The anisotropic power spectrum and bispectrum in the f(phi) F^2 mechanism

Abstract: A suitable coupling of the inflaton phi to a vector kinetic term F^2 gives frozen and scale invariant vector perturbations. We compute the cosmological perturbations zeta that result from such coupling by taking into account the classical vector field that unavoidably gets generated at large scales during inflation. This generically results in a too anisotropic power spectrum of zeta. Specifically, the anisotropy exceeds the 1% level (10% level) if inflation lasted ~5 e-folds (~50 e-folds) more than the minimal amount required to produce the CMB modes. This conclusion applies, among others, to the application of this mechanism for magnetogenesis, for anisotropic inflation, and for the generation of anisotropic perturbations at the end of inflation through a waterfall field coupled to the vector (in this case, the unavoidable contribution that we obtain is effective all throughout inflation, and it is independent of the waterfall field). For a tuned duration of inflation, a 1% (10%) anisotropy in the power spectrum corresponds to an anisotropic bispectrum which is enhanced like the local one in the squeezed limit, and with an effective local f_{NL} ~3 (~30). More in general, a significant anisotropy of the perturbations may be a natural outcome of all models that sustain higher than 0 spin fields during inflation.

• The paper demonstrates that a vector field coupled via f(φ)F² produces scale-invariant yet anisotropic perturbations that challenge CMB isotropy when inflation is extended.
• It derives an anisotropy parameter |g*| reaching up to 0.1, quantifying the deviation from isotropy in the power spectrum of inflationary perturbations.
• The paper analytically computes the bispectrum, revealing a local non-Gaussianity (fₙₗ ≈ 3–30) that offers a testable signature for future cosmological observations.

Anisotropic Power Spectrum and Bispectrum in the $f(\varphi) F^2$ Model

The discussed paper investigates the cosmological perturbations that arise from a vector field coupled to the inflaton via the kinetic term $f(\varphi) F^2$. This coupling generates scale-invariant vector perturbations that can lead to significant anisotropy in the power spectrum of curvature perturbations, denoted by $\zeta$.

Key Contributions and Methodology

The authors provide a comprehensive analysis of the anisotropic power spectrum and bispectrum by considering the inevitability of generating a classical vector field during inflation. This approach diverges from earlier studies by incorporating both primary vector perturbations from the $f(\varphi) F^2$ coupling as well as their role in shaping the cosmological perturbations even before the Cosmic Microwave Background (CMB) modes exit the horizon.

Analytical Insights

1. Anisotropic Contributions:
   • The study reveals that the inflationary expansion generates a classical background vector field that remains scale-invariant. This aspect can lead to excessive anisotropy in the power spectrum, reaching levels that challenge the observed statistical isotropy of the CMB. Specifically, the anisotropy exceeds observational bounds if the inflationary period is extended beyond the minimal requirement necessary for producing observable CMB modes.
2. Power Spectrum Anisotropy:
   • The authors derive expression for the anisotropic power spectrum, taking into account classical field contributions during inflation. The power spectrum shows a parameter $|g_*$|, which quantifies the anisotropy, attaining values as high as 0.1 contingent upon the inflationary duration.
3. Bispectrum Analysis:
   • An analytical computation of the bispectrum is conducted, demonstrating a distinctive anisotropic shape function. Even in the case of statistical isotropy, this model exhibits an enhanced bispectrum characteristic, similar to the local shape in the squeezed limit, with an effective local $f_{\rm NL} \approx 3$ to $f_{\rm NL} \approx 30$, offering potential for empirical validation against CMB data.

Implications

The study underscores two critical insights for future cosmological models involving vector-tensor interactions during inflation:

• Predictive Framework for Anisotropy: The proposed model provides a framework whereby inflationary vector interactions may produce observable anisotropies and non-Gaussian characteristic imprints on primordial perturbations. This calls for tailored observational strategies to detect such signals empirically.
• Constraint on Inflationary Duration: The derived results disclose a stringent limitation on the permissible total number of inflationary e-folds when employing this vector-inflaton coupling, to align with existing isotropy data from CMB measurements.

Future Prospects

This research enriches the current theoretical understanding of inflationary dynamics involving vector fields and marks a pathway forward in probing inflationary models through anisotropic signatures. Future advancements in theoretical cosmology and enhanced precision from observational endeavors such as the Planck Satellite can further elucidate the viability of such models, particularly in relation to primordial magnetogenesis and anisotropic inflation scenarios.

Researchers are encouraged to investigate extensions and alternative implementations, ensuring a refined synthesis between theoretical predictions and empirical sensitivity, paving potential observational ventures into detecting these anisotropic signals across scales.