Cosmology with the line-of-sight shear of strong gravitational lenses

This paper proposes utilizing the line-of-sight shear of strong gravitational lenses detected in Stage-IV surveys as a new cosmological probe, demonstrating through analytical derivations and covariance analysis that this approach significantly enhances the standard $3\times 2$pt correlation method to a$6\times 2$pt scheme with high signal-to-noise ratio and potential for mitigating systematic errors.

The Gist

Imagine the Universe as a giant, invisible ocean. Most of the time, this ocean is calm, but sometimes it has ripples, currents, and hidden underwater mountains made of dark matter. When light from a distant galaxy travels through this ocean to reach our telescopes, its path gets bent and twisted. This phenomenon is called gravitational lensing.

For decades, astronomers have used two main ways to study this ocean:

1. Weak Lensing (The Crowd): They look at billions of faint galaxies. Each one is slightly distorted, like a reflection in a funhouse mirror. By averaging the distortions of the whole crowd, they can map the invisible currents.
2. Strong Lensing (The Spotlight): Occasionally, a massive galaxy sits perfectly in front of a distant one, bending the light so much that it creates a perfect circle or a ring of light (an Einstein Ring). These are rare, spectacular events.

The Problem:
Until now, astronomers mostly treated these "Spotlights" (strong lenses) as just tools to measure the mass of the galaxy causing the ring. They ignored the subtle ripples in the water around the ring. Also, the "Crowd" method (Weak Lensing) has a major flaw: galaxies are naturally lumpy and misshapen. It's hard to tell if a galaxy looks weird because of the ocean's currents or just because it was born that way. This "shape noise" makes it hard to get a perfect map.

The New Idea: The "Line-of-Sight" Shear
This paper proposes a brilliant new trick. The authors say: "Let's look at the Spotlights again, but this time, let's measure the tiny, subtle distortions in the ring caused by the invisible currents between us, the lens, and the source."

They call this the Line-of-Sight (LOS) Shear.

Think of it this way:

• The Main Lens: Imagine a giant, heavy bowling ball sitting on a trampoline. It creates a deep dip.
• The Source: A marble rolling on the edge of that dip.
• The LOS Shear: Imagine there are tiny pebbles scattered on the trampoline around the bowling ball. They don't make a big dip, but they create tiny, subtle ripples in the fabric.

The authors realized that the shape of the Einstein Ring isn't just determined by the bowling ball; it's also slightly warped by those tiny pebbles. By measuring that specific warp, we can learn about the distribution of matter (the pebbles) all along the path of the light.

The Big Upgrade: From 3 to 6
Currently, cosmologists use a standard method called "3x2pt" (3 types of measurements x 2 points of data). It's like trying to solve a puzzle with three pieces:

1. Where galaxies are (Positions).
2. How galaxies are shaped (Ellipticities).
3. How galaxy positions affect galaxy shapes (Lensing).

This paper says: "Let's add a fourth piece to the puzzle!"
They propose adding the LOS Shear from the strong lenses. This creates a "6x2pt" scheme. Now we have:

• The old 3 pieces (Galaxy positions, shapes, and their mix).
• The new 3 pieces:
  1. LL: How the LOS shears of different strong lenses correlate with each other.
  2. LE: How the LOS shear of a strong lens correlates with the shape of a nearby galaxy.
  3. LP: How the LOS shear of a strong lens correlates with the position of a nearby galaxy.

Why is this a game-changer?

1. Different Flaws: The "Crowd" (Weak Lensing) is messed up by the natural shapes of galaxies. The "Spotlight" (Strong Lensing) is messed up by how we model the main lens. By combining them, the errors cancel each other out! It's like using two different types of thermometers to get the exact temperature; if one is stuck on "hot," the other might be stuck on "cold," but the average is perfect.
2. Super Precision: Even though strong lenses are rare (only about 1 in 10,000 galaxies), the signal they give us is incredibly strong. Measuring the LOS shear on one strong lens is like getting a clear, high-definition photo, whereas measuring the weak lensing of one galaxy is like looking at a blurry, grainy photo.
3. The Future: With upcoming telescopes like Euclid and LSST, we will find about 100,000 of these "Spotlights." The authors ran the numbers and found that even in the worst-case scenario, we will be able to detect these new signals with very high confidence.

