Primordial Inflation and Present-Day Cosmological Constant from Extra Dimensions

A semiclassical gravitation model is outlined which makes use of the Casimir energy density of vacuum fluctuations in extra compactified dimensions to produce the present-day cosmological constant as rho_LAMBDA ~ M^8/M_P^4, where M_P is the Planck scale and M is the weak interaction scale. The model is based on (4+D)-dimensional gravity, with D = 2 extra dimensions with radius b(t) curled up at the ADD length scale b_0 = M_P/M^2 ~ 0.1 mm. Vacuum fluctuations in the compactified space perturb b_0 very slightly, generating a small present-day cosmological constant. The radius of the compactified dimensions is predicted to be b_0 = k^{1/4} 0.09 mm (or equivalently M = 2.4 TeV/k^{1/8}), where the Casimir energy density is k/b^4. Primordial inflation of our three-dimensional space occurs as in the cosmology of the ADD model as the inflaton b(t), which initially is on the order of 1/M ~ 10^{-17} cm, rolls down its potential to b_0.


Introduction
Supernova data indicate that the energy density ρ Λ in a present-day cosmological constant is on the order of 0.7ρ c , where the current critical density ρ c ≈ (2.5 × 10 −3 eV) 4 . It is intriguing that ρ Λ ∼ b −4 0 where b 0 ∼ 0.1 mmjust the length scale for compactified extra dimensions predicted by Arkani-Hamed-Dimopoulos-Dvali (ADD) type theories [1] with two extra spatial dimensions.
It is possible that this dark energy derives from vacuum fluctuations in extra compactified dimensions. We outline here a semiclassical gravitation model which makes use of this mechanism to produce the present-day cosmological constant. The model is based on (4 + D)-dimensional gravity, with D = 2 extra dimensions with radius b(t) curled up at the ADD length scale b 0 , where the subscript "0" denotes present-day values.
The ADD model can be realized [2] in type I ten-dimensional string theory, with standard model fields naturally restricted to a 3-brane [3], while gravitons propagate in the full higher dimensional space. For D = 2, two of the six compactified dimensions are curled up with radius ∼ b 0 , while the remaining four are curled up with radius ∼ 1/M I , with the type I string scale M I ∼ 1 TeV. In this picture, the ADD model is formulated within a consistent quantum theory of gravity.
In addition, if supersymmetry is broken only on the 3-brane, then the bulk cosmological constant vanishes (see e.g. Ref. [4]). A single fine tuning of parameters in the potential for b can then cancel the brane tension, setting the usual four-dimensional cosmological constant to zero.
Semiclassical (4 + D)-dimensional gravitation-with a potential for the scale b of the extra compactified dimensions-rapidly becomes a good approximation to the string theory for energies below M I [5]. In the semiclassical gravitation model, we will assume a potential for b(t) which stabilizes b(t 0 ) at b 0 = M P /M 2 and which vanishes 1 at b 0 in the absence of the Casimir effect, where the (reduced) Planck scale M P = 2.4 × 10 18 GeV and the weak interaction scale M ∼ 1 TeV. Vacuum fluctuations in the compactified space will then perturb b(t 0 ) very slightly away from b 0 , generating a small present-day cosmological constant in our three-dimensional world. This mechanism differs from previous cosmological models incorporating the Casimir effect from vacuum fluctuations in extra compactified dimensions (see e.g. Ref. [6]), in which the Casimir energy density in our three-dimensional world is cancelled by a bulk cosmological constant.
Primordial inflation of our three-dimensional space will occur in the model as the inflaton b(t), which initially is on the order of 1/M ∼ 10 −17 cm, rolls down its potential to b 0 [7,8]. Many e-folds of inflation of our 3-space can occur for sufficiently flat potentials.
We will take the spacetime metric to be R 1 × S 3 × T 2 symmetric 2 where S 3 is a 3-sphere and T 2 is a 2-torus: where M, N run from 0 to 5; i, j run from 1 to 3; and m, n run from 4 to 5.g ij is the metric of a unit 3-sphere andg mn is the metric of a unit 2-torus, with a(t) the radius of physical 3-space and b(t) the radius of the compactified space. The nonzero components of the (4 + D)-dimensional Ricci tensor are The generalized Einstein equations are 0 is the volume of the compactified dimensions today,Ω D denotes the volume of the unit D-torus, and T M N is the energymomentum tensor. The gravitational coupling 8πG = 1/(b 2 0 M 4 ) is weak in the ADD picture because b 0 is much greater than the (4 + D)-dimensional Planck length 1/M.
The nonzero components of the energy-momentum tensor are given by Thus T P P = ρ − 3p a − Dp b . Expressed in terms of the radii a and b, the energy density ρ, and the pressures p a and p b , the Einstein equations become After a few e-folds of primordial inflation of our physical 3-space, the curvature term 2/a 2 on the left-hand side of Eq. (6) will be negligible, and we will henceforth set this term to zero. We will be looking for solutions (neglecting matter) in which physical 3-space is inflating at the present epoch during which b(t) is fixed at b 0 , or in the primordial epoch just after the quantum birth of the universe during which b(t) is inflating to its present value. For an inflating 3-space (without matter), p a = −ρ and the Einstein equations become The energy density and pressures on the right-hand sides of Eqs. (8)-(10) are derivable from the internal energy U = U(a, b): where V = Ω 3 a 3Ω 2 b 2 is the volume of (3 + D)-space and Ω 3 denotes the volume of the unit 3-sphere.
We will consider a potential V (b) for the radius b(t) in the internal energy (at zero temperature) which will produce sufficient primordial inflation to solve the horizon, flatness, homogeneity, isotropy, and monopole problems, and which will stabilize b at b 0 = M P /M 2 ∼ 0.1 mm, with a vanishing cosmological constant. Note that if p a is to equal −ρ, then U must be proportional to a 3 , and that V (b) is dimensionless. The potential V (b) will generate a potential B(b) with the right-hand side of the Einstein equation (10)  Quantum fields will be periodic in the compactified space, producing a Casimir effect [6] in the compactified space and in our three-dimensional world. Adding a Casimir (C) term to the internal energy from vacuum fluctuations in the compactified space will perturb b(t 0 ) very slightly away from b 0 and generate a residual present-day cosmological constant ρ Λ = k/b 4 0 . The sign and magnitude 3 of the constant k depend on the particle content and structure of the underlying quantum gravity theory. The magnitude of k may be expected to be roughly in the range 10 −7 -10 −3 based on the analysis of Candelas and Weinberg [6], who calculated the one-loop Casimir contribution from massless scalar and spin-1 2 particles in (4 + D)dimensional gravitation with an odd number of extra dimensions D curled up near the Planck length. In their work, k is positive for a single massless real scalar field for odd dimensions 3 ≤ D ≤ 19, but may be positive or negative. For our model to produce a positive present-day cosmological constant, we will need k > 0.

