Embedded Point-Mass Lens in Cosmology

• Embedded point-mass lens is a gravitational lens model that replaces a comoving volume with a mass condensation from the cosmic mean, ensuring local shielding of the gravitational potential.
• The model utilizes FLRW, Schwarzschild–de Sitter, and McVittie metrics to derive modified lens equations, bending angles, and time-delay functions compared to isolated point-lens models.
• Observable consequences include up to 10% deviations in weak lensing signals and minor time delay corrections, which are critical for accurate astrophysical and cosmological inferences.

An embedded point-mass lens (PL) describes a gravitational lens model in which a point mass is not superimposed on, but rather replaces an equivalent comoving volume of cosmological background, ensuring that in the region outside the “void” centered on the lens, the mean density is unchanged and the gravitational potential is locally shielded. Embedding a point mass in otherwise homogeneous Friedmann–Lemaître–Robertson–Walker (FLRW) or $\Lambda$CDM backgrounds modifies the global geometry, the lensing equation, light deflection, potential, and time-delay structure compared to the standard “isolated” point-lens model. This construction is essential for accurate lens modeling in cosmological contexts, where the lensing object must be considered as a condensation of mass drawn from the mean density.

1. Theoretical Framework and Embedding Metrics

The embedded point-mass lens is formalized by replacing a comoving spherical region in the FLRW universe with a vacuum or Kottler (Schwarzschild–de Sitter) region whose mass matches that removed from the cosmic mean (Chen et al., 2011, Kantowski et al., 2012, Kantowski et al., 2013). The FLRW metric outside the lens is

$$ds^2 = -c^2\,dT^2 + R(T)^2\left[d\chi^2 + \chi^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right],$$

while the embedded region is described by the Schwarzschild–de Sitter metric: $$ds^2 = -\gamma(r)^{-2} c^2 dt^2 + \gamma(r)^2 dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2),$$ where $\gamma^{-1}(r) = \sqrt{1 - \beta^2(r)}$, $\beta^2(r) = r_s/r + \Lambda r^2/3$, $r_s = 2Gm/c^2$, and $r_b = R(T)\chi_b$ defines the comoving boundary. The McVittie metric also provides a model for a point mass embedded in an expanding universe, with the mass parameter $\mu(\rho,t) = M / [2 a(t) \rho] \ll 1$, $a(t)$ the scale factor, and $\mu$ vanishing in the FLRW limit (Piattella, 2016).

Matching at the boundary ensures the gravitational field smoothly transitions from lens-dominated to background-dominated, removing the otherwise long-range $$ds^2 = -c^2\,dT^2 + R(T)^2\left[d\chi^2 + \chi^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right],$$0 tail of the Schwarzschild potential and “shielding” the embedded mass (Kantowski et al., 2012, Kantowski et al., 2013).

2. Embedded Lens Equation and Bending Angle

The lens equation for the embedded point-mass differs from the textbook isolated PL form: $$ds^2 = -c^2\,dT^2 + R(T)^2\left[d\chi^2 + \chi^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right],$$1 to incorporate the finite range of the gravitational field. For a void-centered point mass, the leading-order bending angle acquires a multiplicative factor that tends to zero at the boundary, suppressing the infinite-range behavior: $$ds^2 = -c^2\,dT^2 + R(T)^2\left[d\chi^2 + \chi^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right],$$2 where $$ds^2 = -c^2\,dT^2 + R(T)^2\left[d\chi^2 + \chi^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right],$$3 is the physical void radius, $$ds^2 = -c^2\,dT^2 + R(T)^2\left[d\chi^2 + \chi^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right],$$4, and $$ds^2 = -c^2\,dT^2 + R(T)^2\left[d\chi^2 + \chi^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right],$$5 is the angular-diameter distance to the lens (Kantowski et al., 2013). More generally, for parametric orbit angle $$ds^2 = -c^2\,dT^2 + R(T)^2\left[d\chi^2 + \chi^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right],$$6, the embedded bending is

$$ds^2 = -c^2\,dT^2 + R(T)^2\left[d\chi^2 + \chi^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right],$$7

where $$ds^2 = -c^2\,dT^2 + R(T)^2\left[d\chi^2 + \chi^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right],$$8 is the minimum radius from the lens to the photon trajectory (Chen et al., 2011, Kantowski et al., 2012).

