Power-Law Inflation Survives Observational Constraints

Abstract

Power-law inflation has stood as a classical model in inflationary cosmology since the early 1980s, prized for its exact analytical solutions and ability to naturally resolve the Big Bang theory's horizon and flatness problems through exponential expansion. However, its simplest form appears incompatible with modern precision observations, motivating increasingly complex alternatives. In this work, we demonstrate how previous predictions with power-law inflation considered only a particular solution of the field equations, and derive the complete set of general analytical solutions that satisfy current theoretical and observational constraints. This finding revitalizes power-law inflation as a viable framework, offering new possibilities for cosmological model-building while preserving its original mathematical elegance.

Figures (3)

• Figure 1: Attractor behaviors for various values of $\lambda$
• Figure 2: Numerical solutions of $\omega(t)$, $\phi(t)$, and $a(t)$. The blue curves correspond to the $C>0$ case, while the red curves represent $C=0$. The value of $\lambda$ is fixed to 0.04 because it leads to model parameters that satisfy observational constraints on $n_{\mathrm{s}}$ and $r$. Different $\lambda$ values would merely stretch or compress the curves. The horizontal axis $M_{\rm pl} t / c_0$ is used to removes rescaling caused by $C$ and keeps the time variable dimensionless. For the $C=0$ case we conventionally set $c_0 = 1$; for the $C>0$ case, $c_0$ is defined in Eq. \ref{['eq:ful11']}, and integration constants are chosen as $t_0 = 0$ and $a_0 = 1$, where $t_0$ simply sets the horizontal origin without affecting the physical evolution.
• Figure 3: Parameter space consistent with the current constraints on $n_{\rm s}$ and $r$, as inferred from Planck 2018 data Planck2018 and a combination of BICEP/Keck 2018 and Planck PR4 data Tristram:2021tvhTristram:2020wbiBICEP:2021xfz. The solid contours indicate the 68% (1$\sigma$) and 95% (2$\sigma$) confidence levels.