### August 29

*and*

**Tirthabir Biswas, Anupam Mazumdar***analyze in their paper*

**Warren Siegel***Bouncing Universes in String-inspired Gravity*interesting cosmological questions by putting together general relativity, Newtonian gravity, inflation, Yang-Mills field theory and string theory.

They start by adding to the Einstein-Hilbert action an infinite sum of higher order derivative terms (Eqn. (1.5)). They find approximate and exact solutions such that that (i) the action can give rise to a ghost free and asymptotically free theory of gravity, and (ii) bouncing cosmological solutions for this type of actions.

In the usual FRW cosmology, it is the attractive force of gravity which makes the universe contract, eventually to a singularity. Now in the presence of asymptotically free gravity spacetime contracts but the internal pressure of matter resists, therefore causing the universe to bounce from the contracting phase into an expanding one (Eqn. (4.6), hyperbolic cosine bounce). Their cosmological constant is mainly required for the late-time consistency for their bounce ansatz (4.6), it is not necessary for having a bounce. They argue that its absence solves the graceful exit problem.

They divide the evolution into two distinct regimes: (i) near the bounce when the higher curvature terms dominate the evolution (ii) away from (both before and after) the bounce when the higher order terms can be ignored as compared to ordinary gravity and there is normal FRW evolution. They say they are unable to clearly identify a connection of these constraints to the necessity or sufficiency of asymptotic freedom. The write the most likely reason is that they only looked at a very specific bounce solution, namely the hyperbolic cosine bounce, and therefore are missing more complicated bounces which may be present in some of the asymptotically free theories. "We reserve a more detailed study of these issues to future research." The transition from the bounce should lead to a radiation dominated epoch before and after the bounce. During these phases, they claim to have ordinary gravity coupled to an ideal gas of matter/radiation fluid satisfying the usual omega, rho and Hubble equations (Eqn. (5.13)).

I find the paper very inspiring. A number of interesting calculations have been performed. Perhaps I have not grasped it right so I ask is the addition of higher (infinite) order derivative terms in the Einstein-Hilbert action something like adding resonances and Regge poles (in the 1960's) to get the full amplitude. What would the proper duality entities be, and the relationship between them? I do not have the answer but my favorite thought is in the direction of preons having color-like interaction near Planckian distances (there is only an uncompleted manuscript of it, which only Peter has seen, and one withdrawn from arxiv last year).

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