Model independent bounds for the number of $e$-folds during the evolution of the universe

We present a simple procedure to obtain universal bounds for quantities of cosmological interest, such as the number of $e$-folds during inflation, reheating, and radiation, as well as the reheating temperature. The main assumption is to represent each of the various epochs of evolution of the universe as being due to a single substance changing instantaneously into the next, describing a new era of evolution of the universe. This assumption, commonly used to obtain solutions of the Friedmann equations for simple cosmological models, is implemented here to find model-independent bounds on cosmological quantities of interest. In particular, we find that the bound $N_k\approx 56$ for $-\frac{1}{3} < \omega_{re} < \frac{1}{3}$ is very robust as an upper bound on the number of $e$-folds during inflation and also as a lower bound when $\omega_{re} > \frac{1}{3}$, where $\omega_{re}$ is the effective equation of state parameter during reheating. These are model-independent results that any single-field model of inflation should satisfy. As an example, we illustrate the two approaches with the basic $\alpha$ attractor model and show how they complement each other.