Boundary Terms, Finite FLRW Spacetimes, and the Emergence of the Cosmological Constant Λ

1. Introduction

The origin of the cosmological constant $\mathrm{\Lambda }$ remains one of the deepest open problems in gravitation and cosmology. Observations indicate that the present Universe is undergoing accelerated expansion consistent with a small but non-zero effective cosmological constant, corresponding to an energy density of order ${\rho }_{\mathrm{\Lambda }}\sim {10}^{-29}\mathrm{g}{\mathrm{cm}}^{-3}$. Within the standard cosmological model, this is usually described by Einstein’s field equations with a cosmological constant $\mathrm{\Lambda }$:

$$G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=8\pi G{T}_{\mu \nu }.$$

(1)

Although this phenomenological description agrees remarkably well with observations, the physical origin of $\mathrm{\Lambda }$ remains unclear. Estimates of vacuum energy from quantum field theory exceed the observed value by many orders of magnitude, giving rise to the well-known cosmological constant problem [1,2]. From the perspective of General Relativity, the cosmological constant enters naturally through the Einstein-Hilbert (EH) action:

$$S_{\mathrm{EH}}={\displaystyle \frac{1}{16\pi G}}{\int }_{\mathcal{M}}{d}^{4}x\sqrt{-g}(R-2\mathrm{\Lambda }),$$

(2)

where $\mathrm{\Lambda }$ can be viewed either as a fundamental constant of the action or as an effective contribution arising from vacuum energy, modified gravity, or other microscopic physics.

An often underemphasized aspect of the EH action is that its variational principle depends crucially on boundary terms and on the boundary data held fixed during the variation [3]. In standard textbook derivations, varying the EH action generates a boundary contribution involving variations of the boundary metric and its normal derivatives. To obtain a well-defined variational principle under Dirichlet boundary condition, one usually fixes the induced metric: $\delta h_{ab}=0$, where $h_{ab}$ is the metric induced on the boundary hypersurface. Under such conditions, one could artificially add the Gibbons–Hawking–York (GHY) boundary term to cancel the remaining surface variation [4,5,6,7]. However, the key issue is not whether one adds the GHY term or not. The more fundamental question is: what boundary data are physically fixed?

This distinction becomes crucial when the boundary is not a fixed outer hypersurface but a junction hypersurface $\Sigma$ connecting two spacetime regions. In such problems, one is not imposing ordinary Dirichlet conditions at a finite boundary or at spatial infinity. Instead, one requires that the total action of the joined spacetime remains stationary:

$$\delta S_{\Sigma }=0.$$

(3)

This is a stronger and more specific requirement than the standard fixed-boundary variational problem. In this setting, the natural condition is not simply $\delta h_{ab}=0$, but rather the continuity conditions associated with the junction, involving both the induced metric and the extrinsic curvature. Junction problems therefore belong to a fundamentally different class from the standard Dirichlet boundary problem.

In this work, we study such a junction problem for a finite Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime matched to an exterior vacuum. This setup defines a finite gravitating FLRW region whose boundary is determined dynamically by the global variational principle rather than something that is imposed by hand.

The Openheimer–Snyder solution with pressureless matter [8] remains one of the few exact analytical models of gravitational collapse in General Relativity. Its simplicity relies crucially on the absence of pressure, which ensures that the collapsing matter follows geodesic motion and allows a comoving FLRW interior to be matched exactly to an exterior Schwarzschild spacetime. Once pressure is introduced, the fluid no longer follows geodesics, the matching problem becomes considerably more difficult, and exact analytical FLRW interior solutions are generally believed to be unavailable [9,10,11].

One of the main results of the present work is to show that this conclusion is too restrictive. We show here that the boundary equation admits two physically distinct branches. The first is a trivial comoving branch with vanishing peculiar velocity, $v=0$, which reproduces the standard Oppenheimer–Snyder dust solution and requires the cosmological constant to vanish on both sides of the boundary. This branch is therefore of limited relevance for cosmology, since the observed Universe contains $\mathrm{\Lambda }$, radiation, and pressure components. The physically relevant solution is a second, non-comoving branch satisfying $0<v<1$, which exists whenever pressure or $\mathrm{\Lambda }$ components are present. This branch evolves toward a universal attractor independent of spatial curvature and valid for both expanding and collapsing solutions.

A related approach to the cosmological constant as a boundary condition has been explored in earlier work [12,13]. However, these previous analyses relied on limiting cases of the junction dynamics and on partial use of the full variational principle. In the present work, we derive the full moving-junction solution $v=v\left(a\right)$ for arbitrary spatial curvature and clarify how the finite-Universe realisation emerges as an exact consequence of the variational principle.

A different interpretation of the cosmological constant as a boundary condition was previously proposed in [14,15], where $\mathrm{\Lambda }$ was related to a zero-action condition imposed on a finite causal domain. In that earlier approach, the asymptotic recovery of empty space was effectively implemented by requiring the FLRW interior to satisfy an asymptotic condition equivalent to $\ddot{a}\to 0$, leading to the relation $\mathrm{\Lambda }\simeq 4\pi G\langle \rho +3p\rangle$. The present work clarifies and improves that interpretation in several important ways. First, we show that asymptotic flatness need not be enforced through the asymptotic behaviour of the FLRW interior itself. Instead, the FLRW region may remain finite and be matched consistently to an exterior Schwarzschild (or more generally Schwarzschild–deSitter/Anti-deSitter) spacetime. Empty Minkowski space is then recovered in the exterior region, not as a limit of the interior FLRW solution. Second, the cosmological constant is derived here from the exact variational principle and junction conditions, without invoking volume averages or heuristic zero-action arguments. Thus, the earlier boundary interpretation is not discarded but generalized: the present formulation provides a mathematically precise realization of the same underlying idea, while removing assumptions that were specific to the earlier approximate treatment.

This investigation further advances in fixing the physical understanding of the arbitrary cosmological constant that can always be added to the EH action. The variational boundary value problem in our framework brings this cosmological constant to a specific value associated with the inverse square of the Schwarzschild radius ($r_s=2GM$) below which a finite FLRW Universe evolves with a non-comoving boundary. The larger the asymptotic boundary (i.e., $r_s$) of the finite Universe, the smaller the value of the cosmological constant ($\mathrm{\Lambda }={\displaystyle \frac{3}{r_s^2}}$) a comoving FLRW observer witnesses. We show that the non-comoving boundary of the finite FLRW cloud leads to the understanding of current de Sitter-like accelerated expansion. Thus, we not only explain the smallness of the cosmological constant but also unify the Schwarzschild horizon of the superior observer with the de Sitter horizon of the comoving observer of the FLRW Universe. We also address other conceptual issues associated with the cosmological constant problem under the boundary condition of the variational principle. This brings a new revelation that we can add any number of cosmological constants (such as a combination of standard arbitrary cosmological constant, vacuum energy from matter fields of the standard (particle physics) model, or beyond, or even a negative cosmological constant) to the EH action, and the boundary condition fixes the sum to be inversely proportional to the Schwarzschild radius square. This significantly revises our understanding of the cosmological constant.

