Rolling Tachyon in Brane World Cosmology from Superstring Field

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Provided by CERN DocumentRolling Server Tachyon in Brane World Cosmology from Superstring Field Theory

Gary Shiu1, S.-H. Henry Tye2, and Ira Wasserman3 1 Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104 2 Laboratory of Elementary Particle Physics, Cornell University, Ithaca, NY 14853 3 Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853 (July 15, 2002) The pressureless tachyonic matter recently found in superstring field theory has an over-abundance problem in cosmology. We argue that this problem is naturally solved in the brane inflationary scenario if part of the tachyon energy is drained (via its coupling to the inflaton and other fields) to heating the universe, while the rest of the tachyon energy goes to a network of cosmic strings (lower-dimensional BPS D-branes) produced during the tachyon rolling at the end of inflation.

PACS numbers: 11.25.-q, 11.27.+d, 98.80.Cq

In superstring theory, starting with a non-BPS brane still give detectable signatures in the cosmic microwave or a brane-anti-brane pair, a tachyon is always present. background and the gravitational wave spectral density This tachyon rolls down its potential and tachyon matter [10]. To summarize, the over-abundance of the tachyon appears as energy density without pressure, as pointed matter density problem can be solved rather naturally out by Sen [1]. In the cosmological context [2], it is by draining the tachyon potential energy to (re-)heating easy to estimate that the pressureless tachyonic matter and to the cosmic string production. Below, we discuss density is generically many orders of magnitude too big these two mechanisms. to be compatible with present day cosmological observa- Consider a general Lagrangian of the form (α0 =1) tions [3]. This implies either that tachyon rolling does µν not happen (i.e., irrelevant) in the early universe after = V (T )F (X) X g T,µT,ν , (1) L − ≡ inflation, or, instead of pressureless tachyon matter density, the tachyon potential energy goes somewhere else. where T (x)isthetachyonmode,T,µ = ∂µT ,andF (X) T can be any function at this stage. In Ref. [1], V (T ) e− In this paper, we argue for the latter. ∝ In superstring theory, defects, or stable solitons (i.e., and F (X)=√1+X. lower-dimensional BPS branes), are produced during The energy-momentum of Eq.(1) is tachyon rolling, also first pointed out by Sen [4]. Cos- T = V (T )[2F T T g F (X)] . (2) mologically, these defects will be produced [5]. Averaged µν ,X ,µ ,ν − µν over a region large compared to the typical defect separa- where F,X = ∂F/∂X. We can write this in the form tion, this defect density becomes spatially uniform, and of the energy-momentum tensor for a perfect fluid if we may be identified as a part of the tachyonic matter. identify In the brane world realization of superstring theory, where the standard model particles are open string modes ρ = V (T )[F (X) 2XF ] V (T )D(X) − ,X ≡ that live on the brane while the graviton and other closed p = V (T )F (X) . (3) string modes live in the bulk, the brane inflationary − scenario [6], where the inflaton is simply an interbrane We shall mostly use the tachyon Langrangian sug- separation, looks very robust [7–9]. In this scenario, gested by the boundary (or background-independent) tachyons invariably emerge towards the end of brane in- string field theory [12], in particular, its superstring ver- flation, when the branes approach each other and col- sion [13,14], which has been used to study tachyon mat- lide. (Defects produced before or during inflation are ter recently [15]. The non-BPS brane model (with brane inflated away.) As the tachyon rolls down its potential, tension √2τp)is, it was shown that the only defects copiously produced are those that appear as cosmic strings [9,10]. So we expect T 2/2 V (T )=√2τpe− all of the tachyon potential energy goes to heating the √πΓ(1 + X) X4XΓ(X)2 universe (via its coupling to the inflaton and other light F (X)= = . (4) open string modes) and producing the cosmic string net- Γ(X +1/2) 2Γ(2X) work. Interactions of the cosmic strings will reduce the Here, F (X)=1+X ln 4 for small X. In this model, for very high initial cosmic string network density to an ac- spatially independent solutions, the pressure starts out ceptable level [11], provided the cosmic string tension µ ˙ 2 6 negative when X = T is small and becomes positive satisfies Gµ < 10− , which seems to be always satisfied in when X< 1/2. As− 1 + X =1 T˙ 2 0, F (X) brane inflation [9,10]. The presence of cosmic strings will − 2 − → ' 1/2(X +1),D(X) (X +1)− ,weseethat − '

