# A comparison theorem for cosmological lightcones

aa r X i v : . [ g r- q c ] F e b A COMPARISON THEOREM FOR COSMOLOGICAL LIGHTCONES

MAURO CARFORA ( ⋆ ) AND FRANCESCA FAMILIARI

Abstract.

Let (

M, g ) denote a cosmological spacetime describing the evolution of a universewhich is isotropic and homogeneous on large scales, but highly inhomogeneous on smaller scales.We consider two past lightcones, the ﬁrst, C − L ( p, g ), is associated with the physical observer p ∈ M who describes the actual physical spacetime geometry of ( M, g ) at the length scale L , whereas thesecond, C − L ( p, ˆ g ), is associated with an idealized version of the observer p who, notwithstandingthe presence of local inhomogeneities at the given scale L , wish to model ( M, g ) with a member( M, ˆ 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 twolightcones C − L ( p, g ) and C − L ( p, ˆ g ). In particular, we introduce a scale dependent ( L ) lightcone-comparison functional, deﬁned by a harmonic type energy, associated with a natural map betweenthe physical C − L ( p, g ) and the FLRW reference lightcone C − L ( p, ˆ g ). This functional has a numberof remarkable properties, in particular it vanishes iﬀ, at the given length-scale, the correspondinglightcone surface sections (the celestial spheres) are isometric. We discuss in detail its variationalanalysis and prove the existence of a minimum that characterizes a natural scale-dependent distancefunctional between the two lightcones. We also indicate how it is possible to extend our results tothe case when caustics develop on the physical past lightcone C − L ( p, g ). Finally, we show how thedistance functional is related to spacetime scalar curvature in the causal past of the two lightcones,and brieﬂy illustrate a number of its possible applications. INTRODUCTION

For many among us, the ﬁrst encounter with Boris Dubrovin has been through his classic books G ´ e ometrie Contemporaine: m ´ 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 thelater expanded Springer version [7], we treasured with care. To the best of our knowledge Borisdid not work in general relativity but in his G ´ e ometrie Contemporaine there are two wonderfullittle chapters, just short of a total of ﬁfty tersely written pages, which provide the fastest way toacquaint yourself with general relativity. Thinking of these elegant pages, we hope that it is notinappropriate to dedicate to Boris the present work addressing a long-standing problem in mathe-matical cosmology.Let us recall that the observed universe is described by a spacetime ( M, g ), a 4-dimensional manifold M endowed with a Lorentzian metric g , which is (statistically) isotropic and homogeneous only onsuﬃciently large scales, say L ≥ L , where the current acceptable ﬁgure for the homogeneity scaleis L ≥ h − M pc , where h is the dimensionless Hubble parameter describing the relative uncer-tainty of the true value of the present-epoch Hubble-Lemaitre constant H = 100 h Km/s/M pc .At these homogeneity scales ( M, g ) is described with great accuracy by a member of the homoge-neous and isotropic family of Friedman–Lemaitre–Robertson–Walker (FLRW) spacetimes ( M, ˆ g ).At smaller scales, where inhomogeneities statistically dominate, we should resort to the full-ﬂedgedspacetime geometry of ( M, g ) in order to provide the correct dynamical description of cosmologicalobservations. However, coming to mathematical terms with the geometrical and physical structureof (

M, g ) is a daunting task and typically we keep on modeling the dynamics of the universe overthese inhomogeneity scales with the FLRW model ( M, ˆ g ), thought of as providing a background Date : 7 December 2020. around which the actual spacetime geometry (

M, g ) is perturbatively expanded. If we want to gobeyond perturbation theory, we face the mathematically delicate problem of ﬁnding a way for com-paring the past lightcone region C − L ( p, g ), associated with an (instantaneous) observer p , samplingthe inhomogeneities in ( M, g ) at the given length scale L , with the corresponding past lightconeregion C − L ( p, ˆ g ) in the assumed FLRW background ( M, ˆ g ). In modern high-precision cosmology thisis one of the most delicate issue when modeling of the observed universe. In this paper we providea number of mathematical results that allow to compare the past lightcone regions C − L ( p, g ) and C − L ( p, ˆ g ). In particular, we introduce a scale-dependent lightcone comparison functional E b ΣΣ [ ϕ L ]between the physical and the FLRW reference celestial spheres Σ L ⊂ C − L ( p, g ) and b Σ L ⊂ C − L ( p, ˆ g )probed, at the given length scale L , on the respective lightcones regions C − L ( p, g ) and C − L ( p, ˆ g ). It isimportant to stress that the scale-dependent map ϕ L , is not an abstract map, but it actually relatesthe physical observations on Σ L with those, described with a FLRW bias, on b Σ L . The functional E b ΣΣ [ ϕ L ] is deﬁned by a harmonic map type energy and has a number of remarkable properties. Inparticular it vanishes iﬀ , at the given length-scale L , the corresponding lightcone surface sectionsΣ L and b Σ L (which are topologically 2-spheres, as long as null-caustics are absent) are isometric.Moreover, the inf of E b ΣΣ [ ϕ L ], over a suitable class of extended maps ϕ L (extension necessary inorder to account also for the presence of lightcone caustics), provides a scale-dependent distancefunctional, d L [ b Σ , Σ], between the physical and the FLRW reference lightcones C − L ( p, g ) and C − L ( p, ˆ g ).This distance signiﬁcantly extends the lightcone theorem proved in [5]. Moreover, we show that inthe caustic-free region near the tip p of C − L ( p, g ) and C − L ( p, ˆ g ), namely for L small enough, d L [ b Σ , Σ]is related to the spacetime scalar curvatures R ( g ) and R (ˆ g ), in the interior of these lightcones, arelation that may play an important role in cosmological modeling.2. Cosmological observers and observational coordinates along the PastLightcones

Throughout this paper (

M, g ) denotes a cosmological spacetime where g is a Lorentzian metric (ofsignature (+ , + , + , − )), and where M is a smooth 4-dimensional manifold which for our purposeswe can assume diﬀeomorphic to R . In local coordinates { x i } i =1 , we write g = g ik dx i ⊗ dx k ,where the metric components g ik := g ( ∂ i , ∂ k ) in the coordinate basis { ∂ i := ∂/∂x i } i =1 , havethe Lorentzian signature (+ , + , + , − ), and the Einstein summation convention is in eﬀect. Wedenote by ∇ ( g ) (or ∇ if there is no danger of confusion) the Levi–Civita connection of g , and let R m ( g ) = R iklm ∂ i ⊗ dx k ⊗ dx l ⊗ dx m , R ic ( g ) = R ab dx a ⊗ dx b and R ( g ) be the correspondingRiemann, Ricci and scalar curvature operators, respectively. We assume that ( M, g ) is associatedwith the evolution of a universe which is (statistically) isotropic and homogeneous on suﬃcientlylarge scales, whereas local inhomogeneities dominate over smaller scales. The mass–energy contentin (

M, g ) is phenomenologically described by an energy-momentum tensor T the explicit expressionof which is not needed in our analysis, we only assume that its matter components characterize aHubble ﬂow that generates a family of preferred world-lines parametrized by proper time τγ s : R > −→ ( M, g )(1) τ γ s ( τ ) , and labeled by suitable comoving (Lagrangian) coordinates s . We set c = 1, and denote by˙ γ s := dγ s ( τ ) dτ , with g ( ˙ γ s , ˙ γ s ) = −

1, the corresponding 4-velocity ﬁeld. For simplicity, we assumethat the worldlines (1) are geodesics, i.e. ∇ ˙ γ s ˙ γ s = 0. This is the spacetime within which we canframe the actual cosmological data gathered from our past lightcone observations. If we adopt theweak form of the cosmological principle, ( M, g, γ s ) can be identiﬁed with the phenomenological Details on the notation adopted are explained in full detail in the body of the paper. In particular the regions C − L ( p, g ) and C − L ( p, ˆ g ) are deﬁned by (19). COMPARISON THEOREM FOR COSMOLOGICAL LIGHTCONES 3 background spacetime or

Phenomenological Background Solution (PBS) , according to the notationintroduced in [18]. In the same vein, we deﬁne

Phenomenological Observers the collection ofobservers { γ s } comoving with the Hubble ﬂow.2.1. The phenomenological lightcone metric.

