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D-Branes, String Cosmology and Large Extra Dimensions

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D-Branes, String Cosmology and Large Extra Dimensions View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server D-branes, String Cosmology and Large Extra Dimensions Antonio Riotto Theory Division, CERN, CH-1211, Geneva 23, Switzerland (April, 1999) D-branes are fundamental in all scenarios where there are large extra dimensions and the string scale is much smaller than the four-dimensional Planck mass. We show that this current picture leads to a new approach to string cosmology where inflation on our brane is driven by the large extra dimensions and the issue of the graceful exit becomes inextricably linked to the problem of the stabilization of the extra dimensions, suggesting the possibility of a common solution. We also show that branes may violently fluctuate along their transverse directions in curved spacetime, possibly leading to a period of brane-driven inflation. This phenomenon plays also a crucial role in many other cosmological issues, such as the smoothing out of the cosmological singularities and the generation of the baryon asymmetry on our three brane. PACS: 98.80.Cq; CERN-TH/99-115 D-branes are classical solutions which are known to region. If this is the case, the small ratio Ms/Mp could exist in many string theories [1]. A Dp-brane denotes be a consequence of large compactified dimensions [6,7] a configuration which extends along p spatial directions through a scaling relation like and is localized in all other spatial transverse directions. 2 n+2 From the string point of view, a D-brane is a soliton on Mp = Ms Vn, (1) which open string endpoints may live and whose mass where Vn is the volume of the n large extra dimensions. (or tension) is proportional to the inverse of the string For M s TeV and n 2, the radius of the large extra di- coupling gs. This means that D-branes become light in mensions∼ may be as large≥ as the millimeter. Large extra those regions of the moduli space where the string cou- dimensions are only allowed if the SM fields are confined pling is large. This simple property has led to a new to live on a membrane orthogonal to the large extra di- interpretation for singularities of various fixed spacetime mensions and this hypothesis fits rather nicely with the backgrounds in which strings propagate and provided a current picture of string theory and D-branes. generic and physically intuitive mechanism for smooth- Not surprisingly, one expects many interesting cosmo- ing away cosmological singularities in the strong cou- logical implications of this scenario (some of them have pling/large curvature regime of the evolution of the early already been discussed in [2,8]) and the possibility of Universe [2,3]. large extra dimensions is potentially exciting for several The presence of branes in string theory has also given reasons. rise to the possibility that the Standard Model (SM) In the presence of some matter with energy density ρ, gauge fields live on branes rather than in the bulk of the evolution of the Universe is controlled by the equa- spacetime. This scenario corresponds to a novel com- 2 2 2 tion H (˙a/a) =(8πρ/3Mp ), where a is the scale pactification in which gravity lives in the bulk of the ten factor and≡ H is the Hubble rate. This implies that for dimensional spacetime, but the other observed fields are an expanding Universe, ρ and H2 are proportional, they confined to a brane of lower dimension [4]. In particular, increase or decrease together in time. This is true only if one might imagine the situation where the SM particles Mp is constant. Now, in the weakly coupled string theory live on a three brane and the directions transverse to the 1 where all compactified dimensions have size Ms− , Mp three brane being compact. There might be also other is controlled by the dilaton and it is perfectly∼ possible branes separated from ours in the extra directions. Since to have growing H while ρ decreases, provided the dila- the usual gauge and matter fields live on the brane, the ton φ is also growing. On this peculiar feature is based only long-range interaction which sees the extra dimen- the whole idea of Pre Big-Bang (PBB) cosmology where sions is gravity. there exist superinflationary solutions for the scale fac- In the weakly coupled heterotic string there is a fun- tor a having a growing – rather than a constant – Hubble damental relation between the string scale Ms, the four- parameter and a growing dilaton