Throughout the paper we follow the metric signature $\left(-+++\right)$. We follow Greek indexes $(\mu ,\nu )$ and $x_{\mu }$ coordinates operate within 4D manifold $\mathcal{M}$, while Latin indexes $(a,b)$ and $y_a$ coordinates operate within the 3D hypersurface $\Sigma$ or submanifold $\partial \mathcal{M}$.

2. Einstein–Hilbert Action and the Variational Principle

The fundamental laws of physics are often obtained by requiring an action functional to be stationary under infinitesimal variations of the dynamical fields. For gravity, the configuration is the geometry itself — the metric $g_{\mu \nu }$. The Einstein–Hilbert action encodes the curvature of spacetime governed by the matter and energy content.

Given $\mathcal{M}$, the four-dimensional manifold or bulk, the Einstein–Hilbert action minimally coupled to reads:

$$S_{\mathrm{EH}}\left[\mathcal{M}\right]={\int }_{\mathcal{M}}d\mathcal{M}\left[{\displaystyle \frac{R}{16\pi G}}+{\mathcal{L}}_m\right],$$

(4)

where ${\mathcal{L}}_m$ is the matter Lagrangian and $d\mathcal{M}=\sqrt{-g}{d}^{4}x$ is the invariant volume element. Applying the variational principle to the EH action Eq.4 $\delta S_{\mathrm{EH}}=0$, gives

$$R_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }R=8\pi G{T}_{\mu \nu }\equiv -{\displaystyle \frac{16\pi G}{\sqrt{-g}}}{\displaystyle \frac{\delta \left(\sqrt{-g}{\mathcal{L}}_m\right)}{\delta g^{\mu \nu }}},$$

(5)

which are the well-known Einstein’s field equations. But the standard derivation of Eq.5 ignores or implicitly assumes the boundary variations of the metric to be zero by hand, which we will show shortly in detail.

The variational principle highlights something often overlooked: even though the field equations are local relations, the action itself is global. It integrates curvature and matter over the entire spacetime region under consideration. This global perspective becomes essential when comparing different possible histories of the Universe or when extending gravity into the quantum domain.

3. Boundary Terms

One subtlety arises immediately when varying the Einstein–Hilbert action: total derivatives appear and generate boundary terms. If these terms are ignored, the variational principle becomes incomplete, and the derivation of Einstein’s equations is not mathematically well posed. In some situations the boundary contributions vanish automatically, for example, when fields fall off sufficiently rapidly at infinity. However, this is no longer true when spacetime has nontrivial boundaries or horizons.

This point is the central theme of this work: gravity is not only about local curvature, but also about global structure. Boundaries, whether located at infinity, at black-hole horizons, or at the edge of a finite cosmological region, play a key role in understanding the spacetime geometry. These terms influence the global properties of spacetime and, in cosmology, may even contribute to the effective value of the cosmological constant itself.

All of this understanding emerges from the variation of the Einstein–Hilbert action Eq.4, which contains a "total derivative" term [3]:

$$\delta S_{\mathrm{EH}}={\int }_{\mathcal{M}}{\displaystyle \frac{d\mathcal{M}}{16\pi G}}\left[\left(G_{\mu \nu }-8\pi G{T}_{\mu \nu }\right)\delta g^{\mu \nu }+{\nabla }_{\rho }{V}^{\rho }\right].$$

where $V^{\rho }\equiv g^{\mu \nu }\delta {\mathrm{\Gamma }}_{\mu \nu }^{\rho }-g^{\rho \sigma }\delta {\mathrm{\Gamma }}_{\mu \sigma }^{\mu }$. Using the divergence theorem, this term becomes a surface contribution:

$$\delta S_{\partial \mathcal{M}}={\int }_{\mathcal{M}}d\mathcal{M}{\nabla }_{\mu }{V}^{\mu }={\int }_{\partial \mathcal{M}}{d}^{3}y\sqrt{\left|h\right|}{n}_{\mu }{V}^{\mu }.$$

(6)

Here $\partial \mathcal{M}$ denotes the spacetime boundary, represented by a non-null hypersurface $\Sigma$ with unit normal $n_{\mu }$, and $h_{ab}=h_{ab}\left(y\right)$ is the corresponding induced metric: i.e. $g_{\mu \nu }\equiv g_{\mu \nu }\left(x\right)$ on the boundary. For the timelike boundary ($\left|h\right|=-h$), the normal vector is $n_{\mu }{n}^{\mu }=1$ spacelike, whereas the spacelike boundary ($\left|h\right|=h$) has a timelike normal vector $n_{\mu }{n}^{\mu }=-1$. The tangent basis vectors to $\Sigma$ are defined by

$${e^{\mu }}_a:={\displaystyle \frac{\partial x^{\mu }}{\partial y^a}},n_{\mu }{e^{\mu }}_a=0,$$

(7)

which gives the induced metric $h_{ab}=g_{\mu \nu }{e^{\mu }}_a{e^{\nu }}_b$. The surface variation can be separated into two contributions (see [3]):

$$\delta S_{\partial \mathcal{M}}=\delta S_K+\delta S_h,$$

(8)

where

$$\begin{array}{cc}\hfill \delta S_K& ={\int }_{\partial \mathcal{M}}{d}^{3}y\delta \left(2\sqrt{\left|h\right|}K\right)\hfill \\ \hfill \delta S_h& ={\int }_{\partial \mathcal{M}}{d}^{3}y\sqrt{\left|h\right|}(K{h}_{ab}-K_{ab})\delta h^{ab}\hfill \\ \hfill K_{ab}& =-e_a^{\mu }{e}_b^{\nu }{\nabla }_{\mu }{n}_{\nu },K=h^{ab}{K}_{ab},\hfill \end{array}$$

(9)

The extrinsic curvature tensor of the boundary hypersurface, is given by the covariant derivative ${\nabla }_{\mu }$ of the normal $n_{\mu }$. Since $K_{ab}$ contains the normal derivative of the induced metric, fixing $K_{ab}$ is the gravitational analogue of imposing Neumann boundary conditions, the remaining condition is then $\delta S_h=0$. The term $\delta S_K$ corresponds to the Gibbons–Hawking–York (GHY) boundary term (see [6,16]). It is required when imposing Dirichlet boundary conditions, for which the induced metric is fixed: $\delta h_{ab}=0$ so that $\delta S_h=0$. The role of the GHY term is to cancel the variations involving normal derivatives of the metric (equivalently, variations of the extrinsic curvature), so that the variational principle becomes well defined when the induced metric $h_{ab}$ is held fixed at the boundary. By contrast, Neumann-type boundary conditions correspond to fixing the extrinsic curvature itself rather than the induced metric. In that case the term $\delta S_h=0$ becasue K is fixed.

A well-defined variational principle requires the total boundary, $\partial \mathcal{M}$, variation $\delta$ of the action S to vanish:

$$\delta S_{\partial \mathcal{M}}=0.$$

(10)

The choice of boundary conditions is therefore not merely a mathematical detail. Different physical boundaries impose different geometric constraints on spacetime. In gravitational systems with horizons or finite domains, these consistency conditions can introduce additional global relations between geometry and matter. This is in addition to the regular Einstein field equations.

3.1. Boundary and Junction Conditions

Consider a spacetime formed by the union of two manifolds:

$$\mathcal{M}={\mathcal{M}}^{-}\cup \Sigma \cup {\mathcal{M}}^{+},$$

(11)

where the interior region is ${\mathcal{M}}^{-}$ and the exterior region ${\mathcal{M}}^{+}$, with $\Sigma$ is the junction hypersurface between the two.