Since in our analysis we ﬁx our attention ona given observer, we drop the subscript s in (1)), and describe a ﬁnite portion of the observer’sworld-line with the timelike geodesic segment τ γ ( τ ), − δ < τ < δ , for some δ >

0, where p := γ ( τ = 0) is the selected observational event. To set up the appropriate coordinates along γ ( τ ), let (cid:0) T p M, g p , { E ( i ) } (cid:1) be the tangent space to M at p endowed with a g -orthonormal frame { E ( i ) } i =1 ,..., , g p (cid:0) E ( i ) , E ( k ) (cid:1) = η ik , where η ik is the Minkowski metric, and where E (4) := ˙ γ ( τ ) | τ =0 .Notice that by parallel transport, this basis can be propagated along γ ( τ ). Let us introduce the setof past-directed null vectors and the set of past-directed causal vectors in ( T p M, g p ) according to(2) C − ( T p M, g p ) := (cid:8) X = X i E ( i ) = 0 ∈ T p M | X + r = 0 (cid:9) , (3) C − ( T p M, g p ) := (cid:8) X = X i E ( i ) = 0 ∈ T p M | X + r ≤ (cid:9) , where r := ( P a =1 ( X a ) ) / . We use these sets of vectors in order to introduce observationalcoordinates in (a region of) the causal past of p , J − ( p, g ), by exploiting the exponential mappingbased at p , exp p : C − ( T p M, g p ) −→ M (4) X exp p ( X ) := λ X (1)(5)where λ X : [0 , ∞ ) −→ ( M, g ) is the past-directed causal geodesic emanating from the point p with initial tangent vector ˙ γ X (0) = X ∈ C − ( T p M, g p ). If we assume that the metric is suﬃcientlyregular , then there is a neighborhood N ( g ) of 0 in T p M and a geodesically convex neighborhoodof p , U p ⊂ ( M, g ), deﬁned by all points q ∈ M which are within the domain of injectivity of exp p ,where we can introduce geodesic normal coordinates ( X i ) according to X i := X i ◦ exp − p : M ∩ U p −→ R (6) q X i ( q ) := X i (cid:0) exp − p ( q ) (cid:1) where X i (cid:0) exp − p ( q ) (cid:1) are the components, in the g -orthonormal frame { E ( i ) } , of the vector exp − p ( q ) ∈ T p M . In particular, if we consider the past lightcone C − ( p, g ) with vertex at p , then away fromthe past null cut locus of p , i.e. away from the set of lightcone caustics, normal coordinates can beused to parametrize the past light cone region C − ( p, g ) ∩ U p ,exp p : C − ( T p M, g p ) ∩ N ( g ) −→ C − ( p, g ) ∩ U p (7) X = X i E ( i ) exp p ( X i E ( i ) ) = q ⇒ { X i ( q ) } . Similarly, by restricting exp p to C − ( T p M, g p ) ∩ N ( g ) we can parametrize with normal coordinatesthe region J − ( p, g ) ∩ U p within the causal past J − ( p, g ) of p . In particular, we can foliate J − ( p, g ) ∩ U p with the family of past lightcones C − ( γ ( τ ) , g ) associated with the events γ ( τ ) ∩ U p , − δ < τ ≤ x := r := vuut X a =1 ( X a ) , x := θ ( X a /r ) , x := ϕ ( X a /r ) , x := τ = X + r , A Lipschitz condition for the metric components suﬃces.

M. CARFORA AND F. FAMILIARI where θ ( X a /r ), ϕ ( X a /r ), a = 1 , ,

3, denote the standard angular coordinates of the direction( X a /r ) on the unit 2-sphere S in T p M and where, according to (2), x = 0 corresponds to thelight cone region C − ( p, g ) ∩ U p . Notice that at the vertex p = γ ( τ = 0), the coordinate function x 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 fromits null cut locus, we have that the past lightcone region C − ( p, g ) ∩ U p \ { p } is topologically foliatedby the r -dependent family of 2–dimensional surfaces Σ( p, r ) , the celestial spheres at scale r , reachedby the set of past directed null geodesics as the aﬃne parameter r varies, i.e., (9) Σ( p, r ) := (cid:8) exp p ( r n ) (cid:12)(cid:12) n := ( θ, ϕ ) ∈ S ⊂ T p M (cid:9) . Each Σ( p, r ) is topologically a 2-sphere endowed with the r -dependent family of two-dimensionalRiemannian metrics(10) h ( r ) := (cid:0) exp ∗ p g | Σ( p,r ) (cid:1) αβ dx α dx β (cid:12)(cid:12)(cid:12) r obtained by using the exponential map to pull back to S ⊂ T p M the 2-dimensional metric g | Σ( p,r ) induced on Σ( p, r ) by the embedding Σ( p, r ) ֒ → ( M, g ). We normalize this metric by imposing thatthe angular variables x α = ( θ, ϕ ), in the limit r ց

0, reduce to the standard spherical coordinateson the unit 2-sphere S , i.e. ,(11) lim r ց (cid:12)(cid:12)(cid:12)(cid:12) x =0 h αβ ( r ) dx α dx β r = d Ω := dθ + sin θ dϕ . For a physical interpretation [9], it is convenient to parametrize h ( r ) as a sky-mapping metric(12) h ( r ) = D ( r ) (cid:16) d Ω + L αβ ( r ) dx α dx β (cid:17) , where d Ω is the unit radius round metric on S (see (11)), and the coordinates { x α } α =2 , providethe direction of observation (as seen at p ) of the astrophysical sources on the celestial sphere Σ( p, r ).The function D ( r ) is the observer area distance deﬁned by the relation dµ h ( r ) = D ( r ) dµ S where dµ h ( r ) is the pulled-back (via exp p ) area measure of (cid:0) Σ( p, r ) , g | Σ( p,r ) (cid:1) , (roughly speaking, dµ h ( r ) canbe interpreted [9] as the cross-sectional area element at the source location as seen by the observer at p ) and dµ S is the area element on the unit round sphere S ∈ T p M ( i.e. , the element of solid anglesubtended by the source at the observer location p ). In the same vein, the symmetric tensor ﬁeld L αβ ( r ), describing the distortion of the normalized metric h ( r ) /D ( r ) with respect to the roundmetric d Ω , can be interpreted as the image distortion of the sources on (Σ( p, r ) , h ( r )) as seen bythe observer at p . This term, which in general is not trace-free, involves both the gravitationallensing shear [9] and the gravitational focusing of the light rays generating the local source imagemagniﬁcation. By taking into account these remarks, we have the following characterization of thepast lightcone metric in a neighborhood of the point p . Lemma 2.

In the geometrical coordinates introduced above, the null geodesics generators of C − ( p, g ) ∩ U p have equation x = 0 , x α = const. , and their tangent vector is provided by ∂∂x , with (exp ∗ g ) (cid:0) ∂∂x , ∂∂x (cid:1) = 0 . Since ∂∂x is past-directed we can introduce the normalization (13) lim r ց (exp ∗ g ) (cid:18) ∂∂x , ˙ γ (cid:19) = 1 and write the restriction of the spacetime metric g on C − ( p, g ) ∩ U p according to (14) g | x =0 = g ( dx ) + 2 g dx dx + 2 g α dx dx α + h αβ dx α dx β , A detailed and very informative analysis of geodesic coordinates along the past light cone is provided by [10].

COMPARISON THEOREM FOR COSMOLOGICAL LIGHTCONES 5 where α, β = 2 , , and where the components g ik ( x i ) := (exp ∗ g )( ∂∂x i , ∂∂x k ) , and h αβ ( x i ) are allevaluated for x = 0 . As already stressed, the coordinates { x i } are singular at the vertex γ ( τ = 0) = p of the cone. Adetailed analysis of the limit r ց

0, besides the standard assumptions we already made, is carriedout in detail in the fundational paper [8] (see paragraph 3) and in [6], (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 adaptedto the past lightcone).

Remark 3.

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

The reference FLRW lightcone metric.

Along the physical metric g , we also introduce in M the FLRW metric ˆ g and the family of global Friedmannian observers ˆ γ s that, at the homogeneityscale, we can associate with the cosmological data. This is the Global Background Solution (GBS) according to [18]. In full generality the geodesics τ γ ( τ ), and ˆ τ ˆ γ (ˆ τ ), − δ <, τ, ˆ τ < δ ,associated with the corresponding Hubble ﬂow in ( M, g, γ ) and ( M, ˆ g, ˆ γ ), 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 τ and ˆ τ along γ ( τ ) and ˆ γ (ˆ τ ) so that at τ = 0 = ˆ τ wehave γ (0) = p = ˆ γ (0). Hence, together with the coordinates { x i } in ( M, g, γ s ), describing theobservational metric (14) on the past lightcone C − ( p, g ) ∩ U p , we introduce corresponding (normal)coordinates { Y k } in the reference ( M, ˆ g, ˆ γ ). With an obvious adaptation of the analysis for ( M, g ),carried out in previous subsection, let N (ˆ g ) denote the domain of injectivity of the exponentialmapping d exp p : T p M −→ ( M, ˆ g ) based at the event p = ˆ γ (0). If ˆ U p ⊂ ( M, ˆ g ) denotes the regionof injectivity of d exp p we can consider normal coordinates(15) Y i := Y i ◦ d exp − p : ( M, ˆ g ) ∩ ˆ U p −→ R , where Y i are the components of the vectors Y ∈ T p M with respect to a ˆ g -orthonormal frame { ˆ E ( i ) } i =1 ,..., with ˆ E (4) := ˆ˙ γ (0). Within ˆ U p we can introduce, in full analogy with (7) and (14),the coordinates y := ˆ r = ( P a =1 ( Y a ) ) / , y α | α =2 , = ( θ ( Y a / ˆ r ) , ϕ ( Y a / ˆ r )) and parametrize C − ( p, ˆ g ) ∩ ˆ U p in terms of the 2-dimensional spheres(16) ˆΣ( p, ˆ r ) := (cid:8) ˆexp p (ˆ r n ) (cid:12)(cid:12) n := ( θ, ϕ ) ∈ S ⊂ T p M (cid:9) , endowed with the round metric(17) ˆ h (ˆ r ) := (ˆ g ) αβ dy α dy β (cid:12)(cid:12)(cid:12) ˆ r = a (ˆ r ) ˆ r (cid:0) dθ + sin θdϕ (cid:1) , where a (ˆ r ) is the FLRW expansion factor corresponding to the distance ˆ r . Hence, we can write themetric ˆ g on the reference FLRW past lightcone region C − ( p, ˆ g ) ∩ ˆ U p as(18) ˆ g | y =0 = ˆ g ( dy ) + ˆ h αβ dy α dy β . Comparing lightcones: a scale dependent comparison functional

