A new cosmological constant model

We propose a new cosmological model with a time-dependent cosmological constant (Λ ∝ 1/t), which starting at the Planck time as ΛP l ∼ M P l, evolves to the present-day allowed value of Λ0 ∼ 10M P l. This scenario is supported by non-critical string theory considerations. We compute the age of the Universe and the time-dependence of the scale factor in this model, and find general agreement with recent determinations of the Hubble parameter for substantial values of ΩΛ. This effectively low-density open Universe model differs from the traditional cosmological constant model, and has observable implications for particle physics and cosmology. CERN-TH/95–6 CTP-TAMU-69/94 ACT-25/94 January 1995 The problem of the cosmological constant (Λ) has been with us for over 75 years, ever since Einstein introduced it in order to avoid an expanding Universe. Of course, Einstein was misinformed, although on the other hand, since “whatever is not forbidden is mandatory” one can say that he acted in a physically reasonable way. A lot of effort has been spent tackling this monumental problem, and yet a universally acceptable solution is still lacking. The basic problem is that the vacuum energy of (spontaneously broken) gauge quantum field theories, when coupled to gravity is metamorphosed into a cosmological constant which is usually much larger than the presently allowed one. In this short note we propose yet another cosmological constant model. While our model contains seeds of some ideas that have been developed recently in the framework of non-critical string theory [1], it is motivated by more general, phenomenologically oriented principles, as well as by the recent determination of the Hubble parameter (H0) by the Hubble Space Telescope. Let us start from the Einstein-Friedmann equation of standard Big-Bang Cosmology H = 8πG 3 ρ− k R + Λ 3 , (1) where H = Ṙ/R is the Hubble parameter, R is the scale factor, G is Newton’s constant, ρ is the particle energy density, and k = +1,−1, 0 is the curvature parameter for a closed/open/flat Universe. The present day values of the various parameters (H0 ∼ 1/t0 ∼ 100 km sMpc, with t0 ∼ 10Gyr the age; T0 ≈ 2.73K, Λ0 <∼ 10M P l) are extremely small or large, and make the standard Big-Bang Cosmology rather unnatural. Inflation has been introduced to solve some of these problems with great success: an “instantaneous” blow up of the scale factor (Rafter ∼ 1028Rbefore) creates a smooth, effectively “flat” (i.e., k/R → 0 ⇔ keff = 0) Universe of the right size and entropy to describe our Universe. This natural obliteration of the curvature term is rather effective, and it is interesting to see whether the same can be done to the cosmological “constant” term. Let us suppose that Λ = ΛP l (R/lP l) , (2) where ΛP l ∼ M P l is the “natural” size of the cosmological constant, (R/lP l) is the scale factor in units of the Planck length, and α is a constant to be determined by the present upper bound on Λ: Λ0 <∼ 10M P l, and the present value of R: R0/lP l ∼ ct0/lP l ∼ 10. That is α = 2, and Λ = ΛP l (R/lP l) ∝ 1 R , (3) has the same R dependence as the curvature term, and would give us today an acceptable and perhaps even detectable cosmological “constant”. Let us emphasize the effect of inflation by rewriting Eq. (1) as H ∼ M P l (T/MP l)4 + M P l (lP l/R)2. Before inflation T ∼ MP l, R ∼ lP l and both terms are comparable; after inflation and