What role do the boundary terms discussed previously play in this construction? If we choose ${\mathcal{M}}^{+}$ to be empty space (or a manifold without any other boundaries), the matching hypersurface $\Sigma =\partial \mathcal{M}$ acts as the physical boundary of the ${\mathcal{M}}^{-}$ manifold. A sufficient condition for the total variational principle $\delta S=0$ requires Einstein’s equations inside each region and also consistency of the total boundary/junction variation:

$$G_{\mu \nu }^{(\pm )}=8\pi G{T}_{\mu \nu }^{(\pm )}\mathrm{in}{\mathcal{M}}^{\pm },\delta S_{\Sigma }=0.$$

(12)

This requires: $S_{\Sigma }=\delta S_{\partial \mathcal{M}}=\delta S_K+\delta S_h=0$ in Eq.8. Because the normal to $\Sigma$ has opposite signs in both sides of $\Sigma$, the continuity of the induced metric $h_{ab}$ and its normal derivative $K_{ab}$ implies $\delta S_{\partial \mathcal{M}}=0$ in Eq.8. So in the absence of a surface stress-energy layer, this is equivalent to the Israel junction conditions. In this sense, $\delta S_{\Sigma }=0$ plays a dual role: it is simultaneously the geometric matching condition between the two manifolds and the condition required for a globally consistent variational principle. No additional boundary terms, such as the GHY term, are needed.

This is not necessarily the most general solution, but is a sufficient condition to provides an exact solution to GR. More generally, one could have solutions where $\delta S_{\Sigma }$ compensates $\delta S_{{\mathcal{M}}^{\pm }}$ so that the total variation remains $\delta S=0$.

3.2. The Cosmological Constant $\mathrm{\Lambda }$

We can add a constant $\mathrm{\Lambda }$, called the cosmological constant, as an additional degree of freedom to the EH action. Such a constant is allowed by the symmetries of GR, as first noted by Einstein [17]. We then need to include $\mathrm{\Lambda }$ in the variational principle. This requires replacing: $R\to R-2\mathrm{\Lambda }$ in the Einstein-Hilbert action of Eq.4:

$$S\left[\mathcal{M}\right]={\int }_{\mathcal{M}}d\mathcal{M}\left[{\displaystyle \frac{R}{16\pi G}}-{\displaystyle \frac{\mathrm{\Lambda }}{8\pi G}}+\mathcal{L}\right],$$

(13)

The full variational principle $\delta S=0$ in Eq.12 then change to:

$$G_{\mu \nu }^{(\pm )}+{\mathrm{\Lambda }}^{\pm }{g}_{\mu \nu }=8\pi G{T}_{\mu \nu }^{(\pm )}\mathrm{in}{\mathcal{M}}^{\pm },\delta S_{\Sigma }=0.$$

(14)

where ${\mathrm{\Lambda }}^{+}$ and $T_{\mu \nu }^{+}$ can in principle both be different from ${\mathrm{\Lambda }}^{-}$ and $T_{\mu \nu }^{-}$ as long as $\delta S_{\Sigma }=0$ is fulfilled. Adding $\mathrm{\Lambda }$ does not change the boundary conditions; it only affects the field equations. But the additional freedom can be used to find new solutions where both the field equations and the boundary conditions are jointly fulfilled so that $\delta S=0$.

The value of $\mathrm{\Lambda }$ can have three physically distinct origins or interpretations:

1.

Fundamental constant (${\mathrm{\Lambda }}_F$). This can be view as the simplest form of modified gravity (i.e. modifying the lagrangian of GR). Such additional term needs to be very small to be observationally consistent with the classical, Newtonian, version of gravity.

2.

Vacuum or Ground Energy (${\mathrm{\Lambda }}_G$). A scalar field or quantum vacuum contributes a constant energy density in its ground state (G), behaving like $\mathrm{\Lambda }$. Naive quantum-field estimates often predict such contribution to be very large, which is in conflict with both the Newtonian limit and the cosmological measurements of $\mathrm{\Lambda }$.

3.

A boundary term (${\mathrm{\Lambda }}_B$). The Einstein–Hilbert action permits the addition of a constant term proportional to $\mathrm{\Lambda }$ without violating diffeomorphism invariance. The bulk field equations alone do not determine the physical value of this constant. In the present framework, the boundary-value problem provides an additional global constraint ${\mathrm{\Lambda }}_B$ that can fix the effective observable cosmological constant.

Although these ideas arise from different physical pictures, they could produce the same observational effect in cosmology. So, in general, if all sources are present, we will have that the effective (observed) ${\mathrm{\Lambda }}_o$ value will be the sum of all:

$${\mathrm{\Lambda }}_o={\mathrm{\Lambda }}_B+{\mathrm{\Lambda }}_F+{\mathrm{\Lambda }}_G$$

(15)

Such decomposition should be understood as an effective parametrization rather than a unique microscopic split. Since all constant contributions enter Einstein’s equations through the same tensor structure proportional to $g_{\mu \nu }$, only their total sum is directly observable. We just need to make sure that $\delta S=0$, in accordance with a global variational principle.

For empty space $T_{\mu \nu }=0$ with spherical symmetry, the solution to the field equation gives a metric of the Schwarzschild or deSitter form:

$$d{s}^2=-F\left(r\right)d{t}^2+{\displaystyle \frac{d{r}^2}{F\left(r\right)}}+r^{2}d{\mathrm{\Omega }}^2,F\left(r\right)=1-{\displaystyle \frac{r_s}{r}}-{\displaystyle \frac{{\mathrm{\Lambda }}^{+}}{3}}{r}^2$$

(16)

which is the Schwarzschild solution with a $\mathrm{\Lambda }$ term, which we label ${\mathrm{\Lambda }}^{+}$, a free parameter in the local equations. This results in a deSitter phase at large radius r.

If we require a solution with the boundary condition that the metric is asymptotically flat for the exterior vacuum at $r\to \infty$ then we need to impose the following condition:

$${\mathrm{\Lambda }}_B+{\mathrm{\Lambda }}_F+{\mathrm{\Lambda }}_G={\mathrm{\Lambda }}^{+}=0\Rightarrow {\mathrm{\Lambda }}_B=-{\mathrm{\Lambda }}_F-{\mathrm{\Lambda }}_G$$

(17)

which illustrates, in a practical case, how imposing a boundary condition fixes the value of ${\mathrm{\Lambda }}_B$ and cancels the ${\mathrm{\Lambda }}_F+{\mathrm{\Lambda }}_G$ contributions.

