Comment on "Attractor solutions in scalar-field cosmology" and "How many e-folds should we expect from high-scale inflation?"

TL;DR

The paper challenges the claimed uniqueness of Carroll-Remmen conserved measures for a scalar field with a quadratic potential in spatially flat FRW cosmology. It derives general asymptotic solutions in both low-energy and high-energy regimes, showing that the measure function on phase space can be written as $ f = \Psi(\alpha)/r^3 $ in the low-energy limit and as $ f = \Theta(\mathrm{sgn}(y) r \cos\theta)/( \mathrm{sgn}(y) r^2 \sin\theta ) $ in the high-energy limit, with $\Psi$ and $\Theta$ being arbitrary functions, hence infinitely many conserved measures. Because different choices of these arbitrary functions yield different predictions for the total number of e-folds of inflation, e.g. $ \langle N_{\text{tot}}\rangle = \gamma^2/8 $ for some measures, the claimed universality is undermined. The results emphasize the sensitivity of inflationary inferences to the chosen phase-space measure and argue that previous claims of a unique Carroll-Remmen measure are not supported.

Abstract

In Ref. [1], it was claimed that in the spatially flat cosmological case there exists a unique conserved measure (up to normalization) on the $(φ,\dotφ)$ phase space for scalar field with $m^2φ^2$ potential by finding a unique solution to the differential equation (44) (in Ref. [1]) in the low-energy regime. In Ref. [2], it was also claimed that a unique solution to the same differential equation was found in the high-energy regime and using this solution the authors calculated the expected total number of e-folds of inflation. In this comment, we reanalyze the differential equation (44) and obtain general solutions both in the low-energy and high-energy regime, which can include the solution in Ref. [1] and the solution in Ref. [2] as a special case in the corresponding energy regime. In this way, we find that following the constructions in Ref. [1] there actually exist infinitely many nonequivalent conserved measures for the scalar-field cosmology with $m^2φ^2$ potential on the $(φ,\dotφ)$ phase space. Moreover, through specific calculations, we also show that different choices of measures can lead to quite different predictions of the expected total number of e-folds of inflation.

Figures (4)

• Figure 1: Solutions for the Klein-Gordon equation $\ddot{\phi}+\sqrt{\frac{3\kappa}{2}}\sqrt{\dot{\phi}^2+m^2\phi^2}\dot{\phi}+m^2\phi=0$. Solid lines denote the attractive separatrices which approach $\dot{\phi}=\pm\sqrt{\frac{2}{3\kappa}}m$ at large field values. Plots are in $(\phi,\dot{\phi})$ space, in units where $\kappa=1$ and $m=0.25M_{\text{Pl}}$.
• Figure 2: Dynamical behavior of scalar-field cosmology with $m^2\phi^2$ potential near the origin of $(\phi,\dot{\phi})$ space, in units where $\kappa=1$ and $m=0.25M_{\text{Pl}}$.
• Figure 3: The probability distribution function $P(\theta,r_0)$ at $\sqrt{\kappa}r_{0}/m=10^3$ with $m=10^{-2}M_{\text{Pl}}$.
• Figure 4: The probability distribution function $P(\theta,r_1)$ at $\sqrt{\kappa}r_{1}/m=0.1$. $\theta^{(1)}$ and $\theta^{(2)}=\theta^{(1)}+\pi$ are the angles at which the two attractive separatrices intersect the circle $r=r_1$ respectively in the space $(mx,y)$.