1 w = p/ρ = F (X)/D(X) (1 + X)/2 0 . (5) by cosmic string production and (re-)heating in a realistic − ' → early universe. As 1 + X 0, V (T ) 0, but the energy density ρ One may also consider a stable solitonic solution of stays constant,→ as required→ by energy conservation. The Eq.(6), with T1(x)=u1x1 and T2(x)=u2x2 [14]. eﬀective tachyon sound speed v = F /D also T ,X ,X Since both the second derivatives of TI (x) and the mixed vanishes in this limit, allowing eﬃcient clustering.− p derivatives ∂iT1∂jT2 with i = j vanish in this ansatz, the Next, let us consider the brane-anti-brane pair, which equations of motion reduce to has a complex tachyon T = T1 +iT2. The tachyon action is [14] TI D(XI )=0 I =1, 2 .

p+1 (T 2+T 2)/2 2 = 2τ dx e− 1 2 F (X )F (X )(6)Since D(XI ) only vanishes at XI = uI (this means ST − p 1 2 →∞ Z infinitely thin brane), the codimension-2 brane tension µν (F (X) √πX) becomes where XI g ∂µTI ∂ν TI . Again, let us consider spa- → tially independent≡ solutions. The energy conservation T 2/2 2 2 requires thatρ ˙ =0,where τp 2 =2τp dx1dx2e−| | F (u1)F (u2) − T 2/2 Z 2 2 2 ρ =2τ e−| | [F F 2X F F 2X F F ](7) =2τpF (u )F (u )2π/u1u2 τp(2π) (11) p 1 2 − 1 ,X1 2 − 2 1 ,X2 1 2 → where FI = F (XI )andF,XI = F,X (XI ). First, consider as expected. This is a vortex solution. a simple ansatz of linear tachyon profile T1(t)=t (t) A non-BPS (p 1)-brane may also be produced (u1 t2/4 − − → and T2 = 0. One finds that ˙(t) e− 0and 0,u2 ), but it will quickly decay to BPS D(p 2)- 1+X 0, so w = 0 (i.e., zero pressure).' Since→X =0 branes.→∞ Multi-soliton solutions may also be constructed− 1 → 2 and F (X2) = 1, this essentially reduces to the single T via the introduction of Chan-Paton factors. A moving case. In this solution, pressure approaches zero from the brane with constant velocity v may also be constructed positive side [15]. by generalizing T1 to T1(x)=u1(x1 vt), which gives 2 − Suppose we consider a more symmetric ansatz: the energy E = τp 2/√1 v , as expected. If we start with− a brane-anti-brane− pair, the tachyon T (t)=t (t),T(t)=t (t) . (8) 1 − 1 2 − 2 will roll with non-trivial spatial dependence. This will result in a collection of D(p 2)-branes. However, if An examination of the equations of motion yields ˙1 − 2 we average over a region much larger than the typical ˙ e t /3. However, when these are substituted into ρ∝, 2 − spacing between the D(p 2)-branes, we expect to ob- we∝ find tain a spatially uniform energy− density and isotropized τ e t2 1 1 pressure, that is, a perfect fluid. In this sense, we may ρ p − + . (9) 2 2 interpret the defect density as part of the tachyon mat- '− 8 ˙1˙2 ˙1˙2 ter. However, the fraction of energy that goes to defect A positive constant ρ implies ˙I < 0 (i.e., 1 + XI < 0), a formation is sensitive to the details and difficult to cal- ˙ solution that cannot be reached starting from small TI . culate. The expansion rate of the Universe is simply H2 = The cosmological production of defects in superstring 8πGρ/3. where a(t) is the cosmological scale factor and theory has been studied [5]. In particular, D3-brane-anti- H =˙a/a. For a spatially uniform tachyon field evolving D3-brane annihilation yields D1-branes [4], which appear in an expanding Universe, we haveρ ˙ = 3H (ρ + p). as cosmic strings in our universe. Actually, following 3 − With w =0,ρa = constant, and since vT =0too, from the fact that vortices (instead of domain walls or tachyon matter is dust-like [1,15]. monopoles) are formed, the appearance of only cosmic Let us estimate the tachyon matter density today as a strings is much more general [9,10], As an illustration, 2 2 fraction ΩT of the total density 3H0 MPl of our universe. consider D5-branes with two of its dimensions compact- 4 2 We assume ρinitial Ms , where the string scale Ms = ified on a two-cycle, while the remaining 3 dimensions 2 '2 123 1/α0. Using H0 /MPl 10− and ainitial/atoday span our universe. D5-brane-anti-D5-brane annihilation o ' ' 2.7 K/Ms, we find will produce D3-branes, which will also wrap around the two-cycle, so they also appear as cosmic strings. If there ρ a 3 M Ω = initial initial 1028 s (10) is no one-cycle inside the two-cycle, (e.g., topology of a T 3H2M 2 a ' M 0 Pl today Pl sphere, as in orbifolding a torus), then no domain wall or monopole-like solitons will appear in our universe. where 2.7oK/M 10 31. Since Ω must be less than Pl − T Naively, the cosmic string network has a density that 1, and M > 1 TeV,' we see that the tachyon matter den- s is comparable to and decreases like (for cosmic string sity will be many orders of magnitude too big to be com- loops) that of the tachyon matter (10), or, for cosmic patible with our universe [3]. Fortunately, as we shall ar- strings stretching across the horizon, even slower (i.e., gue below, this tachyon energy can be completely drained