According to our hypotheses, the spacetime (

M, g, γ s ) describes the evolution of a universe whichis isotropic and homogeneous only at suﬃciently large scales L . At these homogeneity scales( M, g, γ s ) is modeled by the FLRW spacetime ( M, ˆ g, ˆ γ s ). Even if at smaller scales, where inhomo-geneities statistically dominate, ( M, g, γ s ) provides the bona ﬁde spacetime describing cosmologicalobservations, we can still use the reference ( M, ˆ g, ˆ γ s ) as a background FLRW model. As the ob-servational length scale L varies from the local highly inhomogeneous regions to the homogeneityscale L , we do not assume a priori that ( M, g, γ s ) is perturbatively near to the reference FLRWspacetime ( M, ˆ g, ˆ γ s ). Rather, we compare ( M, g, γ s ) with ( M, ˆ g, ˆ γ s ) keeping track of the pointwise M. CARFORA AND F. FAMILIARI and global relations among the various geometric quantities involved. In particular, we will com-pare the lightcone region C − ( p, g ) ∩ U p with the reference FLRW lightcone region C − ( p, ˆ g ) ∩ ˆ U p ,assuming that in such a range there are no lightcone caustics. As already emphasized, this is anassumption that makes easier to illustrate some of the technical arguments presented here, in theﬁnal part of the paper we indicate how our main result, concerning the existence and the propertiesof the distance functional described in the introduction, holds also in the more general case whencaustics are present. That said, let us consider the following scale–dependent subsets of the pastlight cones C − ( p, g ) and C − ( p, ˆ g ),(19) C − L ( p, g ) := exp p h C − L ≤ L ( T p M, g p ) i , C − L ( p, ˆ g ) := d exp p h C − L ≤ L ( T p M, ˆ g p ) i , where C − L ≤ L ( T p M, g p ) := (cid:8) X = X i E ( i ) ∈ ( T p M, g p ) | X + r = 0 , − L ≤ X ≤ (cid:9) , (20) C − L ≤ L ( T p M, ˆ g p ) := n Y = Y a ˆ E ( a ) ∈ ( T p M, ˆ g p ) | Y + ˆ r = 0 , − L ≤ Y ≤ o , (21)are the exponential map domains associated with the observational length-scale L up to the ho-mogeneity scale L . Under the stated caustic–free assumption, both C − L ( p, g ) and C − L ( p, ˆ g ) canbe foliated in terms of the 2-dimensional surfaces Σ( p, r ) and ˆΣ( p, ˆ r ) introduced in the previoussection, i.e. , we can write C − L ( p, g ) = [ ≤ r ≤ L Σ( p, r ) , C − L ( p, ˆ g ) = [ ≤ ˆ r ≤ L ˆΣ( p, ˆ r ) . On C − L ( p, g ) and C − L ( p, ˆ g ) the normal coordinates { x i } and { y a } , associated with the observationalmetric (14) and the reference metric (18), cannot be directly identiﬁed since they are deﬁned interms of the distinct exponential mappings exp p and d exp p and, for a given initial tangent vector X ∈ C − L ≤ L ( T p M, g p ) ∩ C − L ≤ L ( T p M, ˆ g p ), we have(22) exp p ( X ) = q = d exp p ( X ) = ˆ q . However, q and ˆ q are in the open spacetime region deﬁned by(23) M p := exp p ( N ( g )) ∩ ˆexp p ( N (ˆ g )) ⊂ M , and since exp p and d exp p are local diﬀeomorphisms from N ( g ) ∩ N (ˆ g ) ⊂ T p M into M p , the mapdeﬁned by ψ : ( M p ∩ C − L ( p, ˆ g ) , ˆ g ) −→ ( M p ∩ C − L ( p, g ) , g )(24) ˆ q ψ (ˆ q ) = q = exp p (cid:16)d exp − p (ˆ q ) (cid:17) is a diﬀeomorphism with ψ ( p ) = id M . In particular, in terms of the coordinates { x i } and { y a } wecan locally write(25) y a (ˆ q ) x i ( q ) = ψ i ( y b (ˆ q )) . In order to describe at a given length scale 0 < L ≤ L , the eﬀect of these diﬀeomorphisms onthe lightcone regions C − L ( p, g ) and C − L ( p, ˆ g ), let us consider the spherical surfaces(26) (Σ L , h ) := [Σ( p, r = L ) , h ] , ( b Σ L , ˆ h ) := [ b Σ( p, ˆ r = L ) , ˆ h ]with their respective metrics h and ˆ h , and where, since the notation wants to travel light, we dropthe explicit reference to the vertex p of the lightcone and where we have replaced the aﬃne param-eters r and ˆ r with the preassigned value L of the probed length scale. The surfaces (Σ L , h ) and We use the letters from the ﬁrst half of the alphabet, a, b, c, d, . . . to index the coordinates { y } ; the letters fromthe second half i, j, k, ℓ, . . . provide the indexing of the coordinates { x } . COMPARISON THEOREM FOR COSMOLOGICAL LIGHTCONES 7 ( b Σ L , ˆ h ) characterize, at the given scale L , the celestial sphere at p as seen by the physical observerand by the reference FLRW observer, respectively.A direct application of the standard geometrical set-up of harmonic map theory (see e.g. [19])provides the following notational lemma directly connecting our analysis to harmonic maps betweensurfaces. Lemma 4.

Let ψ L be the diﬀeomorphism ψ restricted to the surfaces ( b Σ L , ˆ h ) and (Σ L , h ) , (27) ψ L : ( b Σ L , ˆ h ) −→ (Σ L , h ) then we can introduce the pull–back bundle ψ − L T b Σ L whose sections v ≡ ψ − L V := V ◦ ψ L , V ∈ C ∞ ( b Σ , T b Σ L ) , are the vector ﬁelds over b Σ covering the map ψ L . If T ∗ b Σ L denotes the cotangentbundle to ( b Σ L , ˆ h ) , then the diﬀerential dψ L = ∂ψ iL ∂y a dy a ⊗ ∂∂ψ i can be interpreted as a section of T ∗ b Σ L ⊗ ψ − T Σ L , and its Hilbert–Schmidt norm, in the bundle metric (28) h· , ·i T ∗ b Σ L ⊗ ψ − T Σ L := ˆ h − ( y ) ⊗ h ( ψ L ( y ))( · , · ) , is provided by (29) h dψ L , dψ L i T ∗ b Σ L ⊗ ψ − T Σ L = ˆ h ab ( x ) ∂ψ i ( y ) ∂y a ∂ψ j ( y ) ∂y b h ij ( ψ ( y )) = tr ˆ h ( y ) ( ψ ∗ L h ) , where (30) ψ ∗ L h = ⇒ ( ψ ∗ L h ) ab = ∂ψ i ( y c ) ∂y a ∂ψ k ( y d ) ∂y b h ik provides the pull-back of the metric h on b Σ L . The connection between the pulled-back metric ψ ∗ L h and the round metric ˆ h , both deﬁned on b Σ L , is provided by the following proposition where we respectively denote by R L (ˆ h ) and R L ( h )the scalar curvature of ( b Σ L , ˆ h ) and (Σ L , h ), and we let ∆ ˆ h := ˆ h αβ ∇ α ∇ β be the Laplace-Beltramioperator on ( b Σ L , ˆ h ). Notice that the scalar curvature R L (ˆ h ) is associated with the metric (17)evaluated for ˆ r = L and hence is given by the constant R L (ˆ h ) = a ( L ) L . 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 ( L ) and on the lensing distortion L αβ ( L ) dx α dx β . Proposition 5.

