Stability of compactification during inflation

Fermi National Accelerator Laboratory FERMILAB-Pub-90/47-A March 1990 Stability of Compactiflcation Luca Amendola,’ Edward W. Kolb,‘*’ During Marco Litterio,’ Inflation and France Occhionero’ l Osservatorio Astronomico di Roma via de1 Parco Mellini 84 00136 Rome, Italy zNASA/Fermilab Astrophysics Center Fermi National Accelerator Laboratory, Batavia, IL 60510 and Department of Astronomy and Astrophysics and Enrico Fermi Institute The University of Chicago, Chicago, IL 60637 Abstract The possibility that inflation may trigger an instability in compactification of extra spatial dimensions is considered. In old, new, or extended inflation, the false vacuum energy results in a semiclassical instability field representing potential in which the scalar the radius of the extra dimensions may tunnel through barrier leading to an expansion of the internal a space. In chaotic inflation, if the initial value of the scalar field responsible for inflation is large enough, the internal expansion. space becomes classically unstable to ever increasing Restrictions on inflationary models necessary to keep the extra dimensions small are discussed. c Operated by Unlveraities Research Associalion Inc. under contract with the United States Department of Energy I. INTRODUCTION If the fundamental theory of nature is a “higher-dimensional” one with extra spatial dimensions, it is necessary to hide the extra dimensions. The usual mechanism for hiding the extra dimensions is to assume that they form a compact internal space with a physical size small enough to have escaped detection. For currently available accelerator energies, this requires a size smaller than the Fermi length, or about lo-“cm. surprising, since in almost all extra-dimensional set by the Planck length, 1~ G G,“’ This would not be theories the fundamental - 1 616 x lo-sscm. length scale is In the limit that the physical size of the internal space is smaller than the physical size of the external space, it is possible to dimensionally “effective” reduce the system (integrate over the extra dimensions) and obtain an (3 + l)-dimensional theory. The assumption that the extra dimensions form a compact space is quite reasonable since if the Universe is closed (0 > l), the three observed spatial dimensions form a compact space (a 3-sphere, S3). The remarkable thing is that there is such a disparity in the sizes-IO-sscm for the internal space and more than 10’scm for the external space. Theories with extra spatial dimensions are many and varied. However all have common features of relevance for cosmology. In theories with extra dimensions the truly fundamental constants are the ones in the higher-dimensional appear in the effective four-dimensional theory. The constants that theory are the result of integration over the extra dimensions. If the volume of the extra dimensions would change, so would the “observed” constants. This implies that the internal dimensions must be static, or have changed very little since the time of primordial nucleosynthesis.’ The curious cosmology that emerges is one that has some dimensions large and expanding, and some dimensions small and static. Since expansion (or contraction) generic behavior expected, the challenge for cosmologists involves constructing 1 is’the models that have static extra dimensions. dimensional The basic approach is to assume that the higher theory is that of gravity plus a cosmological constant.2 sions are held static due to the interplay classical3 or quantum’ such as superstring The extra dimen- between the cosmological constant and either fields. Although the true mechanism in more complicated theories models might be more complex, there must be some vacuum stress keeping the extra dimensions static and the toy models studied here may very well be relevant. In the models that have been studied, the piesent ground state is stable against small fluctuations tunnelling introduced, of the size of the internal space. Maedas claimed that it is also stable against under the potential barrier. In Section II we show that when other fields are the potential is changed in such a way that a semiclassical instability and there is a non-zero probability appears for the extra dimensions to tunnel out the potential keeping them small. On the other hand, the presence of scalar fields is required during the inflationary era so that their effect on the dynamics of a multidimensional Universe must be considered. In Section III we discuss the stability of internal space when old, new or extended inflation is considered. a calculation of transition In this case the problem has a semiclassical nature: rates is then performed. to Linde’s model of chaotic inflation. II. FROM In Section IV the analysis is extended