**A minimal power-spectrum-based moment expansion for CMB B-mode searches** by S. Azzoni, M. H. Abitbol, D. Alonso, **A. Gough**, N. Katayama, T. Matsumura

Primordial gravitational waves, thought to be sourced by inflation in the very early universe, should imprint themselves on the polarisation signal of the cosmic microwave background (CMB). Direct detection of the “B-mode” polarisation characteristic of these gravitational waves would provide strong evidence for an inflationary period in the early universe, as well as providing insight into the mechanisms driving it. The strength of the tensor perturbations which correspond to gravitational waves is parametrised by the “tensor to scalar ratio”,* r*.

Detecting true B mode signal from the CMB is difficult, as the primordial B mode signal is dominated by foreground elements, such as **thermal dust** emission and **synchrotron** emission within our own galaxy. These galactic foregrounds can be dealt with by masking—cutting the portions of the sky where the galactic plane is out of the signal—though this has the disadvantage of throwing away true signal. The other approach is to model the foreground elements statistically, with some dependence on frequency.

The standard technique is to characterise the foreground elements’ **spectral energy density** (a function of frequency) by an **amplitude** and a **spectral index**. These spectral indices are usually considered to be constant across the sky, which, given that they are empirically fit parameters, rather than motivated by underlying physics, is not well justified. For small sky fraction surveys, the difference is thought to be negligible, but for future CMB experiments, this assumption should be relaxed. This paper is a proof of concept for an extension beyond assuming a constant spectral index.

Allowing the spectral indices of dust and synchrotron emission to vary across the sky, we propagate forward the effect on the total observed power spectrum in a very general way using a moment expansion. A highly simplified case of this model is then tested against simulations of varying degrees of realism, to see whether this kind of expansion tightens the constraints on *r* compared to not using the moment expansion. The result is not possible to significantly detect the effects of this variation on the foreground multi-frequency power spectra for Simon’s-Observatory-like sensitivities, at least using the strong simplifications we used to test against simulations. Nevertheless the methodology set up in this paper, as well as potential extensions, will be useful for analysis of ground based CMB experiments such as Simon’s Observatory and CMB-Stage 4.