SNUTP 01-001,hep-th/0101027 Self-tuning solution of the cosmological constant problem with antisymmetric tensor field Jihn E. Kim, Bumseok Kyae and Hyun Min Lee Department of Physics and Center for Theoretical Physics, Seoul National University, Seoul 151-747, South Korea (Dated: February 1, 2008) Abstract We present a self-tuning solution of the cosmological constant problem with one extra dimension 1 which is curved with a warp factor. To separate out the extra dimension and to have a self- 0 0 tuning solution, a three index antisymmetric tensor field is introduced with the 1/H2 term in the 2 Lagrangian. The standard model fields are located at the y = 0 brane. The existence [1] of the n self-tuning solution (which results without any fine tuning among parameters in the Lagrangian) a J is crucial to obtain a vanishing cosmological constant in a 4D effective theory. The de Sitter and 5 anti de Sitter space solutions are possible. The de Sitter space solutions have horizons. Restricting to the spaces which contain the y = 0 brane, the vanishing cosmological constant is chosen in the 2 v most probable universe. For this interpretaion to be valid, the existence of the self-tuning solution 7 is crucial in view of the phase transitions. In this paper, we show explicitly a solution in case the 2 brane tension shifts from one to another value. We also discuss the case with the H2 term which 0 1 leads to one-fine-tuning solutions at most. 0 1 0 / h t - p e h : v i X r a 1 I. INTRODUCTION The cosmological constant problem [2] is probably the most important clue to the physics at the Planck scale. Most attempts toward solutions of the cosmological constant problem introduceadditionalingredients[3,4,5,6,7,8],exceptthewormholeandanthropicsolutions of the problem [6, 9]. The wormhole solution is based on the probabilistic interpretation that the probability to have a universe with a vanishing cosmological constant is the largest [6]. But in the evolving universe where the true vacuum is chosen as the universe cools down, the probabilistic interpretation in the early universe is in question since at a later epoch additional constant may be generated by spontaneous symmetry breaking. For this interpretation to make sense, there must exists a self-tuning solution. [The self-tuning solution is defined as the flat space solution without any fine-tuning of parameters in the action.] The anthropic interpretation [9] for the small cosmological constant is a working proposal, but does not answer the fundamental question, “Can we explain the vanishingly small constant from the fundamental parameters in the theory?” On the other hand, Hawking’s probably vanishing cosmological constant [5] relies on an undetermined integration constant. He showed that the wave function for a flat universe is infinitely large compared to non-flat universes. Thus, if there exists an undetermined integration constant, the phase transition will end probably to a flat universe. However, his original proposal with a three index antisymmetric tensor field does not introduce any dynamics in the 4D space-time and the integration constant is just another cosmological constant [5, 8]. Witten [3] also argued that the existence of an undetermined integration constant may be a clue to the understanding of the flat universe, since a many-body theory may provide an explanation from a one constant universe to another constant universe [10]. Thus, intheabsence ofaself-tuningsolutionofthecosmologicalconstant, weregardthatthe cosmological constant problem in Hawking’s scenario still remains as an unsolved problem. Therefore, it is worthwhile to search for any new solution of the cosmological constant problem. Since thegravitationallawbeyondthePlanckscaleisnotasettledissue atpresent, we can look for solutions even in models with drastically different gravity beyond the Planck scale. If the cosmological constant problem is solved in models with a different gravity, then the new interaction can be studied extensively whether it leads to another inconsistency in theory or in phenomenology. In this spirit, there have been attempts to understand the cosmological constant in the 5 dimensional (5D) world with the fifth dimension y compact- ified [11] (RSI model) or uncompactified [12] (RSII