Our results are summarized in a concluding section. N TO 4 DIMENSIONS We will start with a theory of gravity in N = D + 4 dimensions with a cosmological constant ;i and some matter fields, for simplicity Upon dimensional reduction, responsible for inflation represented as a single scalar field 4. the scalar field 4 will give rise to a 4-dimensional (called the infiulon), 2 scalar field and the degree of freedom corresponding to dilatations of the internal space will give rise to a second 4-dimensional scalar field known as the diloton. The action is’ [- &R 5 = / dNx a where G is the gravitational GN by G = G,Vj + 2A + z(q) f.. .] , (1) constant in D + 4 dimensions, related to Newton’s constant with V,$ the present volume of the internal assumed to appear as a minimally space. The field 4 is coupled scalar field: _ _ - V(4). l(G) = pNa,&3,q Extra dimensions are assumed to be compactified to a D-sphere of radius b, whose present value is bo. The metric then reads: hfN = diag [i&z) (3) ; b’(t)hij(Y)] After dimensional reduction, fields do not depend on the coordinates of the internal space (hi; is just the metric of a D-sphere of unit radius), so that an integration coordinates yields only a numerical factor. Introducing over these the Newton constant GM, the action (1) becomes: s= (4) where dots stand for other fields needed to obtain Einstein-Hilbert compactification. action may be recovered after a conformal The ordinary transformation of the 4- dimensional metric: !L = w(-D~l~O)gpv, (5) with the dilaton field defined by 3 o=dn(;), ao= it has the ordinary corresponding [q(gDn;$ field dimensions of (length)-I. The desired final state is o = 0, to b = bo, and & = 0. This corresponds to a static internal space. The final 4-dimensional action is R + &7a’~ - U,(u) s = Ids J-s [- 1GrrlG~ +$7”‘ap$a”$ - exp(-D+cl)V(~)] , where Cri(o) and V($) are specified below. In this last expression for the action, the metric tensor CJ~“,and not j,,,,,, appears; furthermore with canonical dimension (length)-’ (length)-‘-“Is]. (7) we introduced the field 4 = (Vi)‘hj [in the D + 4-dimensional In this conformal frame, the gravitational theory 4 has dimensions constant (the coefficient of the Ricci scalar) is constant, but the mass scale associated with the inflaton is not, due to the factor exp(-Do/cc) The potential Vi(u) in front of V($). of Eq.(7) contains contribution The first source is the term in Eq.(4) proportional from (at least) three sources. to iI. The second source is due to the curvature of the internal space, which appears in Eq.(4) as the term proportional bD-‘. to Finally there must be some other source to give a stable ground state. We will consider a general model that encompasses two compactification refer to as either Cusimir, where an extra potential schemes, which we shall is given by the quantization of scalar fields in a compact space,=s4 or monopole, where an extra vector field is considered for which the well known Freund-Rubin ansatz is taken. r Both cases are discussed in details in Ref. (8). The point is that the extra contribution power of the radius of the internal is some (negative and D-dependent) space; thus the curvature term can be balanced and a static solution b = bo (i.e., o = 0) is allowed. Furthermore, energy, so that the N-dimensional this solution has non-zero constant i in the action (1) is tuned to ensure that 4 an effective 4-dimensional cosmological constant does not appear. The potential U*(u), shown in Fig. (1) for the Casimir case has the following expression: u,(g) = Q [ &w+wo + where Q = (D - l)o,l/bi(D ,-D~h _ D + 4 -(~+qo/oa D+2 -e 1 , + 4); in the monopole case it looks like very similar and has the same dynamical properties. When $ is constant and has zero energy, the dilaton field is trapped at the minimum of this potential and is stable from the semiclassical point of view. On the other hand we must introduce a 11 field in order to have inflation. Thus, the evolution of (r will be governed by a potential of the general form: -(D+~)o/co + e-D”‘““v($). I where V($) will be specified below for two different cases. In any inflationary with phase transitions, 11,is initially in a false vacuum state. The potential (9) scenario is of the form: V(tio) = x [+bb - h$ - ;Th?b=] +A, Here X is the dimensionless ratio of the multidimensional to the volume of the internal true-vacuum space Vi. constant X [dimension (length)D] The potential V($), at $ = +T = &[3( 1 + E) + ~/-j/4, shown in Fig. (2), has a and a false-vacuum state at $ = 0. The constant A in Eq.(lO) is now specified to be .