Distance-to-Distance Ratio (DDR) in Cosmology & NLP

• DDR is defined as the ratio of angular diameter distances in gravitational lensing and as a similarity metric in NLP, capturing key geometric and semantic changes.
• In cosmology, DDR links lensing observables to the cosmic expansion rate, enabling constraints on dark energy and other fundamental parameters.
• In NLP, DDR measures token-wise embedding changes to detect subtle semantic variations, offering advantages over traditional cosine similarity.

The Distance-to-Distance Ratio (DDR) is a term that appears in two technically distinct domains: (1) observational cosmology, where it refers to a geometric ratio derived from angular diameter distances in strong gravitational lensing systems, and (2) computational linguistics, where it denotes a similarity metric for LLM embeddings. This article focuses primarily on the cosmological sense, referencing results where the DDR is a fundamental observable in lensing-based cosmological parameter estimation, but also includes a technical description of DDR as a rate-of-change similarity metric for natural language processing, as established in recent literature.

1. Definition and Theoretical Foundations

The distance-to-distance ratio (DDR), in the context of strong gravitational lensing, is defined as the ratio of the observer–source angular diameter distance $D_s$ to the lens–source angular diameter distance $D_{ls}$, both evaluated at the same source redshift: $\mathcal{D}(z_l, z_s) = \frac{D_s}{D_{ls}}$ where $z_l$ and $z_s$ are the redshifts of the lens and source, respectively (Cardone et al., 2015). In standard cosmological models, these angular diameter distances are defined as: $$D_{ij} = \frac{c}{1 + z_j} \int_{z_i}^{z_j} \frac{dz}{H(z)}$$ In most lensing analyses, DDR appears as the inverse of the “distance ratio” $D_{ls}/D_s$ that governs the scaling of the Einstein radius and thus the lensing geometry. The DDR succinctly encodes the effect of cosmological expansion and the background expansion rate $H(z)$.

In computational linguistics, DDR quantifies the local rate of change in semantic similarity induced by an LLM's contextual transformation. For token-embedded sequences $x$, $x'$ and transformation $F$,

$$\mathrm{DDR}(t,t') = \frac{d_{\rm in}(x,x')}{d_{\rm out}(F(x), F(x'))}$$

with $d_{\rm in}, d_{\rm out}$ denoting maximum chordal distances before and after context application (Qureshi et al., 25 Jan 2026).

2. Observational Role in Cosmology

In gravitational lensing, the DDR is not only a theoretical construct but a directly computable observable from empirical data. The inference proceeds from measurement of the Einstein radius $R_E$ and central velocity dispersion $\sigma_{ap}$. The critical surface density for lensing involves the ratio $D_s/D_{ls}$, and observable quantities such as

$$M_{\rm proj}^{\rm obs}(R_E) = \pi\,\Sigma_{\rm crit}(z_l, z_s)\,R_E^2$$

where

$$\Sigma_{\rm crit} = \frac{c^2}{4\pi G} \frac{D_s}{D_l D_{ls}}$$

Here, $D_l$ is the observer–lens distance. Therefore, the DDR explicitly connects lensing observables and cosmological distances through the lens equation (Cardone et al., 2015).

In two-component modeling, the DDR becomes: $$\mathcal{D}(z_l, z_s) = \mathcal{D}_{\rm obs} \times \mathcal{F}_E$$ where $\mathcal{D}_{\rm obs}$ is the directly measured term and $\mathcal{F}_E$ aggregates astrophysical systematics and model-dependent “nuisance” parameters.

3. Methodological Considerations and Error Forecasting

Robust cosmological inference from DDRs requires explicit attention to lens modeling, redshift binning, and systematic marginalization. Cardone et al. (Cardone et al., 2015) propagate distance uncertainties by incorporating statistical error from observational quantities, such as: $$\frac{\sigma_d}{\mathcal{D}_{\rm obs,d}} = \sqrt{4\varepsilon_E^2 + \varepsilon_{eff}^2 + 4\varepsilon_0^2}$$ where $\varepsilon_E$ and $\varepsilon_0$ are fractional errors on Einstein radius and velocity dispersion, respectively.

The nuisance parameter $\mathcal{F}_E$ is averaged in redshift bins, with fractional systematic scatter $\Delta \sim 20\%$ nearly independent of galaxy or halo properties. Forecasting for cosmological parameters employs Fisher matrix techniques, marginalizing over these systematics to deliver uncertainties on $\Omega_M$, $w_0$, and $w_a$, and identifying the pivot equation-of-state parameter $w_p$ where the constraints are optimized.

