How to estimate Fisher information matrices from simulations

The Fisher information matrix is a quantity of fundamental importance for information geometry and asymptotic statistics. In practice, it is widely used to quickly estimate the expected information available in a data set and guide experimental design choices. In many modern applications, it is intractable to analytically compute the Fisher information and Monte Carlo methods are used instead. The standard Monte Carlo method produces estimates of the Fisher information that can be biased when the Monte-Carlo noise is non-negligible. Most problematic is noise in the derivatives as this leads to an overestimation of the available constraining power, given by the inverse Fisher information. In this work we find another simple estimate that is oppositely biased and produces an underestimate of the constraining power. This estimator can either be used to give approximate bounds on the parameter constraints or can be combined with the standard estimator to give improved, approximately unbiased estimates. Both the alternative and the combined estimators are asymptotically unbiased so can be also used as a convergence check of the standard approach. We discuss potential limitations of these estimators and provide methods to assess their reliability. These methods accelerate the convergence of Fisher forecasts, as unbiased estimates can be achieved with fewer Monte Carlo samples, and so can be used to reduce the simulated data set size by several orders of magnitude.Comment: Supporting code available at https://github.com/wcoulton/CompressedFishe

Cosmology with the Thermal-Kinetic Sunyaev-Zel'dovich Effect.

Compton scattering of the cosmic microwave background (CMB) from hot ionized gas produces a range of effects, and the leading order effects are the kinetic and thermal Sunyaev Zel'dovich (kSZ and tSZ) effects. In the near future, CMB surveys will provide the precision to probe beyond the leading order effects. In this Letter, we study the cosmological information content of the next order term which combines the tSZ and kSZ effects, hereafter called the thermal-kinetic Sunyaev Zel'dovich (tkSZ) effect. As the tkSZ effect has the same velocity dependence as the kSZ effect, it will also have many of the useful properties of the kSZ effect. However, it also has its own, unique spectral dependence, which allows it to be isolated from all other CMB signals. We show that with currently envisioned CMB missions the tkSZ effect can be detected and can be used to reconstruct large scale velocity fields, with no appreciable bias from either the kSZ effect or other extragalactic foregrounds. Furthermore, since the tkSZ effect arises from the well-studied pressure of ionized gas, rather than the gas number density as in the kSZ effect, the degeneracy due to uncertain gas physics will be significantly reduced. Finally, for a very low-noise experiment the tkSZ effect will be measurable at higher precision than the kSZ effect

Primordial information content of Rayleigh anisotropies

Anisotropies in the cosmic microwave background (CMB) are primarily generated by Thomson scattering of photons by free electrons. Around recombination, the Thomson scattering probability quickly diminishes as the free electrons combine with protons to form neutral hydrogen off which CMB photons can scatter through Rayleigh scattering. Unlike Thomson scattering, Rayleigh scattering is frequency dependent resulting in the generation of anisotropies with a different spectral dependence. Unfortunately the Rayleigh scattering efficiency rapidly decreases with the expansion of the neutral universe, with the result that only a small percentage of photons are scattered by neutral hydrogen. Although the effect is very small, future CMB missions with higher sensitivity and improved frequency coverage are poised to measure Rayleigh scattering signal. The uncorrelated component of the Rayleigh anisotropies contains unique information on the primordial perturbations that could potentially be leveraged to expand our knowledge of the early universe. In this paper we explore whether measurements of Rayleigh scattering anisotropies can be used to constrain primordial non-Gaussianity (NG) and examine the hints of anomalies found by WMAP and \textit{Planck} satellites. We show that the additional Rayleigh information has the potential to improve primordial NG constraints by $30\%$, or more. Primordial bispectra that are not of the local type benefit the most from these additional scatterings, which we attribute to the different scale dependence of the Rayleigh anisotropies. Unfortunately this different scaling means that Rayleigh measurements can not be used to constrain anomalies or features on large scales. On the other hand, anomalies that may persist to smaller scales, such as the potential power asymmetry seen in WMAP and \textit{Planck}, could be improved by the addition of Rayleigh measurements.Comment: 12 pages, 5 figure

Interpreting the observed UV continuum slopes of high-redshift galaxies

The observed UV continuum slope of star-forming galaxies is strongly affected by the presence of dust. Its observation is then a potentially valuable diagnostic of dust attenuation, particularly at high redshift where other diagnostics are currently inaccessible. Interpreting the observed UV continuum slope in the context of dust attenuation is often achieved assuming the empirically calibrated Meurer et al. relation. Implicit in this relation is the assumption of an intrinsic UV continuum slope (β = −2.23). However, results from numerical simulations suggest that the intrinsic UV continuum slopes of high-redshift star-forming galaxies are bluer than this, and moreover vary with redshift. Using values of the intrinsic slope predicted by numerical models of galaxy formation combined with a Calzetti et al. reddening law we infer UV attenuations (A1500) 0.35–0.5 mag (AV: 0.14 − 0.2 mag assuming Calzetti et al. reddening law) greater than simply assuming the Meurer relation. This has significant implications for the inferred amount of dust attenuation at very high (z ≈ 7) redshift given current observational constraints on β, combined with the Meurer relation, suggesting dust attenuation to be virtually zero in all but the most luminous systems