\ = -A1$(& - &)*- $j; I in order to ensure that V($T) = 0. It will serve as the effective 4-dimensional cosmological constant to drive the de Sitter phase during inflation potential when $ # $T. has the simple form U(o, 0) = Vi(u) + A exp( - Da/go). For 11,= 0, the The effect of the new term is to raise the energy of the minimum of the potential leaving invariant for large cr. Since U(U = 0,O) > U(u = the asymptotic co, 0) = 0, the compactified behaviour to a positive value, while vacuum is semiclassically unstable so long as $ # $JT. There 5 are two true ground states of the system. The first ground state is li, = $r and Q = 0. This is the desired ground state corresponding to a compactified internal space. The other ground state is P = 00, for any $. This is the state to be avoided, corresponding to an expanding internal space. In the second case, the scalar field representing the radius of the extra dimensions tunnels through the barrier, and lowers the energy of the system by ever increasing expansion. Compactification is not stable unless the inflationary stage ends before the internal space can grow. For this to occur, the inflaton must tunnel through the potential faster than the dilaton can tunnel through its own. The 4-dimensional is the result of a competition appearance of the world between the two scalar fields won by the inflaton. section, we calculate the tunnelling In the next rates in $ and Q directions and show that the first one is larger than the second for reasonable choices of parameters, so that compactification of internal space is preserved in new or extended inflation. We will also discuss stability in the context of Linde’s chaotic inflation theory.‘a In this case the dynamics is totally classical, but the guidelines of the discussion are similar to the previous case. Here the introduction the dynamics of the dilaton barrier against evolution of the potential that drives inflation changes field in such a way that for very large values of $, the away from d = 0 disappears, leaving the dilaton free to evolve classically during inflation. The potential assumed for chaotic inflation, Fig. (3), is of the form (12) The relevant potential V,(g). Stability for chaotic inflation of compactification is Uc(a,IC,), which is Eq.(9) with V(g) in chaotic inflation 6 is discussed in Section IV. = III. SEMICLASSICAL Let us turn now to the evaluation directions: of tunnelling STABILITY rates in the two relevant alternative toward an inner-space explosion, or toward the compactified vacuum. Only when the latter results to be much more likely than the former, will the process match the actual observations of an inflated, 4-dimensional decay in flat space is well known,s and in the thin-wall result: potential Universe. The theory of vacuum approximation gives a very simple the probability of transition per unit time per unit volume of a field 4 in a V(q5) is r/V = Aexp(-SE), w h ere SE is the Euclidean action for 4 evaluated along the “bounce” path of the field. The thin-wall approximation is realized when the ratio of the energy difference E between the two vacuum states and the barrier height is much smaller than one. In this case SE is simply9 SE = 2WS’ 2e3 I, (13) where S, = Ji dq%[2V(4)]‘/‘, and 4 = a, 6 are the two vacuum states. In our case we have two fields, the dilaton tr (representing variable) and the inflaton the potential Eq.(lO), is metastable), +, and, in general, several vacuum states. as required for the phase transitions and extended inflationary the inner-space dynamical models,“’ as previously occurring Assuming for $ in the old, new, we have in fact three vacuum states (one of which shown. It is obvious that the tunnelling can occur along any path linking the vacuum states, but in Eq.( 13) we need taking into account only the least-action path. In the same thin-wall approximation we can see that the only possible directions of tunnelling are from $J = 0 to 11,= &- along Q = 0 with bounce action S($) (the desired tunnelling) or from o = 0 to c = 00 along $I = 0 with bounce action SE(U) (the one to be avoided). dilaton tunnelling if We may then state that the inflaton tunnelling overrides’the SE(b) > (14) SE($). Let us finally start with the calculations. Eq.(9) with V(q) (metastable) The complete potential as in Eq.(lO) [Fig. (4)]. F or a small E, the origin v = O,$ = 0 is a vacuum state, with U(O,O) = A. Let us call this vacuum state V,. The other (true) vacuum states lie at (u = O,J, = $r), called VI and V,, respectively). well-known SE(@) for the two fields is and (for any $) at cr = +m (to be The Euclidean Action for the tunnelhng result, and in the thin-wall V, --t VI is a limit (small c) it amounts to” (15) = &. The Euclidean Action in the Q direction can be evaluated in the thin-wall approximation if A < U,, where UM is the maximum of U(U,$ = 0). To first order in l/D, i.e., when we may neglect the first term in Eq.