Primordial Inflation
In this section, we briefly review the cosmological results for primordial inflation of Refs. [7,8] for the ADD model with internal energy U, and check that the Casimir terms in the Einstein equations when U is replaced by U C do not qualitatively change the primordial cosmological picture.
The Einstein equations with the internal energy given by U in Eq. (12) take the form Replacing U by U C in Eq. (13) introduces Casimir terms into the Einstein equations: The Casimir terms do not qualitatively change the primordial inflationary period of the ADD model, since initially and in the intermediate stage of inflation for k < ∼10 −3 , using the estimates in Ref. [8].

Present-Day Cosmological Constant
In the present epoch, the internal dimensions have a fixed radius b(t 0 ) ≫ 1/M and H b = 0. Without the Casimir terms, the static solution for b(t 0 ) requires V (b 0 ) = 0 = V ′ (b 0 ). In our model, vacuum fluctuations in the compactified space perturb b 0 very slightly tob 0 , producing a small cosmological constant in our three-dimensional world. We assume that the potential V (b) is independent of the Casimir effect, so that V (b 0 ) and V ′ (b 0 ) still equal zero. The Einstein equations with Casimir contributions for an inflating 3-space now take the form Eq. (24) then predicts a present-day cosmological term or, in other words, This cosmological term will ≈ 0.7ρ c if b 0 ≈ k 1/4 0.09 mm, or equivalently if M ≈ 2.4 TeV/k 1/8 . Note that the Casimir effect has caused the stabilized radius b 0 to increase slightly, yielding a positive present-day cosmological constant.
The canonically normalized "radion" field ϕ(t) = 2M 2 b(t). The mass squared of the radion field is is positive if m 2 ϕ is, and are globally stable if the respective potentials B(b) and B C (b) are, for example, concave upward (the simplest case), since If the number of extra dimensions D is allowed to be greater than two, the Einstein equations (24) and (25) for an inflating 3-space with static b(t) change to but the result for the present-day cosmological constant has the same form Thus ρ Λ has the right parametric dependence M 8 /M 4 P only for D = 2.

Conclusion
The cosmological picture presented here joins smoothly onto the primordial inflation and big-bang cosmological pictures: The quantum birth of the universe begins with a and b ∼ 1/M. Many (≫ 70) e-folds of primordial inflation occur as the inflaton b(t) rolls down its potential tob 0 . b(t) then undergoes damped oscillations aboutb 0 , heating the universe up to a temperature T above the temperature for big-bang nucleosynthesis (BBN) and creating essentially all the matter and energy we see today. (See Refs. [7] and [10] for two differing views on the maximum value of T , above which the evolution of the universe in ADD-type theories cannot be described by the radiationdominated Friedmann-Robertson-Walker model.) At this point, the universe evolves according to the standard big-bang picture, expanding and cooling, with a fixed small cosmological constant ρ Λ = k/b 4 0 ≈ (2.3 × 10 −3 eV) 4 . This dark energy density is much less than the BBN energy density ∼ (1 MeV) 4 and plays a role in the evolution of the universe only recently, long after the equality of energy density ∼ (1 eV) 4 in matter and radiation. The radius b(t) of the compactified space has not changed since well before BBN.
Finally we note that if the stabilization potential V (b) vanishes at its global minimum, the resolution of the cosmic coincidences of Ref. [11] is naturally realized in the Casimir effect since parametrically ρ Λ ∼ M 8 /M 4 P .