Why We Prefer Concave Shape Inflation Potential Rather Than Convex? (Cosmology / Quantum / Maths)

Summary:

⦿ There are two types of inflation potential we generally prefer: Concave and convex. The Planck data on cosmic microwave background indicates that the Starobinsky-type model with concave inflation potential is favored over the convex-type chaotic inflation. But why? This reason is still unclear.

⦿ Now, Chen and Yeom investigated Euclidean wormholes in the context of the inflationary scenario in order to answer the question on the preference of a specific shape of the inflaton potential.

⦿ They argued that if our universe began with a Euclidean wormhole, then the Starobinsky-type inflation is probabilistically favored.

⦿ They showed that only one end of the wormhole can be classicalized for a convex potential, while both ends can be classicalized for a concave potential. The latter is therefore more probable.

⦿ Their study point towards the fact that its not the universe but the wormhole which is expanding


How did the universe begin? This has long been one of the most fundamental questions in physics. The Big Bang scenario, when tracing back to the Planck time, indicates that the universe should start from a regime of quantum gravity that is describable by a wave function of the universe governed by the Wheeler-DeWitt (WDW) equation. The WDW equation is a partial differential equation and hence it requires a boundary condition. This boundary condition allows one to assign the probability of the initial condition of our universe. As is well known, to overcome some drawbacks of the Big Bang scenario, an era of inflation has been introduced. Presumably, the boundary condition of the WDW equation would dictate the nature of the inflation.

The Planck data on cosmic microwave background (CMB) indicates that certain inflation models are more favored than some others. In particular, the Starobinsky-type model with concave inflation potential (V” < 0 when the inflation is dominant.) appears to be favored over the convex-type (V” > 0) chaotic inflation. Is there any reason for this? Now, Chen and Yeom argued that if our universe began with a Euclidean wormhole, then the Starobinsky-type inflation is probabilistically favored.

One reasonable assumption for the boundary condition of the WDW equation was suggested by Hartle and Hawking, where the ground state of the universe is represented by the Euclidean path integral between two hypersurfaces. The Euclidean propagator can be described as follows:

where gµν is the metric, φ is an inflaton field, SE is the Euclidean action, and h (Sys. (a,b), stat. (µν)) and χ^a,b are the boundary values of gµν and φ on the initial (say, a) and the final (say, b) hypersurfaces, respectively. Using the steepestdescent approximation, this path integral can be well approximated by a sum of instantons, where the probability of each instanton becomes P ∝ e^−S_E. This approach has been applied to different issues with success: (1) It is consistent with the WKB approximation, (2) It has good correspondences with perturbative quantum field theory in curved space, (3) It renders correct thermodynamic relations of black hole physics and cosmology. These provide Chen and Yeom the confidence that the eventual quantum theory of gravity should retain this notion as an effective description.

FIG. 1: Complex time contour and numerical solution of ar, ai, φr, and φi for Vch. The upper figure is a physical interpretation about the wormhole, where Part A (red) and C (green) are Lorentzian and Part B (blue) is Euclidean. © Chen and Yeom

In their original proposal, Hartle and Hawking considered only compact instantons. In that case it is proper to assign the condition for only one boundary; this is the so-called no-boundary proposal. In general, however, the path integral should have two boundaries. If the arrow of time is symmetric between positive and negative time for classical histories, then one may interpret this situation as having two universes created from nothing, where the probability is determined by the instanton that connects the two classical universes. Such a process can be well described by the Euclidean wormholes.

Now, Chen and Yeom investigated Euclidean wormholes in the context of the inflationary scenario in order to answer the question on the preference of a specific shape of the inflaton potential. They showed that only one end of the wormhole can be classicalized for a convex potential, while both ends can be classicalized for a concave potential. The latter is therefore more probable.

“We investigated Euclidean wormholes with a non-trivial inflaton potential. We showed that in terms of probability, the Euclidean path-integral is dominated by Euclidean wormholes, and only the concave potential explains the classicality of Euclidean wormholes. This helps to explain, in our view, why our universe prefers the Starobinsky-like model rather than the convex-type chaotic inflation model.

— told Chen, first author of the study

It should be mentioned that there exist other attempts to explain the origin of the concave inflation potential. For example, it was reported by Hertog in his paper that, the Starobinsky-like concave potential is preferred if a volume-weighted term is added to the measure. Chen and Yeom note that the same principle can be applied not only to compact instantons but also to Euclidean wormholes; hence, this proposal may support their result as well. They must caution, however, that the justification of such a volume-weighted term is theoretically subtle.

This is of course not the end of the story. One needs to further investigate whether this Euclidean wormhole methodology is compatible with other aspects of inflation. It will also be interesting to explore the relation between the probability distribution of wormholes and the detailed shapes of various inflaton potentials. Furthermore, if this Euclidean wormhole creates any bias from the Bunch-Davies state, then it may in principle be confirmed or falsified by future observations. They left these topics for future investigations.

Featured image: Comparing the inflation models with the observational constraints. © Ke Wang


Reference: Chen, P., Yeom, Dh. Why concave rather than convex inflaton potential?. Eur. Phys. J. C 78, 863 (2018). https://doi.org/10.1140/epjc/s10052-018-6357-0 https://link.springer.com/article/10.1140/epjc/s10052-018-6357-0


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