A summary table from (Cardone et al., 2015), for 1000 and 10000 lens samples, illustrates achievable accuracy when treating all parameters as free:

Number of lenses	$\sigma(\Omega_M)$	$\sigma(w_0)$	$\sigma(w_a)$
1000	0.09	0.45	2.5
10000	0.03	0.15	0.8

This quantifies the impact of DDR precision on cosmological parameter estimation, especially at low redshift where lenses concentrate.

4. DDR in the Context of Etherington's Relation and its Extensions

While the DDR of lensing is a specific observable, it must be conceptually distinguished from the distance duality relation (also abbreviated DDR), central to cosmology: $D_L(z) = (1+z)^2 D_A(z)$ Etherington’s reciprocity theorem underpins this, dictating that in a metric spacetime under photon number conservation, the ratio $D_L/(1+z)^2 D_A$ is unity. However, the DDR as defined in lensing analyses $\mathcal{D}(z_l, z_s) = D_s/D_{ls}$ arises purely from geometric considerations and does not, by itself, test for photon number conservation or new physics violating the duality relation (Liao et al., 2015).

Careful cosmological application sometimes aggregates both the lens DDR and the duality relation, where violation of the latter requires modifications to standard modeling and provides a potential probe of new physics.

5. Robustness, Limitations, and Future Prospects

The main technical limitation of DDR-based lensing cosmology arises from uncertainties in lens profile modeling, redshift-dependent calibration of lens and source populations, and unknowns in two-component mass-to-light ratios. As demonstrated in (Cardone et al., 2015), while increasing the sample size (to $10^4$ lenses) and reducing measurement uncertainties can bring sub-percent accuracy, systematic errors—particularly in the nuisance term $\mathcal{F}_E$—remain the dominant limitation.

The DDR in lensing is resilient to many details of the dark matter halo parameterization, as these contribute additively to $\mathcal{F}_E$ and are marginalized in the statistical analysis. However, accurate cosmology requires careful control of evolutionary and instrumental effects across redshift bins.

The prospect of massive strong lensing surveys (e.g., from Euclid, LSST) and improvements in velocity-dispersion measurements imply that DDR-based constraints will become increasingly competitive for low-redshift cosmological inference.

6. DDR in Natural Language Processing

A technically distinct use of DDR appears in natural language similarity assessment (Qureshi et al., 25 Jan 2026). Here, DDR quantifies the contraction or expansion of token-wise distances under the contextual transformation of a LLM. The input and output distances for a pair of sentences (original and perturbed) are computed as maximum token-wise chordal distances. DDR is the ratio: $$\mathrm{DDR}(t,t') = \frac{\max_{i} d_{\rm chord}(x_i, x'_i)}{\max_{i} d_{\rm chord}(y_i, y'_i)}$$ where $x_i$, $x'_i$ are input token embeddings, $y_i$, $y'_i$ are the corresponding contextual embeddings, and $d_{\rm chord}(\cdot, \cdot)$ is chordal (angular) distance on the unit sphere.

Empirical studies indicate that DDR robustly distinguishes small, semantically-preserving edits (such as synonym replacements) from random, disruptive substitutions, surpassing cosine-similarity metrics on pooled embeddings in its ability to separate these cases.

7. Terminological Caution and Research Frontiers

The term DDR or distance-to-distance ratio thus sits at the intersection of gravitational lensing cosmology and data-driven embedding geometry, each with context-sensitive definitions and formulae. In cosmology, DDR provides a vehicle for cosmological parameter inference and enables systematics-aware analyses when statistical marginalizations and astrophysical modeling are carefully implemented (Cardone et al., 2015). In NLP, DDR serves as a theoretically-motivated, practical metric for fine-grained semantic change detection in text representations (Qureshi et al., 25 Jan 2026).

For lensing cosmology, ongoing improvement in lens sample size, redshift coverage, and control of mass profile nuisance will continue to enhance the utility and precision of the distance-to-distance ratio in probing dark energy, spatial curvature, and alternative cosmological models. In embedding-based semantic similarity, refinements of DDR for variable-length texts, alternative distance metrics, and theoretical connections to Lipschitz continuity in neural architectures remain active areas of investigation.