Primordial inflation and present-day cosmological constant from extra dimensions

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1 3 January 22 Physics Letters B ) Primordial inflation and present-day cosmological constant from extra dimensions Carl L. Gardner 1 Department of Mathematics Arizona State University Tempe AZ USA Received 31 May 21; received in revised form 3 August 21; accepted 3 August 21 Editor: J. Frieman Astract A semiclassical gravitation model is outlined which maes use of the Casimir energy density of vacuum fluctuations in extra compactified dimensions to produce the present-day cosmological constant as ρ Λ M 8 /M 4 P wherem P is the Planc scale and M is the wea interaction scale. The model is ased on 4 + D)-dimensional gravity with D = 2 extra dimensions with radius t) curled up at the ADD length scale = M P /M 2.1 mm. Vacuum fluctuations in the compactified space pertur very slightly generating a small present-day cosmological constant. The radius of the compactified dimensions is predicted to e 1/4.9 mm or equivalently M 2.4TeV/ 1/8 ) where the Casimir energy density is / 4. Primordial inflation of our three-dimensional space occurs as in the cosmology of the ADD model as the inflaton t)which initially is on the order of 1/M 1 17 cm rolls down its potential to. 22 Pulished y Elsevier Science B.V. PACS: 98.8.Cq; 4.5.+h; Mj Keywords: Cosmological constant; Inflationary cosmology; Extra dimensions 1. Introduction Supernova data indicate that the energy density ρ Λ in a present-day cosmological constant is on the order of.7ρ c where the current critical density ρ c ev) 4. It is intriguing that ρ Λ 4 where.1 mm just the length scale for compactified extra dimensions predicted y Arani- Hamed Dimopoulos Dvali ADD) type theories [1] with two extra spatial dimensions. It is possile that this dar energy derives from vacuum fluctuations in extra compactified dimen- address: gardner@asu.edu C.L. Gardner). 1 Research supported in part y the National Science Foundation under grant DMS sions. We outline here a semiclassical gravitation model which maes use of this mechanism to produce the present-day cosmological constant. The model is ased on 4 + D)-dimensional gravity with D = 2 extra dimensions with radius t) curled up at the ADD length scale where the suscript denotes present-day values. The ADD model can e realized [2] in type I tendimensional string theory with standard model fields naturally restricted to a 3-rane [3] while gravitons propagate in the full higher-dimensional space. For D = 2 two of the six compactified dimensions are curled up with radius 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 /2/$ see front matter 22 Pulished y Elsevier Science B.V. PII: S )1372-7

2 22 C.L. Gardner / Physics Letters B ) In addition if supersymmetry is roen only on the 3-rane then the ul cosmological constant vanishes see e.g. Ref. [4]). A single fine tuning of parameters in the potential for can then cancel the rane tension setting the usual four-dimensional cosmological constant to zero. Semiclassical 4 + D)-dimensional gravitation with a potential for the scale of the extra compactified dimensions rapidly ecomes a good approximation to the string theory for energies elow M I [5]. In the semiclassical gravitation model we will assume a potential for t) which stailizes t ) at = M P /M 2 and which vanishes 2 at in the asence of the Casimir effect where the reduced) Planc scale M P = GeV and the wea interaction scale M 1 TeV. Vacuum fluctuations in the compactified space will then pertur t ) very slightly away from 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 threedimensional world is cancelled y a ul cosmological constant. Primordial inflation of our three-dimensional space will occur in the model as the inflaton t) which initially is on the order of 1/M 1 17 cm rolls down its potential to [78]. Many e-folds of inflation of our 3-space can occur for sufficiently flat potentials. We will tae the spacetime metric to e R 1 S 3 T 2 symmetric 3 where S 3 is a 3-sphere and T 2 is a 2-torus: g MN = diag { 1 a 2 t) g ij 2 t) g mn } 1) where MN run from 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 and g mn is the metric of a unit 2-torus with at) the radius of physical 3-space and t) the radius of the compactified space. 2 In other words we assume that the 3-rane tension is exactly cancelled in the stailization potential at =. 3 Our treatment through Eq. 11) parallels that of Kol and Turner [9]. The nonzero components of the 4 + D)-dimensional Ricci tensor are R = 3 a + D ) R ij = a + 2 ) a 2 g ij R mn = + D 1)ḃ2 2) ȧ ) ḃ g mn. a The generalized Einstein equations are R MN = 8π G T MN T P ) P 3) g MN where 8π G = 8πGV = V /MP 2 = Ω D /M D+2 is the 4 + D)-dimensional gravitational constant V = Ω 2 2 is the volume of the compactified dimensions today Ω D denotes the volume of the unit D-torus and T MN is the energy momentum tensor. The gravitational coupling 8πG= 1/ 2M4 ) is wea in the ADD picture ecause is much greater than the 4 + D)- dimensional Planc length 1/M. The nonzero components of the energy momentum tensor are given y T = ρ T ij = p a g ij T mn = p g mn. 4) Thus T P P = ρ 3p a Dp. Expressed in terms of the radii a and the energy density ρ andthe pressures p a and p the Einstein equations ecome 3 5) a + D = 8π G [ ] D + 1)ρ + 3pa + Dp a + 2 a 2 = 8π G [ ] ρ + D 1)pa Dp 6) + D 1)ḃ ȧ ḃ a = 8π G [ρ 3p a + 2p ]. 7) After a few e-folds of primordial inflation of our physical 3-space the curvature term 2/a 2 on the lefthand side of Eq. 6) will e negligile and we will henceforth set this term to zero.

