D-Branes, String Cosmology and Large Extra Dimensions

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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 (ä>0) background such∼ that the∼ spatial∼ dimensions of our three brane su- where the dimensions orthogonal to the brane are static. perinflate, whereas the extra dimensions evolve to values 1 We shall consider for definiteness a background where smaller than M − . In string theory, due to the presence s the common scale factor a of the spatial coordinates ξ~ of winding states in the closed string (gravitational) sec- q is inflating, a(η) ( η)− ,where0

2 p a 1 d k a i~k ξ~ a +i~k ξ~ 4 3 3 2 2 2 2 y (η, ξ~)= a h (η)e + a† h †(η)e ,by E Ms d ξa 1+1/2y˙ /a +1/2( y) /a , 1 p/2 k k − · k k · p+1 (2π) ∼ ∇ τ where the ﬁrst term is just the classical stretching. Just Z h i R h i (8) as in (10), we ﬁnd

2 4 where a and a† are the annihilation and creation oper- ( y) H k k . (11) a h ∇ 2 i ators, respectively. The solution is very simple, hk(η)= a ' Ms β 1 1 ( η) c1Hβ( kη)+c2Hβ∗( kη) ,whereβ=(q(p − − − − The term y˙ 2 gives no contribution after the flat space- 1)+1)/h2. If we quantize the modesi and define the vacuum time substraction.h i Thus we deduce that the fractional state by ak 0 = 0, then different choices of vacuum cor- | i energy in the transverse perturbations grows with time respond to different choices of c1 and c2. The adiabatic as H increases with time and their contribution to the vacuum corresponds to c2 = 0 and this (Heisenberg) state energy density becomes sizeable when H Ms. Physical is the state we will assume the brane is in. transverse fluctuations of the three brane∼ – even though The behaviour of a mode ha(η) is very simple. Suppose k initially oscillating – will eventually grow and give rise to from now on to consider the case p =3,i.e. we consider 1 an instability. Therefore, as the three brane is stretched a three brane. As long as the physical wavelength ak− a by the expansion of the Universe, transverse fluctuations of a mode is inside the Hubble radius, hk(η) oscillates grow, the thickness of the brane gets larger and larger with constant physical amplitude aha(η). As the physical k and the brane cannot be considered as a static object wavelength grows it crosses the Hubble radius and the any longer. This growth is accompanied by a huge pro- comoving amplitude ha(η) becomes frozen, so that the k duction of light pseudo-Goldstone bosons of the broken physical amplitude grows like a. translation invariance. Let us try to feed back the effect In this state one can calculate the mean square trans- of this production into the expansion of the Universe. verse displacement As the system evolves towards the singularity, more and d3ξ more energy in transferred into transverse fluctuations of y2 0y2(η, ξ~) 0 the brane till – eventually – they dominate the energy h i≡ V h | | i 2 2 Z 3 density of the Universe. Since ( y) y˙ , the gas D 4 d3k h ∇ ih i = ha 2 , (9) of pseudo-Goldstone bosons is characterized by an energy 1−/2 3 k τ (2π) | | density ρy 3py,wherepy is the pressure of the gas. Z We thus obtain'− an effective negative pressure. Assuming 2 2 3 where y y and V3 = d ξ. The physical dis- a flat Universe, the equation for the accelerationa ¨ alone ≡ a a placement ay has the usual ultraviolet divergence as in becomes flat space: physicalP wavelengthsR with k aH are always well inside the horizon at a given time and their ampli- a/a¨ (ρy +3py) = 0 (12) tude is unaffected by the expansion. Their contribution ∝ 3 1 leads to the usual d kk− divergence. We can subtract and we discover that the back-reaction of the gas of this divergence. Modes with k aiHi,whereai and pseudo-Goldstone bosons onto the evolution of the Uni- R Hi are the values of the scalar factor and Hubble rate at verse is to halt the period of acceleration and – therefore the beginning of the inflationary stage, are always well – to smooth the singularity away when H becomes of the outside the horizon and simply match on adiabatically to order of the string scale! It should be stressed that this the modes before and after inflation. One therefore finds result is only valid when the the stress tensor is of the per- at η ηf 0(forq= 1) fect fluid type, i.e. when it becomes possible to neglect ' ' 6 − the viscosity terms due to mutual and self-interactions. D 4 afHf d3k These terms are not negligible when the approximation a2 y2 a2 − H ha 2 h i' τ1/2 (2π)3 | k| of small transverse fluctuations breaks down. ZaiHi The role played by the transverse fluctuations may be D 4 , (10) even more dramatic, though. Remember that in the con- −4 2 ∼ Ms ηf siderations made so far we have supposed that the directions transverse to our three brane were static. As- wherewehavesetτ M4. One can picture this result ∼ s sume now that these directions are suffering a period by saying that the modes with wavelength of the order of decelerating contraction, while the spatial dimensions 2 2 of the Hubble radius have a physical width y M− h pi' s of the brane are accelerating. This is – for instance – which gets amplified by the expansion after they cross what happens during the stage of radion-driven infla- the Hubble radius. Transverse fluctuations grow in time. tion previously described. From Eq. (6) the fluctua- In the special case of De Sitter with constant rate of tions ya transverse to the brane get a tachyonic mass 2 4 2 expansion H, one finds that y Ms− Hln(EH) 2 ¨ ˙ 4 3 h i' ∼ my =1/3 b/b +3b/ba/a˙ , where now the dots stand for Ms− H t,whereEis the total number of e-foldings. More interestingly, one can calculate the energy ac- derivatives with respect to the cosmic time t.Thesitua- quired by each mode in this process. The energy is given tion is very similar to the thermal tachyon which occurs

