The efficiency gap of two-step forecasts for stochastic gravitational wave background searches

Abstract

The stochastic gravitational-wave background is targeted by the millihertz observatories LISA, Taiji and TianQin, and its forecast precision rests on a Whittle approximation to the periodogram likelihood. The quoted scale is that of a joint fit a two-step forecast need not reach. We show that the residual risk of a two-step forecast lies in the estimator route rather than in the likelihood. At the exact Whittle form we separate the forecast error into a deterministic bias $\mathbf{b}$ and an efficiency gap ${\mathbf{C}}_b$, both in closed form. We find that this gap is governed by the span of the first-step nuisance fit. It decreases substantially as the span increases and vanishes when the first-step fit spans the entire band or when the response exhibits no within-bin structure. The gap does not arise as a cost of the prior. Across the LISA band the gap spans more than an order of magnitude, the amplitude and tilt inflation falling from about thirty and twenty to unity at a band-wide first step. A foreground from the released LDC2a population moves the mid-band gap by an order of magnitude. A two-step forecast should therefore report the span of its first-step fit with its intervals, since that span decides whether they sit at the Fisher scale or an order of magnitude above it.

arXiv comment20 pages, 6 tables, 5 figures, comments welcome