model). It seems that there can be a way to understand the cosmological constant problem [13] in these RS type models since, RSI model for example, the nonvanishing brane tensions k and k and bulk cosmological 1 2 constant k can lead to a flat space solution if these parameters satisfy two fine-tuning con- ditions k = k = k . Therefore, the first step to understand the cosmological constant is 1 2 − to find the flat space solutions without any fine-tuning between parameters in the action. With the fifth dimension compactified (RSI), the attempts to solve the cosmological constant has failed so far. The original try to find a flat space solution without any fine- tuning[14]hadasingularity, andtakingthesingularityintoaccount byputtingabranethere reproduced a fine-tuning condition [15] even though the two conditions have been reduced to just one. Therefore, the first step toward a solution of the cosmological constant in the RS type models is to have flat space solutions without any fine-tuning between parameters in the Lagrangian. Restricting just to an exponentially small cosmological constant, it has 2 been pointed out that it is possible with two or more branes [16, 17, 18]. Recently, theneeded flatspace solutionsintheRSIItypebackground have beenfound[1]. With a smeared-out brane a similar attempt led to a self-tuning solution [19]. In the RS type models, the brane(s) has a special meaning in the sense that the matter fields can reside at the brane only. However, the effective gravitational interaction of the matter fields is obtained after integrating out the fifth dimension y. For this effective theory to make sense, we must require that: (i) the metric is well-behaved in the whole region of the bulk, and (ii) the resulting 4D effective Planck mass is finite. The condition (i) is to find a solution without a singularity in the region defined. In this regard, if the warp factor vanishes at say y = y , then y becomes the horizon and the m m universe connected to the matter brane is up to y = y . In the RSI models, it is a disaster m if 0 < y < 1 is between the two branes located at y = 0 and y = 1/2. This is because m 2 one needs both branes for the consistency in the RSI models. But in the RSII models, if there exists y then one can consider the space only for y < y < y if y is not a naked m m m m − singularity. Indeed, it can be shown that it is consistent to consider the space up to this point only by calculating the effective cosmological constant by integrating y to y . The m localized gravity condition (ii) restricts the solutions severely, since the y integration in some solutions would give a divergent quantity for M or M get more important contributions P P as y . → ∞ The Einstein-Hilbert action with a bulk cosmological constant and a brane tension in the RSII model does not allow a self-tuning solution. Addition of the Gauss-Bonnet term [20] does not improve the situation, and also it does not help to regularize a naked singularity in the self-tuning model with a bulk scalar [14]. But addition of the three index antisymmetric tensor field A (and the field strenth H and H2 H HMNPQ) allows a self- MNP MNPQ MNPQ ≡ tuning solution [1]. In this paper, we discuss the RSII model with antisymmetric tensor field added. The case with 1/H2 term allowing the self-tuning solution is the main motivation for this extensive study. H has been considered before in connection with the cosmological constant MNPQ problem [3] and the possible compactification of the seven internal space in the 11D su- pergravity [21]. Even in the 5D Randall-Sundrum model H is useful to separate the MNPQ 4 dimensional space. The 1/H2 term looks strange, but the consideration of the energy- momentum tensor would require a nonvanishing H , triggering the separation of the MNPQ h i extra dimension from the 4 dimensional space. Below the Planck scale, we consider that the action is an effective theory. Above the Planck scale, we consider that the quantum gravity effects may be very important, but at present the final form for quantum gravity is not known yet. Inthispaper, the self-tuning solution means that it does not need a fine-tuning between the parameters in the action,whichisaprogresstowardunderstandingthecosmologicalconstant problem. The