3 C.L. Gardner / Physics Letters B ) We will e looing for solutions neglecting matter) in which physical 3-space is inflating at the present epoch during which t) is fixed at or in the primordial epoch just after the quantum irth of the universe during which t) is inflating to its present value. For an inflating 3-space without matter) p a = ρ and the Einstein equations ecome 3 8) a + D = 8π G [ ] D 2)ρ Dp 9) a = 8π G [ ] D 2)ρ Dp + D 1)ḃ2 1) ȧ ḃ a = 8π G [4ρ + 2p ]. The energy density and pressures on the right-hand sides of Eqs. 8) 1) are derivale from the internal energy U = Ua): ρ = U V p a = a U/ a p = U/ 3V DV 11) where V = Ω 3 a 3 Ω 2 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) for the radius t) in the internal energy Ua)= Ω 3 a 3 M 4 V) 12) at zero temperature) which will produce sufficient primordial inflation to solve the horizon flatness homogeneity isotropy and monopole prolems and which will stailize at = M P /M 2.1 mm with a vanishing cosmological constant. Note that if p a is to equal ρ thenu must e proportional to a 3 and that V) is dimensionless. The potential V) will generate a potential B) with the right-hand side of the Einstein equation 1) equal to B )/. IfB) is sufficiently flat near 1/M then many e-folds of inflation will occur in our physical 3-space as t) rolls from 1/M to. Quantum fields will e 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 ) U C a ) = Ω 3 a 3 13) 4 + M4 V) from vacuum fluctuations in the compactified space will pertur t ) very slightly away from and generate a residual present-day cosmological constant ρ Λ = / 4. The sign and magnitude 4 of the constant depend on the particle content and structure of the underlying quantum gravity theory. The magnitude of may e expected to e roughly in the range ased on the analysis of Candelas and Weinerg [6] who calculated the one-loop Casimir contriution from massless scalar and spin- 1 2 particles in 4 + D)-dimensional gravitation with an odd numer of extra dimensions D curled up near the Planc length. In their wor is positive for a single massless real scalar field for odd dimensions 3 D 19 ut may e positive or negative. For our model to produce a positive present-day cosmological constant we will need >. 2. Primordial inflation In this section we riefly review the cosmological results for primordial inflation of Refs. [78] for the ADD model with internal energy U and chec that the Casimir terms in the Einstein equations when U is replaced y U C do not qualitatively change the primordial cosmological picture. The Einstein equations with the internal energy given y U in Eq. 12) tae the form 3 14) a + 2 = 3 H + 3H 2 + 2Ḣ + 2H 2 = V ) a ȧ ḃ a = H + 3H 2 + 2HH = V ) + ḃ ȧ ḃ a = H + 2H 2 + 3HH = V) 2 V ) ) B 15) 16) where the Hule parameters H ȧ/a and H ḃ/. For a vanishing present-day cosmological constant V ) = from Eq. 15). Eq. 16) then implies V ) = to stailize t ) at. To summarize the successful phenomenology of Ref. [8]: the ADD model can produce sufficient 4 A logarithmic dependence lnm 2 2 ) can e asored into the definition of without changing the conclusions elow.

