Small-Scale Galaxy-Galaxy Lensing

• Small-scale galaxy-galaxy lensing is a measurement technique that studies distortions in background galaxies to reveal the galaxy–matter connection at scales from kiloparsecs to megaparsecs.
• It combines weak and strong lensing approaches to decompose signals into one-halo and two-halo contributions, enabling detailed investigations of halo profiles, baryonic modifications, and multi-deflector effects.
• Forward modeling methods, including emulator-based analyses and point-mass marginalization, enhance its cosmological applications by controlling non-local uncertainties and sample selection biases.

Searching arXiv for recent and foundational papers on small-scale galaxy-galaxy lensing. arxiv.search({"query":"small-scale galaxy-galaxy lensing excess surface density baryonic effects emulator covariance point-mass marginalization", "max_results": 10}) Small-scale galaxy-galaxy lensing is the regime in which the coherent distortion of background galaxy shapes by foreground galaxies is used to probe the galaxy–matter connection from the inner halo to the few-megaparsec environment, and, in strong-lensing applications, down to sub-galactic scales. In weak-lensing analyses the fundamental observables are the tangential shear $\gamma_t$ and the excess surface density $\Delta\Sigma$, while in strong-lensing analyses the relevant signal can appear as perturbations to Einstein rings, arcs, or highly magnified multiple images. Across current literature, “small-scale” is therefore context-dependent: it includes the $\sim 50$–$500\,\mathrm{kpc}$ regime where baryonic feedback modifies $\Delta\Sigma$, the $0.15$–$1.7\,\mathrm{Mpc}/h$ scales targeted in recent joint clustering+lensing cosmological analyses, and the $\sim 0.5$–$5\,\mathrm{kpc}$ scales accessed statistically through strong-lensing surface-brightness anomalies (Gouin et al., 2019, Mahony et al., 2 Jul 2025, Bayer et al., 2023).

1. Observable structure and scale dependence

In the standard weak-lensing formulation, the critical surface density is

$$\Sigma_c = \frac{c^2}{4\pi G}\,\frac{D_s}{D_\ell D_{\ell s}},$$

and the tangential shear around a lens is related to the excess surface density by

$\Delta\Sigma$0

with

$\Delta\Sigma$1

Operationally, one often measures the galaxy–shear angular correlation function

$\Delta\Sigma$2

which, when averaged over many lens–source pairs in narrow redshift bins, satisfies $\Delta\Sigma$3 (Gouin et al., 2019).

The same observable can be written in terms of the galaxy–matter cross-correlation. One commonly used form is

$\Delta\Sigma$4

so that $\Delta\Sigma$5 (Mahony et al., 2 Jul 2025). This makes explicit that galaxy-galaxy lensing is a filtered projection of $\Delta\Sigma$6.

The label “small-scale” varies across subfields. In joint weak-lensing and clustering analyses it can mean $\Delta\Sigma$7, with the smallest separations sometimes excised in fiducial analyses to avoid baryonic effects (Mahony et al., 2 Jul 2025). In a simulation-based BOSS analysis it denotes lensing from $\Delta\Sigma$8 to $\Delta\Sigma$9, chosen to balance signal-to-noise against modeling and observational systematics (Gao et al., 19 Jan 2026). In strong-lensing work on anomaly power spectra it refers to length scales from $\sim 50$0 to $\sim 50$1, corresponding to $\sim 50$2 to $\sim 50$3 (Bayer et al., 2023). Cluster-member strong lenses can probe projected masses within effective radii of $\sim 50$4 and $\sim 50$5 (Grillo et al., 2014).

2. Halo-scale signal content on tens of kiloparsecs to megaparsecs

Within halo-model descriptions, the galaxy-galaxy lensing signal is decomposed into one-halo and two-halo terms,

$\sim 50$6

In the methodology developed for DES forecasts, the one-halo term is dominant for “small” scales, $\sim 50$7, while the two-halo term governs the linear-bias regime at $\sim 50$8 (Park et al., 2015). This decomposition underlies the interpretation of small-scale measurements as direct probes of halo mass, concentration, and satellite structure.

Hydrodynamical lightcone simulations show that baryons affect these scales at an amplitude relevant for precision analyses. In Horizon-AGN, full-physics predictions differ from dark-matter-only expectations by up to $\sim 50$9–$500\,\mathrm{kpc}$0 on $500\,\mathrm{kpc}$1–$500\,\mathrm{kpc}$2; at $500\,\mathrm{kpc}$3 the steepened stellar cusp boosts the signal by tens of percent, while at intermediate radii $500\,\mathrm{kpc}$4–$500\,\mathrm{kpc}$5 AGN-heated gas suppresses the lensing by $500\,\mathrm{kpc}$6–$500\,\mathrm{kpc}$7. The same simulation finds that the signal for a massive galaxy population at $500\,\mathrm{kpc}$8 agrees with observations from $500\,\mathrm{kpc}$9 to $\Delta\Sigma$0 within the statistical quadrants’ dispersion, and that at $\Delta\Sigma$1 magnification bias can increase the observed $\Delta\Sigma$2 by $\Delta\Sigma$3–$\Delta\Sigma$4 beyond $\Delta\Sigma$5 (Gouin et al., 2019).

