Abstract
ABSTRACT Maximally Smooth Functions (MSFs) are a form of constrained functions in which there are no inflection points or zero crossings in high-order derivatives. Consequently, they have applications to signal recovery in experiments where signals of interest are expected to be non-smooth features masked by larger smooth signals or foregrounds. They can also act as a powerful tool for diagnosing the presence of systematics. The constrained nature of MSFs makes fitting these functions a non-trivial task. We introduce maxsmooth, an open-source package that uses quadratic programming to rapidly fit MSFs. We demonstrate the efficiency and reliability of maxsmooth by comparison to commonly used fitting routines and show that we can reduce the fitting time by approximately two orders of magnitude. We introduce and implement with maxsmooth Partially Smooth Functions, which are useful for describing elements of non-smooth structure in foregrounds. This work has been motivated by the problem of foreground modelling in 21-cm cosmology. We discuss applications of maxsmooth to 21-cm cosmology and highlight this with examples using data from the Experiment to Detect the Global Epoch of Reionization Signature (EDGES) and the Large-aperture Experiment to Detect the Dark Ages (LEDA) experiments. We demonstrate the presence of a sinusoidal systematic in the EDGES data with a log-evidence difference of 86.19 ± 0.12 when compared to a pure foreground fit. MSFs are applied to data from LEDA for the first time in this paper and we identify the presence of sinusoidal systematics. maxsmooth is pip installable and available for download at https://github.com/htjb/maxsmooth.
Highlights
Smooth Functions (MSFs), functions with no inflection points or zero crossings in higher order derivatives, were first proposed by Sathyanarayana Rao et al (2015) for modelling foregrounds in experiments to detect spectral signatures from the Epoch of Recombination
For 21-cm cosmology we find that Completely Smooth Functions (CSFs) and Maximally Smooth Functions (MSFs) are generally equivalent
Derivative Constrained Functions (DCFs) generally are advantageous for experiments in which the desired signal is masked by higher magnitude smooth signals or foregrounds
Summary
Smooth Functions (MSFs), functions with no inflection points or zero crossings in higher order derivatives, were first proposed by Sathyanarayana Rao et al (2015) for modelling foregrounds in experiments to detect spectral signatures from the Epoch of Recombination. In comparison to unconstrained polynomials, DCFs are better able to separate the smooth foreground spectra from the anticipated EoR signals and instrumental systematics (Sathyanarayana Rao et al 2017) This motivates their use in Global 21-cm cosmology experiments such as; REACH (Radio Experiment for the Analysis of Cosmic Hydrogen, de Lera Acedo 2019), SARAS (Shaped Antenna measurement of the background RAdio Spectrum, Singh et al 2018a), EDGES (Experiment to Detect the Global Epoch of Reionization Signature, Bowman et al 2018), LEDA (Largeaperture Experiment to Detect the Dark Ages, Price et al 2018), PRIZM (Probing Radio Intensity at High-Z from Marion, Philip et al 2019), BIGHORNS (Broadband Instrument for Global HydrOgen ReioNisation Signal Sokolowski et al 2015), SCI-HI (Sonda Cosmológica de las Islas para la Detección de Hidrógeno Neutro Voytek et al 2014) and MIST (Mapper of the IGM Spin Temperature, http://www.physics.mcgill.ca/mist/). Experiments that search for the Global 21-cm signal are attempting to detect a signal, according to standard ΛCDM cosmology, approximately 250 mK in foregrounds of up to 104 − 105 times brighter These high-magnitude foregrounds are dominated by synchrotron and free-free emission in the Galaxy and extragalactic radio sources which have smooth power law structures.
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