Non-Gaussian Signatures of Inflationary Gauge Fields: Higher-Order Correlations and Associated Soft Theorems

Abstract

In modern cosmology, the inflationary paradigm offers a natural framework for understanding the large-scale homogeneity and isotropy of our observed universe and the origin of primordial density perturbations. The quantum fluctuations that lasted at the end of the inflationary phase of the early universe serve as the initial conditions for fluctuations in the primordial plasma. Therefore, these initial conditions are subsequently imprinted as the anisotropies in the observed Cosmic Microwave Background (CMB) and the inhomogeneities in large-scale structures. The marvelous CMB measurements of past decades tightly pinned down the initial two-point statistics of primordial density perturbations, often referred to as the scalar power spectrum, with unprecedented precision. Moreover, they also set an upper limit on the amplitude of the primordial gravitational wave (tensor power spectrum). Despite these observational triumphs, initial condition statistics beyond the two‑point level, say bispectra, trispectra, and higher, remain a crucial frontier. Particularly, they carry imprints of the interactions and field dynamics that shaped the very early universe. Thus, investigating the statistical correlations of primordial fluctuations beyond two points provides us with a remarkable opportunity to probe the fundamental laws of physics at extremely high energies. These characteristics of primordial fluctuations beyond two-point statistics are often referred to as primordial non-Gaussianity (PNG).

This thesis investigates the non-Gaussian features of Inflaton-driven gauge‑field fluctuations during the inflationary phase of the early universe. We work in an archetypical model of these inflationary gauge fields, which is inspired by UV-complete frameworks, effective field theories and, more importantly, the phenomenological quest to explain primordial magnetic fields. The thesis begins with Chapter 1, which provides a brief introduction to the subject. Then, in Chapter 2, we compute the three-point cross-correlations of gauge fields with inflationary tensor perturbations using the in-in formalism, and correlation strengths are studied as shape functions in momentum space. Furthermore, using semi-classical methods, we establish that the three-point cross-correlation functions satisfy new consistency relations (soft theorems) in the squeezed limit. In its entirety, we find that the direct coupling between the Inflaton and the gauge field solely determines the local non-linearity parameter associated with the scalar cross-correlation during slow-roll inflation.

Thereafter, in Chapter 3, we establish that these kinds of consistency relations are generic for any spectator field that is directly coupled to the Inflaton. In particular, the scalar consistency relation is derived semi-classically by generalising the consistency relation obtained earlier for cosmic magnetic fields. Notably, we find that the direct coupling between the Inflaton and the spectator solely determines the local non-linearity parameter associated with the scalar cross-correlation during slow-roll inflation, regardless of the specific form of the Lagrangian for the spectator field. Further, we calculate the tensor correlation with spectator fluctuations, explore the associated soft limits, and demonstrate the violation of the conventional tensor consistency relation with a non-minimal derivative coupling. Our analysis stresses that the violation of tensor consistency relations does not necessarily imply the super-horizon evolution of tensor modes. Instead, such violations can arise due to the non-minimal derivative coupling of the spectator field to gravity. Finally, we discuss the wider implications of our results in the context of cosmological soft theorems.

In Chapter 4, we compute the inflationary trispectrum of primordial gauge fields generated through the scalar and tensor exchanges in models with spectator U(1) gauge fields which are kinetically coupled to the Inflaton. Focusing on the connected four-point autocorrelation function of gauge fields, we derive exact analytical expressions for the full trispectrum of both electric and magnetic fields using the in-in formalism and cosmological diagrammatic rules, and explore their respective contributions in specific momentum configurations. For the scalar exchange, we find that the trispectrum signal in the equisided configuration grows with the exchange momentum and reaches its maximum in the flattened limit. However, in the counter collinear limit, we show that the non-linearity parameter associated with the trispectrum scales quadratically with the corresponding parameter of the cross-correlation bispectrum of magnetic fields and curvature perturbations, thereby establishing a hierarchical relation between the higher- and lower-order correlation functions. For the tensor exchange, the trispectrum displays a richer angular dependence, reflecting the sensitivity to the orientation of the momentum quadrilateral with respect to the tensor polarisation, producing characteristic angular modulations in the trispectrum. Detecting such angular signatures in future high-precision cosmological observations would provide a novel window into tensor-mediated interactions in the early universe.

In contrast to the three-point correlation functions involving the scalars and tensors, we observe that the computed three-point cross-correlation between the inflationary perturbations and the magnetic fields exhibits substantial super-Hubble evolution. In Chapter 5, we elucidate whether the behaviour of the correlation function continues for slow-roll scenarios away from the de Sitter approximation of the inflationary phase. We are able to show that for the scale-invariant case of the magnetic field, such slow-roll correction contributions are negligible. Finally, the thesis is concluded in Chapter 6, wherein we summarize the principal findings and discuss their broader implications and future prospects.