A distinct small-scale effect is the prevalence of multiple weak deflections. Monte Carlo calculations with truncated isothermal-sphere halos show that the closest lens in projection is not the only lens for a given source, and that the closest lens is not the most important lens about $\Delta\Sigma$6 of the time. For a broad source-redshift distribution with $\Delta\Sigma$7, the probability of at least two deflections is $\Delta\Sigma$8 for $\Delta\Sigma$9, and full multiple-deflection calculations yield higher net shear for individual sources and larger mean tangential shear than the nearest-lens approximation. Galaxy-galaxy lensing can also contribute substantially to cosmic shear on arcminute scales, with the contribution decreasing rapidly and extrapolating to zero at scales of order $0.15$0 (Brainerd, 2010). A common misconception is therefore that small-scale galaxy-galaxy lensing is determined by a single nearest halo; in these calculations the observable is generically multi-deflector.

3. Forward modeling for cosmology and the galaxy–halo connection

Small-scale galaxy-galaxy lensing is now embedded in forward models that tie cosmological parameters to galaxy occupation, assembly bias, and survey selection. In analytic halo-occupation approaches for DES forecasts, luminosity-threshold samples are described with central and satellite occupations

$0.15$1

and the same HOD controls both the one-halo lensing term and the large-scale bias inferred from clustering (Park et al., 2015).

A key result from BOSS LOWZ and CMASS is that one-halo-scale discrepancies can arise from sample selection rather than from a breakdown of the halo paradigm. In the extended iHOD analysis of the stellar mass function, clustering, and lensing, a halo-mass-dependent satellite detection fraction $0.15$2 eliminates the original $0.15$3 “lensing is low” discrepancy below $0.15$4, whereas above $0.15$5 a residual $0.15$6–$0.15$7 mismatch remains at fixed Planck $0.15$8CDM (Zu, 2020). This indicates that small-scale galaxy-galaxy lensing is acutely sensitive to the detailed mapping between observed galaxies and their host halos.

Simulation-based emulators extend this program into higher-dimensional cosmological spaces. The Mira-Titan emulators are trained on 111 $0.15$9-body simulations spanning eight cosmological parameters, including neutrino mass and dynamical dark energy, and model $1.7\,\mathrm{Mpc}/h$0 together with projected and redshift-space clustering. Their reported accuracy for $1.7\,\mathrm{Mpc}/h$1 is $1.7\,\mathrm{Mpc}/h$2 on $1.7\,\mathrm{Mpc}/h$3–$1.7\,\mathrm{Mpc}/h$4, and realistic mock-catalog tests recover unbiased cosmology within $1.7\,\mathrm{Mpc}/h$5 when scale cuts avoid model-dependent regimes (Kwan et al., 2023). In a separate Aemulus $1.7\,\mathrm{Mpc}/h$6 analysis of BOSS, emulators built from $1.7\,\mathrm{Mpc}/h$7CDM simulations with massive neutrinos combine clustering from $1.7\,\mathrm{Mpc}/h$8 to $1.7\,\mathrm{Mpc}/h$9 and lensing from $\sim 0.5$0 to $\sim 0.5$1; the addition of lensing significantly improves the constraining power on $\sim 0.5$2 while only weakly improving $\sim 0.5$3 (Gao et al., 19 Jan 2026).

Extended subhalo abundance matching provides another route to small-scale inference. In SHAMe, the observable is modeled by measuring $\sim 0.5$4 directly from mock catalogs and then forming $\sim 0.5$5 and $\sim 0.5$6. The fiducial analysis excludes $\sim 0.5$7 to avoid baryonic effects, but an all-scale analysis that includes those separations yields $\sim 0.5$8 and $\sim 0.5$9, improving the precision by $5\,\mathrm{kpc}$0 on $5\,\mathrm{kpc}$1 and $5\,\mathrm{kpc}$2 on $5\,\mathrm{kpc}$3 relative to the fiducial case (Mahony et al., 2 Jul 2025). This suggests that the statistical leverage of small-scale galaxy-galaxy lensing is substantial when the galaxy–halo connection is modeled flexibly enough.

