A Note on the Cosmological Dynamics in Finite-Range Gravity

A NOTE ON THE COSMOLOGICAL DYNAMICS IN FINITE-RANGE GRAVITY M. Sami Inter-University Centre for Astronomy and Astrophysi s, Post Bag 4, Ganeshkhind, Pune-411 007, INDIA. y In this note we onsider the homogeneous and isotropi osmology in the nite-range gravity theory re ently proposed by Babak and Grish huk. In this s enario the universe undergoes late time a elerated expansion if both the massive gravitons present in the model are ta hyons. We arry out the phase spa e analysis of the system and show that the late-time a eleration is an attra tor of the model. PACS numbers: 98.80.Cq, 98.80.Hw, 04.50.+h arXiv:hep-th/0210258 v2 25 Dec 2002 I. INTRODUCTION Re ently there has been renewed interest in the old idea that gravitons ould have a small mass [1℄. The fresh motivation for massive gravity may be attributed to the high pre ision experiments on the dete tion of gravitational waves like LIGO and LISA whi h, in a near future, may result in answering the question about the mass of gravitons[2, 3, 4℄. Se ondly it seems to be natural to look for an alternative to GR to in orporate the re ently observed "late time a elerated expansion" of the universe. Third, the higher dimensional gravity theories seem to naturally mimi the properties of four dimensional massive gravity and should be given serious thought [14℄. Last but not the least it is more than desirable to in orporate the possibility of in ation without opting for an ad ho me hanism to realize it. The massive gravity may or may not address all these issues. Till very re ently, the masses of gravitons were thought to ome from Fierz-Pauli mass term added to Einstein Hilbert a tion. The model fa es the well known mass disontinuity problem alled Van Dam-Veltman-Zakharov dis ontinuity[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17℄.It was very re ently noti ed by Babak and Grish huk that this problem is very spe i to the hoi e of Fierz-Pauli mass term[19℄. In eld theoreti formulation of Babak and Grish huk[19, 20℄, the mass term has the form, k1h h + k2h , where h is the nonlinear gravitational eld propagating in the Minkowski spa e and "h" denotes the tress of h . Linear massive gravity is disussed in Refs.[21, 22, 23℄. For a spe i hoi e of parameters k1 and k2: k1 + k2 = 0, the lo al predi tions of the model have nite di eren e with those of GR though the mass term in the Lagrangian smoothly vanishes for k1 ! 0. Babak and Grish huk have demonstrated that everywhere on the parameter plane (k1; k2) ex luding the Fierz- Pauli straight line( k2 = k1 ), the massive grav- On leave from jamia Millia, New Delhi. Ele troni address: samiiu aa.ernet.in y  ity theory proposed by them is free from dis ontinuity problem[19℄. The parameters k1 and k2 are related to the masses of gravitons: - mass of spin-2 graviton and mass of the spin-0 graviton. The predi tions of BabakGrish huk model are dramati : their model has smooth limit to GR for ; ! 0, the in lusion of a small mass term removes the bla k hole event horizon and makes the osmologi al evolution os illatory. Further, it is remarkable that if both the gravitons in the model are sa ried to ta hyons, the nite range gravity exhibits late time a elerated expansion of the universe. The mirale is done by the presen e of massive s alar graviton. In ase, the mass of s alar graviton is zero, Babak and Grish huk have noti ed that osmology based upon the nite range gravity is identi al to GR osmology independently of the mass of spin-2 graviton[19℄. Hen e the additional spin-0 graviton plays a entral roll in niterange gravity. For earlier referen es on eld theoreti formulation of gravity see Refs[24, 25, 26℄ and the work of Logunov and ollaborators[27℄. The non-linear bigravity theory re ently dis ussed by Damour et al. seems to share many interesting features of the nite range gravity mentioned above [28℄. On e the Fierz-Pauli hoi e for mass parameters is exluded, the su ess of the theory relies on existen e of an additional s alar graviton. This is reminis ent of the manner in whi h a massive s alar boson was introdu ed by Veltman in the ele troweak theory [17℄, and suggestion ontained in Ref. [18℄ that gauge bosons and massive graviton must ne essarily be interpreted as multi-spin obje ts. In parti ular, it is anti ipated that a nite-range gravity should endow graviton with spin 0, 1 and 2 omponents. However, in this work we shall stri tly on ne to the framework of Babak and Grish huk. In this note we investigate the phase spa e behavior of osmologi al evolution in nite range gravity with both the gravitons behaving like ta hyons. We demonstrate that the a elerated expansion is a late time attra tor of the model. 2  A. EVOLUTION EQUATIONS IN FINITE RANGE GRAVITY The gravitational eld in the massive gravity is des ribed by a non-linear tensor eld h propagating in the Minkowski spa e with metri  and with a massive term added to the GR part of the eld Lagrangian whi h is quadrati in h ; ( ovariant derivative is taken with respe t to the metri  ). The e e tive eld equations have the form[19℄ G + M = T (1) where T is the matter energy momentum tensor and the mass tensor M is given by   1 g g (2k h + 2k h) (2) M = Æ Æ 1 2  2 p where gg = p (  + h ). In a homogeneous and isotropi situation h depends upon time alone and the e e tive metri a quires the form
