterizes a conformal diffeomorphism between $(\hat{\Sigma }_L,\,\hat{h})$ and $(\Sigma _L,\, h)$. In components (32) can be written as

$$\begin{aligned} \left( \left( \psi _L\circ \zeta \right) ^* h\right) _{ab}\,=\,\frac{\partial \psi _L^i(\zeta (y))}{\partial y^a}\frac{\partial \psi _L^k(\zeta (y))}{\partial y^b}\,h_{ik}\,=\,{\Phi }_{{\widehat{\Sigma }}\Sigma }^2\,\hat{h}_{ab}\;. \end{aligned}$$

(37)

It follows that by tracing (37) with respect to $\hat{h}^{ab}$, we can express ${\Phi }_{{\widehat{\Sigma }}\Sigma }^2$ in terms of the Hilbert–Schmidt norm of the differential $d\left( \psi _L\circ \zeta \right) \,=\,\frac{\partial \psi _L^i(\zeta (y))}{\partial y^{a}}\text {d}y^{a}\otimes \frac{\partial }{\partial \psi _L^i}$ according to (see (29))

$$\begin{aligned} {\Phi }_{{\widehat{\Sigma }}\Sigma }^2\,=\,tr_{\hat{h}(y)}\,\left( \left( \psi _L\circ \zeta \right) ^* h\right) \,=\, \frac{1}{2}\,\hat{h}^{ab}\,\frac{\partial \psi _L^i(\zeta (y))}{\partial y^a}\frac{\partial \psi _L^k(\zeta (y))}{\partial y^b}\,h_{ik}\;. \end{aligned}$$

(38)

From (37) we get $\det \left( \left( \psi _L\circ \zeta \right) ^* h\right) \,=\, {\Phi }_{{\widehat{\Sigma }}\Sigma }^4\,\det (\hat{h})$; hence, we can equivalently write the conformal factor as the Radon–Nikodym derivative of the Riemannian measure $\text {d}\mu _{\psi ^*{h}}:=\left( \psi _L\circ \zeta \right) ^*\text {d}\mu$, of $(\hat{\Sigma }, \left( \psi _L\circ \zeta \right) ^*h)$, with respect to the Riemannian measure $\text {d}\mu _{\hat{h}}$ of the round metric $(\hat{\Sigma }, \hat{h})$, i.e.,

$$\begin{aligned} {\Phi }_{{\widehat{\Sigma }}\Sigma }^2\,=\,\frac{\text {d}\mu _{\psi ^*{h}}}{\text {d}\mu _{\hat{h}}}\,=\,\frac{\left( \psi _L\circ \zeta \right) ^*\text {d}\mu _h}{\text {d}\mu _{\hat{h}}}\;. \end{aligned}$$

(39)

Equivalently, this states that ${\Phi }_{{\widehat{\Sigma }}\Sigma }^2$ can be interpreted as the Jacobian of the map $\psi _L\circ \zeta$,

$$\begin{aligned} {\Phi }_{{\widehat{\Sigma }}\Sigma }^2\,=\,\mathrm {Jac}(\psi _L\circ \zeta )\,. \end{aligned}$$

(40)

Along the same lines, we can associate with the inverse diffeomorphism

$$\begin{aligned} \left( \psi _L\circ \zeta \right) ^{-1}\,:\,(\Sigma _L,\, h) \,\longrightarrow & {} \, ({\widehat{\Sigma }}_L,\,\hat{h})\nonumber \\ x\,\longmapsto & {} \,\zeta ^{-1}\left( \psi _L^{-1}(x) \right) \end{aligned}$$

(41)

a positive scalar function $\Phi _{\Sigma {\widehat{\Sigma }}}\,\in \,C^\infty ({\Sigma },\,\mathbb {R})$ such that we can write

$$\begin{aligned} \left( \left( \psi _L\circ \zeta \right) ^{-1}\right) ^* \hat{h}\,=\,{\Phi }_{\Sigma {\widehat{\Sigma }}}^2\,{h}\,, \end{aligned}$$

(42)

with

$$\begin{aligned} {\Phi }_{\Sigma {\widehat{\Sigma }}}^2\,= & {} \,\frac{1}{2}\,{h}^{ik}\,\frac{\partial \left( \zeta ^{-1}\left( \psi _L^{-1}(x) \right) \right) ^a}{\partial x^i} \frac{\partial \left( \zeta ^{-1}\left( \psi _L^{-1}(x) \right) \right) ^b}{\partial x^k}\,\hat{h}_{ab}\,=\, \frac{\text {d}\mu _{(\psi ^{-1})^*{\hat{h}}}}{\text {d}\mu _{h}}\nonumber \\= & {} \frac{\left( \left( \psi _L\circ \zeta \right) ^{-1}\right) ^*\text {d}\mu _{\hat{h}}}{\text {d}\mu _h}\;. \end{aligned}$$

(43)

To measure the global deviation of the conformal diffeomorphisms ${\Phi }_{{\widehat{\Sigma }}\Sigma }$ from an isometry between $(\hat{\Sigma }_L,\,\hat{h})$ and $(\Sigma _L,\, h)$, we introduce the following comparison functional where, for later use, we keep track of the $\zeta \,\in \,\mathrm{PSL}(2, \mathbb {C})$ dependence in ${\Phi }_{{\widehat{\Sigma }}\Sigma }$.

Definition 6

(The lightcone comparison functional at scale L) Let ${\Phi }_{{\widehat{\Sigma }}\Sigma }\,\in \,C^\infty (\hat{\Sigma },\,\mathbb {R})$ (or at least $C^2(\hat{\Sigma },\,\mathbb {R})$) be the positive scalar function such that $\psi _L\circ \zeta$ characterizes the conformal diffeomorphism $\left( \psi _L\circ \zeta \right) ^* h\,=\,{\Phi }_{{\widehat{\Sigma }}\Sigma } ^2\,\hat{h}$ between $({\widehat{\Sigma }}_L,\,\hat{h})$ and $(\Sigma _L,\, h)$, then the associated lightcone comparison functional at scale L is defined by

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]\,:=\,\int _{{\widehat{\Sigma }}_L}({\Phi }_{{\widehat{\Sigma }}\Sigma }\,-\,1 )^2\,\text {d}\mu _{\hat{h}}\;. \end{aligned}$$

(44)

The functional $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$ is clearly related to the familiar harmonic map energy associated with the map $\psi _L\circ \zeta \,:\,\hat{\Sigma }\,\longrightarrow \,\Sigma$. Explicitly, if we take into account (38), we can write

$$\begin{aligned} \int _{{\widehat{\Sigma }}_L} \widehat{\Phi }_L^2\,\text {d}\mu _{\hat{h}}\,=\,\frac{1}{2}\,\int _{{\widehat{\Sigma }}_L} \hat{h}^{ab}\,\frac{\partial \psi _L^i(\zeta (y))}{\partial y^a}\frac{\partial \psi _L^k(\zeta (y))}{\partial y^b}\,h_{ik}\,\text {d}\mu _{\hat{h}}\;, \end{aligned}$$

(45)

which provides the harmonic map functional whose critical point is the harmonic maps of the Riemann surface $(\hat{\Sigma }_L,\,[\hat{h}])$ into $(\Sigma _L,\, h)$, where $[\hat{h}]$ denotes the conformal class of the metric $\hat{h}$. Notice that whereas the harmonic map energy (45) is a conformal invariant quantity, the functional $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$ is not conformally invariant. Under a conformal transformation $\hat{h}\,\longrightarrow \, e^{2f}\,\hat{h}$, we get

$$\begin{aligned} \int _{{\widehat{\Sigma }}_L}\left( e^{\,-\,f}{\Phi }_{{\widehat{\Sigma }}\Sigma }\,-\,1 \right) ^2\,e^{2f}\,\text {d}\mu _{\hat{h}}\;. \end{aligned}$$

(46)

It is also clear from its definition that corresponding to large gradients (see (43)), $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$ tends to the harmonic map energy. In this connection, it is important to stress that rather than on the space of smooth maps $C^\infty ({\widehat{\Sigma }},\,\Sigma )$, the functional $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$ is naturally defined on the Sobolev space of maps $W^{1, 2}({\widehat{\Sigma }},\,\Sigma )$ which are, together with their weak derivatives, square integrable. This characterization, familiar when studying weakly harmonic maps [19] and which we discuss in detail below when minimizing $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$, is important in our case when extending our analysis to the low regularity setting when lightcone caustics are present.