4. Non-locality, scale cuts, and localization schemes

A defining theoretical complication of small-scale galaxy-galaxy lensing is non-locality. Because

$5\,\mathrm{kpc}$4

the prediction at separation $5\,\mathrm{kpc}$5 depends on the projected mass distribution at all smaller radii. In the point-mass formulation, the uncertain contribution from unresolved inner scales can be isolated as

$5\,\mathrm{kpc}$6

where $5\,\mathrm{kpc}$7 is proportional to the enclosed projected mass below a chosen $5\,\mathrm{kpc}$8. For Gaussian covariance $5\,\mathrm{kpc}$9, analytic marginalization over $$\Sigma_c = \frac{c^2}{4\pi G}\,\frac{D_s}{D_\ell D_{\ell s}},$$0 updates the covariance to

$$\Sigma_c = \frac{c^2}{4\pi G}\,\frac{D_s}{D_\ell D_{\ell s}},$$1

so that small-scale uncertainty can be carried through likelihood analyses without explicitly sampling large numbers of nuisance parameters (MacCrann et al., 2019).

A complementary strategy is to transform the observable itself. The localized statistic of Baldauf et al. is

$$\Sigma_c = \frac{c^2}{4\pi G}\,\frac{D_s}{D_\ell D_{\ell s}},$$2

with a discrete linear operator $$\Sigma_c = \frac{c^2}{4\pi G}\,\frac{D_s}{D_\ell D_{\ell s}},$$3 acting on the measured $$\Sigma_c = \frac{c^2}{4\pi G}\,\frac{D_s}{D_\ell D_{\ell s}},$$4 or $$\Sigma_c = \frac{c^2}{4\pi G}\,\frac{D_s}{D_\ell D_{\ell s}},$$5 vector. By construction, the transformed quantity depends only on scales $$\Sigma_c = \frac{c^2}{4\pi G}\,\frac{D_s}{D_\ell D_{\ell s}},$$6 and permits more aggressive small-scale cuts for a given physical model (Park et al., 2020).

A third widely used localization is the annular differential surface density or $$\Sigma_c = \frac{c^2}{4\pi G}\,\frac{D_s}{D_\ell D_{\ell s}},$$7 statistic,

$$\Sigma_c = \frac{c^2}{4\pi G}\,\frac{D_s}{D_\ell D_{\ell s}},$$8

which exactly removes contributions from $$\Sigma_c = \frac{c^2}{4\pi G}\,\frac{D_s}{D_\ell D_{\ell s}},$$9 (Prat et al., 2022). A direct comparison of $\Delta\Sigma$00, the $\Delta\Sigma$01-transform, and point-mass marginalization finds equivalent cosmological results in an LSST Year-1-like setup and when applied to DES Y3 data; relative to larger scale cuts without mitigation, the schemes yield $\Delta\Sigma$02 times more constraining $\Delta\Sigma$03 results in the LSST-like case (Prat et al., 2022).

In DES Y6, point-mass marginalization is part of the adopted small-scale control strategy. The nominal galaxy-galaxy lensing signal-to-noise ratio is $\Delta\Sigma$04, reduced to $\Delta\Sigma$05 for the linear-bias scale cuts and $\Delta\Sigma$06 for the nonlinear-bias scale cuts; after point-mass marginalization the corresponding values are $\Delta\Sigma$07 and $\Delta\Sigma$08. Although not used in the main cosmological analysis, the small-angular-scale signal is sufficiently precise to extract geometric shear-ratio measurements that provide an internal consistency test of source-redshift distributions and shear calibration (Giannini et al., 21 Jan 2026). The practical lesson is not that small scales must always be discarded, but that their use depends on explicit localization or controlled marginalization.

5. Covariance, survey systematics, and measurement status

The covariance of small-scale galaxy-galaxy lensing is nontrivial even before observational systematics are considered. In Millennium-XXL mock catalogs, Gaussian plus shot-noise predictions describe covariance well at very small scales, where shot noise dominates, and at large scales, where Gaussian terms dominate, but on intermediate scales $\Delta\Sigma$09 the predicted errors are too small by factors of $\Delta\Sigma$10–$\Delta\Sigma$11. The same study finds a non-zero cross-covariance between galaxy-galaxy lensing and projected clustering, with correlation coefficients reaching $\Delta\Sigma$12 for some large-scale bin pairs (Marian et al., 2014). Neglecting this structure leads to overly optimistic or biased joint constraints.

