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.

令（M，  g）表示描述宇宙演化的宇宙时空，该宇宙在大尺度上是各向同性和同质的，而在小尺度上是高度不均匀的。我们考虑两个过去的光锥，第一个\（{{\ mathcal {C}} _​​ {L} ^ {-}}（p，g）\）与物理观察者\（p \ in \，M \）描述了（M，  g）在长度尺度L上的实际物理时空几何，而第二个\（{\ mathcal {C} _ {L} ^ {-}}（p，\ hat {g} ）\）与观察者p的理想化版本相关联，尽管给定尺度L存在局部不均匀性，希望用Friedmann–Lemaitre–Robertson–Walker时空家族的成员（（M，\ hat {g}）\）来建模（M，  g）。在这样的框架中，我们讨论了许多数学结果，可以对两个光锥\（{\ mathcal {C} _ {L} ^ {-}}（p，g）\）和\（{ mathcal {C} _ {L} ^ {-}}（p，\ hat {g}）\）。特别是，我们引入了由谐波类型能量定义的与尺度相关的（L）光照比较函数，该函数与物理\（{\ mathcal {C} _ {L} ^ {-}}（ p，g）\）和FLRW参考光锥\（{\数学{C} _ {L} ^ {-}}（p，\ hat {g}）\）。该功能具有许多显着的特性，特别是当在给定的长度比例下，相应的圆锥形表面区域（天球）是等距的时，它就消失了。我们将详细讨论其变分分析，并证明存在一个最小值，该最小值表征了两个光锥之间自然比例相关的距离函数。我们还指出了如何将结果扩展到在物理过去的光锥\（{\ mathcal {C} _ {L} ^ {-}}（p，g）\）上发展焦散的情况。最后，通过利用因果菱形理论，我们展示了距离函数如何与两个光锥因果关系中的时空标量曲率（与标度L的前导顺序）相关，并简要说明了其可能的应用。