Remark 7

It must be stressed that energy functionals such as (44) are rather familiar in the problem of comparing shapes of surfaces in relation with computer graphic and visualization problems (see, e.g., [16, 20], to quote two relevant papers in a vast literature). In particular, (44) has been introduced under the name of elastic energy in an inspiring paper by Hass and Koehl [18], who use it as a building block of a more complex functional relevant to surface visualization.

In our particular framework, the functional $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$ has a number of important properties that make it a natural candidate for comparing, at the given length scale L, the physical lightcone region $\mathcal {C}_{L}^{-}(p,{g})$ with the FLRW reference region $\mathcal {C}_{L}^{-}(p,\hat{g})$. To start with, we prove the following general properties (in the smooth setting) .

Lemma 8

The functional $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$ is symmetric

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]\,=\,E_{\Sigma {\widehat{\Sigma }}}[\psi ^{-1}_L,\,\zeta ^{-1}]\;, \end{aligned}$$

(47)

where

$$\begin{aligned} E_{\Sigma {\widehat{\Sigma }}}[\psi ^{-1}_L,\,\zeta ^{-1}]\,:=\,\int _{{\Sigma }_L}({\Phi }_{\Sigma {\widehat{\Sigma }}}\,-\,1 )^2\,\text {d}\mu _{{h}}\;, \end{aligned}$$

(48)

is the comparison functional associated with the inverse map $(\psi _L\circ \zeta )^{-1}\,:\,{\Sigma }_L\,\longrightarrow \,\hat{\Sigma }_L$.

If $(\widetilde{\Sigma }_L,\, \tilde{h})$ is a third surface on the past lightcone $\widetilde{\mathcal {C}}_{L_0}^{-}(p,\tilde{g})$, with vertex at p, associated with yet another reference FLRW metric $\tilde{g}$ on M (say another member of the FLRW family of spacetimes, distinct from $\hat{g}$), and $\sigma _L\,:\Sigma _L\,\longmapsto \,\widetilde{\Sigma }_L$,  $\Phi _{\Sigma \widetilde{\Sigma }}$, respectively, are the corresponding diffeomorphism and conformal factor, then to the composition of maps

$$\begin{aligned} {\widehat{\Sigma }}_L\,\underset{\psi _L}{\longrightarrow }\,\Sigma _L\,\underset{\sigma _L}{\longrightarrow }\,\widetilde{\Sigma }_L \end{aligned}$$

(49)

we can associate the triangular inequality

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]\,+\,E_{\Sigma \widetilde{\Sigma }}[\sigma _L,\,\zeta ]\,\ge \, E_{{\widehat{\Sigma }}\widetilde{\Sigma }}[(\sigma _L\circ \psi _L),\,\zeta ]\,, \end{aligned}$$

(50)

where

$$\begin{aligned} E_{{\widehat{\Sigma }}\widetilde{\Sigma }}[(\sigma _L\circ \psi _L),\,\zeta ]\,:=\,\int _{\hat{\Sigma }_L}({\Phi }_{{\widehat{\Sigma }}\widetilde{\Sigma }}\,-\,1 )^2\,\text {d}\mu _{\hat{h}}\;. \end{aligned}$$

(51)

If $A({\widehat{\Sigma }}_L)\,:=\,\int _{{\widehat{\Sigma }}_L}\text {d}\mu _{\hat{h}}$ and $A({\Sigma }_L)\,:=\,\int _{{\Sigma }_L}\text {d}\mu _{{h}}$, respectively, denote the area of the surfaces $({\widehat{\Sigma }},\,\hat{h})$ and $(\Sigma ,\,{h})$, then we have the upper and lower bounds

$$\begin{aligned} A({\widehat{\Sigma }}_L)\,+\,A({\Sigma }_L)\,\ge \,E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]\,\ge \,\left( \sqrt{\mathcal {A}({\widehat{\Sigma }}_L)}\,-\,\sqrt{A({\Sigma }_L)} \right) ^2\,. \end{aligned}$$

(52)

Finally,

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]\,=\,0 \end{aligned}$$

(53)

iff the surfaces $({\widehat{\Sigma }},\,\hat{h})$ and $(\Sigma ,\,{h})$ are isometric.

Proof

For notational ease, let us temporarily dismiss the action of the linear fractional transformation $\zeta \,\in \,\mathrm{PSL}(2, \mathbb {C})$ and, if there is no chance of confusion, write $E_{{\widehat{\Sigma }}\Sigma }[\psi _L]$ in place of the full $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$. We start with proving the symmetry property (47). To this end, expand the integrand in (44) and rewrite $E_{{\widehat{\Sigma }}\Sigma }[\psi _L]$ as

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\psi _L]\,= & {} \,\int _{{\widehat{\Sigma }}_L}( {\Phi }_{{\widehat{\Sigma }}\Sigma }\,-\,1 )^2\,\text {d}\mu _{\hat{h}}\,=\, \int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }^2\,\text {d}\mu _{\hat{h}}\,+\, \int _{{\widehat{\Sigma }}_L}\,\text {d}\mu _{\hat{h}}\,-\,2\int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}}\nonumber \\= & {} \int _{{\widehat{\Sigma }}_L}\frac{\psi _L^*\text {d}\mu _h}{\text {d}\mu _{\hat{h}}}\,\text {d}\mu _{\hat{h}}\,+\, A\left( {\widehat{\Sigma }}_L\right) \,-\,2\int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}}\nonumber \\= & {} \int _{\psi _L({\widehat{\Sigma }}_L)}{\text {d}\mu _h}\,+\, A\left( {\widehat{\Sigma }}_L\right) \,-\,2\int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}}\nonumber \\= & {} A({\Sigma }_L)\,+\, A\left( {\widehat{\Sigma }}_L\right) \,-\,2\int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}}\;, \end{aligned}$$

(54)

where we have exploited the Radon–Nikodym characterization of $\widehat{\Phi }_{{\widehat{\Sigma }}\Sigma }^2$, (see (39)), the identification $\psi ({\widehat{\Sigma }}_L)\,=\,\Sigma _L$,  and the relation

$$\begin{aligned} \int _{{\widehat{\Sigma }}_L}\frac{\psi _L^*\text {d}\mu _h}{\text {d}\mu _{\hat{h}}}\,\text {d}\mu _{\hat{h}}= \int _{{\widehat{\Sigma }}_L}{\psi _L^*\text {d}\mu _h}=\int _{\psi ({\widehat{\Sigma }}_L)}{\text {d}\mu _h}=\int _{{\Sigma }_L}{\text {d}\mu _h}\,=\,A({\Sigma }_L)\;, \end{aligned}$$