The lens equation in the embedded context becomes, in the small-angle regime,

$$ds^2 = -c^2\,dT^2 + R(T)^2\left[d\chi^2 + \chi^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right],$$9

highlighting the shielding effect.

3. Image Properties: Positions, Magnification, and Ellipticity

Solutions to the embedded lens equation yield image positions ($$ds^2 = -\gamma(r)^{-2} c^2 dt^2 + \gamma(r)^2 dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2),$$0) and signed magnifications ($$ds^2 = -\gamma(r)^{-2} c^2 dt^2 + \gamma(r)^2 dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2),$$1) that differ from the standard lens at the sub-percent level for typical strong lensing, but can have $$ds^2 = -\gamma(r)^{-2} c^2 dt^2 + \gamma(r)^2 dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2),$$2 effects for weak lensing at large impact parameters or for cluster-scale lenses (Kantowski et al., 2012, Chen et al., 2011, Kantowski et al., 2013). The magnification eigenvalues in the tangential and radial directions, $$ds^2 = -\gamma(r)^{-2} c^2 dt^2 + \gamma(r)^2 dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2),$$3 and $$ds^2 = -\gamma(r)^{-2} c^2 dt^2 + \gamma(r)^2 dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2),$$4, encapsulate the embedding corrections via dependence on $$ds^2 = -\gamma(r)^{-2} c^2 dt^2 + \gamma(r)^2 dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2),$$5.

For instance, the two-image solutions (small angles, lowest order) are: $$ds^2 = -\gamma(r)^{-2} c^2 dt^2 + \gamma(r)^2 dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2),$$6 with $$ds^2 = -\gamma(r)^{-2} c^2 dt^2 + \gamma(r)^2 dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2),$$7, and the mapping for a circular source results in an ellipse with principal-axis ratio $$ds^2 = -\gamma(r)^{-2} c^2 dt^2 + \gamma(r)^2 dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2),$$8 and eccentricity

$$ds^2 = -\gamma(r)^{-2} c^2 dt^2 + \gamma(r)^2 dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2),$$9

where deviations from circularity are induced exclusively by the embedding (Chen et al., 2011).

Convergence $\gamma^{-1}(r) = \sqrt{1 - \beta^2(r)}$0 and shear $\gamma^{-1}(r) = \sqrt{1 - \beta^2(r)}$1 in the weak-lensing regime are also modified: $\gamma^{-1}(r) = \sqrt{1 - \beta^2(r)}$2 with negative $\gamma^{-1}(r) = \sqrt{1 - \beta^2(r)}$3 reflecting the absence of background mass within the void for both weak and strong lensing (Kantowski et al., 2012).

4. Time Delay Structure and Potential Corrections

The time-delay function in the embedded PL model includes both geometric and potential terms. The embedded “Fermat potential” (projected lens potential) is given by: $\gamma^{-1}(r) = \sqrt{1 - \beta^2(r)}$4 where to lowest order,

$\gamma^{-1}(r) = \sqrt{1 - \beta^2(r)}$5

In the strong-lensing regime, the geometric part of the delay is nearly unaffected, but the potential part $\gamma^{-1}(r) = \sqrt{1 - \beta^2(r)}$6 can deviate by up to 4% for rich-cluster masses (Kantowski et al., 2012). Embedding always reduces the potential due to the cut-off at radius $\gamma^{-1}(r) = \sqrt{1 - \beta^2(r)}$7 (Kantowski et al., 2013). For time delays between images of fast radio bursts (FRBs) or gravitational waves, the standard PL formulas for $\gamma^{-1}(r) = \sqrt{1 - \beta^2(r)}$8 (via the “Fermat potential” or explicit expressions involving $\gamma^{-1}(r) = \sqrt{1 - \beta^2(r)}$9 and the flux ratio $\beta^2(r) = r_s/r + \Lambda r^2/3$0) can be applied, but corrections are required to interpret the physical meaning of the observed parameters (Chen et al., 2021, Wright et al., 2023, Goyal et al., 19 Dec 2025).

