Einstein-Straus-de Sitter Framework

• The Einstein–Straus–de Sitter framework is a composite model that embeds local vacuum regions (Schwarzschild or Kottler solutions) into a homogeneous, expanding FLRW universe.
• It utilizes matching conditions to ensure continuity of the metric and extrinsic curvature between the local vacuoles and the global cosmological background, avoiding artificial matter shells.
• Generalizations incorporate a cosmological constant and spatial curvature, offering benchmark models for gravitational lensing, perturbation theory, and modified-gravity analyses.

The Einstein–Straus–de Sitter framework is most naturally understood as a composite relativistic construction in which local Schwarzschild solutions, in the form of Einstein–Straus vacuoles, are embedded in a global Einstein–de Sitter (EdS) Friedmann–Lemaître–Robertson–Walker background; in later lensing work, the same label is also used for the corresponding Kottler-vacuole construction in dust-dominated FLRW universes with a cosmological constant and possible non-zero spatial curvature (O'Raifeartaigh et al., 2020, Guenouche, 15 Jul 2025). In that sense, it is not a single historical model so much as a family of closely related frameworks whose common structure is a local vacuum region matched to a homogeneous expanding cosmology, with the EdS model supplying the canonical $k=0,\Lambda=0$ backbone and later extensions incorporating $\Lambda$, $k=\pm1$, perturbations, or de Sitter-invariant local kinematics.

1. Einstein–de Sitter backbone

In modern language, the Einstein–de Sitter universe is the spatially flat, dust-dominated FLRW cosmology with vanishing cosmological constant: homogeneous and isotropic, pressureless matter $p=0$, spatial curvature $k=0$, and $\Lambda=0$ (O'Raifeartaigh et al., 2020). In the 1932 Einstein–de Sitter paper, the flat FLRW line element was written as

$ds^2 = - R^2(dx^2 + dy^2 + dz^2) + c^2 dt^2,$

and the central dynamical relation was

$$\frac{1}{R^2}\left(\frac{dR}{cdt}\right)^2 = \frac{1}{3}\kappa \rho, \qquad \kappa = \frac{8\pi G}{c^2}.$$

In modern notation this is

$H^2 = \frac{8\pi G}{3}\rho,$

with

$$\Omega_m=1,\qquad \Omega_\Lambda=0,\qquad \Omega_k=0.$$

The historical analysis emphasizes that the 1932 article did not explicitly integrate the Friedmann equation or discuss an initial singularity; its focus was the instantaneous relation between cosmic expansion and mean matter density rather than a full temporal history (O'Raifeartaigh et al., 2020). Even so, the EdS model became the canonical flat matter-only reference cosmology, with the implied evolution

$\Lambda$0

and the critical-density relation

$\Lambda$1

Historically and philosophically, the model’s importance lay in providing “the first specific analysis of the case of a dynamic cosmology without a cosmological constant or spatial curvature,” together with a simple, testable relation between $\Lambda$2 and $\Lambda$3 (O'Raifeartaigh et al., 2020). That simplicity is what later made EdS the preferred background for composite constructions, including Einstein–Straus vacuoles.

2. Einstein–Straus vacuoles and the matched spacetime

The Einstein–Straus construction is a “Swiss-cheese” model: one excises a comoving spherical region from a homogeneous FLRW universe and replaces the removed dust with a central mass $\Lambda$4, chosen so that the mass of the central object equals the mass of dust that used to occupy that sphere (Boudjemaa et al., 2010). Inside the vacuole one uses a static vacuum metric; outside one keeps the FLRW background.

With a cosmological constant, the interior is the Schwarzschild–de Sitter, or Kottler, solution

$\Lambda$5

$\Lambda$6

while the exterior is dust-dominated FLRW (Boudjemaa et al., 2010, Guenouche, 15 Jul 2025). In the spatially flat case the exterior line element is

$\Lambda$7

and in the hyperbolic case

$\Lambda$8

The vacuole boundary is the Schücking sphere. In the flat case its physical radius is

$\Lambda$9

and in the open case

$k=\pm1$0

The matching conditions enforce continuity of the metric and of the extrinsic curvature, so there is no matter shell at the boundary (Boudjemaa et al., 2010). The central mass is fixed by mass equivalence with the removed FLRW dust; in the flat case,

$k=\pm1$1

while in the open case

$k=\pm1$2

This matched geometry is the core of the classical Einstein–Straus–de Sitter framework: local vacuum dynamics are treated exactly in the vacuole, whereas the large-scale expansion is carried by the FLRW region.

3. Curved and $k=\pm1$3-generalized Einstein–Straus–de Sitter spacetimes

Later work generalizes the framework from the EdS background to dust-dominated FLRW cosmologies with $k=\pm1$4 and spatial curvature $k=\pm1$5. In the unified notation used for all three cases,

$k=\pm1$6

with

$k=\pm1$7

and

$k=\pm1$8

This provides a single formalism for closed, flat, and open Einstein–Straus–de Sitter lensing calculations (Guenouche, 2024).

In the open, negatively curved case, the current curvature density is

$k=\pm1$9

in the astro-unit normalization used in the lensing analysis (Guenouche, 15 Jul 2025). For the hyperbolic Einstein–Straus–de Sitter solution, a threshold current scale factor

$p=0$0

was identified, corresponding to

$p=0$1

below which the effects of negative spatial curvature on lensing observables become significant (Guenouche, 15 Jul 2025). For $p=0$2, light bending increases slightly by about $p=0$3, while the time delay increases by about $p=0$4; beyond the threshold, the observables converge to those of the spatially flat case (Guenouche, 15 Jul 2025).

