On the accuracy of slow-roll inflation given current observational constraints

TL;DR

Current observational constraints require nearly scale-invariant primordial spectra but permit nonzero running $\alpha_s$; this paper assesses slow-roll accuracy by comparing full numerical solutions with first- and second-order slow-roll formulae across representative potentials and flow-equation simulations. It shows first-order slow-roll generally reproduces the power spectrum to within 1–2% even with sizable running, while second-order offers little improvement; the main source of error is the slope $n_s$ rather than $\alpha_s$ or higher terms. In models with large running and a red-to-blue tilt change, higher derivatives can be important, and the AKS parameterization is not universally superior to a truncated Taylor expansion. Overall, for current data, the simple Taylor-based slow-roll description remains robust, with full numerics reserved for scenarios with pronounced running or nontrivial spectral features.

Abstract

We investigate the accuracy of slow-roll inflation in light of current observational constraints, which do not allow for a large deviation from scale invariance. We investigate the applicability of the first and second order slow-roll approximations for inflationary models, including those with large running of the scalar spectral index. We compare the full numerical solutions with those given by the first and second order slow-roll formulae. We find that even first order slow-roll is generally accurate; the largest deviations arise in models with large running where the error in the power spectrum can be at the level of 1-2%. Most of this error comes from inaccuracy in the calculation of the slope and not of the running or higher order terms. Second order slow-roll does not improve the accuracy over first order. We also argue that in the basis $ε_0=1/H$, $ε_{n+1}={d\ln|ε_n|}/{dN}$, introduced by Schwarz et al. (2001), slow-roll does not require all of the parameters to be small. For example, even a divergent $ε_3$ leads to finite solutions which are accurately described by a slow-roll approximation. Finally, we argue that power spectrum parametrization recently introduced by Abazaijan, Kadota and Stewart does not work for models where spectral index changes from red to blue, while the usual Taylor expansion remains a good approximation.

Figures (8)

• Figure 1: The current constraints (68% and 95% confidence level contours) in the $n_s$--$\alpha_s$ plane from WMAP+SDSSgal (bigger, red) and WMAP+SDSSlya (smaller, green) data 2005PhRvD..71j3515S2004PhRvD..69l3003S2004PhRvD..69j3501T2003ApJS..148..195V. The constrained region clearly allows the value of the spectral scalar index $n_s$ to be around $1$ and the running $\alpha_s$ of the scalar spectral index to be significantly non-zero for either combination of the experiments.
• Figure 2: Panels from the bottom to the top: 1. Potential $V=m^2\phi^2/2$ in the range of $\phi$'s where the inflation occurs around 50 e-folds before the end of inflation. The scale in $V$ is not COBE-normalized on this plot. 2. The dependence of $z"/z$ on the number of the e-folds during the inflation. Number of e-folds $N=0$ corresponds to our arbitrarily chosen pivot scale of $k=0.05$/Mpc. 3. Plots of Hubble slow-roll parameters $\epsilon_{ H}$, $\eta_{ H}$, $\xi_{ H}$. The value of $\epsilon_{ H}$ is non-negligible whereas $\eta_{ H}$ and $\xi_{ H}$ are essentially zeros. 4. Plots of horizon-flow slow-roll parameters $\epsilon_1$, $\epsilon_2$, $\epsilon_3$ and the product of $\epsilon_2\epsilon_3$ for the same inflationary model. Since in this case $\epsilon_{ H}\gg\eta_{ H},\xi_{ H}$, the value of $\epsilon_1$ is essentially $\epsilon_1\approx\epsilon_{ H}$ and $\epsilon_2\approx\epsilon_3\approx2\epsilon_{ H}$. Nothing unexpected is going on here for this model.
• Figure 3: Panels from the bottom to the top: 1. Power spectrum produced by inflation with potential $V=m^2\phi^2/2$. 2. Errors made by approximating the true scalar power spectrum by \ref{['eq:parameterization']} with calculating $n_s$ and $\alpha_s$ from first order slow-roll formulae (\ref{['eq:nsm1_first']}, \ref{['eq:running_first']}), second order slow-roll formulae (\ref{['eq:nsm1']}, \ref{['eq:running']}) and calculating them through numerical derivatives. Numerical third order takes into account third logarithmic derivative of the power spectrum $\beta_s$ in parameterization \ref{['eq:parameterization']}. 3. The evolution of $n_s-1$ is calculated numerically and by the first and second order slow-roll formulae. 4. The evolution of $\alpha_s$ is calculated numerically and by the slow-roll formulae (first and second order slow-roll are the same for $\alpha_s$). The error plot clearly shows that the main error comes from the imprecision of $n_s-1$, whereas the approximation for the $\alpha_s$ works well enough. One can also see that in the case of this potential both first and second order slow-roll formulae for $n_s-1$ overestimate the real value of $n_s-1$.
• Figure 4: Panels from the bottom to the top: 1. Potential \ref{['eq:smoothedpotential']}. 2. Effectively this plot shows the dependence of $z"/z$ on the number of the e-folds during the inflation. 3. Plots of Hubble slow-roll parameters $\epsilon_{ H}$, $\eta_{ H}$, $\xi_{ H}$. Though the potential has a singular behavior, all the flow parameters are smooth. 4. Plots of horizon-flow slow-roll parameters $\epsilon_1$, $\epsilon_2$, $\epsilon_3$ and the product of $\epsilon_2\epsilon_3$ for the same inflationary model. While everything is fine with $\epsilon_1$, $\epsilon_2$ and $\epsilon_2\epsilon_3$, the value of $\epsilon_3$ indeed flips over infinity.
• Figure 5: Panels from the bottom to the top: 1. Power spectrum produced by \ref{['eq:smoothedpotential']}. 2. Errors made by approximating the true scalar power spectrum by \ref{['eq:parameterization']} with calculating $n_s$ and $\alpha_s$ from first order slow-roll formulae (\ref{['eq:nsm1_first']}, \ref{['eq:running_first']}), second order slow-roll formulae (\ref{['eq:nsm1']}, \ref{['eq:running']}) and calculating them through numerical differentiation. Numerical third order takes into account third logarithmic derivative of the power spectrum $\beta_s$ in parameterization \ref{['eq:parameterization']}. 3. The evolution of $n_s-1$ is calculated numerically and by the first and second order slow-roll formulae. 4. The evolution of $\alpha_s$ is calculated numerically and by the slow-roll formulae (first and second order slow-roll are the same for $\alpha_s$). The error plot clearly shows that the main error comes from the imprecision of $n_s-1$, whereas the approximation for the $\alpha_s$ works well enough. We see that in the case of this potential, first order slow-roll formula for $n_s-1$ underestimates the real value, while the second order formula overestimates it.