Constraining a Non-Gaussian Toy Model for CMB Anomalies through Primordial Trispectra

This thesis develops a primordial-trispectrum analysis of the filtered cubic non-Gaussian toy model introduced to reproduce large-angle cosmic microwave background anomaly statistics. The aim is to move from the model's map-level temperature construction to a primordial shape comparison, an amplitude calibration, and an idealized detectability forecast. Starting from the temperature-space construction of Hansen et al. (2019), the work derives a corresponding primordial curvature trispectrum. The template is diagonal-free and can be written as twelve separable terms. Its reduced shape-function overlap with the standard local cubic \(g_{\rm NL}\) trispectrum is non-zero but incomplete, with \(C_{\rm toy,loc}^{\rm RSF}\sim0.2\)--\(0.3\). Thus local \(g_{\rm NL}\) constraints are informative, but they do not by themselves test the filtered model. The thesis then calibrates the map-space Hansen coefficient \(\beta\) against low-power anomaly targets and relates it to a toy primordial amplitude through the Sachs--Wolfe normalization \(g_{\rm NL}^{\rm toy}=\beta/25\). A CLASS-transfer projection gives a large local-template response, \(g_{\rm NL}^{\rm loc,eff}\simeq9.25g_{\rm NL}^{\rm toy}\). A direct ideal full-sky Fisher forecast gives \(\sigma_\beta\simeq1.51\times10^5\), so the anomaly-calibrated amplitudes correspond to signal-to-noise values of order \(30\)--\(60\) in this optimistic benchmark. Taken literally, such a signal should have been visible to a matched trispectrum analysis, while no such detection has been reported. The present calculation therefore disfavors the specific anomaly-calibrated toy model, although it is not a formal exclusion because masks, anisotropic noise, foregrounds, polarization, and the phenomenological filter choice are not treated in a full likelihood.