2 2 worse). Fortunately, the intercommutation of intersect- For fixed φ>φv, T has positive mass mT > 0, so the ing cosmic strings and the decay of cosmic string loops minimum of the potential is at TI = 0 and classically (to gravitational waves) modify the density to decrease ∂µTI =0,soXI =0andF (XI ) = 1. The model reduces like radiation [11]. This adds another power of 2.7oK/M to (φ, T =0)= U(φ) gµν ∂ φ∂ φ/2. This describes s L − − µ ν to Eq.(10), rendering the cosmic string network density the inflationary epoch for φ>φv. to an acceptable level. Based on the above analysis, let only one of the To drain the tachyon energy to heat the universe, we tachyons roll, say T = T1, while T2 is essentially frozen, need to couple the tachyon field to other fields, the most with F (X2) = 1. For a spatially uniform scalar field and obvious one being the scalar field φ that describes the tachyon, separation between the brane and the anti-brane (φ =0 ¨ ˙ in the above discussion). In brane inflation, φ is the φ +(3H +Γφ)φ = F (X)[V ( T ,φ)],φ − | | inflaton. Let us take a simple generalization of the above V V T¨ +3HA(X)T˙ = ,T B(X) ,φ A(X)T˙ φ,˙ (15) tachyon model to include the inflaton φ, − 2V − V 1 µν where A(X)=F,X /(F,X +2XF,XX)andB(X)= (T,φ)= V ( T ,φ)F (X1)F (X2) g φ,µφ,ν . (12) L − | | − 2 (F 2XF )/(F +2XF ); Γ is a phenomenological − ,X ,X ,XX φ For brane inflation, the inflaton potential is known to term that models the decay of φ to ordinary particles and d 2 radiation, whose density is ρ . If the inflaton decays have the form U(φ) U0 U1/φ − for large φ,when matter there are d extra dimensions' − transverse to the brane primarily to relativistic particles (“radiation”), then [7,16]. For a particular brane pair, U are easily cal- i ˙2 culable [17]. For example, for a brane-anti-brane pair, ρ˙matter +4Hρmatter =Γφφ . (16) U =2τ V where V is the compactification volume 0 p p 3 p 3 The total energy density in this situation is of the extra− (p 3) dimensions− of the branes and U is − 1 simply twice that of the gravitational force. When the ˙2 ˙ 2 φ brane separation φ is comparable to the compactification ρ = V ( T ,φ)(F (X)+2T F,X )+ + ρmatter . (17) | | 2 size, the compactification effect becomes important and it flattens the potential further [7,9], allowing inflation to Here, we are not interested in the entire process of brane take place for many e-foldings. inflation, but rather what happens at the end, when the When the branes are far apart, T is a normal scalar tachyon starts to roll. We assume that there is a prior 2 field with positive mass squared mT . As the branes move period of slow rolling (φ decreasing slowly), during which 2 closer, mT decreases, and becomes negative at the bifur- almost all of the inflation occurs, and density fluctuations cation point φv,whereT becomes tachyonic. This sug- are generated [7–9]. During the inflationary period, when 1/2 gests the following potential: T φ is small and (let λ =1)φ>2− , the evolution of |T (|t) resembles a massive field, with a mass φ,andwe 1 ∼ V ( T ,φ)=U(φ)exp T 2 λφ2 (13) therefore expect T (t) to remain small, and dominated | | −| | 2 − by quantum fluctuations. When the tachyon becomes where the bifurcation point λφ2 =1/2. unstable, it begins with an amplitude T HI /2π√U0 v M 1 and T˙ H T ,whereH =(8πGU∼ /3)1/2 is the∼ Although the potential U(φ) for small φ remains to Pl− I I 0 expansion rate∼ during slow roll. We follow the evolution be calculated, it is expected to be positive and finite as 1/2 φ 0. In terms of open strings, U(φ =0)isaquan- implied by Eq. (15) and (16) after φ =2− . → As an illustration, numerical results are shown in Fig. tum one-loop correction, so 2τp is renormalized to 2τpv0, 3 whereweshalltake1>v > 0. So, for φ =0,were- 1, assuming (in terms of string scale) 1/MPl =10− , 0 U =1,1 v =0.1, d =4,φ =1,andΓ =0 cover the tachyonic model (6). Notice that the minimum 0 − 0 0 φ or 0.1. In fact, because HI 0.003, cosmological ex- of the potential V ( T ,φ)(φ<φv and T )isauto- ' matically zero, independent| | of the details→∞ of U(φ). As a pansion is relatively unimportant. The simulations start √ ˙ consequence, the dynamics is insensitive to the particular with φ(t)=1/ 2andφ =0,att 0anda(t) 1. The initial data can be varied, but the≡ behavior shown≡ form of U(φ)forφ<φv, since tachyon rolling happens ˙ very fast, and V ( T ,φ) 0 rapidly. For phenomenol- in Fig. 1 is generic. At first, φ(t) decreases, and T in- | | → creases. Ultimately, as T˙ 1, T t grows, and the ogy, we choose a particularly simple form of U(φ), where → ' 2 U(φ) has a simple harmonic form for small φ, product V (T,φ)F (X) decreases rapidly for φ < 1/2. In the nondissipative case, Γ =0,φa˙ 3 constant, and φ → (1 v ) since a(t) is only changing slowly, φ increases or decreases U(φ)=U 1 − 0 . (14) 0 2 1 (d 2) linearly with time. However, since the total energy is con- − (φ2/φ +1)2 " 0 − # served, and V (T,φ) would grow exponentially large for In summary, Eqs.(12,13,14) comprise the model, which φ2 > 1/2, the growth of φ cannot continue indefinitely, | | may be considered as a novel version of hybrid inflation. so φ(t) bounces near φ2 =1/2. Associated with these

3 bounces are short-lived dips in T˙ , which are evident but production of cosmic strings, which intersect, intercom- not completely resolved in Fig. 1. Eventually, cosmologi- mute, and decay, as proposed above and in Ref. [9–11] cal expansion alone would cause φ˙ to decay, and φ would and (2) heating the universe via the tachyon coupling to settle to some constant, nonzero value. Dissipation accel- the inﬂaton and other ﬁelds. erates the decay of the nonlinear oscillations. The bottom We thank Nick Jones, Asad Naqvi, Ashoke Sen and panel in Fig. 1 shows what happens for Γφ =0.1, which Samson Shatashvili for discussions. This research is par- may be unrealistically large, but illustrates how dissipa- tially supported by the DOE grant EY-76-02-3071 (G.S.), tion suppresses the oscillations, and causes φ to settle to UPenn SAS Dean’s fund (G.S.), NSF (S.-H.H.T.), and a nonzero value asymptotically. For either Γ , ρ a3 NASA (I.W.). φ T → constant 1 eventually. For Γφ =0.1, the Universe does not become∼ radiation dominated.

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