(55)

where $A({\Sigma }_L)$ and $A\left( {\widehat{\Sigma }}_L\right)$, respectively, denote the area of $(\hat{\Sigma }_L,\,\hat{h})$ and $(\Sigma _L,\, h)$. Along the same lines, let us compute the lightcone comparison functional $E_{\Sigma {\widehat{\Sigma }}}[\psi _L^{-1}]$ associated with the inverse diffeomorphism $\psi _L^{-1}\,:\,(\Sigma _L,\, h) \,\longrightarrow \, (\hat{\Sigma }_L,\,\hat{h})$ and the corresponding conformal factor $\Phi _{\Sigma {\widehat{\Sigma }}}\,\in \,C^\infty ({\Sigma },\,\mathbb {R})$- (see (42)),

$$\begin{aligned} E_{\Sigma {\widehat{\Sigma }}}[\psi _L^{-1}]\,:=\,\int _{\Sigma _L}\left( {\Phi }_{\Sigma {\widehat{\Sigma }}}\,-\,1 \right) ^2\,\text {d}\mu _{h}\;. \end{aligned}$$

(56)

We have

$$\begin{aligned} E_{\Sigma {\widehat{\Sigma }}}[\psi _L^{-1}]\,:=\,A\left( {\widehat{\Sigma }}_L\right) \,+\,A({\Sigma }_L)\, -\,2\int _{\Sigma _L}{\Phi }_{\Sigma {\widehat{\Sigma }}}\,\text {d}\mu _{h}\;. \end{aligned}$$

(57)

Since

$$\begin{aligned} \int _{\Sigma _L}{\Phi }_{\Sigma {\widehat{\Sigma }}}\,\text {d}\mu _{h}\,= & {} \,\int _{\Sigma _L} \sqrt{\frac{\text {d}\mu _{(\psi ^{-1})^*{\hat{h}}}}{\text {d}\mu _{h}}}\, \text {d}\mu _{h}=\int _{\Sigma _L}\sqrt{\frac{\text {d}\mu _{(\psi ^{-1})^*{\hat{h}}}}{\text {d}\mu _{h}}}\,\frac{\text {d}\mu _{h}}{\text {d}\mu _{(\psi ^{-1})^*{\hat{h}}}}\,\text {d}\mu _{(\psi ^{-1})^*{\hat{h}}}\nonumber \\= & {} \,\int _{\Sigma _L}\,\sqrt{\frac{\text {d}\mu _{h}}{\text {d}\mu _{(\psi ^{-1})^*{\hat{h}}}}}\,(\psi ^{-1})^*\text {d}\mu _{\hat{h}}\;. \end{aligned}$$

(58)

On the other hand, if we take the pullback, under the action of $\psi _L^{-1}\,:\,(\Sigma _L, h)\,\longrightarrow \, ({\widehat{\Sigma }}_L, \hat{h})$, of the relation ${\Phi }_{{\widehat{\Sigma }}\Sigma }^2\,\text {d}\mu _{\hat{h}}\,=\,\psi _L^*\text {d}\mu _h$, (see (39)), we have

$$\begin{aligned} \left( \psi ^{-1} \right) ^*\left( {\Phi }_{{\widehat{\Sigma }}\Sigma }^2\,\text {d}\mu _{\hat{h}}\right) \,= & {} \,\left( \psi _L^{-1} \right) ^*\left( \psi _L^*\text {d}\mu _h\right) \, \nonumber \\&\Longrightarrow&{\Phi }_{{\widehat{\Sigma }}\Sigma }^2\left( \psi _L^{-1}(x) \right) \left( \left( \psi _L^{-1} \right) ^*\text {d}\mu _{\hat{h}}\right) (x)\,=\,\text {d}\mu _h(x),\nonumber \\ \end{aligned}$$

(59)

from which we get

$$\begin{aligned} {\Phi }_{{\widehat{\Sigma }}\Sigma }^2\left( \psi _L^{-1}(x) \right) \,=\,\frac{\text {d}\mu _h(x)}{\left( \left( \psi _L^{-1} \right) ^*\text {d}\mu _{\hat{h}}\right) (x)}\;. \end{aligned}$$

(60)

Hence, we can rewrite (58) as

$$\begin{aligned} \int _{\Sigma _L}{\Phi }_{\Sigma {\widehat{\Sigma }}}\,\text {d}\mu _{h}\,= & {} \,\int _{\Sigma _L}\, \sqrt{\frac{\text {d}\mu _{h}}{\text {d}\mu _{(\psi ^{-1})^*{\hat{h}}}}}\,(\psi _L^{-1})^*\text {d}\mu _{\hat{h}}\,=\, \int _{\Sigma _L}\,{\Phi }_{{\widehat{\Sigma }}\Sigma }\left( \psi _L^{-1} \right) \,(\psi _L^{-1})^*\text {d}\mu _{\hat{h}}\nonumber \\= & {} \,\int _{\Sigma _L}\,(\psi _L^{-1})^*\left( {\Phi }_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}}\right) \,=\, \int _{\psi ^{-1}(\Sigma _L)}\,{\Phi }_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}} \nonumber \\= & {} \, \int _{\hat{\Sigma }_L}\,{\Phi }_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}}, \end{aligned}$$

(61)

and

$$\begin{aligned} E_{\Sigma {\widehat{\Sigma }}}[\psi ^{-1}]\,:= & {} \,A\left( {\widehat{\Sigma }}_L\right) \,+\,A({\Sigma }_L)\, -\,2\int _{\Sigma _L}{\Phi }_{\Sigma {\widehat{\Sigma }}}\,\text {d}\mu _{h}\nonumber \\= & {} \,A\left( {\widehat{\Sigma }}_L\right) \,+\,A({\Sigma }_L)\, -\,2\int _{\hat{\Sigma }_L}\,{\Phi }_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}}\,=\,E_{{\widehat{\Sigma }}\Sigma }[\psi ]\;. \end{aligned}$$

(62)

Hence, the comparison functional is symmetric.

In order to prove the triangular inequality (50), let us consider the sum

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\psi _L]\,+\,E_{\Sigma \widetilde{\Sigma }}[\sigma _L]\,=\, \,\int _{\hat{\Sigma }_L}({\Phi }_{{\widehat{\Sigma }}\Sigma }\,-\,1 )^2\,\text {d}\mu _{\hat{h}}\,+\, \,\int _{{\Sigma }_L}({\Phi }_{\Sigma \widetilde{\Sigma }}\,-\,1 )^2\,\text {d}\mu _{{h}}\;. \end{aligned}$$

(63)

From the relation (59), we have $\text {d}\mu _h\,=\,{\Phi }_{{\widehat{\Sigma }}\Sigma }^2\left( \psi _L^{-1} \right) \left( \psi _L^{-1} \right) ^*\text {d}\mu _{\hat{h}}$, and we can write

$$\begin{aligned} \int _{{\Sigma }_L}({\Phi }_{\Sigma \widetilde{\Sigma }}\,-\,1 )^2\,\text {d}\mu _{{h}}\,=\, \int _{{\widehat{\Sigma }}_L}({\Phi }_{\Sigma \widetilde{\Sigma }}\,-\,1 )^2{\Phi }_{{\widehat{\Sigma }}\Sigma }^2\,\text {d}\mu _{\hat{h}}\;. \end{aligned}$$

(64)

