Testing Two-Field Inflation

TL;DR

This work develops a fully covariant, two-field inflation framework with arbitrary potentials and non-canonical kinetic terms, introducing a slow-turn extension to the familiar slow-roll paradigm and expressing perturbations in a kinematical basis. It derives semi-analytic, second-order SRST expressions for the curvature, isocurvature, and cross power spectra, relating their features to background kinematics and the geometry of the field manifold via the mass matrix and the effective entropy mass. The authors show how horizon-exit and end-of-inflation spectra, transfer functions, and observables like $r_T$, $n_T$, $n_{ ext{R}}$, $n_{ ext{S}}$, and cross-correlations depend on a small set of core quantities $H$, $oldsymbol{ abla}oldsymbol{ abla}oldsymbol{ abla}oldsymbol{V}$, and $R$, enabling reconstruction of inflationary dynamics from data. They apply the formalism to four model classes, demonstrating how initial conditions, mass ratios, and non-canonical terms can render certain two-field models viable or ruled out by the WMAP-era constraints, and revealing that multi-field effects are governed by the turn rate and its ratio to the speed-up rate. The study provides a practical, transfer-matrix-based toolkit for testing and constraining two-field inflation against observations, including how to account for initial-condition sensitivity and reheating uncertainties in interpreting isocurvature and cross spectra.

Abstract

We derive semi-analytic formulae for the power spectra of two-field inflation assuming an arbitrary potential and non-canonical kinetic terms, and we use them both to build phenomenological intuition and to constrain classes of two-field models using WMAP data. Using covariant formalism, we first develop a framework for understanding the background field kinematics and introduce a "slow-turn" approximation. Next, we find covariant expressions for the evolution of the adiabatic/curvature and entropy/isocurvature modes, and we discuss how the mode evolution can be inferred directly from the background kinematics and the geometry of the field manifold. From these expressions, we derive semi-analytic formulae for the curvature, isocurvature, and cross spectra, and the spectral observables, all to second-order in the slow-roll and slow-turn approximations. In tandem, we show how our covariant formalism provides useful intuition into how the characteristics of the inflationary Lagrangian translate into distinct features in the power spectra. In particular, we find that key features of the power spectra can be directly read off of the nature of the roll path, the curve the field vector rolls along with respect to the field manifold. For example, models whose roll path makes a sharp turn 60 e-folds before inflation ends tend to be ruled out because they produce strong departures from scale invariance. Finally, we apply our formalism to confront four classes of two-field models with WMAP data, including doubly quadratic and quartic potentials and non-standard kinetic terms, showing how whether a model is ruled out depends not only on certain features of the inflationary Lagrangian, but also on the initial conditions. Ultimately, models must possess the right balance of kinematical and dynamical behaviors, which we capture in a set of functions that can be reconstructed from spectral observables.

Figures (11)

• Figure 1: The exact (colored lines) and approximate (black dashed lines) solutions are depicted for both the (a) first-order and (b) second-order SRST approximations. Shown are the field trajectory and the three kinematical scalars (with $\epsilon$ used in place of $v$) for six different values of the mass ratio $\frac{m_2}{m_1}$ for the double quadratic potential $V = \frac{1}{2} m_1^2 \phi_1^2 + \frac{1}{2} m_2^2 \phi_2^2$ with canonical kinetic terms. The same initial conditions were assumed $60$$e$-folds before the end of inflation, and the $x$-axis for plots (ii) - (iv) represents the number of $e$-folds before inflation ends. Only the trajectory corresponding to $\frac{m_2}{m_1}=8$ violates the slow-roll and slow-turn conditions and only for less than 2 $e$-folds. Overall, the SRST approximation is a good approximation as long as the gradient of $\ln V$ is not too large and is not changing rapidly in magnitude or direction.
• Figure 2: The accuracy of three different approximations for the super-horizon evolution of entropy modes for six different values of the mass ratio $\frac{m_2}{m_1}$ for the double quadratic potential $V = \frac{1}{2} m_1^2 \phi_1^2 + \frac{1}{2} m_2^2 \phi_2^2$ with canonical kinetic terms. The same initial conditions were assumed $60$$e$-folds before the end of inflation, and the $x$-axis for plots (b) - (f) represents the number of $e$-folds before inflation ends. In Figures 2(a)-(d), the exact solutions (thick colored lines) and the second-order SRST approximation (dashed black lines) are shown for (a) the field vector trajectory, (b) $\epsilon$, (c) the speed up rate, and (d) the turn rate. In Figure 2(e), the exact solutions (thick colored lines) and the second-order SRST approximation (dashed black lines) are shown for the amplitude of entropy modes that exit the horizon $N_*=10,20,30,40,50,60,$ and $70$$e$-folds before the end of inflation. The dimensionless effective entropy mass (thin brown line) is overlaid for comparison. In Figure 2(f), the exact solution (thick colored lines) and three different approximations (dashed lines) for the post-horizon damping of the entropy mode that exits the horizon $N_*=60$$e$-folds before the end of inflation is shown for four of the six different trajectories. The three approximations are the assumption that the effective entropy mass is constant after horizon exit (dotted green line); the first-order slow-turn approximation for both the background and perturbed field vectors (dashed blue line); and the second-order SRST approximation for both the background and perturbed field vectors (longer dashed purple line). Here, by the end of inflation, and often much sooner, the assumption that the effective entropy mass can be treated as constant greatly over-estimates the amplitude of entropy modes.
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