field [9]. The sugges- dimensional Planck mass Mp, the value of the dilaton tion from weakly coupled string theory is therefore rather 2 field φ and the gauge coupling constant αg,(Ms/Mp) compelling, since it is possible to generate a scenario con- φ 18∼ αg e and the string scale is fixed to be about 10 ∼ taining a dilaton-driven PPB inflation. GeV. However, in other regions of the moduli space the These simple considerations together with the relation string scale can be much smaller [4] and it is tempting to (1) holding in the current picture of string theory suggest imagine [5] that the string scale is not far from the TeV a simple way of getting inflation on our three brane and a 1 fresh approach to string cosmology: since Mp is now gov- the sizes of our observed Universe and of the extra dimen- erned by the size b of the extra dimensions, it is possible sions if γ γ . The cosmological solution stops being | 1|| 2| to obtain superinflationary solutions provided that the valid when the Hubble rate becomes of the order of Ms. extra dimensions change in time, that is a radion-driven However, differently from the dilaton-driven PPB cos- PPB cosmology. The Einstein action in 4 +n dimensions mology where one has to face a regime of strong coupling 2 2 and metric gMN =(1, a δij , b δmn)reads when approaching the big-bang singularity at t 0, in − − the radion-driven PPB scenario the singularity may→ take ¨b a˙ b˙ b˙2 place well inside the pertubative regime and be under- S d4xa3bn R 2n 6n n(n 1) , (2) ∼ − b − ab − − b2 stood as the process of decompactifying the extra dimen- Z " # sions. The issue of the graceful exit in string cosmology where R is the usual four-dimensional Ricci scalar. Set- becomes therefore inextricably linked to the problem of ting bn = ecσ, with c =(n/n 1)1/2, and integrating by the stabilization of the extra dimensions, suggesting the − possibility of a common solution. parts the term ¨b in (2), the action for the radion field σ There is – however – another feature of D-branes that becomes may help in solving the radion-driven PPB singular- 4 3 cσ µ ity problem as well as give new insight into other fun- S d xa e [R + ∂µσ∂ σ] . (3) ∼ damental cosmological issues: this is the peculiar be- Z haviour of fluctuations transverse to the brane in curved Were c = 1, string cosmology experts would recognize spacetime. The action governing the dynamics of a the action of the dilaton field coupled to gravity (with generic Dp-brane (neglecting the gauge fields) is [1] S = p+1 φ 1 φ = σ). For us c = 1 in the limit of n and Tp d ξe− detGαβ ,whereTp− is the brane ten- in this− regime we can recover the more familiar→∞ results sion,− ξα (α =0,...,p− ) parametrize the D-brane world- R p µ ν of dilaton-driven PPB cosmology. The evolution of the volume and Gαβ = gµν (X)∂αX ∂β X is the induced radion is governed by the equation obtained by varying metric. Given a brane configuration X¯ µ(ξ) in a spacetime the lapse function of D-dimensions, arbitrary fluctuations transverse to this configuration (the would-be Goldstone bosons of the bro- σ˙ = H, = 3c 9c2 6, (4) ken translational invariance along the directions perpen- ± ± − ± − diclar to the brane) are parametrized by (D p) world- p a µ ¯µ −µ a while the equation for the Hubble parameter becomes volume scalars y (ξ)asX (ξ)=X(ξ)+¯na(ξ)y (ξ), wheren ¯µ(ξ)(a=1,...,(D p)) are the (D p) vectors ˙ 2 a H = α H ,α=3+c . (5) normal to the brane at the− point X¯(ξ). The− expansion − ± ± ± of the metric in Riemann coordinates is [10] For each value of n, α is negative and it generates a su- − perinflationary solution, that is H grows until a singular- 1 a b ρ σ 3 gµν (X)=¯gµν y y R¯µρνσ n¯ n¯ + (y ), (6) ity is reached, say at t = 0. In terms of the scale factors − 3 a b O γ1 a and b, the solutions are given by a(t)=a0(t/t0)− γ where the overbar indicates that the quantity is evaluated and b(t)=b (t/t ) 2 where t and t are negative and 0 0 0 at X¯. If the extrinsic curvature of the membrane is small, γ1 = 1/α , γ2 = c /(nα ) .Since <0, the volume ρ ρ − − − − the normal vectors aren ¯ δ , and therefore g (X)= of the| internal| dimensions| is contracting.| Suppose that a a µν g¯ 1 yaybR¯ + (y3)[2].' a and b start with comparable values at low curvature at µν 3 µaνb Let− us begin, for simplicity,O by considering a single Dp- t t ,saya b M 1; the subsequent evolution is 0 s− brane in an expanding accelerating (¨a>0) background such∼ that the∼ spatial∼ dimensions of our three brane su- where the dimensions orthogonal to the brane are static.
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