Holography, the cosmological constant and the upperlimit of the number of e-foldings

Abstract

If the source of the current accelerating expansion of theuniverse is a positive cosmological constant, Banks andFischler argued that there exists an upper limit of the totalnumber of e-foldings of inflation. We further elaborate on theupper limit in the sense of viewing the cosmological horizon asthe boundary of a cavity and of the holographic D-bound in a deSitter space. Assuming a simple evolution model of inflation, weobtain an expression of the upper limit in terms of thecosmological constant, the energy density of the inflaton when theinflation starts, the energy density as the inflation ends andreheating temperature. We discuss how the upper limit ismodified in the different evolution models of the universe. Theholographic D-bound gives a higher upper limit than the entropythreshold in the cavity. For the most extreme case where theinitial energy density of inflation is as high as the Planckenergy, and the reheating temperature is as low as the energyscale of nucleosynthesis, the D-bound gives the upper limit as146 and the entropy threshold as 122. For a reasonable assumptionin the simplest cosmology, the holographic D-bound leads to avalue of about 85, while the cavity model gives a value of around 65for the upper limit, which is close to the value needed tosolve the flatness problem and the horizon problem in the hot bigbang cosmology.