Cosmology with standard sirens: the importance of the shape of the lensing magnification distribution

Abstract

The gravitational waves (GWs) emitted by inspiralling binary black holes, expected to be detected by the Laser Interferometer Space Antenna (LISA), could be used to determine the luminosity distance to these sources with the unprecedented precision of 1 per cent. We study cosmological parameter constraints from such standard sirens, in the presence of gravitational lensing by large-scale structure. Lensing introduces magnification with a probability distribution function (PDF) whose shape has significant skewness and kurtosis, and depends on cosmological parameters. We use Monte Carlo simulations to generate mock samples of standard sirens, including a small intrinsic scatter, as well as the additional, larger scatter from lensing, in their inferred distances. We derive constraints on cosmological parameters, by simultaneously fitting the mean and the distribution of the residuals on the distance versus redshift (d L -z) Hubble diagram. We find that for standard sirens at redshift z ≈ 1, the sensitivity to a single cosmological parameter, such as the matter density m , or the dark energy equation of state w, is ∼50-80 per cent tighter when the lensing PDF is used, compared to the sensitivity derived from a Gaussian PDF with the same variance. When these two parameters are constrained simultaneously, the non-Gaussian shape yields a further enhanced improvement (by ∼120 per cent), owing to the correlation between the parameters. The sensitivity to the amplitude of the matter power spectrum, σ 8 from the cosmological dependence of the PDF alone, however, is ∼20 per cent worse than that from the Gaussian PDF. The improvements for m and w arise purely from the non-Gaussian shape of the lensing PDF and can be attributed specifically to the sharpness of the peak of this PDF (i.e. to a finite kurtosis); the dependence of the PDF on these parameters does not improve constraints relative to those available from the mean d L -z relation. At higher redshifts, the PDF resembles a Gaussian more closely, and the effects of the non-Gaussianities become less prominent. These results highlight the importance of obtaining an accurate and reliable PDF of the lensing convergence, in order to realize the full potential of standard sirens as cosmological probes.

Effect of lensing magnification on the apparent distribution of black hole mergers

“…Before we consider specific models of the rate density in the literature, we will demonstrate some features of the cumulative rate in Eq. (14), both generally and within a toy model for the evolution of the merger rate that we tune to emphasize the effects of lensing.…”

Tracing the redshift evolution of Hubble parameter with gravitational-wave standard sirens

“…Holz and Linder [30] estimated the lensing error on the distance measurement by using Monte Carlo simulation, and, assuming the Gaussian form of lensing magnification probability, they derived a fitting formula for the systematic error. Later, the significance of the non-Gaussian tail has been recognized [33,34], and it turns out that this effect reduces the lensing error by a factor of 1.5-2, compared to the Gaussian distribution. More recently, Hirata, Holz, and Cutler [33] adopted a log-normal distribution for the magnification probability and obtained the fitting formula for the (averaged) distance error:…”

Observational probes of cosmic acceleration

“…A third complication, important for the high S/N observations expected from LISA, is that weak lensing magnification becomes a dominant source of noise at z 1, inducing a scatter in distance of several percent per observed source (Markovic, 1993;Holz and Linder, 2005;Jonsson et al, 2007). By taking advantage of the non-Gaussian shape of the lensing scatter, one can reduce the error on the mean by a factor ∼ 2 − 3 below the naive σ/ √ N expectation Shang and Haiman, 2011), so samples of a few dozen well observed sources could yield sub-percent distance scale errors. Nissanke et al (2010) forecast constraints on H 0 from next-generation ground-based gravitational wave detectors, including Monte Carlo simulations of parameter recovery from neutron starneutron star and neutron star-black hole mergers.…”