self-tuning solutions are found from time independent Einstein equations. However, the existence of the self-tuning solution alone does not solve the cosmological constant problem completely. It is because if the ansatz for a time dependent metric allows, for example, the de Sitter space solutions with the antisymmetric tensor field added then choosing the flat space is simply choosing a boundary condition. However, the existence of the self-tuning solution and the probabilistic interpretation for the wave function of the universe can provide a logical understanding of the vanishing cosmological constant even in this case [5]. Suppose that we start from a flat universe from the beginning a´ la the 3 wormhole interpretation of the vanishing cosmological constant. But this interpretation alone may encounter a difficulty when the phase transitions such as the electroweak phase transition or the QCD phase transition add a nonvanishing cosmological constant at a later epoch. However, the existence of the self-tuning solution chooses the flat space solution out of numerous possibilities when the universe goes through these phase transitions. If there exists a self-tuning solution, then the wormhole interpretation chooses the vanishing cosmological constant even after these phase transitions. We find that the 1/H2 term does always allow de Sitter space solutions with localized gravity. In the RS model, however, a negative brane tension (Λ < 0) does not allow a 1 localized gravity in the de Sitter space [22], but allows only a nonlocalized gravity. There are arguments excluding these nonlocalized gravity [23], and in the RSII model for a negative brane tension one may exclude the de Sitter space solutions. However, the nonlocalized gravity cannot become a strong argument for the vanishing cosmological constant. It simply means that the de Sitter space solution with nonlocalized gravity cannot materialize to our universe. At some point, we may invoke an anthropic principle or turn to a probabilistic interpretation. Namely, as long as there exist solutions for nonzero cosmological constants whether there results a localized gravity or not, we need a probabilistic interpretaion. For a probabilistic interpretation, the existence of the self-tuning solution is crucial in choosing the flat universe. In Sec. II, we present the flat space solutions with 1/H2 term and with H2 term. It is shown that 1/H2 term allows the flat space self-tuning solution but H2 term allows at most one-fine-tuning solutions. De Sitter space and anti de Sitter space solutions are also commented. In Sec. III, it is shown that the 1/H2 term allows the anti de Sitter and de Sitter space solutions. We discuss the horizons appearing in our solutions. We also discuss how the universe chooses the vanishing cosmological constant. In Sec. IV, we present a time-dependent solution such that the 4D space time remains flat when the brane tension shift instantaneously to another value. Sec. V is a conclusion. II. THE STATIC SOLUTIONS The five dimensional space is composed of the bulk and a 3-brane located at y = 0 where y is the fifth coordinate. We assume that matter fields live in the brane. For studying the gravity sector, we include the three index antisymmetric tensor field A whose field MNP strength is denoted as H , where M,N, = 0,1,2,3,5( y). We find that there MNPQ ··· ≡ exist solutions for different bulk cosmological constants at y < 0 and y > 0. But for simplicity of the discussion, we will introduce a Z symmetry so that the bulk cosmological 2 constant isuniversal. Inthissection, wesummarize theself-tuningsolution [1]withthetime independent metric. But for comparison we briefly comment the time dependent metric, i.e. the de Sitter space and anti de Sitter space solutions with the 1/H2 term. We also present one-fine-tuning solutions for H2 term with time independent metric, and compare with the other known tuning solutions [14, 19]. 