Hence,

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\psi _L]\,+\,E_{\Sigma \widetilde{\Sigma }}[\sigma _L]\,= & {} \, \int _{{\widehat{\Sigma }}_L}\left[ ({\Phi }_{{\widehat{\Sigma }}\Sigma }\,-\,1 )^2\,+\, ({\Phi }_{\Sigma \widetilde{\Sigma }}\,-\,1 )^2{\Phi }_{{\widehat{\Sigma }}\Sigma }^2 \right] \,\text {d}\mu _{\hat{h}}\nonumber \\\ge & {} \, \int _{{\widehat{\Sigma }}_L}\left[ ({\Phi }_{{\widehat{\Sigma }}\Sigma }\,-\,1 )\,+\, ({\Phi }_{\Sigma \widetilde{\Sigma }}\,-\,1 ){\Phi }_{{\widehat{\Sigma }}\Sigma } \right] ^2\,\text {d}\mu _{\hat{h}}\nonumber \\= & {} \,\int _{{\widehat{\Sigma }}_L}({\Phi }_{{\widehat{\Sigma }}\Sigma }{\Phi }_{\Sigma \widetilde{\Sigma }}\,-\,1 )^2\,\text {d}\mu _{\hat{h}}\,=\, E_{{\widehat{\Sigma }}\widetilde{\Sigma }}[(\sigma _L\circ \psi _L)],\nonumber \\ \end{aligned}$$

(65)

where we have exploited the relation

$$\begin{aligned} {\Phi }_{\Sigma \widetilde{\Sigma }}\left( \psi _L \right) \,{\Phi }_{{\widehat{\Sigma }}\Sigma }\,=\,{\Phi }_{{\widehat{\Sigma }}\widetilde{\Sigma }}\;, \end{aligned}$$

(66)

which follows from observing that the positive functions ${\Phi }_{{\widehat{\Sigma }}\widetilde{\Sigma }}\in \,C^\infty ({\widehat{\Sigma }}, \mathbb {R})$ and ${\Phi }_{\Sigma \widetilde{\Sigma }}\in \,C^\infty (\Sigma , \mathbb {R})$ are such that

$$\begin{aligned} {\Phi }^2_{{\widehat{\Sigma }}\widetilde{\Sigma }}\,\hat{h}=\left( \sigma _L\circ \psi _L \right) ^*\tilde{h} =\psi _L^*\left( {\Phi }^2_{\Sigma \widetilde{\Sigma }}\,h \right) \,=\,{\Phi }^2_{\Sigma \widetilde{\Sigma }}\left( \psi _L \right) \,\psi _L^*\,h\, =\,{\Phi }^2_{\Sigma \widetilde{\Sigma }}\left( \psi _L \right) \,{\Phi }^2_{{\widehat{\Sigma }}\Sigma }\,\hat{h},\nonumber \\ \end{aligned}$$

(67)

where we have set ${\Phi }_{\Sigma \widetilde{\Sigma }}\left( \psi _L \right) =\psi _L^*{\Phi }_{\Sigma \widetilde{\Sigma }}:={\Phi }_{\Sigma \widetilde{\Sigma }}\,\circ \,\psi _L$.

From (54) and the Schwarz inequality

$$\begin{aligned} \int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}}\,\le \, \left( \int _{{\widehat{\Sigma }}_L}{\Phi }^2_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}} \right) ^{1/2}\, \left( \int _{{\widehat{\Sigma }}_L}\,\text {d}\mu _{\hat{h}} \right) ^{1/2}\,=\, \sqrt{A\left( {\widehat{\Sigma }}_L\right) A\left( {\Sigma }_L\right) }\,, \end{aligned}$$

(68)

we get the lower bound

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]\,= & {} \,A({\Sigma }_L)\,+\, A\left( {\widehat{\Sigma }}_L\right) \,-\,2\int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}}\nonumber \\\ge & {} \,\,A({\Sigma }_L)\,+\,A\left( {\widehat{\Sigma }}_L\right) \,-\,2\sqrt{A\left( {\widehat{\Sigma }}_L\right) A\left( {\Sigma }_L\right) }\,\nonumber \\= & {} \, \left( \sqrt{\mathcal {A}({\widehat{\Sigma }}_L)}\,-\,\sqrt{A({\Sigma }_L)} \right) ^2\,, \end{aligned}$$

(69)

where we have exploited (55). The upper bound in (52) easily follows from (44)

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]\,:= & {} \,\int _{{\widehat{\Sigma }}_L}({\Phi }_{{\widehat{\Sigma }}\Sigma }\,-\,1 )^2\,\text {d}\mu _{\hat{h}}\, \le \,\int _{{\widehat{\Sigma }}_L}{\Phi }^2_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}}\,+\, \int _{{\widehat{\Sigma }}_L}\,\text {d}\mu _{\hat{h}}\nonumber \\= & {} A({\widehat{\Sigma }}_L)\,+\,A({\Sigma }_L)\,. \end{aligned}$$

(70)

The proof of the last part of the lemma follows observing that the integrand in $\int _{\hat{\Sigma }_L}({\Phi }_{{\widehat{\Sigma }}\Sigma }\,-\,1 )^2\,\text {d}\mu _{\hat{h}}$ is non-negative and, as long as ${\Phi }_{{\widehat{\Sigma }}\Sigma }$ is a smooth function on $(\hat{\Sigma }_L, \hat{h})$, the condition

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\psi _L]\,=\, \,\int _{\hat{\Sigma }_L}({\Phi }_{{\widehat{\Sigma }}\Sigma }\,-\,1 )^2\,\text {d}\mu _{\hat{h}}\,=\,0 \end{aligned}$$

(71)

implies ${\Phi }_{{\widehat{\Sigma }}\Sigma }\,=\,1$, hence the isometry between $(\hat{\Sigma }_L, \hat{h})$ and $({\Sigma }_L, {h})$. $\square$