For the infinite FLRW homogeneous and isotropic solution $H\equiv \dot{a}/a$:

$$\begin{array}{ccc}\hfill d{s}^2& =& -d{\tau }^2+a{\left(\tau \right)}^2\left[{\displaystyle \frac{d{\chi }^2}{1-k{\chi }^2}}+{\chi }^{2}d{\mathrm{\Omega }}^2\right],\hfill \\ \hfill H^2+{\displaystyle \frac{1}{R_k^2}}& =& {\displaystyle \frac{1}{R_G^2}}+{\displaystyle \frac{1}{R_{\mathrm{\Lambda }}^2}}\hfill \end{array}$$

(18)

where $R_k=a{\chi }_k$ is the expanding spatial curvature radius corresponding to a comoving radius ${\chi }_k\equiv k^{-1/2}>\chi$, and $R_G$ is the gravitational radius: the radius within which the FLRW density would form a black hole, i.e. $R_G^{-2}\equiv 8\pi G\rho /3$. As with the Schwarzschild solution, the $\mathrm{\Lambda }$ term (labeled here $R_{\mathrm{\Lambda }}^2=3/{\mathrm{\Lambda }}^{-}$) becomes a free parameter of Einstein field equations. Measurements of $H^2$ today indicate: $R_{\mathrm{\Lambda }}\underset{\sim }{<}{R}_G<R_k$, while in the early universe $R_G<R_k\ll R_{\mathrm{\Lambda }}$ (with $R_G\simeq R_k$ near a cosmic bounce [18]). A key question is why the observed $\mathrm{\Lambda }$ is so small, but not zero, dominating $H^2$ today ($R_{\mathrm{\Lambda }}\underset{\sim }{<}{R}_G$).

This results in a deSitter phase and late time cosmic acceleration. Even if spacetime is infinite, we are trapped inside an event horizon, where the area radius in Eq.18 is:

$$r\equiv a\chi <R_{\mathrm{\Lambda }}\equiv \sqrt{3/{\mathrm{\Lambda }}^{-}}.$$

(19)

All observers outside $R_{\mathrm{\Lambda }}$ measure the very same expansion, $\rho$ and trapped surface despite being causally disconnected from each other. Moreover, because $R_{\mathrm{\Lambda }}<R_G$, such radius corresponds to a black hole interior. This is a very puzzling causal structure.

As in the Schwarzschild solution, unless we impose ${\mathrm{\Lambda }}^{-}=0$, this does not result in an asymptotically flat metric. So the boundary conditions $\delta S_{\partial \mathcal{M}}=0$ need to be carefully checked in this particular case. We will do that by defining a finite FLRW cloud (FLRW*) and later explore what happens when we take the limit to infinity.

4. The FLRW Cloud (FLRW*)

If every observer inhabits only a finite causal domain—even in an infinite FLRW universe—what prevents the manifold we live in from being finite?

Einstein’s equations do not require an infinite cosmos; they determine only how local curvature responds to the local energy–momentum distribution. Nothing in General Relativity forbids all matter around us from occupying a bounded region of space. Indeed, once causal domains are understood to be finite, the possibility that our local manifold itself is a finite gravitating system becomes not only permissible, but natural. By contrast, a truly infinite manifold can never be observed in its entirety and therefore lies outside direct empirical verification.

This possibility is realised in a remarkably simple construction: a homogeneous spherical region of matter evolving according to the FLRW metric and smoothly matched at its boundary to empty space. The spacetime is then formed by the union of two manifolds:

$$\mathcal{M}={\mathcal{M}}^{-}\cup {\mathcal{M}}^{+},$$

(20)

where the interior region ${\mathcal{M}}^{-}$ is described by the FLRW metric and the exterior region ${\mathcal{M}}^{+}$ by the Schwarzschild solution. The resulting joined spacetime is the FLRW cloud, or FLRW*.

The FLRW* solution is simply the standard FLRW metric (Eq.18) restricted to a finite moving radius $0<r<R\left(\tau \right)$, with the exterior region matched to a Schwarzschild vacuum. Inside, the Universe expands or contracts homogeneously; outside, spacetime contains no matter. The construction is therefore the cosmological analogue of a star or black-hole interior: a finite world evolving under its own gravity and bounded by vacuum space. In modern language, this finite region may also be interpreted as a large nonlinear metric perturbation embedded within a larger background spacetime.

The idea dates back to the original work of Lemaître, who recognised that a homogeneous spherical region satisfying Einstein’s equations naturally produces a distance–redshift relation of the form $\dot{r}=Hr$ [19]. Friedmann, Tolman, Oppenheimer, Snyder, Misner, and Sharp each developed important aspects of this picture [8,20,21,22,23].

Matched FLRW–Schwarzschild spacetimes now form a well-studied class of exact solutions [10,11]. The standard Oppenheimer–Snyder solution [8,9] corresponds to a highly special case: pressureless matter, comoving boundary, and vanishing effective cosmological constant ${\mathrm{\Lambda }}^{-}=0$. It should therefore not be interpreted as the most general FLRW–Schwarzschild matching solution. Once pressure, radiation, or $\mathrm{\Lambda }$ are present, the physically relevant junction generically becomes non-comoving.

4.1. Boundary Conditions

The boundary conditions (see §Section 3) require both the induced metric and its normal derivative (the extrinsic curvature in Eq.9) to match continuously across the junction hypersurface $\Sigma$. When these conditions are satisfied, the joined spacetime becomes a valid solution of Einstein’s equations and satisfies the boundary condition in Eq.14.

The first boundary condition requires continuity of the induced metric across the hypersurface $\partial \mathcal{M}$. Comparing the areal radius of the FLRW metric (Eq.18) with that of the Schwarzschild metric (Eq.16) gives

$$R\left(\tau \right)=a\left(\tau \right)\chi .$$

(21)

A natural matching therefore consists of defining a moving boundary $R\left(\tau \right)$ such that the FLRW solution applies for $r\le R$ and the Schwarzschild solution for $r\ge R$.

During matter domination, the comoving boundary remains fixed, $\chi =R_0$, so the junction simply follows the Hubble flow: $\dot{R}=HR$. This corresponds to a timelike hypersurface moving along the geodesic flow of a pressureless FLRW fluid. More generally, the boundary may move relative to the Hubble flow:

$$\dot{R}\equiv HR-v$$

(22)

where $v\equiv -a\dot{\chi }$ is the peculiar velocity of the junction hypersurface relative to the local FLRW frame. In units where $c=1$, the flat case allows $\left|v\right|\le 1$. As $\left|v\right|$ approaches unity, the timelike boundary approaches a null (lightlike) evolution.

Using Eq.22, the second boundary condition becomes (see §Appendix A):

$$\begin{array}{|c|}\hline {\dot{R}}^2={\gamma }^2{\displaystyle \frac{{(1-k{\chi }^2-vHR)}^2}{1-k{\chi }^2}}-{\gamma }^{-2}F\left(R\right),\\ \hline\end{array}$$

(23)

where $F\left(R\right)=1-{\displaystyle \frac{r_s}{R}}-{\displaystyle \frac{{\mathrm{\Lambda }}^{+}}{3}}{R}^2$ and $\gamma$ is a Lorentz boost defined in Eq.A2. Eq.23 is the most general boundary condition for the FLRW* geometry embedded within an exterior vacuum spacetime. The entire dynamics of the matching surface is encoded in the angular component of the extrinsic curvature, whose continuity in Eq.A7 yields Eq.23.

Given the evolution of $H\left(\tau \right)$ from the Friedmann equation, Eq.23 determines the evolution of the boundary radius $R\left(\tau \right)=a\left(\tau \right)\chi \left(\tau \right)$ and therefore of both $\dot{R}$ and the peculiar velocity $v=HR-\dot{R}$. Consistent solutions only exist for some particular cases.