Let q ( i ) , i = 1 , , three distinct points intercepted, on the observer celestial sphere (Σ L , h ) , by three past-directed null geodesics on C − L ( p, g ) , and let ˆ q ( i ) , i = 1 , , , three distinguishedpoints on the reference FLRW celestial sphere ( b Σ L , ˆ h ) , characterizing three corresponding past-directed null directions on C − L ( p, ˆ g ) . If ζ ∈ P SL (2 , C ) denotes the fractional linear transformationin the projective special linear group, describing the automorphism of ( b Σ L , ˆ h ) that brings { ψ − ( q i ) } into ˆ q ( i ) , i = 1 , , , then there is a positive scalar function Φ b ΣΣ ∈ C ∞ ( ˆΣ , R ) , solution of theelliptic partial diﬀerential equation (31) − ∆ ˆ h ln(Φ b ΣΣ ) + R L (ˆ h ) = R L ( h ) Φ b ΣΣ , such that ψ L ◦ ζ characterizes a conformal diﬀeomorphism between ( b Σ L , ˆ h ) and (Σ L , h ) , i.e. (32) ( ψ L ◦ ζ ) ∗ h = Φ b ΣΣ ˆ h . Proof.

This is a direct consequence of the Poincare-Koebe uniformization theorem which impliesthat the 2-sphere with the pulled back metric ( b Σ L , ψ ∗ L h ) can be mapped conformally, in a one-to-oneway, onto the round 2-sphere ( b Σ L , ˆ h ). Recall that on the unit sphere S , with its canonical roundmetric d Ω , there is a unique conformal class [ d Ω ] and that the metric (17) on b Σ L ≃ S , rescaled M. CARFORA AND F. FAMILIARI according to ˆ h/ ( a (ˆ r ) ˆ r ), is isometric to d Ω . Hence, by the uniformization theorem, all metricson b Σ L ≃ S may be pulled back by diﬀeomorphisms to the conformal class [ˆ h ] of the round metricwith the chosen radius a (ˆ r ) ˆ r . Since ( b Σ L , ˆ h/ ( a (ˆ r ) ˆ r )) ≃ S , the pullback is unique modulo theaction of the conformal group group of the sphere Conf( S ). If we denote by P S the stereographicprojection (from the north pole (0 , ,

1) of S := { ( x, y, z ) ∈ R | x + y + z = 1 } )(33) P S : S ⊂ R −→ C ∪ {∞} , P S ( x, y, z ) = x + i y − z , then we can identify Conf( S ) with the 6-dimensional projective special linear group P SL (2 , C )describing the automorphisms of S ≃ C ∪ {∞} . The elements of P SL (2 , C ) are the fractionallinear transformations the Riemann sphere S ≃ C ∪ {∞} C ∪ {∞} −→ C ∪ {∞} (34) z ζ ( z ) := az + bcz + d , a, b, c, d ∈ C , ad − bc = 0 . These transformations act on the diﬀeomorphism (27) according to

P SL (2 , C ) × ( b Σ L , ˆ h ) −→ (Σ L , h )(35) ( ζ, y ) ψ L ( ζ ( y ))where, abusing notation, we have denoted by ζ ( y ) the action that the fractional linear transfor-mation ζ ( z ) deﬁnes on the point y ∈ b Σ L corresponding, via stereographic projection, to the point z ∈ C ∪ {∞} . This action may be a potential source of a delicate problem since P SL (2 , C ) is non-compact and Φ b ΣΣ is evaluated on the composition ψ L ◦ ζ deﬁned by (35). This is not problematicas long as ζ varies in the maximal compact subgroup of P SL (2 , C ) generated by the isometriesof ( b Σ , ˆ h ). However, if we consider a sequence { ζ k } k ∈ N ∈ P SL (2 , C ) deﬁned by larger and largerdilation (corresponding to larger and larger (local) Lorentz boosts of the surface b Σ in the referencespacetime ( M, ˆ g )), then the composition ψ L ◦ ζ k may generate a sequence of conformal factors { Φ k ) b ΣΣ } converging to a non-smooth function. To avoid these pathologies we exploit the fact thata linear fractional transformation is fully determined if we ﬁx its action on three distinct points ofthe sphere. In our setting this corresponds to ﬁxing the action on three distinct null direction in thelightcone region C − L ( p, ˆ g ). In physical terms this is equivalent to require that the FLRW referenceobserver at p has to adjust his velocity and orientation in such a way that three given astrophysicalsources of choice are in three speciﬁed position on the celestial sphere ( b Σ L , ˆ h ) at scale L . Thisis a gauge ﬁxing of the action of P SL (2 , C ) that corresponds in a very natural way to adjust thelocation of three reference observations in order to be able to compare the data on the physical pastlightcone C − L ( p, g ) with the data on the reference past lightcone C − L ( p, ˆ g ). By ﬁxing in this way the P SL (2 , C ) action, the pullback ( ψ L ◦ ζ ) ∗ h on ˆΣ L of the metric h is well deﬁned. By the Poincare-Koebe uniformization theorem the metric ( ψ L ◦ ζ ) ∗ h is in the same conformal class of ˆ h . Let usdenote by Φ b ΣΣ ∈ C ∞ ( ˆΣ , R ) the corresponding conformal factor such that ( ψ L ◦ ζ ) ∗ h = Φ b ΣΣ ˆ h . Ifwe set e f := Φ b ΣΣ , then the properties of the scalar curvature under the conformal transformation h = e f ˆ h (see e.g. , [1]) provide the relation(36) R (( ψ L ◦ ζ ) ∗ h ) = e − f h R (ˆ h ) + ∆ ˆ h f i . If for notational ease we keep on writing R ( h ) for R (( ψ L ◦ ζ ) ∗ h ) = R ( h ( ψ L ◦ ζ )), then it followsfrom (36) that Φ b ΣΣ necessarily is a solution on ( b Σ L , ˆ h ) of the elliptic partial diﬀerential equation(31), solution that under the stated hypotheses always exists [1]. (cid:3) COMPARISON THEOREM FOR COSMOLOGICAL LIGHTCONES 9

According to the above result, there is a positive scalar function Φ b ΣΣ ∈ C ∞ ( ˆΣ , R ) such that ψ L ◦ ζ characterizes a conformal diﬀeomorphism between ( ˆΣ L , ˆ h ) and (Σ L , h ). In components (32)can be written as(37) (( ψ L ◦ ζ ) ∗ h ) ab = ∂ψ iL ( ζ ( y )) ∂y a ∂ψ kL ( ζ ( y )) ∂y b h ik = Φ b ΣΣ ˆ h ab . It follows that by tracing (37) with respect to ˆ h ab , we can express Φ b ΣΣ in terms of the Hilbert–Schmidt norm of the diﬀerential d ( ψ L ◦ ζ ) = ∂ψ iL ( ζ ( y )) ∂y a dy a ⊗ ∂∂ψ iL according to (see (29))(38) Φ b ΣΣ = tr ˆ h ( y ) (( ψ L ◦ ζ ) ∗ h ) = 12 ˆ h ab ∂ψ iL ( ζ ( y )) ∂y a ∂ψ kL ( ζ ( y )) ∂y b h ik . From (37) we get det (( ψ L ◦ ζ ) ∗ h ) = Φ b ΣΣ det(ˆ h ), hence we can equivalently write the confor-mal factor as the Radon-Nikodym derivative of the riemannian measure dµ ψ ∗ h := ( ψ L ◦ ζ ) ∗ dµ , of( ˆΣ , ( ψ L ◦ ζ ) ∗ h ), with respect to the riemannian measure dµ ˆ h of the round metric ( ˆΣ , ˆ h ), i.e. ,(39) Φ b ΣΣ = dµ ψ ∗ h dµ ˆ h = ( ψ L ◦ ζ ) ∗ dµ h dµ ˆ h . Equivalently, this states that Φ b ΣΣ can be interpreted as the Jacobian of the map ψ L ◦ ζ ,(40) Φ b ΣΣ = Jac( ψ L ◦ ζ ) . Along the same lines, we can associate to the inverse diﬀeomorphism( ψ L ◦ ζ ) − : (Σ L , h ) −→ ( b Σ L , ˆ h )(41) x ζ − (cid:0) ψ − L ( x ) (cid:1) a positive scalar function Φ Σ b Σ ∈ C ∞ (Σ , R ) such that we can write(42) (cid:16) ( ψ L ◦ ζ ) − (cid:17) ∗ ˆ h = Φ b Σ h , with(43) Φ b Σ = 12 h ik ∂ (cid:0) ζ − (cid:0) ψ − L ( x ) (cid:1)(cid:1) a ∂x i ∂ (cid:0) ζ − (cid:0) ψ − L ( x ) (cid:1)(cid:1) b ∂x k ˆ h ab = dµ ( ψ − ) ∗ ˆ h dµ h = (cid:16) ( ψ L ◦ ζ ) − (cid:17) ∗ dµ ˆ h dµ h . To measure the global deviation of the conformal diﬀeomorphisms Φ b ΣΣ from an isometry between( ˆΣ L , ˆ h ) and (Σ L , h ) we introduce the following comparison functional where, for later use, we keeptrack of the ζ ∈ P SL (2 , C ) dependence in Φ b ΣΣ . Deﬁnition 6. (The lightcone comparison functional at scale L )Let Φ b ΣΣ ∈ C ∞ ( ˆΣ , R ) (or at least C ( ˆΣ , R ) ) be the positive scalar function such that ψ L ◦ ζ characterizes the conformal diﬀeomorphism ( ψ L ◦ ζ ) ∗ h = Φ b ΣΣ ˆ h between ( b Σ L , ˆ h ) and (Σ L , h ) ,then the associated lightcone comparison functional at scale L is deﬁned by (44) E b ΣΣ [ ψ L , ζ ] := Z b Σ L (Φ b ΣΣ − dµ ˆ h . The functional E b ΣΣ [ ψ L , ζ ] is clearly related to the familiar harmonic map energy associatedwith the map ψ L ◦ ζ : ˆΣ −→ Σ. Explicitly, if we take into account (38) we can write(45) Z b Σ L b Φ L dµ ˆ h = 12 Z b Σ L ˆ h ab ∂ψ iL ( ζ ( y )) ∂y a ∂ψ kL ( ζ ( y )) ∂y b h ik dµ ˆ h , which provides the harmonic map functional whose critical point are the harmonic maps of theRiemann surface ( ˆΣ L , [ˆ h ]) into (Σ L , h ), where [ˆ h ] denotes the conformal class of the metric ˆ h .Notice that, whereas the harmonic map energy (45) is a conformal invariant quantity, the functional E b ΣΣ [ ψ L , ζ ] is not conformally invariant. Under a conformal trasformation ˆ h −→ e f ˆ h we get(46) Z b Σ L (cid:16) e − f Φ b ΣΣ − (cid:17) e f dµ ˆ h . It is also clear from its deﬁnition that corresponding to large gradients (see (43)), E b ΣΣ [ ψ L , ζ ] tendsto the harmonic map energy. In this connection, it is important to stress that rather than on thespace of smooth maps C ∞ ( b Σ , Σ), the functional E b ΣΣ [ ψ L , ζ ] is naturally deﬁned on the Sobolevspace of maps W , ( b Σ , Σ) which are, together with their weak derivatives, square integrable. Thischaracterization, familiar when studying weakly-harmonic maps [16] and which we discuss in detailbelow when minimizing E b ΣΣ [ ψ L , ζ ], is important in our case when extending our analysis to thelow regularity setting when lightcone caustics are present. Remark 7.