3 at the Hagedorn temperature in string theory [11,12]. and certainly leave many questions unanswered. For in- In that case a condensate of winding states forms when stance, what are the initial conditions for the Universe (or the periodic imaginary times becomes too small. In the perhaps just for our region of the Universe) right before present case the tachyon forms because of the nontrivial the onset of inflationary PBB stage driven by the radion geometry of the spacetime. As a given physical wave- field? What is the subsequent dynamics of the gas of 1 length ak− of a transverse fluctuation ya crosses the pseudo-Goldstone bosons which is generated at H Ms a ∼ horizon, the comoving amplitude hk does not freeze out; and whose presence may help avoiding the singularity? instead the transverse physical amplitude grows faster How does the system get into the familiar radiation- than the scale factor a and therefore faster than the phys- dominated stage? At this point open strings attached to ical coordinates defining the brane. In fact the brane the branes have to play a role since the expansion time 1 tends to loose its own identity as the thickness of the H− is of the order of the typical scale of the stringy net- 1 brane grow faster than the physical coordinates defin- work Ms− . It seems safe therefore to assume that this gas ing the three brane itself; the brane violently fluctuate of strings attached to the branes and pseudo-Goldstone along the transverse directions. This phenomenon holds bosons reaches thermal equilibrium. The fact that fun- for branes of any dimensionality. If one goes beyond the damental strings have a limiting temperature Ms may approximation of small transverse fluctuations and com- be turn out to be crucial since it implies that∼ – if radia- putes the higher order terms in the tachyon effective po- tion is created and is in thermal contact with the strings 4 tential, one can show that – along with these huge fluc- –it cannot attain a density higher than Ms ,andthe tuations – branes become almost tensionless [2]. This is Universe necessarily is dominated by strings.∼ hardly surprising since the brane can fluctuate along the We are grateful to T. Banks, R. Brustein, E. Copeland, transverse directions paying no price in energy. When K. Dienes, M. Dine, S. Dimopoulos, T. Gherghetta, G.F. the tension becomes negative, branes become unstable Giudice, D. Lyth, R. Rattazzi, G. Veneziano and espe- and – at least in the regime of strong coupling – it be- cially to M. Maggiore for many useful discussions. comes easier and easier for the gravitational background to produce further massless branes. On the other hand, when the transverse sizes of the branes become larger than the horizon, the branes presumably break down and decay, but branes are continuously created. The rate of brane creation due to these quantum effects may be so fast to balance the dilution of the brane density due to [1] See, for instance, J. Polchinski, TASI Lectures on D- the expansion. One might be therefore led into a phase of branes, hep-th/9611050. constant brane density and exponentially expanding Uni- [2] M. Maggiore and A. Riotto, hep-th/9811089. verse, phase of brane-driven inflation [13] reminescent of [3] T. Banks, W. Fischler and L. Motl, hep-th/9811194. the old idea of string-driven inflation [14]. [4] E. Witten, Nucl. Phys. B471, 135 (1996). The huge transverse fluctuations of branes in curved [5] J. Lykken, Phys. Rev. D54, 135 (1996). spacetime may be also relevant to the issue of baryon [6] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. , 263 (1998). asymmetry production on our three brane if we accept B429 [7] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. the idea that there might be other branes separated from Dvali, Phys. Lett. B436, 257 (1998). ours in the extra directions. Even though on our brane [8] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, hep- global charges, such as the baryon and the lepton num- ph/9807344; K.R. Dienes, E. Dudas, T. Ghergetta, A. ber, are conserved with a great accuracy, this is not nec- Riotto, hep-ph/9809406; N. Arkani-Hamed, S. Dimopou- essarily the case on other branes. This breaking might be los and J. March-Russell, hep-th/9809124; K. Benakli communicated to our brane by messenger fields that leave and S. Davidson, hep-ph/9810280; D.H. Lyth, hep- in the bulk [15]. However, this communication is sup- ph/9810320; N. Kaloper and A. Linde, hep-th/9811141; pressed by the enormous distance separating the branes G. Dvali and S.-H. H. Tye, hep-ph/9812483; T. Banks, in the bulk and this is the reason why today the baryon M. Dine, A. Nelson, hep-th/9903019; N. Arkani-Hamed number is conserved in SM interactions. Nevertheless, at et al., hep-ph/9903224; G. Dvali and G. Gabadadze, hep- early epochs in the evolution of the Universe the thick- ph/9904221; G. Dvali and M. Shifman, hep-th/9904021. ness of a brane is highly fluctuating. This makes it pos- [9] G. Veneziano, Phys. Lett. 265, 287 (1991); M. Gasperini sible for two branes separated in the bulk to overlap. and G. Veneziano, Astropart. Phys. 1, 317 (1993). One can also envisage various other dynamical phenom- [10] See for instance A.Z. Petrov, Einstein spaces (Pergamon, ena like the melting of two or more branes [2]. If our Oxford, 1969). brane was considerably overlapping with another brane [11] B. Sathiapalan, Phys. Rev. D35, 3277 (1987). where the breaking on the baryon number is (1), a sig- [12] Ya.I.Kogan, JETP Lett. 45, 709 (1987). nificant amount of baryon number might haveO been de- [13] A. Riotto, to appear. posited on our three brane, thus explaining the observed [14] N. Turok, Phys. Rev. Lett. 60, 549 (1988). baryon asymmery of our Universe [16]. [15] N. Arkani-Hamed and S. Dimopoulos, hep-ph/9811353. The considerations reported here are very preliminary [16] See also G. Dvali and G. Gabadadze in [8].