4.2. Solutions

The equation admits two qualitatively distinct branches. The first is a comoving solution with $v=0$, corresponding to a dust universe whose energy density scales as $a^{-3}$. The second is a non-comoving solution with $v\ne 0$, which evolves toward the attractor ${\gamma }^{-2}\to 0$ and asymptotically approaches $R=r_s$:

$$H^2+{\displaystyle \frac{1}{R_k^2}}=\left\{\begin{array}{cc}{\displaystyle \frac{r_s}{R^3}}+{\displaystyle \frac{{\mathrm{\Lambda }}^{+}}{3}},\hfill & v=0,R=R_{0}a,\hfill \\ {\displaystyle \frac{1}{R^2}},\hfill & {\gamma }^{-2}\to 0,R\to r_s.\hfill \end{array}\right.$$

(24)

For the first branch, the boundary follows the Hubble flow, so that $R=R_{0}a$. For the second branch, the junction evolves toward the asymptotic attractor ${\gamma }^{-2}\to 0$.

Although Eq.24 resembles the Friedmann equation, Eq.18, its origin is fundamentally different. It arises from matching a finite gravitating region to an exterior vacuum geometry through the Israel junction conditions rather than from solving Einstein’s equations under the assumption of exact homogeneity and isotropy. Both equations need to be solved simultaneously to fulfil the stationary action principle.

For the comoving branch $v=0$, Eq.24 reduces to

$$H^2+{\displaystyle \frac{1}{R_k^2}}={\displaystyle \frac{r_s}{R^3}}+{\displaystyle \frac{{\mathrm{\Lambda }}^{+}}{3}}.$$

(25)

Since $v=0$ the boundary follows the Hubble flow, $R=a{R}_0$ and $H=\dot{R}/R$. Compare to the Fridmann equation, the matter contribution $\rho$ scales as $a^{-3}$, which is the characteristic behaviour of pressureless dust. This only works for ${\mathrm{\Lambda }}^{+}=0$. Thus the comoving solution $v=0$ is not merely a particular branch of the junction condition: it is precisely the branch corresponding to a dust-dominated universe with ${\mathrm{\Lambda }}^{+}=0$.

The observed Universe, however, cannot be described purely by dust. Radiation satisfies $p=\rho /3$ and evolves as $\rho \propto a^{-4}$, while more general fluids obey $\rho \propto a^{-3(1+w)}$. These scalings are incompatible with $v=0$, which contains only the dust behaviour $a^{-3}$. Consequently, a realistic cosmology containing radiation, pressure, or multiple fluid components cannot remain on the comoving branch $v=0$ and must evolve onto the non-trivial branch $v\ne 0$. In this branch, the boundary acquires a peculiar velocity relative to the Hubble flow and evolves toward the attractor ${\gamma }^{-2}\to 0$. Equation 24 then implies the asymptotic condition

$$H^2+{\displaystyle \frac{1}{R_k^2}}\to {\displaystyle \frac{1}{R^2}},$$

(26)

independently of the values of $r_S$, ${\mathrm{\Lambda }}^{+}$ or ${\mathrm{\Lambda }}^{-}$.

This solution satisfies $\dot{R}\to 0$, so the boundary asymptotically approaches a constant radius $R\to R_c$. Since $r_s=2GM={\displaystyle \frac{8\pi G}{3}}\rho R^3$ is fixed by the mass enclosed within the junction, the Friedmann equation Eq.18 can be written as

$$H^2+{\displaystyle \frac{1}{R_k^2}}={\displaystyle \frac{8\pi G\rho \left(a\right)}{3}}+{\displaystyle \frac{1}{R_{\mathrm{\Lambda }}^2}}={\displaystyle \frac{r_s}{R^3}}+{\displaystyle \frac{1}{R_{\mathrm{\Lambda }}^2}},$$

(27)

where $3/R_{\mathrm{\Lambda }}^2={\mathrm{\Lambda }}^{-}$. In the asymptotic limit this gives

$${\displaystyle \frac{1}{R_c^2}}={\displaystyle \frac{r_s}{R_c^3}}+{\displaystyle \frac{1}{R_{\mathrm{\Lambda }}^2}},$$

(28)

or equivalently

$$R_c=r_s+{\displaystyle \frac{R_c^3}{R_{\mathrm{\Lambda }}^2}}.$$

(29)

In the special case where the original bulk FLRW equation contains no explicit cosmological constant, ${\mathrm{\Lambda }}^{-}\to 0$ (equivalently $R_{\mathrm{\Lambda }}\to \infty$), Eq.29 reduces to

$$R_c=r_s.$$

(30)

Thus an effective cosmological constant $R_c=r_s$ emerges dynamically from the asymptotic boundary condition, even when no explicit cosmological constant is present in the bulk FLRW equation.

An important feature of this attractor branch is its universality. The asymptotic condition

$$H^2+{\displaystyle \frac{1}{R_k^2}}\to {\displaystyle \frac{1}{R_c^2}}$$

(31)

does not depend on the sign of H, the spatial curvature k, or the value of the exterior cosmological constant ${\mathrm{\Lambda }}^{+}$. The same attractor is obtained for expanding ($H>0$) and collapsing ($H<0$) solutions, as well as for open, flat, and closed FLRW geometries.

For the flat case ($k=0$), the attractor corresponds to $v\to 1$. For $k\ne 0$, however, the relevant causal limit is not simply $v=1$, but the curvature-corrected null condition

$$v^2\to 1-k{\chi }^2\iff {\gamma }^{-2}\to 0.$$

(32)

Consistency of ${\gamma }^{-2}\to 0$ with the Friedmann equation Eq.18 then requires

$${\mathrm{\Omega }}_{\mathrm{\Lambda }}{H}_0^2={\displaystyle \frac{{\mathrm{\Lambda }}^{-}}{3}}={\displaystyle \frac{1}{r_s^2}},⟹R_{\mathrm{\Lambda }}=r_s.$$

(33)

Thus the boundary condition forces the cosmological horizon scale to coincide with the Schwarzschild radius associated with the enclosed mass, $R_{\mathrm{\Lambda }}=r_s$. In other words, the interior FLRW solution requires ${\mathrm{\Lambda }}^{-}={\displaystyle \frac{3}{r_s^2}}$, even when the exterior spacetime is asymptotically Minkowski (${\mathrm{\Lambda }}^{+}=0$). This solution is illustrated in Figure 1.

The identification $R_{\mathrm{\Lambda }}=r_s$ appears to be a geometric property of the junction dynamics itself rather than a special feature of a particular cosmological model.

4.3. Implications for $\mathrm{\Lambda }$

We have found that the variational principle admits a consistent FLRW solution with a non-zero interior cosmological constant. This is a central result of the present work. The fact that ${\mathrm{\Lambda }}^{+}\ne {\mathrm{\Lambda }}^{-}$ does not imply a discontinuity of either the induced metric or the extrinsic curvature. Continuity follows from the boundary (or Israel) junction conditions themselves, which remain satisfied because the matching conditions depend on the geometry of the hypersurface rather than on the individual values of $\mathrm{\Lambda }$ appearing in the bulk field equations.