DES Y3 revealed an additional phenomenological issue. In the fiducial redMaGiC sample, the ratio of large-scale bias inferred from lensing to that inferred from clustering is $\Delta\Sigma$13, a $\Delta\Sigma$14 departure from unity interpreted as a de-correlation between galaxies and mass on large scales. The effect is traced to a color-dependent photometric issue in the redMaGiC selection; redefining the lens sample with a broader $\Delta\Sigma$15 cut yields $\Delta\Sigma$16 across bins and cosmology consistent with MagLim (Pandey et al., 2021). This is a cautionary example that apparent small-scale or intermediate-scale anomalies need not be of cosmological origin.

DESI DR1 provides a multi-survey measurement baseline for current-generation small-scale analyses. Tangential shear and $\Delta\Sigma$17 are measured in 15 logarithmic $\Delta\Sigma$18 bins from $\Delta\Sigma$19 to $\Delta\Sigma$20, with a small-scale cut $\Delta\Sigma$21 to avoid blending. Tests find no significant trends with DESI lens-galaxy properties, but a significant trend with average source redshift appears and vanishes once shifts $\Delta\Sigma$22 and $\Delta\Sigma$23 are applied to the HSC-Y3 redshift distributions. The observed scatter exceeds the theoretical covariance primarily on small scales $\Delta\Sigma$24; on larger scales the scatter is consistent with theory at the $\Delta\Sigma$25 level (Heydenreich et al., 26 Jun 2025).

DES Y6 illustrates the present statistical maturity of the field. The survey covers $\Delta\Sigma$26, uses a MagLim++ lens sample of $\Delta\Sigma$27 million galaxies in six redshift bins and a Metadetection source sample of $\Delta\Sigma$28 million galaxies in four bins, and reaches total galaxy-galaxy lensing signal-to-noise ratio $\Delta\Sigma$29, a $\Delta\Sigma$30 improvement over DES Y3. Validation includes boost-factor corrections, random-point subtraction, PSF residual tests, and cross-shear null tests, all consistent with the statistical requirements of the Y6 analysis (Giannini et al., 21 Jan 2026). Small scales carry roughly half of the total signal-to-noise, which explains both their statistical appeal and their outsized sensitivity to covariance and calibration errors.

6. Strong-lensing, sub-galactic structure, and higher-order extensions

On sub-galactic scales, galaxy-galaxy strong lensing provides a complementary statistical probe of matter structure. In the anomaly-power-spectrum approach, the observed residual field is

$\Delta\Sigma$31

and its azimuthally averaged power spectrum is

$\Delta\Sigma$32

Applied to high-resolution HST/WFC3 F390W imaging of four SLACS lenses, the corrected residual power exceeds the noise only in the lowest three $\Delta\Sigma$33 bins from $\Delta\Sigma$34 to $\Delta\Sigma$35, corresponding to length scales from $\Delta\Sigma$36 to $\Delta\Sigma$37. No significant high-$\Delta\Sigma$38 anomalies are detected. Interpreted through Gaussian-random-field potential-perturbation models, the measurements imply $\Delta\Sigma$39 (2$\Delta\Sigma$40) at $\Delta\Sigma$41, corresponding to a few-percent mass fraction in substructures of $\Delta\Sigma$42–$\Delta\Sigma$43 within the Einstein region (Bayer et al., 2023).

A different strong-lensing application is the use of cluster-member galaxies as low-mass lenses. In MACS J1206.2–0847, a double background source at $\Delta\Sigma$44 is strongly lensed into ten images by two early-type cluster members at $\Delta\Sigma$45. Modeling the galaxies and the cluster with singular isothermal profiles yields effective velocity dispersions of $\Delta\Sigma$46 and $\Delta\Sigma$47, projected total masses of $\Delta\Sigma$48 within $\Delta\Sigma$49 and $\Delta\Sigma$50 within $\Delta\Sigma$51, and projected stellar-to-total mass fractions $\Delta\Sigma$52 and $\Delta\Sigma$53 (Grillo et al., 2014). These measurements extend galaxy-galaxy strong-lensing mass studies to lower velocity dispersions and luminous masses than typical SLACS field lenses.

Higher-order generalizations aim to capture non-Gaussian galaxy–mass correlations not present in the standard two-point signal. Fourth-order galaxy-galaxy lensing defines a four-point function between the shear and triplets of foreground galaxies and the corresponding aperture statistic $\Delta\Sigma$54. Using compensated filters

$\Delta\Sigma$55

one can evaluate the statistic either through multidimensional integration or through a direct FFT-based estimator on pixelized data. In a mock Stage IV survey with $\Delta\Sigma$56, the connected part of $\Delta\Sigma$57 is detected with a signal-to-noise ratio of roughly nine on small aperture scales (Oel et al., 20 Apr 2026). This extends the small-scale galaxy-galaxy lensing program from halo profiles and two-point cosmography toward direct measurements of non-Gaussian galaxy–mass structure.