4 A scale-dependent distance functional

The properties of the comparison functional $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$ indicate that we can associate with it a corresponding distance functional $d_L\left[ {\widehat{\Sigma }}_L,\,\Sigma _L\right]$. To put the characterization of this distance in perspective, let us recall that a fractional linear transformations $\zeta \in \,\mathrm{PSL}(2, \mathbb {C})$ is fully determined if, given three distinct points of ${\widehat{\Sigma }}\simeq \mathbb {S}^2$, we specify their images. We exploited this in Proposition 5, where we assigned three distinct points ${q}_{(i)}$, $i\,=1,2,3$ on the observer celestial sphere $({\Sigma }_L,\,{h})$, and we fixed the action of $\mathrm{PSL}(2, \mathbb {C})$ by choosing that particular automorphism $\zeta \in \,\mathrm{PSL}(2, \mathbb {C})$ that identifies the inverse images $\{\psi ^{-1}(q_i)\}\,\in \,{\widehat{\Sigma }}_L$ with three chosen points $\hat{q}_{(i)}$, $i\,=1,2,3$ on the reference FLRW celestial sphere $({\widehat{\Sigma }}_L,\,\hat{h})$. Since $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$ is not conformally invariant, the particular choice of the automorphism $\zeta \in \,\mathrm{PSL}(2, \mathbb {C})$, or which is the same, the particular choice of the points ${q}_{(i)}$, $i\,=1,2,3$ on $({\Sigma }_L,\,{h})$, affects $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$; hence, it is natural to inquire if there is a choice of the automorphism $\zeta$ that minimizes $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$. Given the reference points $\hat{q}_{(i)}$, $i\,=1,2,3$ on $({\widehat{\Sigma }}_L,\,\hat{h})$, this optimal choice for $\zeta \in \,\mathrm{PSL}(2, \mathbb {C})$, say $\zeta =\zeta _0$, will induce the proper selection of the alignment points ${q}_{(i)}$, $i\,=1,2,3$ on the observer celestial sphere $({\Sigma }_L,\,{h})$ by setting ${q}_{(i)}\,:=\,\psi _L\left( \zeta _0(\hat{q}_{(i)})\right)$. In order to characterize this optimal choice, we need to minimize $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$ over a suitable class of functions, and a natural strategy, according to these remarks, is to keep fixed the diffeomorphism⁶$\psi _L$ as well the points $\hat{q}_{(i)}$, $i\,=1,2,3$ on the reference $({\widehat{\Sigma }}_L,\,\hat{h})$, and let vary in a controlled way the automorphism $\zeta \in \,\mathrm{PSL}(2, \mathbb {C})$, so as to minimize $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$. We also need a slightly more general setting that will allow us to deal with celestial spheres $(\Sigma _L,\, h)$ on a lightcone region $\mathcal {C}_{L}^{-}(p,g)$ where caustics develop (hence, relaxing in a controlled way the regularity of $\psi _L$ allowing for exponential mappings which are no longer injective). In other words, we need to extend $\psi _L\circ \zeta \,:\,{\widehat{\Sigma }}_L\longrightarrow \Sigma _L$ to be a member of a more general space of maps which allow for the low regularity setting associated with the possible presence of (isolated) caustics. We start with a more precise characterization of the Sobolev space of maps $W^{1,2}({\widehat{\Sigma }},\,\Sigma )$, mentioned on passing in commenting Definition 6. To define $W^{1,2}({\widehat{\Sigma }},\,\Sigma )$ we follow a standard approach in harmonic map theory and use Nash embedding theorem [17, 28], by considering the compact surface $(\Sigma _L, h)$ isometrically embedded into some Euclidean space $\mathbb {E}^m\,:=\,(\mathbb {R}^m, \delta )$ for m sufficiently large. In particular, if $J: (\Sigma _L, h)\hookrightarrow \mathbb {E}^m$ is any such an embedding, then we define the Sobolev space of maps

$$\begin{aligned} {W}^{1, 2}_{(J)}({\widehat{\Sigma }}, \Sigma )\, :=\,\{\varphi \in {W}^{1, 2}({\widehat{\Sigma }},\,\mathbb {R}^m)\left. \right| \,\varphi (\hat{\Sigma }_L)\subset J(\Sigma _L) \}\;, \end{aligned}$$

(72)

where ${W}^{1, 2}({\widehat{\Sigma }},\mathbb {R}^m)$ is the Hilbert space of square summable $\varphi :{\widehat{\Sigma }} \rightarrow \mathbb {R}^m$, with (first) distributional derivatives in $L^2({\widehat{\Sigma }},\mathbb {R}^m)$, endowed with the norm

$$\begin{aligned} \parallel \varphi \parallel _{{W}^{1,2}}\,:=\,\int _{{\widehat{\Sigma }}}\,\left( \varphi ^a(x)\,\varphi ^b(x)\,\delta _{ab}\,+\, \hat{h}^{\mu \nu }(x)\,\frac{\partial \varphi ^{a}(x)}{\partial x^{\mu }} \frac{\partial \varphi ^{b}(x)}{\partial x^{\nu }}\,\delta _{ab} \right) \,\text {d}\mu _{\hat{h}}\;, \end{aligned}$$

(73)

where, for $\varphi (x)\in \,J(\Sigma _L)\subset \mathbb {R}^m$,  $a,b=1,\ldots ,m$ label coordinates in $(\mathbb {R}^m,\,\delta )$, and $\text {d}\mu _{\hat{h}}$ denotes the Riemannian measure on $({\widehat{\Sigma }}, \hat{h})$. This characterization is independent of J since $\Sigma _L$ is compact, and in that case for any two isometric embeddings $J_1$ and $J_2$, the corresponding spaces of maps ${W}^{1, 2}_{(J_1)}({\widehat{\Sigma }}, \Sigma )$ and ${W}^{1, 2}_{(J_2)}({\widehat{\Sigma }}, \Sigma )$ are homeomorphic [19]. For this reason, in what follows we shall simply write ${W}^{1, 2}({\widehat{\Sigma }}, \Sigma )$. The set of maps ${W}^{1, 2}({\widehat{\Sigma }}, \Sigma )$ provides the minimal regularity allowing for the characterization of the energy functional $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$. Maps of class ${W}^{1, 2}({\widehat{\Sigma }}, \Sigma )$ are not necessarily continuous and, even if the space of smooth maps ${C}^{\infty }({\widehat{\Sigma }}, \Sigma )$ is dense [26] in ${W}^{1, 2}({\widehat{\Sigma }}, \Sigma )$, to carry out explicit computations, in what follows we must further require that $\varphi \in {W}^{1, 2}({\widehat{\Sigma }}, \Sigma )$ is localizable (cf. [22], Sect. 8.4) and keeps track both of the given $\psi _L$ and of the three alignment points $\hat{q}_{(i)}$, $i\,=1,2,3$ on $({\widehat{\Sigma }}_L,\,\hat{h})$. The only freedom remaining is in the conformal group automorphisms $\zeta \in \,\mathrm{PSL}(2, \mathbb {C})$ acting on $({\widehat{\Sigma }}_L,\,\hat{h})$, and in terms of which we need to control that the images of the reference points $\hat{q}_{(i)}\in {\widehat{\Sigma }}_L$ stay separated and do not concentrate in a small neighborhood of ${\Sigma }_L$. Hence, and for a fixed $\psi _L\,\in \,{W}^{1, 2}({\widehat{\Sigma }}, \Sigma )$, we define the space of maps over which $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$ is minimized according to the following definition.

Definition

Let us assume that $\psi _L\,:\,({\widehat{\Sigma }}_L,\,\hat{h})\,\longrightarrow \,(\Sigma _L,\, h)$ is, for almost all points of ${\widehat{\Sigma }}_L$, a ${W}^{1, 2}({\widehat{\Sigma }}, \Sigma )$ diffeomorphism between the two celestial spheres⁷, and let $\hat{q}_{(i)}$, $i\,=1,2,3$,  be the three distinguished points on $({\widehat{\Sigma }}_L,\,\hat{h})$, characterizing the three reference past-directed null directions on $\mathcal {C}_{L}^{-}(p,\hat{g})$ introduced in Proposition 5. A map $\varphi \,:=\,\psi _L\circ \zeta \,\in \,{W}^{1, 2}({\widehat{\Sigma }}, \Sigma )$, with $\zeta \in \,\mathrm{PSL}(2, \mathbb {C})$,  is said to be $\varepsilon$-localizable if: (i) For every $\hat{q} \in {\widehat{\Sigma }}_L$ there exists a metric disks $D(\hat{q},\,\delta ):=\{y\in {\widehat{\Sigma }}_L\,|\,d_{\gamma }(\hat{q},y)\,\le \,\delta \} \subset {\widehat{\Sigma }}_L$, of radius $\delta >0$, with smooth boundary $\partial \,D$, and containing at most one of the three points $\hat{q}_{(i)}$, $i\,=1,2,3$; and (ii) corresponding to each of these disks, there exists a metric disk $B(q,\,\varepsilon )\,=\,\varphi (D(\hat{q},\,\delta ))\,:=\,\{x\in \Sigma _L\,\,|\,d_h(q, x)\le \varepsilon \}\,\subset \, (\Sigma _L, h)$ centered at $\varphi (x_0)\,:=\,q\,\in \, M$, of radius $r>0$ such that $\varphi (D(x_0,\,\delta ))\subset B(q,\,r)$, with $\varphi (\partial D)\,\subset \,B(q,\,\varepsilon )$. Under such assumptions, we consider, for fixed $\psi _L$, the space of maps

$$\begin{aligned}&\mathrm {Map}_\psi ({\widehat{\Sigma }}_L, \Sigma _L)\,:=\,\left\{ \varphi \,:=\,\psi _L\circ \zeta \,\in \,{W}^{1, 2}({\widehat{\Sigma }}, \Sigma )\cap \,{C}^{0}({\widehat{\Sigma }}, \Sigma )\,,\,\,\right. \nonumber \\&\quad \left. \left. \zeta \in \,\mathrm{PSL}(2, \mathbb {C})\,\,\,\right| \,\,\varphi \,:=\,\psi _L\circ \zeta \,\,\,\,\mathrm {is\,\, \varepsilon -localizable\,\,and}\,\,\Phi _{{\widehat{\Sigma }}\Sigma }(\hat{q})\ge 0 \right\} ,\nonumber \\ \end{aligned}$$

