# Does History Repeat Itself? Periodic Time Cosmology

## TL;DR

This paper proposes a model of the universe with a perfectly repeating cosmic history, where the primordial power spectrum is reconstructed from the transfer function and observational data, offering a novel cyclic cosmology framework.

## Contribution

It introduces Periodic Time Cosmology, a model where the universe's history repeats exactly, linking the primordial power spectrum to the transfer function and observational constraints.

## Key findings

- Minimal model is disfavoured at 5.1σ compared to power-law spectrum.
- Extensions with modified dark energy equation of state improve fit.
- Periodic Time Cosmology remains viable with certain parameter adjustments.

## Abstract

It has been suggested that the cosmic history might repeat in cycles, with an infinite series of similar aeons in the past and the future. Here, we instead propose that the cosmic history repeats itself exactly, constructing a universe on a periodic temporal history, which we call Periodic Time Cosmology. In particular, the primordial power spectrum, convolved with the transfer function throughout the cosmic history, would form the next aeon's primordial power spectrum. By matching the big bang to the infinite future using a conformal rescaling (a la Penrose), we uniquely determine the primordial power spectrum, in terms of the transfer function up to two free parameters. While nearly scale invariant with a red tilt on large scales, using Planck and Baryonic Acoustic Oscillation observations, we find the minimal model is disfavoured compared to a power-law power spectrum at $5.1\sigma$. However, extensions of $\Lambda$CDM cosmic history change the large scale transfer function and can provide better relative fits to the data. For example, the best fit seven parameter model for our Periodic Time Cosmology, with $w=-1.024$ for dark energy equation of state, is only disfavoured relative to a power-law power spectrum (with the same number of parameters) at $1.8\sigma$ level. Therefore, consistency between cosmic history and initial conditions provides a viable description of cosmological observations in the context of Periodic Time Cosmology.