Inflation with a constant rate of roll (1411.5021v3)

Abstract: We consider an inflationary scenario where the rate of inflaton roll defined by $\ddot\phi/H\dot \phi$ remains constant. The rate of roll is small for slow-roll inflation, while a generic rate of roll leads to the interesting case of `constant-roll' inflation. We find a general exact solution for the inflaton potential required for such inflaton behaviour. In this model, due to non-slow evolution of background, the would-be decaying mode of linear scalar (curvature) perturbations may not be neglected. It can even grow for some values of the model parameter, while the other mode always remains constant. However, this always occurs for unstable solutions which are not attractors for the given potential. The most interesting particular cases of constant-roll inflation remaining viable with the most recent observational data are quadratic hilltop inflation (with cutoff) and natural inflation (with an additional negative cosmological constant). In these cases even-order slow-roll parameters approach non-negligible constants while the odd ones are asymptotically vanishing in the quasi-de Sitter regime.

• The paper derives exact inflaton potentials under constant roll conditions, providing an alternative to conventional slow-roll inflation.
• It distinguishes between power-law and quadratic hilltop inflation scenarios, highlighting unique dynamics and potential observational challenges.
• The study demonstrates that specific α values yield stable scalar perturbations with red-tilted spectra, consistent with current CMB data.

Analysis of Inflation with a Constant Rate of Roll

The paper "Inflation with a Constant Rate of Roll" by Hayato Motohashi, Alexei A. Starobinsky, and Jun'ichi Yokoyama presents an exploration of an inflationary model characterized by a constant rate of inflaton field roll. The authors propose a variation of the standard slow-roll inflation paradigm by considering scenarios where the rate of roll, given by the expression $\ddot{\phi}/H\dot{\phi} = -3 - \alpha$, is constant. This approach allows for a broad spectrum of inflationary behaviors, thus providing potential alternatives to conventional inflationary models based on slow-roll conditions.

Overview of Inflationary Dynamics

The authors derive a general exact solution for the inflaton potential that supports constant-roll behavior. The parameter $\alpha$ acts as a tunable element that influences the dynamics of the inflationary phase. The classical slow-roll inflation corresponds to $\alpha \approx -3$, while ultra-slow-roll conditions arise when $\alpha = 0$. The integration of the constant-roll conditions yields solutions that encompass both traditional slow-roll and fast-roll situations, expanding the potential landscape of inflationary cosmology.

Examination of Specific Solutions

Three notable solutions emerge from the paper of this framework:

1. Power-law Inflation: This occurs when $-2 < \alpha < -3$, where the potential takes an exponential form, leading to a power-law expansion of the universe. Power-law inflation has been explored previously; however, it faces observational challenges, particularly concerning the predicted amplitude of gravitational waves.
2. Quadratic Hilltop Inflation: For $\alpha<-3$, the analysis reveals a viable model that encompasses scenarios akin to natural inflation with an additional negative cosmological constant. In this case, the inflaton potential assumes a form akin to an inverted parabola near the origin, characteristic of hilltop models.
3. Constant-roll with Viable Observational Data: Among the constant-roll solutions, those for which $\alpha\lesssim -3$ exhibit stable scalar perturbations with red-tilted spectra. This aligns well with observations, thus identifying conditions under which the constant-roll inflationary scenario can coherently describe the early universe.

Scalar and Tensor Perturbations

A significant focus of the paper is the behavior of perturbations. The authors conduct a thorough analysis of scalar perturbations, determining that for certain values of $\alpha$, those particularly less than or equal to -3, the super-Hubble evolution of curvature perturbations is stable. The work also includes a treatment of tensor perturbations, ensuring that the computed tensor-to-scalar ratio is consistent with observational data, providing a measure of the inflation model’s viability.

Implications and Speculation on Future Research

The model proposed in this paper is emblematic of a growing interest in revisiting standard assumptions in inflationary theory. By offering a set of exact solutions that are observationally viable, this paper proposes alternatives that can be empirically tested against cosmic microwave background data. Future research may focus on refining the parameter $\alpha$ and further exploring connections to reheating, dark energy, and the nuanced details of scalar and tensor modes during inflation.

In summary, the work by Motohashi, Starobinsky, and Yokoyama contributes a new dimension to the field of cosmological inflationary models by introducing and exploring the implications of a constant rate of roll. The focus on exact solutions and their compatibility with current observational constraints makes it a significant addition to ongoing efforts to understand the early universe's dynamics.