(74)

where the non-negativity requirement $\Phi _{{\widehat{\Sigma }}\Sigma }(\hat{q})\ge 0$ is assumed to hold for almost all points of ${\widehat{\Sigma }}_L$.

As in harmonic map theory, there is a further delicate issue related to the fact that maps in $\mathrm {Map}_\psi ({\widehat{\Sigma }}_L, \Sigma _L)$ are partitioned in different homotopy classes. Recall that every map from $\mathbb {S}^2$ into itself is characterized by the degree of the map [24], measuring how many times the map wraps $\mathbb {S}^2$ around itself. In particular through the action of a sequence of conformal dilations $\in \,\mathrm{PSL}(2, \mathbb {C})$ of the form $\zeta \longmapsto \zeta '_{(k)}\,:=\,\omega _{(k)}\zeta$ where $\omega _{(k)}\in \mathbb {R}$ we can easily construct sequences of mappings $\{\varphi _{(k)}\}$ that tend to focus all points of a disk D in $\mathbb {S}^2$ toward a given point (say the north pole). Physically this corresponds to the effect of acting with a sequence of Lorentz boosts (with rapidity $\log \,\omega _{(k)}$) on an observer P who is looking at the giving region D of the celestial sphere. From the point of view of P, one can also interpret this as a focusing of the past null geodesics eventually leading to the formation of a caustic point. Regardless of the physical interpretation, in harmonic map theory this sort of behavior leads to the phenomenon of bubble convergence when discussing the minimization problem for the harmonic map energy functional [25]. In our case, we can exploit the analogous of bubbling convergence to our advantage in order to extend our analysis to the case when caustics are present. We can model the generation of caustic points as the results of a focusing mapping, such as the $\{\varphi _{(k)}\}$ described above, converging in $\mathrm {Map}_\psi ({\widehat{\Sigma }}_L, \Sigma _L)$, to a $\mathrm {deg}\,\varphi \,=\,h\,>\,0$ map (if there are $h-1$ caustic points in $(\Sigma _L, h)$). We postpone the details of such analysis to a paper in preparation [4], and limit here our analysis to show that $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$ can be minimized over diffeomorphisms in $\mathrm {Map}_\psi ({\widehat{\Sigma }}_L, \Sigma _L)$.

Theorem 9

(The lightcone comparison distance at scale L). The functional $E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ]$ achieves a minimum on $\mathrm {Map}_\psi ({\widehat{\Sigma }}_L, \Sigma _L)$, and

$$\begin{aligned} d_L\left[ {\widehat{\Sigma }}_L,\,\Sigma _L\right] \,:=\,\inf _{\psi _L\circ \zeta \in \mathrm {Map}_\psi ({\widehat{\Sigma }}_L, \Sigma _L)}\,E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ] \end{aligned}$$

(75)

defines a scale-dependent distance between the celestial spheres $({\widehat{\Sigma }}_L,\,\hat{h})$ and $(\Sigma _L,\, h)$ on the lightcone regions $\mathcal {C}_{L}^{-}(p,\hat{g})$ and $\mathcal {C}_{L}^{-}(p,g)$.

Proof

To simplify notation let us set $\varphi \,:=\,\psi _L\circ \zeta$. Since we have the upper bound $E_{{\widehat{\Sigma }}\Sigma }[\varphi ] \le \,C_L\,:=\,A({\widehat{\Sigma }}_L)\,+\,A({\Sigma }_L)$, (see (52)), we can limit our analysis to the subset of maps

$$\begin{aligned} \mathrm {Map}_{\psi , C_L}({\widehat{\Sigma }}_L, \Sigma _L) \,:=\,\left\{ \varphi \,\in \,\mathrm {Map}_{\psi }({\widehat{\Sigma }}_L, \Sigma _L)\,\,|\,\,\mathrm {s.t.}\,\, E_{{\widehat{\Sigma }}\Sigma }[\varphi ] \,\le \,C_L \right\} \;. \end{aligned}$$

(76)

According to Definition 4, the space of maps $\mathrm {Map}_{\varepsilon , C_L}({\widehat{\Sigma }}, \Sigma )$ is equicontinuous, namely for any point $\hat{q}\in \,\hat{\Sigma }$ we can choose the disk $D(\hat{q}_1, \delta )$ (for notation see Definition 4) in such a way that for a given $\varepsilon \,>\,0$,   $\varphi (\hat{q}_1)$ and $\varphi (\hat{q}_2)$ are such that $d_h\left( \varphi (\hat{q}_1),\,\varphi (\hat{q}_2) \right) \,<\,\varepsilon$, for all $\hat{q}_2\,\in \,D(\hat{q}_1, \delta )$ and all $\varphi \,\in \,\mathrm {Map}_{\psi , C_L}({\widehat{\Sigma }}_L, \Sigma _L)$. Hence, a minimizing sequence $\{\varphi _{(k)} \}_{k\in \mathbb {N}}\,\in \,\mathrm {Map}_{\varepsilon , C_L}({\widehat{\Sigma }}, \Sigma )$ for $E_{{\widehat{\Sigma }}\Sigma }[\varphi ]$ is equicontinuous. By selecting a subsequence we may assume that $\{\varphi _{(k)} \}$ converges to a continuous map $\varphi$ which is also the weak limit of $\{\varphi _{(k)} \}$ in $W^{1, 2}({\widehat{\Sigma }}, \Sigma )$, since this latter is a weakly compact space of maps.