4 24 C.L. Gardner / Physics Letters B ) inflation 7 e-folds) to solve the cosmological prolemsfora class of potentials V) which satisfy H 1 H ) M at the eginning of inflation at the quantum irth of the universe when a 1/M and H H H H 2 18) during the initial stages of inflation. The correct magnitude and approximate scale invariance of density perturations δρ/ρ = are created if at an intermediate stage of inflation when t) 1 3/2 /M H H/1. There may e a period of contraction similar to the vacuum Kasner solutions) of our physical 3-space ut for D = 2 the amount of contraction of at) is at most 7 e-folds so the contraction phase does not invalidate the solution of the flatness prolem. Replacing U y U C in Eq. 13) introduces Casimir terms into the Einstein equations: 3Ḣ + 3H 2 + 2H + 2H 2 = M V ) 19) H + 3H 2 + 2HH = M V ) 2) H + 2H 2 + 3HH = 2 M V) 2 V ) B C ). 21) The Casimir terms do not qualitatively change the primordial inflationary period of the ADD model since initially M 4 6 M2 M 2 H 2 H 2 and in the intermediate stage of inflation M M M 2 H H 2 for 1 3 using the estimates in Ref. [8]. 3. Present-day cosmological constant 22) 23) In the present epoch the internal dimensions have afixedradiust ) 1/M and H =. Without the Casimir terms the static solution for t ) requires V ) = = V ). In our model vacuum fluctuations in the compactified space pertur very slightly to producing a small cosmological constant in our three-dimensional world. We assume that the potential V) is independent of the Casimir effect so that V ) and V ) still equal zero. The Einstein equations with Casimir contriutions for an inflating 3-space now tae the form 3H 2 = 24) M V ) 4 = 2 25) M V ) 2 V ). 4 Setting = 1 + δ) and solving Eq. 25) to order δ M 4 /MP 4 yields δ 26) 4 V ) + O δ 2) = 2 M 4 6 or ) 8 M 4 )) 1 + M 4 6V = 1 + O ) MP 4 27) where V ) 1/ 2 = M4 /MP 2. Eq. 24) then predicts a present-day cosmological term 3H 2 = δ 4 V ) M O δ 2) = 28) M O δ 2) or in other words H 2 = 8πG 3 ρ Λ ρ Λ = 4 = M8 MP 4. 29) This cosmological term will.7ρ c if 1/4.9 mm or equivalently if M 2.4 TeV/ 1/8. Note that the Casimir effect has caused the stailized radius to increase slightly yielding a positive present-day cosmological constant. The canonically normalized radion field ϕt) = 2M 2 t). The mass squared of the radion field is m 2 ϕ = M4 d2 V dϕ 2 M4 ϕ M 2 P 3)

5 C.L. Gardner / Physics Letters B ) which must e positive at ϕ = 2M 2 = 2M P to have a linearly stale solution [5]. The staility properties of B C ) in Eq. 21) are the same as of B) in Eq. 16): the respective solutions with t ) = and are linearly stale if the radion mass squared is positive since the radion mass squared including the Casimir contriution m 2 ϕc = m2 ϕ + 5M8 M 6 P M4 M 2 P ) 1 + 5M4 M 4 P 31) is positive if m 2 ϕ is and are gloally stale if the respective potentials B) and B C ) are for example concave upward the simplest case) since B C ) = B) + 32) 2M const is concave upward if B is. If the numer of extra dimensions D is allowed to e greater than two the Einstein equations 24) and 25) for an inflating 3-space with static t) change to 3H 2 = D 2 MD+2 D+4 V ) + )M D 2 D 1 V ) M D 2 D 33) 4 = + 4 V ) DMD+2 D+4 M D 2 D 2 V ) 34) D) MD 2 D 1 ut the result for the present-day cosmological constant has the same form ρ Λ = 35) 4 = M4+8/D M 8/D P where now satisfies DMD+2 = MP 2. Thus ρ Λ has the right parametric dependence M 8 /MP 4 only for D = Conclusion The cosmological picture presented here joins smoothly onto the primordial inflation and ig-ang cosmological pictures: the quantum irth of the universe egins with a and 1/M. Many 7) e-folds of primordial inflation occur as the inflaton t) rolls down its potential to. t) then undergoes damped oscillations aout heating the universe up to a temperature T aove the temperature for ig-ang nucleosynthesis BBN) and creating essentially all the matter and energy we see today. See Refs. [7] and [1] for two differing views on the maximum value of T aove which the evolution of the universe in ADDtype theories cannot e descried y the radiationdominated Friedmann Roertson Waler model.) At this point the universe evolves according to the standard ig-ang picture expanding and cooling with a fixed small cosmological constant ρ Λ = / ev) 4. This dar 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 t) of the compactified space has not changed since well efore BBN. Finally we note that if the stailization potential V) vanishes at its gloal 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. References [1] N. Arani-Hamed S. Dimopoulos G. Dvali Phys. Lett. B ) 263 hep-ph/ [2] I. Antoniadis N. Arani-Hamed S. Dimopoulos G. Dvali Phys. Lett. B ) 257 hep-ph/ [3] J. Polchinsi Phys. Rev. Lett ) 4724 hepth/ [4] N. Arani-Hamed L. Hall D. Smith N. Weiner Phys. Rev. D 62 2) 152 hep-ph/ [5] N. Arani-Hamed S. Dimopoulos J. March-Russell Phys. Rev. D 63 21) 642 hep-th/ [6] P. Candelas S. Weinerg Nucl. Phys. B ) 397. [7] N. Arani-Hamed S. Dimopoulos G. Dvali Phys. Rev. D ) 864 hep-ph/ [8] N. Arani-Hamed S. Dimopoulos N. Kaloper J. March- Russell Nucl. Phys. B 567 2) 189 hep-ph/ [9] E.W. Kol M.S. Turner The Early Universe Addison Wesley Reading MA 199. [1] N. Kaloper J. March-Russell G.D. Starman M. Trodden Phys. Rev. Lett. 85 2) 928 hep-ph/21. [11] N. Arani-Hamed L.J. Hall C. Kolda H. Murayama Phys. Rev. Lett. 85 2) 4434 astro-ph/5111.