Setting initial conditions for inflation with reaction–diffusion equation
We discuss the issue of setting appropriate initial conditions for inflation. Specifically, we consider natural inflation model and discuss the fine tuning required for setting almost homogeneous initial conditions over a region of order several times the Hubble size which is orders of magnitude larger than any relevant correlation length for field fluctuations. We then propose to use the special propagating front solutions of reaction–diffusion equations for localized field domains of smaller sizes. Due to very small velocities of these propagating fronts we find that the inflaton field in such a field domain changes very slowly, contrary to naive expectation of rapid roll down to the true vacuum. Continued expansion leads to the energy density in the Hubble region being dominated by the vacuum energy, thereby beginning the inflationary phase. Our results show that inflation can occur even with a single localized field domain of size smaller than the Hubble size. We discuss possible extensions of our results for different inflationary models, as well as various limitations of our analysis (e.g. neglecting self gravity of the localized field domain).
KeywordsInflation Initial conditions Reaction–diffusion equation Phase transition
We are very grateful to Raghavan Rangarajan for very useful and detailed comments on the manuscript. We also thank Subhendra Mohanty, Nirupam Dutta, Oindrila Ganguly, Pranati Rath, and Biswanath Layek for useful discussions. Some of the results here were presented by AMS at the International conference “Saha Theory Workshop: Aspects of Early Universe Cosmology”, SINP, Kolkata, 16–20 Jan, 2017. We thank the participants of this conference, especially Arjun Berera, Koushik Dutta, and L. Sriramkumar for very useful comments and suggestions. We thankfully acknowledge Robert Brandenberger for informing us about important previous works relating to the issue of initial conditions.
- 1.Gravitation and Cosmology: Principles and Applications of the General Theory Of Relativity. S. Weinberg, Wiley (1972)Google Scholar
- 2.Riotto, A.: arXiv:hep-ph/0210162
- 8.Starobinsky, A.A.: Field theory. In: de Vega, H.J., Sanchez, N. (eds.) Quantum Gravity and Strings. Springer, Berlin (1986)Google Scholar
- 14.Freese, K.: arXiv:astro-ph/9310012
- 32.Rangarajan, R.: arXiv:1506.07433
- 35.Bradshaw-Hajek, B.: Reaction–diffusion Equations for Population Genetics. Ph.D. thesis, School of Mathematics and Applied Statistics, University of Wollongong (2004). http://ro.uow.edu.au/thesis/201
- 43.Sengupta, S.: Aspects of QCD Phase Transition with Reaction–Diffusion Equations. Ph.D. thesis, Homi Bhabha National Institute, India (2015)Google Scholar
- 44.Kolb, E.W., Turner, M.S.: The Early Universe. Addison-Wesley Publishing company (1990)Google Scholar
- 48.Greiner, C., Xu, Z., Biro, T.S.: arXiv:hep-ph/9809461
- 51.Bjorken, J.D., Kowalski , K.L., Taylor, C.C.: in Results and Perspectives in Particle Physics 1993; Proceedings of the 7th Rencontres de Physique de la Vallee dAoste, La Thuile, Italy, (1993) edited by M. Greco (Editions Frontieres, Gif-sur-Yvette, France, 1993)Google Scholar