Cyclic inflation

We present an inflationary model that is geodesically complete and does not suffer from the trans-Planckian problem. In most inflationary models, massless (conformal) scalar field fluctuations in a de Sitter background gives rise to a scale-invariant spectrum. In this work, we realize scale-invariant perturbations from thermal fluctuations in (conformal) radiation during a radiation dominated contraction era prior to inflation. As the modes exit the Hubble radius during the contraction phase, scale-invariant fluctuations are generated. After many cycles, we enter into a power-law inflationary phase, that stretches the modes produced in the previous contraction phase to scales that we observe today.

Figure 1

Qualitative plot of how the scale factor evolves as a function of the conformal time $\tau$. The cycles have the same period in $\tau$, but they grow as a function of proper time. The blue curve denotes the transition between the Hagedorn (below the curve) and the nonthermal (above the curve) phase. The evolution ends in an inflating expanding branch.

Figure 2

The red curve corresponds to $V(\varphi )$, while the blue curves correspond to $\rho (\varphi )$ at the bounce point (the uppermost curve) and turnaround points (the lower curves). The scalar field (represented by the black ball) approximately tracks the minimum formed between $V(\varphi )$ and $\rho (\varphi )$. As the entropy and the volume of the Universe increases from cycle to cycle, the turnaround happens at lower and lower energies, until the penultimate cycle (the lowest blue curve). Before the next turnaround can happen, we enter the inflationary phase, which ultimately leads to our Universe.