4 A. A self-tuning solution of the cosmological constant with 1/H2 A self-tuning solution exists for the following action, 1 2 4! S = d4x dy√ g R+ · Λ + δ(y) (1) Z Z − 2 HMNPQHMNPQ − b Lm ! which will be called the KKL model [1]. Here we set the fundamental mass parameter M as 1 and we will recover the mass M wherever it is explicitly needed. We assume a Z 2 symmetry of the warp factor solution, β( y) = β(y). The sign of the 1/H2 term is chosen − such that at the vacuum the propagating field A has a standard kinetic energy term. MNP The action contains the 1/H2 term which does not make sense if H2 does not develop a vacuum expectationvalue. Sincethe cosmologicalconstantproblem isatthebottomofmost cosmological application of particle dynamics, it is worthwhile to study any solution to the cosmological constant problem. We note that this problem has led to so many interesting but unfamiliar ideas [3, 4, 5, 7]. Therefore, any new idea in the possible interpretation of the cosmological constant problem is acceptable at this stage. In fact, we found a very nice solution with the above action and hence we propose the action (1) as the fundamental one in gravity. Being a part of gravity, we do not worry about the renormalizability at this stage. Flat space solution The ansatz for the metric is taken as ds2 = β2(y)η dxµdxν +dy2 (2) µν where (η ) = diag.( 1,+1,+1,+1). Then Einstein tensors are, µν − 2 β β ′ ′′ G = g 3 +3 , µν µν  β ! β !  2  β ′ G = 6 . (3) 55 β ! where prime denotes differentiation with respect to y. With the brane tension Λ at the 1 y = 0 brane and the bulk cosmological constant Λ , the energy momentum tensors are b 4 1 1 T = g Λ g δµ δν Λ δ(y)+4 4! H H PQR + g . (4) MN − MN b − µν M N 1 · H4 MPQR N 2 MNH2 (cid:18) (cid:19) The specific form for H2 H HMNPQ in Eq. (1) makes sense only if H2 develops a MNPQ ≡ vacuum expectation value at the order of the fundamental mass scale. Because of the gauge invariant four index H , four space-time is singled out from the five dimensions [21]. MNPQ The four form field is denoted as H , µνρσ ǫ H = √ g µνρσ (5) µνρσ − n(y) where µ, run over the Minkowski indices 0, 1, 2, and 3. With the above ansatz, the field ··· equation for the four form field is satisfied, HMNPQ ∂ √ g = 0. (6) M − H4 ! 5 There exists a solution for Λ < 0. The two relevant Einstein equations are the (55) and b (µµ) components, β 2 β8 ′ 6 = Λ (7) b β ! − − A β 2 β β8 ′ ′′ 3 +3 = Λ Λ δ(y) 3 (8) b 1 β ! β ! − − − A where A is a positive constant in view of Eq. (5). It is easy to check that Eq. (8) in the bulk is obtained from Eq. (7) for any Λ ,Λ , and A. This property is of the specialty of the b 1 H field. Near B1(the y = 0 brane), the δ function must be generated by the second MNPQ drivative of β. The Z2 symmetry, β( y) = β(y), implies (d/dy)β(y) 0+ = (d/dy)β(y) 0−. − | − | Thus, d2 d2 d β( y ) = β( y ) +2δ(y) β( y ). (9) dy2 | | dy2 | | y=0 d y | | (cid:12) 6 | | (cid:12) This δ-function condition at B1 leads to a bou(cid:12)ndary condition β ′ k , (10) 1 β y=0+ ≡ − (cid:12) (cid:12) where we define k’s in terms of the bulk(cid:12)cosmological constant and the brane tension, Λ Λ b 1 k , k . (11) 1 ≡ s− 6 ≡ 6 Let us find a solution for the bulk equation Eq. (7) with the boundary condition Eq. (10). We define a in terms of A, 1 a = . (12) s6A The solution of Eq. (7) consistent with the Z symmetry is 2 1/4 k β( y ) = [cosh(4k y +c)] 1/4, (13) − | | a! | | where c is an integration constant to be determined by the boundary condition Eq. (10). This solution, consistent with Condition (i), is possible for any value of the brane tension Λ . Note that c can take any sign. This solution gives a localized gravity consistent with 1 the above Condition (ii). The boundary condition (10) determines c in terms of Λ and Λ , b 1 k Λ c = tanh 1 1 = tanh 1 1 . (14) − − k ! √ 6Λb! − A schematic shape of β(y) is shown in Fig. 1. The effective 4D Planck mass is finite k 1/2 1 M3 1 ∞ M2 = 2M3 ∞dy = F α, P,eff a! Z0 cosh(4ky +c) √2ka " √2#0 q 6 β(y) 0.8 0.6 0.4 0.2 0 1 2 3 4 5 y FIG. 1: β(y) as a function of y for the flat space ansatz. It is plotted for k = 1 and a = 1. M3 1 dx = . (15) √2ka Z√1−(cosh(c))−1 (1−x2)(1− 12x2) q Here F(α,r) is the elliptic integral of the first kind and α = sin 1 (cosh(4ky +c) 1)/(cosh(4ky +c)). (16) − − q Note that the Planck mass is given in terms of the integration constant a, or the inte- gration constant is expressed in terms of the fundamental mass M and the 4D Planck mass M , P,eff 2 M3 1 ∞ a = F α, . (17) M2 √2k " √2#  P,eff 0   Curved space solution The curved space solution is a time-dependent solution which will be