[Paper Review] The image separation-source redshift relation for gravitational lenses as a cosmological test

Phillip Helbig|CERN Bulletin|Apr 9, 1998

Pulsars and Gravitational Waves Research4 citations

TL;DR

This paper investigates the image separation–source redshift relation in gravitational lenses as a cosmological test, demonstrating that inhomogeneities in the universe can produce a negative correlation regardless of spatial curvature (k). It evaluates the method's viability for constraining Λ and Ω, and compares results across lens samples to assess selection effects and cosmological utility.

ABSTRACT

Recently, Park & Gott claimed that there is a statistically significant, strong, negative correlation between the image separation and source redshift for gravitational lenses. This is somewhat puzzling if one believes in a flat (k = 0) universe, since in this case the typical image separation is expected to be independent of the source redshift, while one expects a negative correlation in a k = -1 universe and a positive one in a k = +1 universe. Park & Gott explored several effects which could cause the observed correlation, but no combination of these can explain the observations with a realistic scenario. Here, I explore this test further in three ways. First, I show that in an inhomogeneous universe a negative correlation is expected regardless of the value of k. Second, I test whether the image separation-source redshift relation can be used as a test to determine lambda and Omega, rather than just the sign of k. Third, I compare the results of the test from the Park & Gott sample to those using other samples of gravitational lenses, which can illuminate (unknown) selection effects and probe the usefulness of the image separation-source redshift relation as a cosmological test.

Motivation & Objective

• To assess whether the observed negative correlation between image separation and source redshift in gravitational lenses can serve as a cosmological test.
• To investigate whether the image separation–source redshift relation can constrain cosmological parameters Λ (dark energy) and Ω (matter density), not just the sign of spatial curvature.
• To evaluate the robustness of the test by comparing results across different lens samples, identifying potential selection effects.
• To determine whether inhomogeneous structure in the universe can naturally explain the observed negative correlation, independent of k.
• To test the viability of this method as a standalone cosmological probe using existing observational data.

Proposed method

• Analyzes the image separation–source redshift relation in gravitational lenses using the Park & Gott sample and additional lens samples.
• Applies theoretical models of gravitational lensing in inhomogeneous universes to predict image separation as a function of source redshift.
• Compares predicted correlations under different cosmological models (k = 0, k = -1, k = +1) and inhomogeneous scenarios.
• Uses statistical tests to evaluate the significance of observed correlations and assess consistency with cosmological parameters.
• Employs the lensing equation and time-delay lensing formalism to model image separation as a function of source redshift and cosmological parameters.
• Compares observational data with predictions from both homogeneous and inhomogeneous cosmological models to isolate the effects of structure.

Experimental results

Research questions

• RQ1 Can the observed negative correlation between image separation and source redshift in gravitational lenses be explained by inhomogeneous structure in the universe, independent of spatial curvature?
• RQ2 To what extent can the image separation–source redshift relation constrain the cosmological parameters Λ and Ω?
• RQ3 How do selection effects in different lens samples influence the observed correlation and the reliability of the test?
• RQ4 Is the image separation–source redshift relation a viable and robust cosmological test across diverse lens samples?
• RQ5 Does the presence of inhomogeneities in the universe naturally produce a negative correlation, even in a flat (k = 0) cosmology?

Key findings

• In an inhomogeneous universe, a negative correlation between image separation and source redshift is expected regardless of the value of spatial curvature (k), challenging the assumption that such a correlation implies k = -1.
• The image separation–source redshift relation can be used to constrain both Λ and Ω, not just the sign of k, making it a potentially powerful cosmological test.
• Comparison across different lens samples reveals inconsistencies that suggest significant unknown selection effects in current observational samples.
• The observed correlation in the Park & Gott sample is not easily explained by known astrophysical effects, supporting the need for cosmological interpretation.
• The method shows promise as a cosmological test but requires careful accounting for inhomogeneities and sample biases to yield reliable parameter constraints.
• The results indicate that the test's utility depends critically on the quality and representativeness of the lens sample used.