Weak-lensing magnification bias in the gamma-ray burst Hubble diagram and the robustness of high-redshift dark-energy constraints

Long gamma-ray bursts (GRBs), standardizable through empirical luminosity relations, can extend the extragalactic Hubble diagram to redshift $z\sim8$ and are in principle the only distance indicators able to test the dark-energy equation of state $w(z)$ at redshifts far beyond the reach of Type Ia supernovae. Their observed fluxes are, however, distorted by weak gravitational lensing: the non-Gaussian distribution of magnifications along the line of sight both adds scatter to, and — through flux-limited selection — systematically biases, the inferred distances. Wang & Dai (2011) showed with 116 GRBs that this magnification bias is non-negligible, shifting the best-fit matter density ($\Omega_M$ from 0.30 to 0.26) and biasing the dark-energy equation of state toward more negative values, using magnification probability distribution functions (PDFs) built on the Peacock & Dodds (1996) nonlinear power spectrum and an analytic universal PDF (UPDF) parametrization. The open problem is to quantify how uncertainties in the magnification PDFs (nonlinear matter clustering on small scales, baryonic effects, the UPDF parametrization) and in the standardization of the luminosity relations propagate into GRB-derived cosmological constraints, and to determine whether definitive evidence for $w(z)$ evolution at high redshift ($z\gtrsim2$, out to $z\sim8$) can be established from GRBs once these systematics are controlled. No subsequent work has redone the lensing analysis with modern ray-tracing-based magnification PDFs, and current GRB dark-energy analyses generally omit lensing entirely.

Background

A gamma-ray burst (GRB) is a brief, extremely luminous flash of gamma rays produced by the collapse of a massive star (long GRB) or the merger of compact objects (short GRB); long GRBs have been observed out to $z\sim9$, making them potential distance indicators beyond the range of Type Ia supernovae (SNe Ia), which fade beyond $z\approx2$. A "standardizable candle" is a source whose intrinsic luminosity can be inferred from other measured properties: for GRBs, several empirical luminosity relations connect a brightness-like quantity (e.g., the isotropic-equivalent radiated energy $E_{\rm iso}$) to an independently observable quantity (e.g., the spectral peak energy $E_{\rm p}$). Because GRB luminosity relations cannot be calibrated against known distances at high redshift, Wang & Dai (2011) calibrated five such relations at $z\leq1.4$ by cosmography — a Taylor-expansion fit of the distance–redshift relation to an SNe Ia sample that assumes no specific cosmological model — and extrapolated them to build a GRB Hubble diagram (distance modulus versus redshift) for 116 GRBs, discarding one relation whose intrinsic scatter was too large.

Weak gravitational lensing by the intervening matter distribution magnifies or demagnifies the flux of a distant source. For point sources like GRBs, the magnification $\mu=(1-\kappa)^{-2}$ (with $\kappa$ the convergence, the projected mass contrast along the line of sight) has a broad, strongly non-Gaussian probability distribution whose width grows with redshift. Wang, Holz & Munshi (2002) introduced a universal probability distribution function (UPDF) — a fitted stretched-Gaussian form for the lensing amplification that depends on redshift and cosmology through the variance of the convergence — which Wang & Dai (2011) used to compute the magnification PDFs $p(\mu)$ entering their likelihood. The convergence variance is an integral over the nonlinear matter power spectrum $P(k)$ (the fluctuation power as a function of wavenumber on scales where clustering has left the linear regime); Wang & Dai (2011) adopted the Peacock & Dodds (1996) fitting formula, based on the N-body simulations of the 1990s. Because GRB samples are effectively flux-limited (there is a detection threshold in burst brightness), lensing induces a magnification bias, a Malmquist-type selection effect: bursts just below the threshold may be magnified into detection, while bursts just above it may be demagnified out of the sample, so the observed distance distribution is skewed. Predecessors had addressed this with smaller samples: Oguri & Takahashi (2006) computed lensing effects on the GRB Hubble diagram analytically, and Schaefer (2007) found lensing and Malmquist biases smaller than the intrinsic error bars for 69 GRBs.

In a flat $\Lambda$CDM universe (matter density parameter $\Omega_M$ plus a cosmological constant), Wang & Dai (2011) found that including the lensing magnification bias shifts the best-fit $\Omega_M$ from 0.30 to 0.26 and biases the dark-energy equation of state, the ratio $w$ of pressure to energy density, toward more negative values; using a binned, redshift-piecewise-constant parametrization of $w(z)$ they constrained the equation of state out to $z\sim8$ for the first time, finding deviations from $w=-1$ (the cosmological-constant value) at $1\sigma$ but consistency at $2\sigma$. The authors concluded by emphasizing that "we need detailed studies of new correlations with a much greater number of GRBs and an examination of systematic errors to be able to regard GRBs as more accurate standardizable candles." The open limitations that subsequent work identifies as defining this problem include: the magnification PDFs rest on the adopted nonlinear $P(k)$ and the UPDF parametrization, both of which are affected by uncertainties in the small-scale matter distribution including baryonic effects (gas physics driven by star formation and feedback that suppresses the matter power spectrum on small scales); the extrapolation to high redshift assumes, on the basis of a companion analysis asserting that the calibrated luminosity relations do not evolve with redshift, that this no-evolution assumption remains adequate, an assumption still debated rather than settled in the later literature; and the sample size and intrinsic scatters limit the statistical power, so GRBs cannot yet serve as precise high-redshift standard candles.

Problem Statement

Establish whether gamma-ray bursts can deliver trustworthy dark-energy constraints at high redshift once weak-lensing magnification bias and standardization systematics are properly modeled, as left open by Wang & Dai (2011). Concretely, the problem is to quantify and correct the lensing magnification bias in the GRB Hubble diagram — the analysis of Wang & Dai (2011) rested on the Peacock & Dodds (1996) nonlinear power spectrum and the analytic UPDF parametrization, and the sensitivity of the results to these inputs, including uncertainties in the small-scale matter distribution such as baryonic effects, has not been quantified — and, propagating this bias through realistic flux-limited selection and GRB likelihoods, to determine: (a) how much the best-fit matter density $\Omega_M$ and the dark-energy equation of state $w(z)$ (binned or parametrized over $z\gtrsim2$ out to $z\sim8$) shift relative to the statistical errors of current and planned GRB samples; (b) whether the assumption that the calibrated luminosity relations do not evolve with redshift is adequate, or whether residual evolution and lensing bias are degenerate; and (c) whether definitive evidence for or against evolution of $w(z)$ at high redshift can be established from GRBs once these systematics are controlled. These components jointly serve one objective — validating GRBs as precise high-redshift standardizable candles for dark energy — and a complete answer supplies a quantified systematic-error budget showing which conclusions of the GRB Hubble diagram survive it.

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