[Paper Review] Distinguishing between inflationary models from cosmic microwave background

Shinji Tsujikawa|arXiv (Cornell University)|Jan 19, 2014

Cosmology and Gravitation Theories12 references3 citations

TL;DR

This paper analyzes observational constraints on single-field inflationary models using Planck CMB data, focusing on scalar spectral index $n_s$, tensor-to-scalar ratio $r$, and primordial non-Gaussianity $f_{\text{NL}}$. It demonstrates that Starobinsky and non-minimally coupled Higgs inflation are well within Planck bounds, while many k-inflation and natural inflation models are excluded or tightly constrained by combined $n_s$, $r$, and $f_{\text{NL}}^\text{equil}$ measurements.

ABSTRACT

In this paper, inflationary cosmology is reviewed, paying particular attention to its observational signatures associated with large-scale density perturbations generated from quantum fluctuations. In the most general scalar-tensor theories with second-order equations of motion, we derive the scalar spectral index $n_s$, the tensor-to-scalar ratio $r$, and the nonlinear estimator $f_{ m NL}$ of primordial non-Gaussianities to confront models with observations of Cosmic Microwave Background (CMB) temperature anisotropies. Our analysis includes models such as potential-driven slow-roll inflation, k-inflation, Starobinsky inflation, and Higgs inflation with non-minimal/derivative/Galileon couplings. We constrain a host of inflationary models by using the Planck data combined with other measurements to find models most favored observationally in the current literature. We also study anisotropic inflation based on a scalar coupling with a vector (or, two-form) field and discuss its observational signatures appearing in the two-point and three-point correlation functions of scalar and tensor perturbations.

Motivation & Objective

• To systematically evaluate the observational viability of a broad class of single-field inflationary models using the latest CMB data.
• To constrain models based on scalar-tensor theories with second-order equations of motion, particularly within the Horndeski framework.
• To assess the compatibility of potential-driven slow-roll, k-inflation, Starobinsky, and Higgs inflation models with Planck measurements of $n_s$, $r$, and $f_{\text{NL}}^\text{equil}$.
• To investigate anisotropic inflation models with vector or two-form field couplings and their imprints on CMB statistics.
• To determine which inflationary scenarios satisfy the tight observational bounds from Planck+WP+BAO+high-$\ell$ data, especially regarding non-Gaussianity and statistical anisotropy.

Proposed method

• Derives general expressions for the scalar spectral index $n_s$, tensor-to-scalar ratio $r$, and equilateral non-Gaussianity $f_{\text{NL}}^\text{equil}$ in the most general scalar-tensor theories with second-order equations of motion (Horndeski theory).
• Applies these general results to specific models: potential-driven slow-roll, k-inflation, Starobinsky, and Higgs inflation with non-minimal, derivative, and Galileon couplings.
• Uses Planck 2013 data (up to $\ell \sim 2500$), WMAP large-angle polarization (WP), BAO, and high-$\ell$ CMB measurements to constrain model parameters.
• Computes the anisotropy parameter $g_*$ in models with vector or two-form field couplings to the inflaton, and evaluates its impact on the scalar power spectrum and bispectrum.
• Evaluates the observational signatures in two-point and three-point correlation functions of scalar and tensor perturbations, particularly focusing on statistical isotropy and non-Gaussianity.
• Applies the Planck bounds: $n_s = 0.9603 \pm 0.0073$ (68% CL), $r < 0.11$ (95% CL), and $f_{\text{NL}}^\text{equil} = -42 \pm 75$ (68% CL) to test model compatibility.

Experimental results

Research questions

• RQ1 Which single-field inflationary models are most consistent with the Planck 2013 CMB data, particularly in terms of $n_s$, $r$, and $f_{\text{NL}}^\text{equil}$?
• RQ2 How do k-inflation and models with $c_s^2 \ll 1$ fare under the combined constraints of $n_s$, $r$, and non-Gaussianity bounds?
• RQ3 To what extent can anisotropic inflation models with vector or two-form field couplings survive observational constraints from $g_*$ and $f_{\text{NL}}$?
• RQ4 How do non-minimal, derivative, and Galileon couplings in Higgs inflation affect the model's compatibility with Planck data compared to minimal or Starobinsky-like models?
• RQ5 What are the observational signatures of anisotropic hair in the CMB power spectrum and bispectrum, and how do they constrain the coupling strength and $g_*$ parameter?

Key findings

• Starobinsky inflation, with $n_s = 1 - 2/N$ and $r = 12/N^2$, is well within the 68% CL region of Planck constraints for $N \approx 50$ to $60$ e-folds.
• Non-minimally coupled Higgs inflation with $|\xi| \gg 1$ reproduces the Starobinsky prediction and is consistent with Planck data, provided quantum corrections are suppressed.
• The monomial potential $V(\phi) = \lambda_n \phi^n / n$ lies outside the 68% CL region, while very small-field potentials like $V(\phi) = \Lambda^4(1 - e^{-\phi/M})$ are consistent due to suppressed $r$.
• Natural inflation with $V(\phi) = V_0[1 + \cos(\phi/f)]$ is constrained to $5.1M_{\text{pl}} < f < 7.9M_{\text{pl}}$ at 68% CL.
• K-inflation models are tightly constrained by the Planck bound on $f_{\text{NL}}^\text{equil}$, with the dilatonic ghost condensate model excluded due to $|f_{\text{NL}}^\text{equil}|$ being too large in the allowed $c_s$ range.
• Anisotropic inflation models with vector or two-form couplings yield $|g_*| < 0.01$ under Planck constraints, and $f_{\text{NL}}$ remains below order 1, satisfying non-Gaussianity bounds.