discussed in the KKL model of Sec. III. Localization of gravity and no tachyon The perturbed metric is ds2 = (β2η +h )dxµdxν +dy2 (18) µν µν where we chose the Gaussian normal condition, h = h = 0. With the transverse traceless 5µ 55 gauge, ∂µh = hµ = 0, and by a separation of variables such as h = ǫ eipxψ(y) (p2 = µν µ µν µν m2), we obtain the following linearized equations without matter on the brane, − 1 1 β 2m2 ∂2 +2k2 2k δ(y) 6a2β8 ψ(y) = 0 (19) − 2 − − 2 y − 1 − (cid:20) (cid:21) where Λ 1 k , (20) 1 ≡ 6 1 a . (21) ≡ √6A 7 Here we make a change of variable by z = y dy ydue C(u) and ψ(y) = β1/2ψˆ(z) and β(y) ≡ − then obtain the Schr¨odinger-like equation as follows, R R ∂2 +V(z) ψˆ(z) = m2ψˆ(z) (22) − ∂z2 (cid:20) (cid:21) where 15 39 V(z) = k2β2 3k βδ(z) a2β10 (23) 1 4 − − 4 9 ∂C 2 3 ∂2C = + . (24) 4 ∂z 2 ∂z2 (cid:18) (cid:19) (cid:18) (cid:19) Therefore, we can see that m2 0, i.e., there is no tachyon state near the background, by ≥ regarding the above equation as a supersymmetric quantum mechanics [19], 3∂C 3∂C Q Qψˆ(z) ∂ + ∂ + ψˆ(z) = m2ψˆ(z). (25) † z z ≡ 2 ∂z − 2 ∂z (cid:18) (cid:19)(cid:18) (cid:19) Moreover, there appears the localization of gravity on the brane as expected from the finite 4D Planck mass, due to one bound state of massless graviton, ψˆ (z) e3C/2 = β3/2, that is, 0 ∼ ψ (y) β2. Note that the zero mode solution ψ (y) automatically satisfies the boundary 0 0 ∼ condition (∂ +2k )ψ = 0 from Eq. (19). y 1 y=0+ | B. One-fine-tuning solutions of the cosmological constant with H2 term In this section, we obtain just the flat space solution for the H2 term in the action, M3 M S = d4x dy√ g R H HMNPQ Λ + (i)δ(y y ) . (26) − 2 − 2 4! MNPQ − b Lm − i ! Z Z · Xi The ansatze for the metric and the four form field are taken also as given in Eq. (2) and Eq. (5), respectively. Thus the field equation for A is trivially satisfied again, MNP ∂ √ gHMNPQ = 0. (27) M − (cid:20) (cid:21) The two relevant Einstein equations are the (55) and (µµ) components, 2 β A ′ 6 = Λ + (28) β ! − b β8 2 β β A ′ ′′ 3 +3 = Λ Λ δ(y) Λ δ(y y ) (29) β ! β ! − b − 1 − 2 − c − β8 where A/β8 1/2n2 expressed in terms of a ‘positive’ constant A. ≡ The solutions of Eq. (28) and (29) are a 1/4 for Λ < 0 : β( y ) = [sinh( 4k y +c )]1/4 (30) b | | k | | | | (cid:18) (cid:19) a 1/4 for Λ > 0 : β( y ) = [sin( 4k y +c )]1/4 (31) b | | k | | | | (cid:18) (cid:19) for Λ = 0 : β( y ) = ( 4a y +c )1/4, (32) b | | | | | | 8 where the a is defined in terms of A, A a . (33) ≡ s6 We note that for a positive c in Eqs. (30)– (32) β’s do not give localized graviton solutions near the brane B1, and for a negative c β’s have naked singularities at y = c/4k or | | − c/4a. Therefore, to get the effective four dimensional gravity or to avoid the sigularities − in the bulk, it is indispensable to cut the extra dimension such that it has a finite length size by introducing another brane, say B2. Thus, we need at least two branes and the situation is similar to that of the RSI except for the 4 form field contributions. Since the extra dimension is finite, the effective four dimensional Planck mass M M3 dyβ2 is also P ≡ finite. If we introduce two branes, we should satisfy the boundary conditions at the two R branes, consistently with the S1/Z orbifold symmetry, 2 d2 d2 d β( y ) = β( y ) +2(δ(y) δ(y y )) β( y ). (34) c dy2 | | dy2 | | y=0 − − d y | | (cid:12) 6 | | (cid:12) (cid:12) Then the boundary conditions for the above three cases are k k for Λ < 0 : c = coth 1 1 = 4ky coth 1 2 (35) b − c − k ! − k ! k k for Λ > 0 : c = cot 1 1 = 4ky cot 1 2 (36) b − c − k ! − k ! a 1 for Λ = 0 : c = = a 4y , (37) b c k − k 1 (cid:18) 2(cid:19) where k is defined in terms of the brane tension Λ at B2, 2 2 Λ 2 k . (38) 2 ≡ 6 We note that in the case of the H2 term (not 1/H2) in the action, the one-fine-tuning relations between k and k appear always, e.g. the relations (35,36,37), while in the case of 1 2 RSI model, the two-fine-tuning relations k = k = k were inevitable. 1 2 − If we introduce both 1/H2 and H2 in the action, there does not exist a flat space self- tuning solution. In this case, the derivative of the warp factor satisfies Λ β2 β8 A β = Λ¯ + b + . (39) ′ ±s 6 − 6A 6β8 A necessary condition for the existence of a flat space solution is β 0 and β 0 as ′ → → y . Certainly, Eq. (39) does not satisfy this necessary condition. On the other hand, → ∞ a necessary condition for the de Sitter space regular horizon is that β = finite as β 0. ′ → Also, this condition is not met in Eq. (39) and hence there does not exist de Sitter space regular horizon. 