Since

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\varphi ]=A({\Sigma }_L)\,+\, A\left( {\widehat{\Sigma }}_L\right) \,-\,2\int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }(\varphi )\,\text {d}\mu _{\hat{h}}\,, \end{aligned}$$

(77)

a minimizing (sub)sequence $\{\varphi _{(k)} \}$ for $E_{{\widehat{\Sigma }}\Sigma }[\varphi ]$ corresponds to a maximize sequence for the functional $\int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }(\varphi )\,\text {d}\mu _{\hat{h}}$. Hence, given $\delta \,>\,0$,  there exists $k_0$ such that for all $k\,\ge \,k_0$, we have

$$\begin{aligned} \int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }(\overline{\varphi })\,\text {d}\mu _{\hat{h}}\,\ge \, \int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }(\varphi _{(k)})\,\text {d}\mu _{\hat{h}}\,-\delta , \end{aligned}$$

(78)

along a minimizing sequence $\{\varphi _{(k)} \}\,\longrightarrow \,\overline{\varphi }$ for the functional $E_{{\widehat{\Sigma }}\Sigma }[\varphi ]$, and where ${\Phi }_{{\widehat{\Sigma }}\Sigma }(\varphi _{(k)})$ is non-negative for almost all points of ${\widehat{\Sigma }}_L$. By adding and subtracting $\int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }(\varphi _{(k)})\,\text {d}\mu _{\hat{h}}$ to (77), (evaluated for $\overline{\varphi }$), and by taking into account (78), we get

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\overline{\varphi }]\,=\, E_{{\widehat{\Sigma }}\Sigma }[\varphi _{(k)}]\,-\, 2\int _{{\widehat{\Sigma }}_L}\left( {\Phi }_{{\widehat{\Sigma }}\Sigma }(\overline{\varphi })\,-\, {\Phi }_{{\widehat{\Sigma }}\Sigma }(\varphi _{(k)}) \right) \,\text {d}\mu _{\hat{h}}\,\le \,E_{{\widehat{\Sigma }}\Sigma }[\varphi _{(k)}]\,+\,2\delta \;, \end{aligned}$$

(79)

for all $k\,\ge \,k_0$. Since the choice of $\delta \,>\,0$ is arbitrary, (79) implies that the functional $E_{{\widehat{\Sigma }}\Sigma }[\varphi ]$ is lower semicontinuous, i.e.,

$$\begin{aligned} E_{{\widehat{\Sigma }}\Sigma }[\overline{\varphi }]\,\le \,\lim _k\inf \,E_{{\widehat{\Sigma }}\Sigma }[\varphi _k] \end{aligned}$$

(80)

for all $\varphi \in \,\mathrm {Map}_{\varepsilon , C_L}({\widehat{\Sigma }}, \Sigma )$ with $\varphi _k$ weakly converging, in the above sense, to $\overline{\varphi }$. Hence, $\{\varphi _{(k)} \}\,\longrightarrow \,\overline{\varphi }$ minimizes $E_{{\widehat{\Sigma }}\Sigma }[\varphi ]$ in the space of maps $\mathrm {Map}_{\varepsilon }({\widehat{\Sigma }}, \Sigma )$, as stated.

If we set

$$\begin{aligned} d_L\left[ {\widehat{\Sigma }}_L,\,\Sigma _L\right] \,:=\,\inf _{\psi _L\circ \zeta \in \mathrm {Map}_\varepsilon ({\widehat{\Sigma }}, \Sigma )}\,E_{{\widehat{\Sigma }}\Sigma }[\psi _L,\,\zeta ] \end{aligned}$$

(81)

then as a consequence of the properties of the functional $E_{{\widehat{\Sigma }}\Sigma }[\varphi ]$, described in Lemma 8, we have that $d_L\left[ {\widehat{\Sigma }}_L,\,\Sigma _L\right]$ provides a scale-dependent distance function between the physical celestial sphere $({\Sigma }_L,\,h)$ and the reference FLRW celestial sphere $({\widehat{\Sigma }}_L,\,\hat{h})$, as the scale L varies. In particular, with the notation of Lemma 8 we have (i) Non-negativity   $d_L\left[ {\widehat{\Sigma }}_L,\,\Sigma _L \right] \,\ge \,0$;  (ii)   $d_L\left[ {\widehat{\Sigma }}_L,\,\Sigma _L \right] \,=\,0$  iff $({\widehat{\Sigma }}_L,\,\hat{h})$ and $(\Sigma _L,\, h)$ are isometric; (iii)  Symmetry    $d_L\left[ {\widehat{\Sigma }}_L,\,\Sigma _L \right] = d_L\left[ {\Sigma }_L,\,{\widehat{\Sigma }}_L \right]$; (iv)  Triangular inequality   $d_L\left[ {\widehat{\Sigma }}_L,\,\widetilde{\Sigma }_L \right] \le d_L\left[ \hat{\Sigma }_L,\,\Sigma _L \right] +d_L\left[ {\Sigma }_L,\,\widetilde{\Sigma }_L \right]$. $\square$

5 Causal diamonds and the physical meaning of $d_L\left[ {{\widehat{\Sigma }}}_L,\,{\Sigma }_L \right]$

The distance functional $d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right]$ is a geometric quantity that we can associate with the observer who wishes to describe with a Friedmannian bias the cosmological region where inhomogeneities may dominate. To appreciate what this role implies, let us briefly discuss the physical interpretation of $d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right]$, when we probe the light cone regions $\mathcal {C}_{L}^{-}(p,\hat{g})$ and $\mathcal {C}_{L}^{-}(p,g)$ over a sufficiently small length scale L. If $\bar{\varphi }$ denotes the minimizing map characterized in Theorem 9, we can write

$$\begin{aligned} d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right] \,=\,E_{{\widehat{\Sigma }}\Sigma }[\bar{\varphi }]\,:=\,A\left( {\widehat{\Sigma }}_L\right) \,+\,A({\Sigma }_L)\, -\,2\int _{{\widehat{\Sigma }}_L}{\Phi }_{{\widehat{\Sigma }}\Sigma }\,\text {d}\mu _{\hat{h}}\,. \end{aligned}$$

(82)

To simplify matters, we assume that at the given length scale L the corresponding region $\mathcal {C}_{L}^{-}(p,g)$ is caustic-free, and parametrize ${\Phi }_{\Sigma {\widehat{\Sigma }}}(\bar{\varphi })$ as

$$\begin{aligned} {\Phi }_{\Sigma {\widehat{\Sigma }}}(\bar{\varphi })\,=\,1\,+\,F(\bar{\varphi })\;, \end{aligned}$$

(83)

where $F(\bar{\varphi })$ is a smooth function (not necessarily positive) which, by discarding the traceless lensing shear, may be thought of as describing the (small) local isotropic focusing distortion of the images of the astrophysical sources on $(\Sigma , h)$ due to gravitational lensing (see the expression (12) of the sky-mapping metric h). Under these assumptions, we can write (82) as

$$\begin{aligned} d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right] \,=\,A\left( {\Sigma }_L\right) \,-\,A({\widehat{\Sigma }}_L)\, -\,2\int _{{\widehat{\Sigma }}_L}\,F(\bar{\varphi })\,\text {d}\mu _{\hat{h}}\,. \end{aligned}$$

(84)

This expression can be further specialized if we exploit the asymptotic expressions of the area $A\left( {\widehat{\Sigma }}_L\right)$ and $A\left( {\Sigma }_L\right)$ of the two surfaces $({\widehat{\Sigma }}_L, \,\hat{h})$,  $({\Sigma }_L, \,{h})$ on the corresponding lightcones $\mathcal {C}_{L}^{-}(p, \hat{g})$ and $\mathcal {C}_{L}^{-}(p, {g})$. These asymptotic expressions can be obtained if we consider the associated causal past regions $\mathcal {J}_{L}^{-}(p, \hat{g})$ and $\mathcal {J}_{L}^{-}(p, {g})$ sufficiently near the (common) observation point p, in particular when the length scale L 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 $({\Sigma }_L, \,{h})$ as its boundary. This surface intersects the worldline $\gamma (\tau )$ of the observer p at the point $q=\gamma (\tau _0\,=\,-\,L)$ defined by the given length scale L. For the reference FLRW the analogous setup is associated with the constant-time slicing of the FLRW spacetime $(M, \hat{g})$ considered. The corresponding three-dimensional region $\widehat{V}_L^3(p)$, embedded in $\mathcal {J}_{L}^{-}(p, \hat{g})$, has the surface $({\widehat{\Sigma }}_L, \,\hat{h})$ as its boundary. The FLRW observer $\hat{\gamma }(\hat{\tau })$ will intersect $\widehat{V}_L^3(p)$ at the point $\hat{q}=\hat{\gamma }(\hat{\tau }_0\,=\,-\,L)$. By introducing geodesic normal coordinates $\{X^i\}$ in $\mathcal {J}_{L}^{-}(p, {g})$ and $\{Y^k\}$ in $\mathcal {J}_{L}^{-}(p, \hat{g})$, respectively, based at the point q and $\hat{q}$, we can pull back the metric tensors g and $\hat{g}$ to $T_{q}M$ and $T_{\hat{q}}M$, and obtain the classical normal coordinate development of the metrics g and $\hat{g}$ valid in a sufficiently small convex neighborhood of q and $\hat{q}$. Explicitly, for the (more relevant case of the) metric g, we have (see, e. g., Lemma 3.4 (p. 210) of [27] or [24])

