[Paper Review] Strong Lensing Constraints on Small-Scale Linear Power

Neal Dalal, C. S. Kochanek|arXiv (Cornell University)|Feb 15, 2002

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TL;DR

This paper uses strong gravitational lensing in massive elliptical galaxies to constrain small-scale linear power in the universe, leveraging satellite substructure abundance to probe cosmological parameters. It finds strong agreement with ΛCDM, setting tight limits: n > 0.94 (95% CL), mν < 0.74 eV, and m > 5.2 keV for dark matter particle mass, providing the tightest constraints on primordial power spectrum tilt and warm dark matter from small-scale structure.

ABSTRACT

We place limits on the linear power spectrum on small scales (k &gt; 50 h/Mpc) using measurements of substructure in gravitational lens galaxies. We find excellent agreement with the simplest LambdaCDM models, and in conjunction with other cosmological probes, place constraints on the neutrino mass m_nu, tilt of the primordial power spectrum n, and mass of the dark matter particle m. We find n&gt;0.94, and for a Harrison-Zeldovich spectrum, find m_nu&lt;0.74 eV and m&gt;5.2 keV, at 95% confidence.

Motivation & Objective

• To constrain the small-scale linear power spectrum using gravitational lensing substructure in massive galaxies.
• To test the consistency of ΛCDM models with observations of satellite subhalos in lensing systems.
• To place limits on cosmological parameters such as primordial power spectrum tilt (n), neutrino mass (mν), and dark matter particle mass (m).
• To provide a clean, mass-based probe of small-scale structure, avoiding biases from baryonic physics in star-forming dwarfs.
• To improve constraints on warm dark matter and inflationary physics using high-resolution lensing data.

Proposed method

• Measures the fraction of mass in satellite subhalos (fsat) in lens galaxies using flux and position shifts in multiply-imaged radio sources.
• Applies the Sheth-Tormen (ST) conditional mass function to relate fsat to the linear rms density fluctuation σ(Msat) on subgalactic scales.
• Uses the conditional mass function: fsat(>Msat) = A[erfc(√(aν/2)) + Γ(1/2−p, aν/2)/(2^p√π)], with ν defined via overdensity and growth factor.
• Relies on the Press-Schechter formalism with ST parameters (a=0.707, p=0.3, A≈0.322) to model subhalo abundance from the linear power spectrum.
• Compares observed σ(M) constraints to theoretical predictions across cosmological models (ΛCDM, WDM, varying n and mν).
• Applies COBE normalization and fitting functions for transfer functions (Eisenstein & Hu, WDM) to compute power spectra for different cosmologies.

Experimental results

Research questions

• RQ1 How well do ΛCDM models with standard parameters reproduce the observed abundance of substructure in strong lensing systems?
• RQ2 What constraints can be placed on the tilt of the primordial power spectrum (n) using small-scale linear power from lensing?
• RQ3 What upper limits can be set on the sum of neutrino masses (mν) based on suppression of small-scale power?
• RQ4 What lower limit can be placed on the mass of dark matter particles (m) assuming warm dark matter (WDM) models?
• RQ5 How do observational systematics (e.g., tidal stripping, beam smearing) affect the inferred substructure constraints?

Key findings

• The observed substructure abundance in lens galaxies is consistent with the simplest ΛCDM model, with no evidence for suppression of small-scale power.
• The paper finds n > 0.94 at 95.5% confidence, ruling out significant deviations from a Harrison-Zeldovich (n=1) primordial power spectrum.
• For a Harrison-Zeldovich spectrum, the 95.5% upper limit on neutrino mass is mν < 0.74 eV, tightening previous bounds from Lyα forest data.
• With a logarithmic prior, the upper limit improves to mν < 0.53 eV, showing robustness to prior choice.
• The analysis sets a lower limit of m > 5.2 keV on the dark matter particle mass at 95.5% confidence, ruling out very light warm dark matter.
• Systematic effects such as tidal stripping and beam smearing are shown to bias results toward lower substructure abundance, making the constraints conservative.