5. Observable Consequences and Astrophysical Inference

Embedding has several key observational consequences:

• Leading-order image positions and fluxes are nearly unaffected ($\beta^2(r) = r_s/r + \Lambda r^2/3$1 for galaxy-scale lenses), but can differ at the $\beta^2(r) = r_s/r + \Lambda r^2/3$2–$\beta^2(r) = r_s/r + \Lambda r^2/3$3 level for clusters or at large angular separations (Chen et al., 2011, Kantowski et al., 2012).
• For weak lensing, shape distortions and the negative surface mass density $\beta^2(r) = r_s/r + \Lambda r^2/3$4 become detectable; fractional corrections to shear and ellipticity exceed $\beta^2(r) = r_s/r + \Lambda r^2/3$5 for impact angles $\beta^2(r) = r_s/r + \Lambda r^2/3$6 (Kantowski et al., 2012).
• Time-delay potential terms differ at the few-percent level for massive cluster lenses, potentially shifting lag estimates in lensed FRBs or GW events (Kantowski et al., 2012, Goyal et al., 19 Dec 2025).
• Embedding reduces the inferred microlens mass compared to isolated PL analysis in lensed GW events; for example, GW231123 is more consistently explained with an embedded PL, yielding typical microlens masses $\beta^2(r) = r_s/r + \Lambda r^2/3$7–$\beta^2(r) = r_s/r + \Lambda r^2/3$8 (Goyal et al., 19 Dec 2025).
• Shielding and finite-field effects appear as a global cut-off of the lens potential, meaning lensing does not accumulate outside the void region (Kantowski et al., 2012, Kantowski et al., 2013).

A table summarizes characteristics of the embedded and conventional PL models:

Feature	Conventional PL	Embedded PL (Kottler/Swiss-cheese)
Potential	$\beta^2(r) = r_s/r + \Lambda r^2/3$9 to $r_s = 2Gm/c^2$0	$r_s = 2Gm/c^2$1 cutoff at $r_s = 2Gm/c^2$2
Deflection	$r_s = 2Gm/c^2$3 ($r_s = 2Gm/c^2$4)	$r_s = 2Gm/c^2$5
Surface mass	Always $r_s = 2Gm/c^2$6	Negative $r_s = 2Gm/c^2$7 in the void
Time delay	Standard	Extra $r_s = 2Gm/c^2$8 corrections

6. Generalizations and Mathematical Extensions

The mathematical study of embedded PL models extends naturally to $r_s = 2Gm/c^2$9-point mass systems, leading to complex lens equations that incorporate shear and distribution asymmetries: $r_b = R(T)\chi_b$0 or with external shear $r_b = R(T)\chi_b$1,

$r_b = R(T)\chi_b$2

with detailed theorems quantifying the increase in image multiplicity upon adding small masses to maximal lenses, reaching established rigorous upper bounds such as $r_b = R(T)\chi_b$3 (no shear) or $r_b = R(T)\chi_b$4 (with shear) images for $r_b = R(T)\chi_b$5 point masses (Sète et al., 2014).

Theoretical advances detail conditions for achieving maximal image numbers, the effect of sense-reversing/sense-preserving images, and provide constructive recipes for asymmetric, generic maximal $r_b = R(T)\chi_b$6-point lenses.

7. Cosmological Corrections and Significance

Cosmological expansion enters at sub-dominant order in the deflection angle. Calculations using the McVittie metric demonstrate that leading-order bending matches the standard Schwarzschild value ($r_b = R(T)\chi_b$7), with expansion- or Hubble-flow-induced corrections appearing only at $r_b = R(T)\chi_b$8 for typical strong-lens configurations—orders of magnitude below any current observational uncertainty (Piattella, 2016).

A plausible implication is that “Swiss-cheese” lensing constructions, in which embedded condensed masses replace comoving spheres in FLRW backgrounds, validate the use of conventional lensing observables for most practical purposes. Only ultra-high-precision weak lensing or time-delay cosmology approaches in cluster environments, or next-generation observatories, may be able to detect tiny cosmological imprints unique to truly embedded mass profiles.

• (Piattella, 2016): Analysis of effects of cosmological expansion on embedded point-mass lensing
• (Chen et al., 2011): Embedded lens equation and analytic corrections to image properties
• (Kantowski et al., 2012): Detailed image properties, negative convergence, time-delay analysis
• (Kantowski et al., 2013): Simplified time-delay function, shielding, explicit lens-law corrections
• (Sète et al., 2014): Mathematical theory of image creation by embedding new point masses
• (Chen et al., 2021): Embedded PL modeling in FRB lensing, mass estimation from observables
• (Wright et al., 2023): Point-mass lens formalism in GW lensing
• (Goyal et al., 19 Dec 2025): Embedded PL in wave-optics analysis for black hole GW lensing events