The non-trivial branch therefore, generates an interior cosmological constant ${\mathrm{\Lambda }}^{-}={\displaystyle \frac{3}{r_s^2}}$ independently of the exterior ${\mathrm{\Lambda }}^{+}$. Within the decomposition introduced in Eq.15, this fixes the required boundary contribution to:

$${\mathrm{\Lambda }}_B={\displaystyle \frac{3}{r_s^2}}-{\mathrm{\Lambda }}_F-{\mathrm{\Lambda }}_G$$

(34)

where ${\mathrm{\Lambda }}_F$ and ${\mathrm{\Lambda }}_G$ are other possible contributions to $\mathrm{\Lambda }$ in Eq.15. This shows how ${\mathrm{\Lambda }}_F$ and ${\mathrm{\Lambda }}_G$ are cancelled by the boundary term.

The effective cosmological constant ${\mathrm{\Lambda }}^{-}$ is inversely proportional to the square of the Schwarzschild radius of the FLRW cloud. For sufficiently large systems, the resulting value is naturally small while remaining non-zero.

In the present framework, we can recover the standard infinite FLRW interpretation as a limiting case by taking $r_s\to \infty$. In this limit, the boundary-selected contribution satisfies ${\mathrm{\Lambda }}^{-}\to 0$, so that the boundary contribution disappears and the FLRW region becomes effectively infinite.

Conversely, a finite value of the interior cosmological constant requires $r_s<\infty$. Since the junction condition implies ${\mathrm{\Lambda }}^{-}={\displaystyle \frac{3}{r_s^2}}$, the observed non-zero value of ${\mathrm{\Lambda }}^{-}$ corresponds, within the present framework, to a finite gravitating FLRW region bounded by a causal horizon.

4.4. Early Times

In the limit $R\to 0$ (equivalently $a\to 0$), for a fluid with non-zero pressure satisfying $p=\omega \rho$ with $\omega >0$, the Friedmann equation gives

$$H^2\propto a^{-3(1+\omega )}.$$

(35)

Since $R\propto a$ at early times, this implies

$$HR\propto a^{-(1+3\omega )/2},$$

(36)

which diverges as $a\to 0$. Therefore

$$HR\gg v,$$

(37)

and the junction evolution satisfies

$$\dot{R}=HR-v\simeq HR.$$

(38)

Thus the boundary approximately follows the Hubble expansion in its radial motion, even though it remains intrinsically non-comoving ($v^2>0$). In this regime, Eq.23 simplifies and yields the unique asymptotic solution

$$v^2={\displaystyle \frac{1-k{\chi }^2}{2}}.$$

(39)

Remarkably, this early-time attractor is universal, completely independent of the detailed Friedmann evolution $H\left(a\right)$ and of both the interior and exterior cosmological constants, ${\mathrm{\Lambda }}^{-}$ and ${\mathrm{\Lambda }}^{+}$.

For the flat case ($k=0$), this gives $v=1/\sqrt{2}$. Thus the physically relevant non-comoving branch does not approach the comoving Oppenheimer–Snyder solution ($v=0$) at early times. Instead, it begins at a finite peculiar velocity and evolves monotonically toward the late-time attractor

$$v\to 1,R\to r_s.$$

(40)

The numerical solutions presented below confirm this behavior, showing that $v\left(a\right)$ grows from $1/\sqrt{2}$ at early times to unity at late times, while the variationally selected boundary tracks the causal horizons of the FLRW spacetime, interpolating between the observable horizon at early times and the future event horizon at late times.

4.5. Numerical Solutions and Approximaitons

Figure 2 shows the numerical solutions $v=v\left(a\right)$ and $R=R\left(a\right)$ obtained from the boundary equation (Eq. 23) for a flat $\mathrm{\Lambda }$CDM background (${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.75$, ${\mathrm{\Omega }}_R=2.5\times {10}^{-5}$, and ${\mathrm{\Omega }}_m=1-{\mathrm{\Omega }}_{\mathrm{\Lambda }}-{\mathrm{\Omega }}_R$).

The upper panel shows that the peculiar velocity $v\left(a\right)$ never vanishes. Instead, the junction remains on the non-comoving branch throughout the entire cosmic evolution. During the matter-dominated era, the solution approaches the asymptotic limit $v^2\to 1/2$ (independent of the precise form of H), as discussed above, rather than the comoving Oppenheimer–Snyder branch ($v=0$). The numerical solution therefore confirms the analytical expectation that the physically relevant branch is non-comoving at all times.

The magenta curve shows the evolution of the normalised comoving radius, ${\chi }_0/\chi \left(a\right)$, where ${\chi }_0\equiv \chi (a=1)$ is the present-day comoving position of the junction. The slow evolution of this ratio demonstrates that the boundary remains approximately comoving throughout most of cosmic history before rapidly departing from the Hubble flow at late times as it approaches the asymptotic causal attractor.

The lower panel shows the corresponding evolution of the proper boundary radius, illustrating how this nearly comoving evolution at early times naturally leads to the asymptotic approach $R\to r_s$ discussed in the following. The values of $R\left(a\right)$ are compared to the Hubble horizon $1/H$ (in blue) and the radius of the observable Universe in green), which corresponds to the past null cone:

$$R_{obs}\left(\tau \right)=a\left(\tau \right){\int }_{\tau }^{{\tau }_0}{\displaystyle \frac{d\tau }{a\left(\tau \right)}}=a{\int }_a^1{\displaystyle \frac{da}{H{a}^2}},$$

(41)

also known as particle horizon, proper angular diameter distance or conformal lookback time. A null past geodesic in the FLRW metric ($ds=0$) satisfies $\dot{\chi }=-1/a$, hence

$$\dot{r}=\dot{a}\chi +a\dot{\chi }=Hr-1.$$

(42)

which corresponds to $v=1$ in the flat case. The maximum observable radius occurs when $\dot{r}=0$, i.e. where the light path intersects the Hubble horizon $r_H=1/H$. In a flat LCDM cosmology, this maximum appears at $z\simeq 1.5$, corresponding to $R_{obs}\simeq 0.4c/H_0$. Thus, even if the Universe were infinite, we only ever receive photons from within $\sim 40\%$ of today’s Hubble radius.

The numerical solution shows that the variationally selected boundary closely tracks the causal horizons of the FLRW spacetime, approaching the observable horizon at early times and the future event horizon at late times. This strongly suggests that the junction condition selects a globally causal boundary rather than an arbitrary matching radius.

We can also see that $R\left(a\right)$ always lies outside the observable Universe. This is also true for any off-centred observer within the FLRW cloud (see Appendix §Appendix B). But even if we cannot see $R\left(a\right)$ directly, we can measure its effect in H.

The values of $R\left(a\right)$ are close to the event horizon $R_{*}$, which corresponds to the outgoing null geodesic in the FLRW metric:

$$R_{*}\left(a\right)=a{\chi }_{*}=a{\int }_a^{\infty }{\displaystyle \frac{da}{H{a}^2}}<R_{\mathrm{\Lambda }},$$

(43)

For early times ($a\ll 1$), $R_{*}\left(a\right)$ grows approximately linearly with a. As $a\to \infty$, H tends to a constant and $R_{*}$ freezes at $R_{*}\to R_{\mathrm{\Lambda }}$. Although $R=a\chi$ appears to increase indefinitely with a, the comoving integral decreases as $\chi \propto a^{-1}$, leaving the product $R_{*}$ finite. Thus $R_{*}$ defines a physical trapped surface in an expanding $\mathrm{\Lambda }$ universe, closely analogous to the event horizon of a black hole.