$$\begin{aligned}&\left( (\mathrm {exp}_q)^*\,g \right) _{ef}\,=\,\eta _{ef}\,-\,\frac{1}{3}\,\mathrm {R}_{eabf}|_qX^aX^b\,-\,\frac{1}{6}\,\nabla _c\mathrm {R}_{eabf}|_qX^aX^bX^c\\&\quad +\,\left( -\,\frac{1}{20}\,\nabla _c\nabla _\text {d}\mathrm {R}_{eabf}\,+\,\frac{2}{45}\,\mathrm {R}_{eabm}\,\mathrm {R}^m_{fcd} \right) _q\,X^aX^bX^cX^\text {d}\,+\,\ldots \;, \end{aligned}$$

where $\mathrm {R}_{abcd}$ is the Riemann tensor of the metric g (evaluated at the point q). The induced expansion in the pulled-back measure $\left( (\mathrm {exp}_{s(\eta )})^*\text {d}\mu _{g}\right)$ provides the Lorentzian analog of the familiar Bertrand–Puiseux formulas associated with the geometrical interpretation of the sectional, Ricci and scalar curvature for a Riemannian manifold in terms of the length, area, and volume measures of small geodesic balls. In the Lorentzian case the relevant formulas are more delicate to derive, and their analysis has given rise to a deep line of research where the role of Riemannian geodesic balls is taken over by the spacetime geometry of small causal diamonds. The study of these Lorentzian avatars of Riemannian balls was presciently initiated by Myrheim [23], and their properties are finely described in [13] (see also [2, 14]). That said, small causal diamonds asymptotics [13] provides, to leading order in L, the following expressions for the area of $({\Sigma }_L, \,{h})$ and $({\widehat{\Sigma }}_L, \,\hat{h})$,

$$\begin{aligned} A\left( {\Sigma }_L\right) \,=\,{\pi }\,L^2\,\left( 1\,-\,\frac{1}{72}\,L^2\,\mathrm {R}(q)\,+\,\ldots \right) \;, \end{aligned}$$

(85)

and

$$\begin{aligned} A\left( {\widehat{\Sigma }}_L\right) \,=\,{\pi }\,L^2\,\left( 1\,-\,\frac{1}{72}\,L^2\,\widehat{\mathrm {R}}(\hat{q})\,+\,\ldots \right) \;, \end{aligned}$$

(86)

Introducing these expressions in (84), we get

$$\begin{aligned} d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right] \,=\,\frac{\pi }{72}\,L^4\,\left( \widehat{\mathrm {R}}(\hat{q})\,-\,{\mathrm {R}}({q}) \right) \, -\,2\int _{{\widehat{\Sigma }}_L}\,F(\bar{\varphi })\,\text {d}\mu _{\hat{h}}\,+\,\ldots \,. \end{aligned}$$

(87)

We can rewrite this equivalently as

$$\begin{aligned} \widehat{\mathrm {R}}(\hat{q})\,=\,{\mathrm {R}}({q})\,+\,\frac{72}{\pi }\frac{d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right] }{L^4}\, +\,\frac{144}{\pi L^4}\,\int _{{\widehat{\Sigma }}_L}\,F(\bar{\varphi })\,\text {d}\mu _{\hat{h}}\,+\,\ldots \,. \end{aligned}$$

(88)

The asymptotics (87) (but also the very characterization of $d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right]$), shows clearly that the lightcone comparison functional $E_{{\widehat{\Sigma }} \Sigma }$ and the associated distance $d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right]$ provide a generalization of the lightcone theorem [7] proved by Y. Choquet-Bruhat, P. T. Chrusciel, and J. M. Martin-Garcia. As suggested in Introduction, the normal coordinates asymptotics (88) explicitly relating $d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right]$, the average lensing shear and the scalar curvatures $\widehat{\mathrm {R}}(\hat{q})$  and ${\mathrm {R}}({q})$, is a strong indication of the existence of a deeper relation between the geometric analysis approach to lightcones comparison discussed here and the comparison theorems for causal diamonds discussed by Berthiere et al. [2]. Motivations and techniques characterizing the two approaches are clearly very different, but the analysis of Bishop’s inequality in [2] is strongly indicative of the possibility of introducing a lightcone distance functional also within the causal diamond formalism. In Riemannian geometry, Bishop’s inequality is indeed instrumental to the characterization of the Gromov–Hausdorff distance between Riemannian manifolds [15], and currently there are optimal transport approaches to establishing a Lorentzian Bishop–Gromov inequality (see, e.g., [6]). In this connection is perhaps worthwhile to stress that the variational characterization of our distance functional $d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right]$, which remains valid also for lightcone sections with caustics, complies with optimal transport techniques since the harmonic type functional $E_{{\widehat{\Sigma }} \Sigma }$ can be rephrased within such a formalism (see, e.g., [5] for an optimal transport transcription of a functional similar to $E_{{\widehat{\Sigma }} \Sigma }$). This varied landscape is indicative of a potential common ground between our approach and that causal diamond theory. This remains, however, as an open problem worthwhile of further investigation.

Further interest in the normal coordinates asymptotics (88) lies in the fact that it directly connects $d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right]$ to the fluctuations in the spacetime scalar curvature: if we decide to keep on in modeling with a FLRW solution a cosmological spacetime, homogeneous on large scale but highly inhomogeneous at smaller scale, then the associated scalar curvature $\widehat{\mathrm {R}}(\hat{q})$ can be approximately identified with the physical scalar curvature ${\mathrm {R}}(\hat{q})$, with a rigorous level of scale dependence precision, only if we take into account the contribution provided by the lightcone distance functional $d_L\left[ {\widehat{\Sigma }}_L,\,\Sigma _L \right]$ (and by the average of the local focusing term $F(\bar{\varphi })$). This can be of relevance in addressing backreaction problems in cosmology (see, e.g., [3]). Finally, $d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right]$ can be also of some use in providing a rigorous way of addressing some aspect of the best-fitting problem in cosmology (see [10, 11]), roughly speaking, the strategy is to vary the family of model spacetimes $(M, \hat{g})$ (for instance, the family of FLRW solutions, or the larger family of homogeneous spacetimes) in such a way to minimize (over the relevant interval of length scales L) the distance functional $d_L\left[ {\widehat{\Sigma }}_L,\,{\Sigma }_L \right]$ between the physical celestial spheres $({\widehat{\Sigma }}_L, \hat{h})$ and the family of reference celestial spheres $({\widehat{\Sigma }}_L, \hat{h})$ associated with the model spacetimes $(M, \hat{g})$ adopted.