9 C. Comparison with other tuning solutions There appeared several self-tuning solutions since early eighties [3, 5, 14, 19]. The early stage scenario [3, 5] used field strength of three index antisymmetric tensor field H to µνρσ introduce an integration constant. The vacuum value of H = ǫ c introduces an µνρσ µνρσ h i integration constant c which contributes to the cosmological constant c2. Therefore, there ∝ exists a value of c such that the effective cosmological constant vanishes for a range of bare cosmological constant. Once c is determined to give the zero effective cosmological constant, c cannot change since it is a constant. In this sense, the four dimensional example is not a working model. Thus, the self-tuning solution needed a dynamical field to propagate. The three index antisymmetric field is not a dynamical field in 4D spacetime. Introduction of the extra dimension opened a new game in the self-tuning solutions of the cosmological constants [1, 14, 17, 18, 19, 25]. Here, we briefly comments on the key points of these solutions. The California self-tuning solution [14, 15] must use the RSI model set up, i.e. introduce two branes B1 and B2, and introduce a specific form for the potential of a bulk scalar field φ coupling to thebrane matter with adesired form. But the modelhas a naked singularity and to cure the problem, one has to introduce a brane at this singular point. Then, one needs a condition at the new brane and leads to one fine-tuning there [15]. If one satisfies this one-fine-tuning condition between parameters in the Lagrangian, there always exists a flat space solution. Therefore, the solution is not a self-tuning solution originally anticipated. One interesting or (disastrous to some) point of this model is that it does not allow the de Sitter or anti de Sitter space solutions. In this sense, it is an improvement over the original proposal [3, 5] for the self-tuning solutions. However, it may be difficult to obtain a period of inflation since the de Sitter space exponential expansion is not possible. One should see whether thebulkenergymomentum tensor satisfies ρ = p = p /2tohave theexponential 5 − − expansion of the effective 4D space [24]. However, at present it is not known what matter satisfies this kind of equation of state. There appeared another interesting self-tuning solution [19, 26], which does not introduce any brane. This model assumes a specific form of the 5D scalar potential multiplied to the matter Lagrangian in 5D. Even though the metric gives a localized gravity, the matter fields propagate in the full 5D space since the potential tends to a constant value as y . It is → ∞ not certain how we are forbidden to realize the extra dimension in this model. Then, there are proposals that the de Sitter space solution is phenomenologically ac- ceptable as far as the curvature at present is sufficiently small [16, 17, 18, 25]. In a sense, it also tries to accomodate the current small vacuum energy [27] with the de Sitter space solutions. [On the other hand, note that the quintessence idea is based on the solution of the cosmological constant problem [28].] The way the proposals make the vacuum energy small is to separate the distance between branes sufficiently large since the vacuum energy is exponentially decreasing with the separating distance. The one fine-tuning at B1 is met but the second fine tuning at B2 is not satisfied and allows the de Sitter space solutions. To have the vacuum energy decreased down to the current energy density, one needs that the separation distance increases as t increases. Namely, the horizon point at the y axis is required to increase. Then it is possible to relate the current vacuum energy with the current mass energy. In Ref. [25], the relation Ω = (5/2)Ω is obtained for Ω = 1. Λ m total The model presented in [1] allows a self-tuning flat space solution, self-tuning de Sitter and anti de Sitter space solutions. In general it is possible to introduce a period of inflation. 10