4.6. Causal Interpretation

The junction radius admits an even more direct causal interpretation. Using Eq.22 together with $R=a\chi$, we obtain

$$a\dot{\chi }=-v.$$

(44)

Thus the comoving position of the junction is determined by

$$\chi \left(\tau \right)=-{\int }_0^{\tau }{\displaystyle \frac{vd\tau }{a\left(\tau \right)}}.$$

(45)

Using the asymptotic boundary condition ${\chi }_{\infty }=0$, this can be equivalently written as

$$\chi \left(a\right)={\int }_a^{\infty }{\displaystyle \frac{vda}{H{a}^2}}.$$

(46)

Therefore

$$R\left(a\right)=a\chi \left(a\right)=a{\int }_a^{\infty }{\displaystyle \frac{v\left(a\right)da}{H{a}^2}}<R_{\mathrm{\Lambda }}.$$

(47)

Here $v\left(a\right)$ is the peculiar velocity of the junction hypersurface relative to the local FLRW frame. This expression shows that, once the late-time Friedmann evolution $H\left(a\right)$ is specified, the late-time attractor solution to Eq.23:

$$R\to r_s,v\to 1$$

(48)

is imposed, and the non-comoving branch is uniquely fixed. Other algebraic roots of the junction equation Eq.23 may exist locally, but they do not define distinct physical branches. The solution is therefore identical to a generalized future causal horizon whose effective propagation speed is given by the peculiar velocity $v\left(a\right)$ of the junction hypersurface. Because $1/2<v^2<1$, the boundary remains timelike at finite cosmic time. As the attractor is approached, $v\to 1$, the boundary becomes asymptotically null and converges to the future event horizon.

This provides a geometric interpretation of the variationally selected boundary. It is not an arbitrary matching surface but the unique causal horizon $R\left(a\right)$ generated by the junction dynamics itself. The logical chain can be summarised as:

$$\begin{array}{|c|}\hline \delta S_{\Sigma }=0⟹v\ne 0⟹R\left(a\right)=a{\int }_a^{\infty }{\displaystyle \frac{vda}{H{a}^2}}.\\ \hline\end{array}$$

(49)

The appearance of $R_{\mathrm{\Lambda }}=r_s$ is therefore no longer mysterious. It is not imposed as an additional assumption but emerges from the causal structure selected by the variational principle.

The causal-horizon branch identified above also has important implications for gravitational collapse. In particular, the asymptotic condition $R\to r_s$ suggests that trapped regions arise naturally from the boundary dynamics and can persist through both collapsing and expanding phases. This raises the possibility that bounce cosmologies remain inside a trapped region throughout their evolution [18]. A detailed analysis of collapse, bounce dynamics, and their relation to black-hole and white-hole spacetimes will be presented elsewhere.

4.7. Closed Curvature

In terms of a finite FLRW cloud, $r_s$ is fixed by the enclosed mass, while ${\chi }_k$ is related to the classical binding energy of the system. The comoving radius of the boundary is

$$\chi \left(a\right)={\displaystyle \frac{R\left(a\right)}{a}}.$$

(50)

For a closed FLRW geometry, the coordinate range is limited by the curvature radius ${\chi }_k$, so the finite FLRW patch must satisfy

$$\chi \left(a\right)<{\chi }_k$$

(51)

at all times. This gives

$$|{\mathrm{\Omega }}_k|\equiv {\displaystyle \frac{1}{H_0^2{\chi }_k^2}}<{\displaystyle \frac{1}{H_0^2\chi {\left(a\right)}^2}}.$$

(52)

As shown by the magenta curve in Figure 2, the comoving boundary decreases monotonically with cosmic time. Therefore the strongest bound comes from its largest value, attained at early times. Using the numerical solution at recombination,

$${\chi }_{\mathrm{CMB}}={\displaystyle \frac{R_{\mathrm{CMB}}}{a_{\mathrm{CMB}}}},$$

(53)

one obtains

$$|{\mathrm{\Omega }}_k|<{\displaystyle \frac{1}{H_0^2{\chi }_{\mathrm{CMB}}^2}}={\mathrm{\Omega }}_{\mathrm{\Lambda }}{\left({\displaystyle \frac{r_s}{{\chi }_{\mathrm{CMB}}}}\right)}^2,$$

(54)

where we used $R_{\mathrm{\Lambda }}=r_s$ and ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=1/\left(H_0^2{r}_s^2\right)$. From the magenta line in Figure 2, ${\chi }_{\mathrm{CMB}}\simeq 3.3r_s$. We therefore find

$$|{\mathrm{\Omega }}_k|\lesssim 0.063\left({\displaystyle \frac{{\mathrm{\Omega }}_{\mathrm{\Lambda }}}{0.7}}\right).$$

(55)

This gives a consistency condition on the allowed curvature radius of the FLRW cloud. The condition does not require exact flatness, but it does require the curvature radius to exceed the largest comoving extent of the boundary. In this sense, the finite-boundary solution naturally favours a curvature radius larger than the observable FLRW domain, although the resulting bound is not by itself as strong as current observational limits on ${\mathrm{\Omega }}_k$.

Consequently, even if the Universe possesses a small positive spatial curvature, its effects remain subdominant over the entire causal FLRW region, except on very early times where curvature can drive a bouncing start [18,24]. Although the resulting bound is weaker than current observational constraints on ${\mathrm{\Omega }}_k$, it provides a simple geometric consistency condition implied by the finite-boundary solution.

4.8. The Maximum Causally Connected Angular Scale

Although the boundary always satisfies $R_{\mathrm{obs}}<R$, observations are not restricted to radial distances. Structures may also be measured across the sky on transverse scales larger than the observable radius. One therefore does not expect to observe the boundary itself, but rather a maximum angular scale corresponding to the largest causally connected region.

This provides a direct observational prediction of the present framework. If all causally connected perturbations are contained within the junction radius $R\left(a\right)$, then the maximum observable angular scale at redshift $1+z=a^{-1}$ is

$${\theta }_{\max }\left(z\right)={\displaystyle \frac{R\left(a\right)}{R_{\mathrm{obs}}\left(a\right)}}={\displaystyle \frac{\chi \left(a\right)}{{\chi }_{\mathrm{obs}}\left(a\right)}}={\displaystyle \frac{{\displaystyle {\int }_a^{\infty }\frac{vda}{H{a}^2}}}{{\displaystyle {\int }_a^1\frac{da}{H{a}^2}}}}.$$

(56)

Figure 3 shows the resulting prediction. At recombination ($z\simeq 1100$), the numerical solution gives

$${\theta }_{\max }=81.0^{\circ },65.6^{\circ },52.1^{\circ }\mathrm{for}{\mathrm{\Omega }}_{\mathrm{\Lambda }}=0.40,0.70,1.0,$$

while Camacho–Quevedo et al. [25] obtained the independent observational estimate

$${\theta }_{\max }=65.9\pm 9.2^{\circ },$$

(57)

from the Hausdorff dimension of the observed CMB angular correlation function.