The Bottom Line
This paper is a blueprint for a new era of cosmology. It suggests that by treating strong gravitational lenses not just as isolated objects, but as sensitive detectors of the invisible universe around them, we can build a much clearer, more accurate map of the Universe's dark matter and energy. It's like upgrading from a blurry, grainy black-and-white photo of the cosmos to a sharp, 4K color movie.

Technical Summary

Here is a detailed technical summary of the paper "Cosmology with the line-of-sight shear of strong gravitational lenses" by Fleury et al.

1. Problem Statement

Current Stage-IV photometric galaxy surveys (e.g., Euclid, LSST) are designed to constrain cosmological parameters using the $3 \times 2$pt correlation method. This method combines:

1. Galaxy Clustering (PP): Two-point correlation of galaxy positions.
2. Cosmic Shear (EE): Two-point correlation of galaxy ellipticities (weak lensing).
3. Galaxy-Galaxy Lensing (EP): Cross-correlation between galaxy positions and shapes.

While these surveys will detect billions of galaxies, they are also expected to detect over 100,000 strong gravitational lenses (multiple images, Einstein rings). Currently, strong lenses are primarily used for time-delay cosmography or mass modeling, but their potential as a statistical probe of large-scale structure is underutilized.

The core problem addressed is that strong lenses are subject to Line-of-Sight (LOS) shear ($\gamma_{LOS}$), a tidal distortion caused by matter inhomogeneities between the observer, the main deflector, and the source. While often treated as a nuisance parameter in lens modeling, $\gamma_{LOS}$ encodes cosmological information about the matter distribution. The paper asks: Can $\gamma_{LOS}$ be measured from strong lens images and used as a new cosmological observable to extend the standard $3 \times 2$pt scheme?

2. Methodology

A. Theoretical Framework: LOS Shear

The authors define the LOS shear as a complex parameter $\gamma_{LOS} = \gamma_{od} + \gamma_{os} - \gamma_{ds}$, where subscripts $o, d, s$ denote observer, deflector, and source.

• Physical Origin: It represents the net weak-lensing distortion of a strong-lensing image caused by tidal fields along the line of sight.
• Measurability: Unlike the main lens potential, $\gamma_{LOS}$ acts as a homogeneous tidal field. The authors argue that with sufficiently detailed lens models (beyond simple elliptical power-laws), $\gamma_{LOS}$ can be inferred from the distortion of Einstein rings or arcs, independent of the source morphology.

B. The $6 \times 2$pt Correlation Scheme

The paper proposes extending the standard observables by adding the LOS shear ($L$) to the existing set of galaxy positions ($P$) and shapes ($E$). This creates a $6 \times 2$pt scheme comprising six correlation functions:

1. LL: Autocorrelation of LOS shear.
2. LE: Cross-correlation between LOS shear and galaxy ellipticities.
3. LP: Cross-correlation between LOS shear and galaxy positions.
4. EE, EP, PP: The standard weak-lensing and clustering terms.

C. Estimators and Covariance Calculation

• Estimators: The authors define unbiased estimators for the LL, LE, and LP correlation functions in angular bins, accounting for noise ($\eta_{LOS}$) and intrinsic ellipticity ($\epsilon_0$).
• Analytical Covariance Matrix: A major technical contribution is the analytical derivation of the full covariance matrix for the $6 \times 2$pt scheme.
  • They distinguish between three types of uncertainty:
    1. Cosmic Covariance (CCov): Irreducible variance due to observing a single realization of the Universe.
    2. Noise Covariance (NCov): Arising from measurement errors ($\eta_{LOS}$) and intrinsic shape noise ($\epsilon_0$).
    3. Sparsity Covariance (SCov): A generalization of shot noise arising from the finite number of discrete tracers (lenses and galaxies).
  • Key Insight: For standard cosmic shear, shape noise dominates. However, for LOS shear, the measurement uncertainty is comparable to the signal ($\sigma_{LOS} \sim \gamma_{LOS}$). Consequently, Sparsity Covariance becomes a dominant and crucial term, comparable to noise covariance, which must be explicitly modeled.

