New Universal Local Feature in the Inflationary Perturbation Spectrum

TL;DR

This work identifies a universal local feature in the inflationary perturbation spectrum caused by a fast, small change in the inflaton's effective mass, implemented as a step in $m_{ m eff}^2$ due to a rapid second-order phase transition of a coupled scalar field. The authors provide a general treatment showing how a discontinuity in $V''(\varphi)$ translates into a jump in $m_{ m eff}^2$ in the Mukhanov–Sasaki equation and derive the resulting perturbation spectrum, including a step in the spectral index $n_s(k)$ with superimposed decaying oscillations. They connect this to a concrete microphysical model based on a hybrid-inflation-like potential, deriving the pre- and post-transition indices $n_1$ and $n_2$ in terms of model parameters and showing how a transition near $k_0$ can produce the observed running of $n_s$ over a limited range of scales. The findings suggest that the WMAP-reported running could originate from such a fast phase transition during inflation, with the feature's position and amplitude constrained by cosmological normalization; further work is needed to compare to Planck data and map parameters to observations.

Abstract

A model is developed in which the inflaton potential experiences a sudden small change in its second derivative (the effective mass of the inflaton). An exact treatment demonstrates that the resulting density perturbation has a quasi-flat power spectrum with a break in its slope (a step in n_s). The step in the spectral index is modulated by characteristic oscillations and results in large running of the spectral index localised over a few e-folds of scales. A field-theoretic model giving rise to such behaviour of the inflationary potential is based on a fast phase transition experienced by a second scalar field weakly coupled to the inflaton. Such a transition is similar to that which terminates inflation in the hybrid inflationary scenario. This scenario suggests that the observed running of the spectral index in the WMAP data may be caused by a fast second order phase transition which occurred during inflation.

Figures (4)

• Figure 1: The transfer function $\left\vert\alpha-\beta\right\vert^2$ is shown as a function of $x=k/k_0$. The exact expression for $\left\vert\alpha-\beta\right\vert^2$ given in (\ref{['eq:bog3']}) is represented by the solid line in the left and right panels while the asymptotic expression (\ref{['eq:asymp1']}) is shown dot-dashed in the right panel which shows the oscillations in the transfer function in greater detail. Note that the asymptotic expression provides an excellent approximation to the results for $x=k/k_0 \hbox{$\buildrel > \over \sim$} ~ 2$. The feature associated with the step in $V"(\phi)$ occurs at $x \sim 1$. The relevant values of the parameters are $\mu_1 = 1.49, \,\mu_2 = 1.52$.
• Figure 2: The primordial spectral index $n_s$ is shown as a function of $x = k/k_0$ for an inflationary model in which the potential has a sudden change in its second derivative. Such a discontinuity in $V"$ leads to step in $n_s$ at $x \sim 1$ which is followed by oscillations with decreasing amplitude described by (\ref{['eq:index2']}). The parameters of our model are $\mu_1 = 1.49,\, \mu_2 = 1.52$ which correspond to $n_1 = 1.02$, $n_2 = 0.96$.
• Figure 3: The spectral index just before ($n_1$) the phase transition in hybrid inflation and immediately after it ($n_2$), is shown as a function of the parameter $\kappa = 2\lambda m^2/g^2M^2$ in the left and right hand panel of this figure. The red (solid) line corresponds to 50 e-folds of inflationary expansion occuring after the phase transition in hybrid inflation, while the dashed (blue) line corresponds to 60 e-folds.
• Figure 4: Spectral indices for perturbations generated just before ($n_1$) and immediately after ($n_2$) the phase transition in hybrid inflation are shown. Note that distinct values of the pair $\lbrace n_1,n_2\rbrace$ correspond to different values of the parameter $\kappa$, as shown in figure \ref{['fig:n1-alpha']}. Results are shown for two possible number of e-folds after the phase transiton: ${\cal N} = 50$ (red, solid), and ${\cal N} = 60$ (blue, dashed).