The agreement with ${\mathrm{\Omega }}_{\mathrm{\Lambda }}\simeq 0.7\pm 0.2$ is remarkable because the prediction is not adjusted to reproduce the CMB observations. Once the observed value of the cosmological constant is specified, the variationally selected boundary uniquely determines ${\theta }_{\max }$. Conversely, within the present framework, the measured value of ${\theta }_{\max }$ provides an independent determination of the cosmological constant which is conceptually different from standard probes such as Type Ia supernovae, baryon acoustic oscillations, or the integrated Sachs–Wolfe effect. Equation (56) therefore defines a new cosmological probe of the cosmological constant based on the maximum causally connected angular scale rather than on the cosmic expansion history.

The prediction is not restricted to recombination. Equation (56) determines the full evolution ${\theta }_{\max }\left(z\right)$ and may therefore be tested over a broad range of cosmic time. Current and future surveys such as DESI, Euclid, LSST, SPHEREx, SKA, and forthcoming 21-cm experiments should progressively improve measurements of the homogeneity scale and the largest correlated structures as a function of redshift. Such observations have the potential to provide a direct test of the finite-boundary scenario proposed here and, consequently, of the variational origin of the cosmological constant.

5. Conclusions

We have shown that requiring a well-defined variational principle for a finite FLRW spacetime matched to an exterior vacuum leads to a unique non-trivial junction solution with important consequences for gravitation and cosmology.

The first main result is that the junction equation admits two qualitatively distinct branches. The familiar comoving branch ($v=0$) reproduces the Oppenheimer–Snyder dust solution and exists only for pressureless matter with vanishing effective cosmological constant. Once radiation or any fluid with non-zero pressure is present, this branch ceases to satisfy the Friedmann evolution and the physically relevant solution is necessarily the non-comoving branch ($v\ne 0$). The boundary remains timelike throughout cosmic history, approaches a null hypersurface asymptotically, and evolves toward the universal attractor

$$H^2+{\displaystyle \frac{1}{R_k^2}}\to {\displaystyle \frac{1}{r_s^2}},$$

independent of the sign of H and valid for open, flat and closed FLRW geometries.

The second main result is that this attractor uniquely fixes the asymptotic boundary radius,

$$R_{\mathrm{\Lambda }}=r_s,$$

which immediately implies

$${\mathrm{\Lambda }}^{-}={\displaystyle \frac{3}{r_s^2}}.$$

In this framework the observed cosmological constant is therefore not an arbitrary parameter of the Einstein equations but a geometric consequence of the causal boundary selected by the variational principle. The junction radius satisfies

$$R\left(a\right)=a{\int }_a^{\infty }{\displaystyle \frac{vda}{H{a}^2}},$$

showing that the variationally selected boundary is itself a causal horizon generated dynamically by the junction evolution.

A particularly important consequence is that the variational principle remains perfectly well defined even when the effective cosmological constant differs across the junction,

$${\mathrm{\Lambda }}^{-}\ne {\mathrm{\Lambda }}^{+}.$$

The induced metric and extrinsic curvature remain continuous, so the difference between the two cosmological constants does not represent a physical discontinuity of spacetime. The interior FLRW region and the parent spacetime may therefore possess different effective vacuum energies. The exterior need not be asymptotically de Sitter; it may equally well be asymptotically Minkowski, de Sitter or Anti-de Sitter. This considerably enlarges the class of physically admissible embeddings of finite cosmological regions and may prove relevant to approaches based on holography, string theory or other higher-dimensional constructions, where the natural vacuum of the parent spacetime differs from that observed inside our cosmological domain.

Within the present framework the observable cosmological constant may be decomposed as

$${\mathrm{\Lambda }}^{-}={\mathrm{\Lambda }}_F+{\mathrm{\Lambda }}_G+{\mathrm{\Lambda }}_B,$$

where ${\mathrm{\Lambda }}_F$ represents possible modifications of the Einstein–Hilbert action, ${\mathrm{\Lambda }}_G$ denotes vacuum-energy contributions, and ${\mathrm{\Lambda }}_B$ is the boundary contribution fixed by the variational principle. Since the junction condition uniquely determines the total observable value,

$${\mathrm{\Lambda }}^{-}={\displaystyle \frac{3}{r_s^2}},$$

the boundary contribution automatically compensates any additional constant terms. In this sense, the cosmological constant problem is reformulated as a boundary-value problem rather than a fine-tuning problem.

The third main result is observational. The causal boundary predicts a maximum causally connected angular scale,

$${\theta }_{\max }\left(z\right),$$

which provides a direct observational signature of the finite-boundary solution. For the observed value of $\mathrm{\Lambda }$, the model predicts

$${\theta }_{\max }(z\simeq 1100)\simeq {64}^{\circ },$$

in excellent agreement with the independent observational estimate $65.9\pm 9.2^{\circ }$ derived from the large-angle CMB correlation function. The prediction contains no additional free parameters beyond those already fixed by the background cosmology and can be tested over a broad range of redshifts with forthcoming surveys such as DESI, Euclid, LSST, SPHEREx, SKA and future 21-cm observations. Conversely, measurements of ${\theta }_{\max }\left(z\right)$ provide an independent determination of the cosmological constant itself.

The non-comoving branch also provides an exact extension of the Oppenheimer–Snyder interior solution to fluids with non-zero pressure. The solution applies equally to expanding and collapsing phases and to open, flat and closed spatial geometries, demonstrating that General Relativity admits exact FLRW interior solutions beyond the pressureless case. In this sense, the black-hole Universe picture emerges naturally from the variational principle rather than being introduced as an independent assumption.

The standard infinite FLRW solution is recovered in the limit

$$r_s\to \infty ,$$

for which the boundary contribution disappears and ${\mathrm{\Lambda }}^{-}\to 0$. Thus, within the present framework, the observed non-zero cosmological constant points naturally to a finite gravitating FLRW region bounded by a causal horizon rather than to an exactly infinite cosmological spacetime.

Finally, although the proper boundary radius asymptotically approaches the finite value $R\to r_s$, the corresponding comoving coordinate satisfies $\chi =R/a\to 0$. This does not signal the formation of a physical singularity. Rather, it reflects the exponential stretching of the FLRW coordinate grid, exactly as occurs when static and expanding coordinate systems are compared in de Sitter spacetime. The attractor is therefore fundamentally a horizon solution rather than a comoving one.

More generally, the present work suggests that observable cosmological parameters may ultimately be determined not only by local field equations but also by the global causal structure selected by the variational principle. If so, the cosmological constant may represent the first example of a physical parameter whose observed value is fixed by the geometry of the finite causal domain in which observations are possible.

An important conceptual consequence concerns the cosmological principle. A finite FLRW region bounded by a causal horizon does not imply a preferred position or observable edge. Since all cosmological observations are restricted to the observer’s past light cone, no present observation can distinguish an infinite FLRW spacetime from a sufficiently large finite FLRW region with identical local matter content. As shown in Appendix B, the causal boundary always remains outside the past light cone of every comoving observer located inside the horizon. Consequently every such observer measures the same homogeneous and isotropic background despite the finite extent of the spacetime.

The boundary therefore reveals itself only indirectly through global causal effects. Within the present framework these include the observed value of the cosmological constant and the existence of a maximum causally connected angular scale, rather than any observable anisotropy associated with a physical edge. In this sense the causal boundary enhances large-scale homogeneity rather than violating the cosmological principle.