D. Numerical Implementation

The authors developed a Python code, loscov, to compute the correlation functions and the full covariance matrix using Monte Carlo integration for high-dimensional integrals. They utilize Limber's approximation for angular power spectra and spherical harmonic decompositions.

3. Key Contributions

1. Formal Extension to $6 \times 2$pt: The paper provides the first complete theoretical framework for incorporating strong lens LOS shear into standard cosmological correlation analyses.
2. Analytical Covariance Derivation: They derived the full covariance matrix analytically, explicitly handling the "coincident indices" problem (where the same lens or galaxy appears in multiple pairs) which leads to the sparsity covariance term.
3. Identification of Sparsity Variance: The work highlights that for strong lenses, the "sparsity variance" (due to the finite number of lenses) is as significant as the measurement noise, a regime distinct from standard weak lensing.
4. Synergy and Systematics Mitigation: The authors argue that LOS shear is immune to intrinsic alignments (IA), a major systematic in cosmic shear. Combining $L$ with $E$ and $P$ offers a pathway to mitigate IA biases.

4. Results and Forecasts

The authors tested the detectability of the new observables under two scenarios based on the Euclid survey specifications ($\Omega = 15,000$ deg$^2$):

• Optimistic Scenario:

  • $L = 10^5$ lenses with measurement uncertainty $\sigma_{\eta_{LOS}} = 0.05$.
  • Result: All three new correlations (LL, LE, LP) are detectable with very high Signal-to-Noise Ratios (SNR).
  • The SNR for LP exceeds 100, and for LE exceeds 40 on arcminute scales.
  • The uncertainty budget is dominated by noise and sparsity, with cosmic variance being subdominant.

• Conservative Scenario:

  • $L = 10^4$ lenses (10x fewer) with higher uncertainty $\sigma_{\eta_{LOS}} = 0.1$.
  • Result: The autocorrelation LL becomes undetectable (SNR $\approx$ 0.5). However, the cross-correlations LE and LP remain detectable with high SNR due to the massive size of the galaxy sample ($G \approx 2 \times 10^9$).
  • Thresholds: A $5\sigma$detection of LP is possible with fewer than 200 lenses (if$\sigma_{\eta_{LOS}}=0.05$), and LE with$\approx 300$lenses. Even with extremely poor measurements ($\sigma_{\eta_{LOS}} \approx 1$), a Euclid-like sample of$10^5$lenses yields a$5\sigma$ detection of cross-correlations.

• Limiting Factors:

  • For realistic sample sizes ($L < 10^6$), the measurements are limited by noise or sparsity, not cosmic variance.
  • Improving measurement precision ($\sigma_{\eta_{LOS}}$) yields diminishing returns once the sparsity floor is reached; increasing the number of lenses is the only way to improve SNR further.

5. Significance

• New Cosmological Probe: This work establishes strong lenses not just as individual tools for measuring $H_0$ or mass profiles, but as a statistical ensemble capable of constraining the matter power spectrum.
• Systematics Control: By adding a probe (LOS shear) that is physically distinct from galaxy shapes (immune to intrinsic alignments), the $6 \times 2$pt method offers a robust way to break degeneracies and control systematics in Stage-IV surveys.
• Feasibility: The results demonstrate that even with conservative assumptions about lens modeling and measurement precision, the LOS shear signal is readily detectable in upcoming surveys like Euclid and LSST.
• Future Work: The authors note that this paper focuses on methodology and detectability. A subsequent paper will present Fisher forecasts quantifying the specific gain in constraints on cosmological parameters ($\Omega_m, \sigma_8$) when adding the LOS shear to the standard $3 \times 2$pt analysis.

In summary, the paper successfully argues that the "noise" of strong lensing (LOS shear) is actually a rich cosmological signal that, when combined with standard weak lensing, significantly enhances the statistical power of next-generation surveys.