(9), 1‘t is easy to see that the maximum is attained at ye = (1 - 2/D), with VM = 2a/(De’). Then the thin-wall condition is equivalent to A/a < 2/( D er) and is fulfilled for D > 4neZ+b;GNb;. (16) We will comment later on this inequality. The calculation of SE(O) involves the integral s, = o- da[HJ(u,+ / = o)]‘/~, that can be recast in the form (neglecting Sl=UOVG A/a) z(D+l) + yD-’ _ -0-14 D+2Y where we have defined y = exp(-~/as). a,F&i/D, (17) D “= 1 ’ Again, to first order in l/D, we have St = where F is a geometric dimensionless factor of order unity, with a very mild dependence on D. When D = 6, for example, a numerical integration while for D = 20 we get F = 0.57. Eq.(13) now reads 8 gives F = 0.966, SE(U) = - “,‘% (16~:~6~)~ The parameter A can be expanded in a power series in c, and at lowest order is A = cX$t/2. Putting everything D ->>PX’ Gdo together, the inequality of Eq.(14) gives 4 where we introduced (20) the li, mass, m$ = X$,2/2. The constants appearing in Eq.(20) are all free parameters of the theory (except of course GN), but they are in principle observable quantities. Notice that although we assumed that $ =const = 0, we do not expect the calculation to be changed much if ?c,is slowly rolling as in new inflation. inflation, i.e., one which does not violate the constraint fluctuations, either in the form of gravitational In particular, a successful new on the production waves or scalar perturbations, of primordial must have very small n+ and X. For example, in Planck units, it is often assumed m+ -., 10m6 and X - 10-i*. The natural, yet unknown, value for bo is the Planck length, so that Eq.(20) is expected to be satisfied even for D = 1. Moreover, one can see that Eq.( 16) is consistent with Eq.(20) when the same values as above are assumed, rendering inflation a good mechanism for having dimensional stability, 9 at least in the thin-wall limit. IV. In Linde’s chaotic inflation,” CLASSICAL STABILITY the field ti need not have a potential of the form Eq.( 10). Indeed, it is possible to have inflation for any V, field evolving classically to zero starting from an initial value of a few Planck units (at least three Planck masses for producing 70 e-folds of inflation). From the modified potential [coupled now with Eq.(12)-see for the dilaton field v in Eq.(9) Fig. (5)], one sees that, for large +, the potential barrier that makes Q = 0 a stable solution could disappear. Including $-dependent term, the total potential assumes the following form 2(D+Z) + [l + W($)]yD with W($) = X$4/4a. minimum as in Fig. (l)] or no local extrema at all, depending on the This can be seen most easily by splitting into two functions (21) It is not difficult to show that CJc has either two local extrema [a and a maximum, value of W(g). D+4 - D+ZyD+r}, of y, Q.(y) = ayD-‘[fi(y) (D + 4)~’ - D( 1 + W). It is clear that fr(y) the derivative U’ = aUc/ay - f*(y)], where fi = 4yD+’ and fr = crosses fl(y) at most two times (for y > 0), and that there must exist some W., and hence some $., for which fr is tangent to fi. The value + = +. signals that the barrier has vanished, and that starting from y above this critical value the classical evolution will be toward Q = +cc (which we want to avoid). The critical value 111.can be determined f: = fL fl = exactly by solving the system in y and $ fi. (22) From the first equation we learn that the barrier disappears when y = y. = 2-‘l(n+rl, and from the seccnd one that this happens when W($J) has the value w* = y.2( 1.; > -1. Then, we may state that the condition (23) for the existence of a barrier between the corn-- pactified Universe (u = 0) and the unfolded one (u = +w) 10 is, for large D, D 8nGN bi The last term in Eq.(24) is of order unity in Planck units if bo is close to the Planck length. In this case, the condition V($) Eq.(24) is similar to the “quantum-boundary” < M$,, and both inequalities may be satisfied assuming a very weakly coupled inflaton, as usually done in current inflationary scenarios. Equation (24) has another, very interesting, the initial implication. value pi of the field and its self-coupling from the requirement fluctuations constraint of sufficient inflation In Linde’s chaotic inflation constant X are given a lower bound ($; 2 3M~l), to drive the subsequent large-scale structure and of enough initial formation12J3 (X 1 10-t’). seed In this case, Eq.(24) implies that bf,< D 8rG~V(+,i) z D x 10’” 1’Ply where Ipr is the Planck length. If one considers that the experimental upper bound on bo is not better than b. < 10” Ipr, the purely theoretical speculations lead to an improvement of more than ten orders of magnitude. Notice that most theoretical bounds on the inner dimensions deal only with the rate of change of the inner radius, i.e., with h/b or with some compactification ratio b/b s ( see, for example, upper bounds from nucleosynthesis’ or microwave background (d.,$.) anisotropy14). Here, in contrast, the very existence of a point at which the barrier disappears allows a direct upper bound on the absolute value of the present inner radius bu. A similar constraint but there it rests on the hypothesis of thin-wall can be derived from Eq.