It must be stressed that energy functionals such as (44) are rather familiar in the prob-lem of comparing shapes of surfaces in relation with computer graphic and visualization problems(see e.g. [17] and [13], to quote two relevant papers in a vast literature). In particular, (44) hasbeen introduced under the name of elastic energy in an inspiring paper by J. Hass and P. Koehl [15],who use it as a building block of a more complex functional relevant to surface visualization.

In our particular framework, the functional E b ΣΣ [ ψ L , ζ ] has a number of important propertiesthat make it a natural candidate for comparing, at the given length scale L , the physical lightconeregion C − L ( p, g ) with the FLRW reference region C − L ( p, ˆ g ). To start with, we prove the followinggeneral properties (in the smooth setting) . Lemma 8.

The functional E b ΣΣ [ ψ L , ζ ] is symmetric (47) E b ΣΣ [ ψ L , ζ ] = E Σ b Σ [ ψ − L , ζ − ] , where (48) E Σ b Σ [ ψ − L , ζ − ] := Z Σ L (Φ Σ b Σ − dµ h , is the comparison functional associated with the inverse map ( ψ L ◦ ζ ) − : Σ L −→ ˆΣ L .If ( e Σ L , ˜ h ) is a third surface on the past lightcone e C − L ( p, ˜ g ) , with vertex at p , associated with yetanother reference FLRW metric ˜ g on M (say another member of the FLRW family of spacetimes,distinct from ˆ g ), and σ L : Σ L e Σ L , Φ Σ e Σ respectively are the corresponding diﬀeomorphism andconformal factor, then to the composition of maps (49) b Σ L −→ ψ L Σ L −→ σ L e Σ L we can associate the triangular inequality (50) E b ΣΣ [ ψ L , ζ ] + E Σ e Σ [ σ L , ζ ] ≥ E b Σ e Σ [( σ L ◦ ψ L ) , ζ ] , where (51) E b Σ e Σ [( σ L ◦ ψ L ) , ζ ] := Z ˆΣ L (Φ b Σ e Σ − dµ ˆ h . COMPARISON THEOREM FOR COSMOLOGICAL LIGHTCONES 11 If A ( b Σ L ) := R b Σ L dµ ˆ h and A (Σ L ) := R Σ L dµ h respectively denote the area of the surfaces ( b Σ , ˆ h ) and (Σ , h ) , then we have the upper and lower bounds (52) A ( b Σ L ) + A (Σ L ) ≥ E b ΣΣ [ ψ L , ζ ] ≥ (cid:18)q A ( b Σ L ) − p A (Σ L ) (cid:19) . Finally, (53) E b ΣΣ [ ψ L , ζ ] = 0 iﬀ the surfaces ( b Σ , ˆ h ) and (Σ , h ) are isometric.Proof. For notational ease, let us temporarily dismiss the action of the linear fractional transfor-mation ζ ∈ P SL (2 , C ) and, if there is no chance of confusion, write E b ΣΣ [ ψ L ] in place of the full E b ΣΣ [ ψ L , ζ ]. We start with proving the symmetry property (47). To this end, expand the integrandin (44) and rewrite E b ΣΣ [ ψ L ] as E b ΣΣ [ ψ L ] = Z b Σ L (Φ b ΣΣ − dµ ˆ h = Z b Σ L Φ b ΣΣ dµ ˆ h + Z b Σ L dµ ˆ h − Z b Σ L Φ b ΣΣ dµ ˆ h (54) = Z b Σ L ψ ∗ L dµ h dµ ˆ h dµ ˆ h + A (cid:16) b Σ L (cid:17) − Z b Σ L Φ b ΣΣ dµ ˆ h = Z ψ L ( b Σ L ) dµ h + A (cid:16) b Σ L (cid:17) − Z b Σ L Φ b ΣΣ dµ ˆ h = A (Σ L ) + A (cid:16)b Σ L (cid:17) − Z b Σ L Φ b ΣΣ dµ ˆ h , where we have exploited the Radon-Nikodyn characterization of b Φ b ΣΣ , (see (39)), the identiﬁcation ψ ( b Σ L ) = Σ L , and the relation(55) Z b Σ L ψ ∗ L dµ h dµ ˆ h dµ ˆ h = Z b Σ L ψ ∗ L dµ h = Z ψ ( b Σ L ) dµ h = Z Σ L dµ h = A (Σ L ) , where A (Σ L ) and A (cid:16) b Σ L (cid:17) respectively denote the area of ( ˆΣ L , ˆ h ) and (Σ L , h ). Along the same lines,let us compute the lightcone comparison functional E Σ b Σ [ ψ − L ] associated with the inverse diﬀeo-morphism ψ − L : (Σ L , h ) −→ ( ˆΣ L , ˆ h ) and the corresponding conformal factor Φ Σ b Σ ∈ C ∞ (Σ , R )-(see (42)),(56) E Σ b Σ [ ψ − L ] := Z Σ L (cid:0) Φ Σ b Σ − (cid:1) dµ h . We have(57) E Σ b Σ [ ψ − L ] := A (cid:16)b Σ L (cid:17) + A (Σ L ) − Z Σ L Φ Σ b Σ dµ h . Since Z Σ L Φ Σ b Σ dµ h = Z Σ L s dµ ( ψ − ) ∗ ˆ h dµ h dµ h = Z Σ L s dµ ( ψ − ) ∗ ˆ h dµ h dµ h dµ ( ψ − ) ∗ ˆ h dµ ( ψ − ) ∗ ˆ h (58) = Z Σ L s dµ h dµ ( ψ − ) ∗ ˆ h ( ψ − ) ∗ dµ ˆ h . On the other hand, if we take the pull back, under the action of ψ − L : (Σ L , h ) −→ ( b Σ L , ˆ h ), of therelation Φ b ΣΣ dµ ˆ h = ψ ∗ L dµ h , (see (39)), we have(59) (cid:0) ψ − (cid:1) ∗ (cid:16) Φ b ΣΣ dµ ˆ h (cid:17) = (cid:0) ψ − L (cid:1) ∗ ( ψ ∗ L dµ h ) = ⇒ Φ b ΣΣ (cid:0) ψ − L ( x ) (cid:1) (cid:16)(cid:0) ψ − L (cid:1) ∗ dµ ˆ h (cid:17) ( x ) = dµ h ( x ) , from which we get(60) Φ b ΣΣ (cid:0) ψ − L ( x ) (cid:1) = dµ h ( x ) (cid:16)(cid:0) ψ − L (cid:1) ∗ dµ ˆ h (cid:17) ( x ) . Hence, we can rewrite (58) as Z Σ L Φ Σ b Σ dµ h = Z Σ L s dµ h dµ ( ψ − ) ∗ ˆ h ( ψ − L ) ∗ dµ ˆ h = Z Σ L Φ b ΣΣ (cid:0) ψ − L (cid:1) ( ψ − L ) ∗ dµ ˆ h = Z Σ L ( ψ − L ) ∗ (cid:0) Φ b ΣΣ dµ ˆ h (cid:1) = Z ψ − (Σ L ) Φ b ΣΣ dµ ˆ h (61) = Z ˆΣ L Φ b ΣΣ dµ ˆ h , and E Σ b Σ [ ψ − ] := A (cid:16) b Σ L (cid:17) + A (Σ L ) − Z Σ L Φ Σ b Σ dµ h (62) = A (cid:16) b Σ L (cid:17) + A (Σ L ) − Z ˆΣ L Φ b ΣΣ dµ ˆ h = E b ΣΣ [ ψ ] . Hence, the comparison functional is symmetric.In order to prove the triangular inequality (50) let us consider the sum(63) E b ΣΣ [ ψ L ] + E Σ e Σ [ σ L ] = Z ˆΣ L (Φ b ΣΣ − dµ ˆ h + Z Σ L (Φ Σ e Σ − dµ h . From the relation (59) we have dµ h = Φ b ΣΣ (cid:0) ψ − L (cid:1) (cid:0) ψ − L (cid:1) ∗ dµ ˆ h , and we can write(64) Z Σ L (Φ Σ e Σ − dµ h = Z b Σ L (Φ Σ e Σ − Φ b ΣΣ dµ ˆ h . Hence, E b ΣΣ [ ψ L ] + E Σ e Σ [ σ L ] = Z b Σ L h (Φ b ΣΣ − + (Φ Σ e Σ − Φ b ΣΣ i dµ ˆ h (65) ≥ Z b Σ L (cid:2) (Φ b ΣΣ −

