Taller in the saddle: constraining CMB physics using saddle points

Abstract

The statistics of extremal points in the cosmic microwave background (CMB) temperature (hot and cold spots) have been well explored in the literature, and have been used to constrain models of the early Universe. Here, we extend the study of critical points in the CMB to the set that remains after removing extrema, namely the saddle points. We perform stacks of temperature and polarization about temperature saddle points in simulations of the CMB, as well as in data from the Planck satellite. We then compute the theoretical profile of saddle-point stacks, given the underlying power spectra of the CMB. As an example of the utility of such stacks, we constrain models of cosmic birefringence, and compare the constraining power of the saddle points with that of extremal points. We find that, in the specific example of birefringence, we can place tighter constraints using saddle points in our analysis than using extrema. In fact, we find our saddle-point analysis yields close to optimal constraints, as seen by comparing to a power spectrum analysis. We, therefore, suggest that stacking on saddle points may, in general, be a useful way of testing for non-standard physics effects that change the CMB power spectra.Essential to the process of extracting cosmological information from the cosmic microwave background (CMB) radiation is a full characterization of its statistics. The temperature fluctuations in the CMB are consistent with an isotropic Gaussian random field, with some minor exceptions (e.g. due to gravitational lensing, and the integrated Sachs-Wolfe effect) [1]. Thus, to appreciate the CMB we must understand the statistics of a 2D Gaussian random field on a 2-sphere (i.e. the sky).The study of 1D Gaussian random fields originated in the study of telecommunications [2,3]. This approach was adapted for cosmological applications in the seminal 1986 paper by Bardeen, Bond, Kaiser & Szalay, in which the perturbations of the evolving early Universe were modelled as a 3D Gaussian field [4]. These results were particularized to the 2D case of Gaussian fields on a sphere by Bond & Efstathiou in 1987 [5]. The focus of these papers was the statistical properties of extremal points, i.e. local maxima and minima in Gaussian random fields, which has led to a fruitful avenue of study of the statistics of collapsing peaks in large-scale structure. The average profiles of the temperature and polarization fields around temperature peaks were calculated in the flat-sky limit by the WMAP team [6], and a more general formalism for calculating these profiles, valid at large scales (i.e. with no flat-sky approximation) and allowing for biased peak eccentricity, was developed by Marcos-Caballero et al. [7]. Stacks of temperature and polarization patches around temperature peaks in CMB data can be compared to these calculations to test cosmological models. Importantly, the theoretical predictions for the profiles of these stacks rely on the assumptions of Gaussianity and statistical isotropy, and, therefore, they can be used as genera...

Planck intermediate results

“…Our specific implementation of the peak method follows earlier studies, for example Planck Collaboration VII (2020). The weighting scheme has not been shown to be optimal, but a similar approach was used for determining constraints on cosmic birefringence Contreras et al (2017) and gave similar results to using the power spectra (and the issue of weighting is further discussed in Jow et al 2019). Intuitively we would expect that sharper peaks have a higher signal, and hence that influences our choice for the weighting scheme described below.…”

“…We evaluate the Laplacian numerically in pixel space at pixel p. The weighting scheme closely resembles the bias factors that come about when relating peaks to temperature fluctuations (Bond & Efstathiou 1987), as used in Komatsu et al (2011), Planck Collaboration Int. XLIX (2016 and Jow et al (2019). Combining Eqs.…”