(20), bubbles, and it is a less stringent bound. Let us conclude this section observing that the shrinking general feature, provided the self-coupling that the shape of U(u,$ of the barrier is a quite potential for rj is monotonically = 0) for Q is as in Fig. (1). 11 growing, and V. CONCLUSIONS In multidimensional theories, there exists an internal space of radius b. that is assumed to be very small and static. semiclassically-by This configuration an appropriate potential. potential is modified so that an instability incompatible classically and However, in any inflationary scenario this appears. Is, then, multidimensional cosmology with inflation? We took in consideration inflation is made stable-both on the other. nature: stability old, new and extended inflation on one hand and chaotic In the first case the problem turns out to be of a semiclassical is preserved if the probability for the dilaton to tunnel through its po- tential is smaller than that for the inflaton to do the same under its own. A calculation the transition rates in the thin-wall choices of the mass of the inflaton instability limit shows that this is actually the case; reasonable and of its self-coupling for any number of internal dimensions. constant do not give rise to In chaotic inflation, totally classical; for very large values of the inflaton field 4, the potential pears and the internal space can grow without limit. the problem is barrier disap- Nevertheless, the initial conditions and the parameters of the model adjust themselves naturally for a successful inflation, of in such a way as to allow and at the same time to meet the conditions to exist. For both cases, the result is then that the internal for the barrier space remains stable during inpation. Extensions of Linde’s modeIt nal inflationary predict that there are regions of the Universe in an eter- stage. This happens when, in one of the causally disconnected verses,” the scalar field 11 is initially “miniuni- greater than X-‘/s 44~1. In this case II, grows larger and larger climbing the potential in Fig. (3) rather than rolling down to zero. However, the maximum value that li, can reach is $0~ s X- ‘/‘Mpt suppressed.‘s In our multidimensional environment, 12 at which its growth becomes this could imply that eventually $ becomes larger than $J. where the compactification that 11. z (D/bi)“‘&~; breaks. Now, we see from Eq.(24) then, depending on the values of b. and D, $,. lies either in the classical or in the quantum region. In the latter case li, never reaches $J. where the barrier disappears, and we may conclude that in eternally inflating domains internal dimensions cannot be unfolded; in the former case, on the contrary, unfolding takes place, with the consequence that most part of the physical volume of the Universe lives in a multidimensional state. Of course one must have in mind that these considerations for compactification hold true only schemes of the kind we discussed in Section II, and that chaoticity allows in principle all kinds of dimensional dynamics in different miniuniverses. One last result is worth of mention. The knowledge of the physical point in the (u,$J) plane at which the barrier disappears, allows a direct bound on the present radius of the internal space 6; < D x lO’s$.r, while, usually, only limits on the value of the ratio b/b0 are given. ACKNOWLEDGMENTS This work was supported by DOE and NASA (grant # NAGW-1340) EWK would like to thank CNR for financial support at Osservatorio at Fermilab. Astronomico di Roma. REFERENCES 1. E. W. Kolb, M. .I. Perry, and T. P. Walker, Phys. Rev. D 35, 869 (1986). 2. There are very interesting models of gravity including higher-derivative terms. The cosmological significance of such models have been studied by Q. Shafi and C. Wetterich, Phys. Mt. 12BB, 387 (1983), and by D. Bailin, A. Love, and D. Wong, Phys. Lett. 165B, 270 (1985). 13 3. S. Randjbar-Daemi, A. Salam, and J. Strathdee, Phys. Letl. 135B, 388 (1984); K. Maeda, Class. @ant. 4. T. Applequist Gravity S, 233 (1986); 651 (1986). and A. Chodos Phys. Rev. Lett. 50, 141 (1983); Phys. Rev. D 28, 772 (1983); P. Candelas and S. Weinberg, Nucl. Phys. B237, 397 (1094). 5. K. Maeda. Phys. Lett. 186B, 33 (1987). 6. 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Fig. 5: Potential of Eq.(9) in chaotic infiation. o direction is seen to disappear for large +. 16 The barrier against evolution in the 3 G’ x I I I I I I I 0 0 I I I . bs . ‘;; P s: I I I I I 0 0 I I I I I I I I I ~11 w $ ;;; + B’ I-* cf