A complementary analysis treating both closed and open backgrounds concluded that, within the current error bar

$p=0$5

the effect of spatial curvature on strong lensing may safely be neglected, although larger positive curvature decreases bending and delay and larger negative curvature increases them (Guenouche, 2024). The combined implication is that the framework is sensitive to curvature in principle, but only appreciably so when $p=0$6 is larger than presently favored cosmological values.

4. Null geodesics, light bending, and time delay

In the lensing applications, photon trajectories are integrated across three regions: FLRW from the observer to the vacuole boundary, the SdS interior across the vacuole, and FLRW again from the boundary to the source (Guenouche, 15 Jul 2025). In the Kottler region, null geodesics satisfy

$p=0$7

with $p=0$8 fixed by the perilens $p=0$9. Inside the vacuole, the angular integral yields the local bending accumulated near the lens, dominated by the mass term $k=0$0 (Guenouche, 15 Jul 2025).

The total time delay splits conceptually into a geometrical contribution, from the difference in total path length, and a gravitational or Shapiro-type contribution, from the slower coordinate speed in the lens potential (Guenouche, 15 Jul 2025). Earlier flat-background Einstein–Straus calculations for the lensed quasar SDSS J1004+4112 obtained delays of order $k=0$1–$k=0$2 years and found compatibility with the observational lower bound on the C–D delay (Boudjemaa et al., 2010). A later curved analysis forecasted five delays between the four bright images of the same system in the flat case and then tracked their dependence on $k=0$3 (Guenouche, 2024).

A recurrent issue in this literature is the role of a positive cosmological constant in lensing. Within the hyperbolic extension, the conclusion is that the effect of the cosmological constant remains present and acts to reduce light bending, corroborating the claim of Rindler and Ishak (Guenouche, 15 Jul 2025). At the same time, a broader curved-spacetime calculation concluded that the changes due to a positive $k=0$4 are seemingly too small to be appreciable on cosmological scales, both for bending and for time delay (Guenouche, 2024). Those results are not contradictory: they identify a definite sign for the $k=0$5-effect, but also find it to be numerically small for realistic cosmological parameters.

5. Benchmark role in cosmology and perturbation theory

The Einstein–de Sitter backbone gave the framework much of its enduring utility. The EdS model marked the case in which cosmic expansion is precisely balanced by the critical density of matter, enabling the later textbook classification of $k=0$6 cosmologies into closed, flat, and open cases and fixing the benchmark value

$k=0$7

for a Euclidean matter-dominated universe (O'Raifeartaigh et al., 2020). For much of the twentieth century it functioned as the prototype “big bang” model and as the default reference cosmology against which observations were interpreted (O'Raifeartaigh et al., 2020).

That benchmark status persists in perturbation theory. In a parameterized modified-gravity framework, an EdS universe was treated as a “safe testbed” in which the background remains matter-dominated and flat while modifications enter only in the linear perturbation sector (Baker, 2011). In conformal Newtonian gauge,

$k=0$8

with EdS background evolution

$k=0$9

For metric-only parameterizations, subhorizon growth remains power-law in $\Lambda=0$0, the dominant superhorizon mode has constant $\Lambda=0$1, and the gauge-invariant curvature perturbation $\Lambda=0$2 is conserved on large scales, provided the modified-gravity functions are not pathological in $\Lambda=0$3 (Baker, 2011). The same analysis showed that modified gravity generically generates a nonzero integrated Sachs–Wolfe signal even during an EdS-like matter era (Baker, 2011).

This suggests a broader significance of the framework: EdS is not only a background to be matched to local vacuoles, but also a canonical reference spacetime for perturbative, observational, and modified-gravity constructions.

6. Broader reinterpretations and contemporary variants

Several later lines of work extend the framework beyond its classical Swiss-cheese meaning. One direction replaces local Minkowskian kinematics with de Sitter-invariant special relativity. In that setting, local inertial frames are described by the Beltrami metric

$\Lambda=0$4

and the cosmological constant is written geometrically as

$\Lambda=0$5

In an Einstein–Straus–de Sitter-type picture, local physics is then taken to be de Sitter-like rather than exactly Minkowskian, but present bounds imply

$\Lambda=0$6

so the deviations from standard special relativity are effectively unobservable (Tretyakova, 2016).

A second direction is de Sitter-invariant cosmology, in which $\Lambda=0$7 is constitutive rather than added by hand and the Friedmann equations are modified so that the source term involves a difference $\Lambda=0$8 rather than the standard matter-plus-$\Lambda=0$9 structure (Araujo et al., 2022). A third direction is the local-to-global inhomogeneous Einstein–de Sitter universe, described as conceptually close in spirit to Einstein–Straus vacuoles but promoted to a dynamical, volume-weighted global description rather than a static Swiss-cheese construction (Raffai et al., 5 Nov 2025). In that proposal, a quasilinear coasting evolution carries the universe from an Einstein–de Sitter to a Milne state without invoking dark energy, and two realizations, iEdS(1) and iEdS(2), were tested against CMB, BAO, and SN Ia data (Raffai et al., 5 Nov 2025).

There is also a distinct gauge-theoretic use of “Einstein-Strauss” in the expression “Einstein-Strauss Hermitian gravity,” where the tangent-group symmetry is unitary and the construction concerns Hermitian gravity rather than Einstein–Straus vacuoles (Chamseddine et al., 2010). That usage is conceptually separate from the cosmological Swiss-cheese framework.

Taken together, these developments show that the Einstein–Straus–de Sitter framework has evolved from a specific matched spacetime into a broader family of local-to-global constructions. Its classical core remains the exact embedding of a local vacuum region in an EdS or $ds^2 = - R^2(dx^2 + dy^2 + dz^2) + c^2 dt^2,$0-FLRW background; its broader legacy is the continued use of EdS and de Sitter structures as canonical backgrounds for matching problems, perturbation theory, and alternative cosmological kinematics.