1) + (Φ Σ e Σ − b ΣΣ (cid:3) dµ ˆ h = Z b Σ L (Φ b ΣΣ Φ Σ e Σ − dµ ˆ h = E b Σ e Σ [( σ L ◦ ψ L )] , where we have exploited the relation(66) Φ Σ e Σ ( ψ L ) Φ b ΣΣ = Φ b Σ e Σ , which follows from observing that the positive functions Φ b Σ e Σ ∈ C ∞ ( b Σ , R ) and Φ Σ e Σ ∈ C ∞ (Σ , R )are such that(67) Φ b Σ e Σ ˆ h = ( σ L ◦ ψ L ) ∗ ˜ h = ψ ∗ L (cid:16) Φ e Σ h (cid:17) = Φ e Σ ( ψ L ) ψ ∗ L h = Φ e Σ ( ψ L ) Φ b ΣΣ ˆ h , COMPARISON THEOREM FOR COSMOLOGICAL LIGHTCONES 13 where we have set Φ Σ e Σ ( ψ L ) = ψ ∗ L Φ Σ e Σ := Φ Σ e Σ ◦ ψ L .From (54) and the Schwarz inequality(68) Z b Σ L Φ b ΣΣ dµ ˆ h ≤ (cid:18)Z b Σ L Φ b ΣΣ dµ ˆ h (cid:19) / (cid:18)Z b Σ L dµ ˆ h (cid:19) / = r A (cid:16) b Σ L (cid:17) A (Σ L ) , we get the lower bound E b ΣΣ [ ψ L , ζ ] = A (Σ L ) + A (cid:16)b Σ L (cid:17) − Z b Σ L Φ b ΣΣ dµ ˆ h (69) ≥ A (Σ L ) + A (cid:16)b Σ L (cid:17) − r A (cid:16)b Σ L (cid:17) A (Σ L )= (cid:18)q A ( b Σ L ) − p A (Σ L ) (cid:19) , where we have exploited (55). The upper bound in (52) easily follows from (refenfunct1)(70) E b ΣΣ [ ψ L , ζ ] := Z b Σ L (Φ b ΣΣ − dµ ˆ h ≤ Z b Σ L Φ b ΣΣ dµ ˆ h + Z b Σ L dµ ˆ h = A ( b Σ L ) + A (Σ L ) . The proof of the last part of the lemma follows observing that the integrand in R ˆΣ L (Φ b ΣΣ − dµ ˆ h is non-negative and, as long as Φ b ΣΣ is a smooth function on ( ˆΣ L , ˆ h ), the condition(71) E b ΣΣ [ ψ L ] = Z ˆΣ L (Φ b ΣΣ − dµ ˆ h = 0implies Φ b ΣΣ = 1, hence the isometry between ( ˆΣ L , ˆ h ) and (Σ L , h ). (cid:3) A scale-dependent distance functional

The properties of the comparison functional E b ΣΣ [ ψ L , ζ ] indicate that we can associate with ita distance functional d L hb Σ L , Σ L i . To put the characterization of this distance in perspective, letus recall that a fractional linear transformations ζ ∈ P SL (2 , C ) is fully determined if, given threedistinct points of b Σ ≃ S , we specify their images. We exploited this in Proposition 5, where weassigned three distinct points q ( i ) , i = 1 , , L , h ), and we ﬁxedthe action of P SL (2 , C ) by choosing that particular automorphism ζ ∈ P SL (2 , C ) that identiﬁesthe inverse images { ψ − ( q i ) } ∈ b Σ L with three chosen points ˆ q ( i ) , i = 1 , , b Σ L , ˆ h ). Since E b ΣΣ [ ψ L , ζ ] is not conformally invariant, the particular choiceof the automorphism ζ ∈ P SL (2 , C ), or which is the same, the particular choice of the points q ( i ) , i = 1 , , L , h ), aﬀects E b ΣΣ [ ψ L , ζ ], hence it is natural to inquire if there is a choice of theautomorphism ζ that minimizes E b ΣΣ [ ψ L , ζ ]. Given the reference points ˆ q ( i ) , i = 1 , , b Σ L , ˆ h ),this optimal choice for ζ ∈ P SL (2 , C ), say ζ = ζ , will induce the proper selection of the alignmentpoints q ( i ) , i = 1 , , L , h ) by setting q ( i ) := ψ L (cid:0) ζ (ˆ q ( i ) ) (cid:1) . Inorder to characterize this optimal choice, we need to minimize E b ΣΣ [ ψ L , ζ ] over a suitable class offunctions, and a natural strategy, according to these remarks, is to keep ﬁxed the diﬀeomorphism ψ L as well the points ˆ q ( i ) , i = 1 , , b Σ L , ˆ h ), and let vary in a controlled way theautomorphism ζ ∈ P SL (2 , C ), so as to minimize E b ΣΣ [ ψ L , ζ ]. We also need a slightly more generalsetting that will allow us to deal with celestial spheres (Σ L , h ) on a lightcone region C − L ( p, g ) where The reason for keeping ψ L ﬁxed is directly related to the fact that ψ L it is constructed by using the null geodesicsalong the past lightcones by exploiting the exponential maps (see (24) ), and this is the way actual cosmologicalobservations are carried out. caustics develop (hence, relaxing in a controlled way the regularity of ψ L allowing for exponentialmappings which are no longer injective). In other words, we need to extend ψ L ◦ ζ : b Σ L −→ Σ L tobe a member of a more general space of maps which allow for the low regularity setting associatedwith the possible presence of (isolated) caustics. We start with a more precise characterizationof the Sobolev space of maps W , ( b Σ , Σ), mentioned on passing in commenting Deﬁnition 6. Todeﬁne W , ( b Σ , Σ) we follow a standard approach in harmonic map theory and use Nash embeddingtheorem [14], [25], by considering the compact surface (Σ L , h ) isometrically embedded into someEuclidean space E m := ( R m , δ ) for m suﬃciently large. In particular, if J : (Σ L , h ) ֒ → E m is anysuch an embedding then we deﬁne the Sobolev space of maps(72) W , J ) ( b Σ , Σ) := { ϕ ∈ W , ( b Σ , R m ) | ϕ ( ˆΣ L ) ⊂ J (Σ L ) } , where W , ( b Σ , R m ) is the Hilbert space of square summable ϕ : b Σ → R m , with (ﬁrst) distributionalderivatives in L ( b Σ , R m ), endowed with the norm(73) k ϕ k W , := Z b Σ (cid:18) ϕ a ( x ) ϕ b ( x ) δ ab + ˆ h µν ( x ) ∂ϕ a ( x ) ∂x µ ∂ϕ b ( x ) ∂x ν δ ab (cid:19) dµ ˆ h , where, for ϕ ( x ) ∈ J (Σ L ) ⊂ R m , a, b = 1 , . . . , m label coordinates in ( R m , δ ), and dµ ˆ h denotesthe Riemannian measure on ( b Σ , ˆ h ). This characterization is independent of J since Σ L is compact,and in that case for any two isometric embeddings J and J , the corresponding spaces of maps W , J ) ( b Σ , Σ) and W , J ) ( b Σ , Σ) are homeomorphic [16]. For this reason, in what follows we shallsimply write W , ( b Σ , Σ). The set of maps W , ( b Σ , Σ) provides the minimal regularity allowingfor the characterization of the energy functional E b ΣΣ [ ψ L , ζ ]. Maps of class W , ( b Σ , Σ) are notnecessarily continuous and, even if the space of smooth maps C ∞ ( b Σ , Σ) is dense [23] in W , ( b Σ , Σ),to carry out explicit computations, in what follows we must further require that ϕ ∈ W , ( b Σ , Σ) islocalizable (cf. [19], Sect. 8.4) and keeps track both of the given ψ L and of the three alignment pointsˆ q ( i ) , i = 1 , , b Σ L , ˆ h ). The only freedom remaining is in the conformal group automorphisms ζ ∈ P SL (2 , C ) acting on ( b Σ L , ˆ h ), and in terms of which we need to control that the images of thereference points ˆ q ( i ) ∈ b Σ L stay separated and do not concentrate in a small neighborhood of Σ L .Hence, and for a ﬁxed ψ L ∈ W , ( b Σ , Σ), we deﬁne the space of maps over which E b ΣΣ [ ψ L , ζ ] isminimized according to the following deﬁnition. Deﬁnition.

Let us assume that ψ L : ( b Σ L , ˆ h ) −→ (Σ L , h ) is, for almost all points of b Σ L , a W , ( b Σ , Σ) diﬀeomorphism between the two celestial spheres , and let ˆ q ( i ) , i = 1 , , , be the threedistinguished points on ( b Σ L , ˆ h ) , characterizing the three reference past-directed null directions on C − L ( p, ˆ g ) introduced in Proposition 5. A map ϕ := ψ L ◦ ζ ∈ W , ( b Σ , Σ) , with ζ ∈ P SL (2 , C ) ,is said to be ε -localizable if: (i) For every ˆ q ∈ b Σ L there exists a metric disks D (ˆ q, δ ) := { y ∈ b Σ L | d γ (ˆ q, y ) ≤ δ } ⊂ b Σ L , of radius δ > , with smooth boundary ∂ D , and containing at most oneof the three points ˆ q ( i ) , i = 1 , , ; and (ii) Corresponding to each of these disks, there exists a metricdisk B ( q, ε ) = ϕ ( D (ˆ q, δ )) := { x ∈ Σ L | d h ( q, x ) ≤ ε } ⊂ (Σ L , h ) centered at ϕ ( x ) := q ∈ M , ofradius r > such that ϕ ( D ( x , δ )) ⊂ B ( q, r ) , with ϕ ( ∂D ) ⊂ B ( q, ε ) . Under such assumptions,we consider, for ﬁxed ψ L , the space of maps Map ψ ( b Σ L , Σ L ) := n ϕ := ψ L ◦ ζ ∈ W , ( b Σ , Σ) ∩ C ( b Σ , Σ) , (74) ζ ∈ P SL (2 , C ) | ϕ := ψ L ◦ ζ is ε − localizable and Φ b ΣΣ (ˆ q ) ≥ (cid:9) , Hence, we are assuming that there can be a ﬁnite collections of points for which the exponential mapping exp p along C − L ( p, g ) may be not injective. COMPARISON THEOREM FOR COSMOLOGICAL LIGHTCONES 15 where the non-negativity requirement Φ b ΣΣ (ˆ q ) ≥ is assumed to hold for almost all points of b Σ L . As in harmonic map theory, there is a further delicate issue related to the fact that maps inMap ψ ( b Σ L , Σ L ) are partitioned in diﬀerent homotopy classes. Recall that every map from S intoitself is characterized by the degree of the map [21], measuring how many times the map wraps S around itself. In particular through the action of a sequence of conformal dilations ∈ P SL (2 , C )of the form ζ ζ ′ ( k ) := ω ( k ) ζ where ω ( k ) ∈ R we can easily construct sequences of mappings { ϕ ( k ) } that tend to focus all points of a disk D in S toward a given point (say the north pole).Physically this corresponds to the eﬀect of acting with a sequence of Lorentz boosts (with rapiditylog ω ( k ) ) on an observer P who is looking at the giving region D of the celestial sphere. From thepoint of view of P , one can also interpret this as a focusing of the past null geodesics eventuallyleading to the formation of a caustic point. Regardless of the physical interpretation, in harmonicmap theory this sort of behavior leads to the phenomenon of bubble convergence when discussingthe minimization problem for the harmonic map energy functional [22]. In our case, we can exploitthe analogous of bubbling convergence to our advantage in order to extend our analysis to the casewhen caustics are present. We can model the generation of caustic points as the results of a focusingmapping, such as the { ϕ ( k ) } described above, converging in Map ψ ( b Σ L , Σ L ), to a deg ϕ = h > h − L , h )). We postpone the details of such analysis to apaper in preparation [4], and limit here our analysis to show that E b ΣΣ [ ψ L , ζ ] can be minimizedover diﬀeomorphisms in Map ψ ( b Σ L , Σ L ). Theorem 9. (The lightcone comparison distance at scale L ). The functional E b ΣΣ [ ψ L , ζ ] achievesa minimum on Map ψ ( b Σ L , Σ L ) , and (75) d L hb Σ L , Σ L i := inf ψ L ◦ ζ ∈ Map ψ ( b Σ L , Σ L ) E b ΣΣ [ ψ L , ζ ] deﬁnes a scale-dependent distance between the celestial spheres ( b Σ L , ˆ h ) and (Σ L , h ) on the lightconeregions C − L ( p, ˆ g ) and C − L ( p, g ) .Proof. To simplify notation let us set ϕ := ψ L ◦ ζ . Since we have the upper bound E b ΣΣ [ ϕ ] ≤ C L := A ( b Σ L ) + A (Σ L ), (see (52)), we can limit our analysis to the subset of maps(76) Map ψ,C L ( b Σ L , Σ L ) := n ϕ ∈ Map ψ ( b Σ L , Σ L ) | s . t . E b ΣΣ [ ϕ ] ≤ C L o . According to Deﬁnition 4, the space of maps Map ε,C L ( b Σ , Σ) is equicontinuous, namely, for anypoint ˆ q ∈ ˆΣ we can choose the disk D (ˆ q , δ ) (for notation see Deﬁnition 4) in such a way that fora given ε > ϕ (ˆ q ) and ϕ (ˆ q ) are such that d h ( ϕ (ˆ q ) , ϕ (ˆ q )) < ε , for all ˆ q ∈ D (ˆ q , δ ) and all ϕ ∈ Map ψ,C L ( b Σ L , Σ L ). Hence, a minimizing sequence { ϕ ( k ) } k ∈ N ∈ Map ε,C L ( b Σ , Σ) for E b ΣΣ [ ϕ ] isequicontinuous. By selecting a subsequence we may assume that { ϕ ( k ) } converges to a continuousmap ϕ which is also the weak limit of { ϕ ( k ) } in W , ( b Σ , Σ), since this latter is a weakly compactspace of maps.Since(77) E b ΣΣ [ ϕ ] = A (Σ L ) + A (cid:16)b Σ L (cid:17) − Z b Σ L Φ b ΣΣ ( ϕ ) dµ ˆ h , a minimizing (sub)sequence { ϕ ( k ) } for E b ΣΣ [ ϕ ] corresponds to a maximize sequence for the functional R b Σ L Φ b ΣΣ ( ϕ ) dµ ˆ h . Hence, given δ >

0, there exists k such that for all k ≥ k , we have(78) Z b Σ L Φ b ΣΣ ( ϕ ) dµ ˆ h ≥ Z b Σ L Φ b ΣΣ ( ϕ ( k ) ) dµ ˆ h − δ, , along a minimizing sequence { ϕ ( k ) } −→ ϕ for the functional E b ΣΣ [ ϕ ], and where Φ b ΣΣ ( ϕ ( k ) ) isnon-negative for almost all points of b Σ L . By adding and subtracting R b Σ L Φ b ΣΣ ( ϕ ( k ) ) dµ ˆ h to (77),(evaluated for ϕ ), and by taking into account (78), we get(79) E b ΣΣ [ ϕ ] = E b ΣΣ [ ϕ ( k ) ] − Z b Σ L (cid:0) Φ b ΣΣ ( ϕ ) − Φ b ΣΣ ( ϕ ( k ) ) (cid:1) dµ ˆ h ≤ E b ΣΣ [ ϕ ( k ) ] + 2 δ , for all k ≥ k . Since the choice of δ > E b ΣΣ [ ϕ ] islower semicontinuous, i.e. ,(80) E b ΣΣ [ ϕ ] ≤ lim k inf E b ΣΣ [ ϕ k ]for all ϕ ∈ Map ε,C L ( b Σ , Σ) with ϕ k weakly converging, in the above sense, to ϕ . Hence, { ϕ ( k ) } −→ ϕ minimizes E b ΣΣ [ ϕ ] in the space of maps Map ε ( b Σ , Σ), as stated.If we set(81) d L hb Σ L , Σ L i := inf ψ L ◦ ζ ∈ Map ε ( b Σ , Σ) E b ΣΣ [ ψ L , ζ ]then as a consequence of the properties of the functional E b ΣΣ [ ϕ ], described in Lemma 8, we havethat d L hb Σ L , Σ L i provides a scale dependent distance function between the physical celestial sphere(Σ L , h ) and the reference FLRW celestial sphere ( b Σ L , ˆ h ), as the scale L varies. In particular, withthe notation of Lemma 8 we have (i) Non-negativity d L hb Σ L , Σ L i ≥ (ii) d L hb Σ L , Σ L i = 0 iﬀ( b Σ L , ˆ h ) and (Σ L , h ) are isometric; (iii) Symmetry d L hb Σ L , Σ L i = d L h Σ L , b Σ L i ; (iv) Triangularinequality d L hb Σ L , e Σ L i ≤ d L h ˆΣ L , Σ L i + d L h Σ L , e Σ L i . (cid:3) The physical meaning of d L hb Σ L , Σ L i The distance functional d L hb Σ L , Σ L i is a geometric quantity that we can associate with theobserver who wish to describe with a Friedmannian bias the cosmological region where inhomo-geneities may dominate. To appreciate what this role implies, let us brieﬂy discuss the physicalinterpretation of d L hb Σ L , Σ L i , when we probe the light cone regions C − L ( p, ˆ g ) and C − L ( p, g ) over asuﬃciently small length scale L . If ¯ ϕ denotes the minimizing map characterized in Theorem 9, wecan write(82) d L hb Σ L , Σ L i = E b ΣΣ [ ¯ ϕ ] := A (cid:16)b Σ L (cid:17) + A (Σ L ) − Z b Σ L Φ b ΣΣ dµ ˆ h . To simplify matters, we assume that at the given length scale L the corresponding region C − L ( p, g )is caustic free, and parametrize Φ Σ b Σ ( ¯ ϕ ) as(83) Φ Σ b Σ ( ¯ ϕ ) = 1 + F ( ¯ ϕ ) , where F ( ¯ ϕ ) is a smooth function (not necessarily positive) which, by discarding the traceless lensingshear, may be thought of as describing the (small) local isotropic focusing distortion of the imagesof the astrophysical sources on (Σ , h ) due to gravitational lensing, (see the expression (12) of thesky-mapping metric h ). Under these assumptions, we can write (82) as(84) d L hb Σ L , Σ L i = A (Σ L ) − A ( b Σ L ) − Z b Σ L F ( ¯ ϕ ) dµ ˆ h . This expression can be further specialized if we exploit the asymptotic expressions of the area A (cid:16)b Σ L (cid:17) and A (Σ L ) of the two surfaces ( b Σ L , ˆ h ), (Σ L , h ) on the corresponding lightcones C − L ( p, ˆ g ) COMPARISON THEOREM FOR COSMOLOGICAL LIGHTCONES 17 and C − L ( p, g ). These asymptotic expressions can be obtained if we consider the associated causalpast regions J − L ( p, ˆ g ) and J − L ( p, g ) suﬃciently near the (common) observation point p , in particularwhen the length scale L we are probing is small with respect to the ”cosmological” curvaturescale. Under such assumption, there is a unique maximal 3-dimensional region V L ( p ), embedded in J − L ( p, g ), having the surface (Σ L , h ) as its boundary. This surface intersects the world line γ ( τ ) ofthe observer p at the point q = γ ( τ = − L ) deﬁned by the given length scale L . For the referenceFLRW the analogous set up is associated to the constant-time slicing of the FLRW spacetime( M, ˆ g ) considered. The corresponding 3-dimensional region b V L ( p ), embedded in J − L ( p, ˆ g ), hasthe surface ( b Σ L , ˆ h ) as its boundary. The FLRW observer ˆ γ (ˆ τ ) will intersect b V L ( p ) at the pointˆ q = ˆ γ (ˆ τ = − L ). By introducing geodesic normal coordinates { X i } in J − L ( p, g ) and { Y k } in J − L ( p, ˆ g ), respectively based at the point q and ˆ q , we can pull back the metric tensors g and ˆ g to T q M and T ˆ q M , and obtain the classical normal coordinate development of the metrics g and ˆ g valid in a suﬃciently small convex neighborhood of q and ˆ q . Explicitly, for the (more relevant caseof the) metric g , we have (see e. g. Lemma 3.4 (p. 210) of [24] or [21])(( exp q ) ∗ g ) ef = η ef −

13 R eabf | q X a X b − ∇ c R eabf | q X a X b X c + (cid:18) − ∇ c ∇ d R eabf + 245 R eabm R mfcd (cid:19) q X a X b X c X d + . . . , where R abcd is the Riemann tensor of the metric g (evaluated at the point q ). The induced expansionin the pulled-back Lorentzian measure (cid:0) ( exp s ( η ) ) ∗ dµ g (cid:1) and a rather delicate analysis (related to thespacetime geometry of causal diamonds ) described (at various levels of rigor) in [2], [11], [12], [20])provides, to leading order in L , the following expressions for the area of (Σ L , h ) and ( b Σ L , ˆ h ),(85) A (Σ L ) = π L (cid:18) − L R( q ) + . . . (cid:19) , and(86) A (cid:16)b Σ L (cid:17) = π L (cid:18) − L b R(ˆ q ) + . . . (cid:19) , Introducing these expressions in (84) we get(87) d L hb Σ L , Σ L i = π L (cid:16)b R(ˆ q ) − R( q ) (cid:17) − Z b Σ L F ( ¯ ϕ ) dµ ˆ h + . . . . We can rewrite this equivalently as(88) b R(ˆ q ) = R( q ) + 72 π d L hb Σ L , Σ L i L + 144 πL Z b Σ L F ( ¯ ϕ ) dµ ˆ h + . . . . The asymptotics (87), (but also the very characterization of the distance functional d L hb Σ L , Σ L i ),shows clearly that the lightcone comparison functional E b ΣΣ and the associated distance d L hb Σ L , Σ L i provide a generalization of the lightcone theorem [5] proved by Y. Choquet-Bruhat, P. T. Chrusciel,and J. M. Martin-Garcia. The normal coordinates asymptotics (88) is also interesting since itdirectly connects d L hb Σ L , Σ L i to the ﬂuctuations in the spacetime scalar curvature: if we decideto keep on in modeling with a FLRW solution a cosmological spacetime, homogeneous on largescale but highly inhomogeneous at smaller scale, then the associated scalar curvature b R(ˆ q ) canbe approximately identiﬁed with the physical scalar curvature R(ˆ q ), with a rigorous level of scaledependence precision, only if we take into account the contribution provided by the lightcone distance functional d L hb Σ L , Σ L i (and by the average of the local focusing term F ( ¯ ϕ )). This can be ofsome interest in addressing backreaction problems in cosmology (see e.g. , [3]). Finally, d L hb Σ L , Σ L i can be also of some use in providing a rigorous way of addressing some aspect of the best-ﬁttingproblem in cosmology (see [8] and [9]), roughly speaking, the strategy is to vary the family of modelspacetimes ( M, ˆ g ) (for instance, the family of FLRW solutions, or the larger family of homogeneousspacetimes) in such a way to minimize (over the relevant interval of length scales L ) the distancefunctional d L hb Σ L , Σ L i between the physical celestial spheres ( b Σ L , ˆ h ) and the family of referencecelestial spheres ( b Σ L , ˆ h ) associated with the model spacetimes ( M, ˆ g ) adopted.6. Acknowledgments

We wish to thank Thomas Buchert and Dennis Stock for valuable discussions. This work hasbeen partially supported by the European project ERC-2016-ADG advanced grant ”arthUS”: ad-vances in the research on theories of the dark Universe . References [1] M. S. Berger,

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University of Pavia (GNFM and INFN)

Italian National Group of Mathematical Physics, and INFN Pavia Section

Email address : [email protected] (Corresponding author) (Department of Physics, University of Pavia) University of Pavia (GNFM and INFN)

Italian National Group of Mathematical Physics, and INFN Pavia Section

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