ds2 = b2(t)dt2 a2(t) dx2 + dy2 + dz 2 (3) where a(t) and b(t) are given by the timt-time and spa espa e omponents of the tensor eld h . The evolution equation for a(t) in massive gravity has the form _ 3 aa((tt)) !2 2 0 + 8(3 + 2) [y3 (1 4 ) y1 + 2a2 (y2 3)℄ = a83(G !+1) (4) where 2 = 4k1, 2 = 2k (k1 + 4k2) =(k1 + k2),  = 2 = 2. The perfe t uid form of T is assumed and the equation of state is taken in the simple form, p = ! with ! onstant. In ontrast to GR the fun tion b(t) is not arbitrary and gets determined through a(t) itself ; there is an algebrai relation between a(t) and b(t) [19℄. This relation a quires a simple form for  = 1=4 p a ( 1 + 7 + 9a4 ) y = (5) b 3a2 II. PHASE SPACE ANALYSIS In this se tion we investigate the phase spa e behavior of osmologi al evolution [29℄ in nite range gravity for both the gravitons behaving like ta hyons, i.e, 2 ; 2 < 0. For simpli ity, we will be working with  = 1=4. The evolution equation ( 4 ) in this ase for a >> 1 and ! = 0, where interesting things are expe ted to happen, simpli es to 7 7 a 1 2 2 4 m (6) a02( ) = 3 + m a + 6 m a + 18 m a and for a << 1 a 1  (7) a02 ' m a 4 + 3 3  j 1 , and prime denotes the where m = 8(j +2) 8G0 p derivative with respe t to  = (8G0 )t. In order to obtain the expression ( 6 ) we have used the following series expansion for y(a >> 1) 2 7 9 y = 1 + a 4 + higher order terms Before pro eeding further, a omment about Eq.( 6 ) is in order. From the point of view of GR ,Equation ( 6 ) an be thought as an e e tive Friedmann equation where the rst term on the right hand side of this equation is the usual term ontributed by the matter density and the rest of the terms ontaining m originate from the Mass tensor appearing in the nite range gravity theory. The se ond term behaves like osmologi al onstant and is responsible for drawing the late time a elerated expansion in Babak-Grish huk theory. It is remarkable that su h a term automati ally appears at late stages of evolution in massive gravity. The terms ontaining m smoothly vanish in the mass-zero limit leaving behind the usual Freedmann equation. It is surprising that an arbitrarily small mass of gravitons an hange the hara ter of evolution, turning de eleration into a eleration.The third term in Eq. 6 mimi s radiation whereas the last term looks like urvature term. It is interesting to noti e that Eq. 6 looks formally similar to the e e tive Friedmann equation in brane-world osmology [30℄. It should however be kept in mind that unlike the FRW osmology where the numeri al value of the s ale fa tor does not arry any physi al signi an e and the s ale fa tor an be made larger or smaller than one at any given epo h, su h a hoi e no longer exists in massive gravity. As already mentioned, the fun tion b(t) is not arbitrary in Babak-Grish huk theory, it gets determined through a(t) via Eq. 4 whi h learly shows that the numeri al value of s ale fa tor is indeed important. Eq. 6 is valid for large values of "a" and in this regime out of all the terms ontaining m , the se ond term is most dominant whi h is to ompete with the rst term allowing ultimately the transition to a eleration. It is further important to mention, as pointed out by Babak and Grish huk[19℄, that for no values of parameters, a_ = 0 in the present ase ( however, for 2; 2 > 0, the s ale fa tor goes through regular maximum and minimum). At early times, the evolution of universe is des ribed by Eq. 7. The rst term on the right hand side (RHS) of this equation dominates at early epo hs and mimi s sti equation of state. It would be suggestive to re ast the expression ( 6 ) to look like Newton equation in one dimension[31℄ a ( ) = 00 d V ( a) da (8) with the rst integral of motion E= a02 2 + V ( a) = 0 (9) 3 0 40 -1e-04 35 30 -0.0002 1/2 25 u /Λ m V(a) -0.0003 -0.0004 20 15 -0.0005 10 -0.0006 -0.0007 5 5000 10000 15000 20000 25000 30000 0 0 0.002 0.004 a FIG. 1: Plot of the potential V(a) given by Eq.10 for m = 10 12 . This is a generi form of potential whi h in ase of spe ially at universe allows de eleration to hange into a eleration. where V (a) is given by the expression, a 1 m 2 7 2 7  a 4 + m V (a) = 6 2 a 12 m a 36 m 2 (10) The potential V(a) is plotted in Fig.1. The system, in the present ase, has suÆ ient kineti energy to surmount the potential barrier and, onsequently, the s ale fa tor passes through a point of in e tion at whi h the de eleration hanges into a eleration. In order to analyze the dynami s of the system we re ast the equation ( 8 ) in the following form v0 = uv (11) u0 = m where 7  v4 7  v6 6 m 9 m 1 v= ; a u= v3 6 u2 (12) a0 a . The onstraint equation ( 9 ) takes the form u2 0.008 v 0.01 0.012 FIG. 2: Phase portrait of osmologi al evolution in massive gravity for 2 ; 2 < 0 , i.e., for both the gravitons in the theory behaving like ta hyons.Traje tories, orresponding to di erent values of parameter m , starting anywhere in the phase spa e, onsistent with the onstraint given by Eq. (13), end up at the stable riti al point (0,1). v r = (m )1=2 is not relevant here. At the rst riti al point, a0( ) = 0 and this is not possible in the present ase i .e, for 2 ; 2 < 0. Indeed, omitting terms of higher order than v3 in equation ( 14 ), one gets the approximate value of v0 ' (6m )1=3. This riti al point violates the onstraint equation ( 13 ) and therefore should be disarded. The se ond riti al point whi h satis es the onstraint equation ( 13 ) is very interesting, u r = +(m )1=2 means that 1=2 a0( )=a( ) = m a( ) / em 1=2  or and v r = 0 means that the exponential behavior is approa hed asymptoti ally for large values of a( ). Thus, this s enario des ribes a rapidly a elerating universe eternally at late times and like any other generi model of quintessen e, the nite-range theory of gravity will also be fa ed with the problem of future event horizon[32℄. In order to he k the stability of the riti al point we perturb about xed point (0; +1=2) v = v r + Æv u = u r + Æu v3 = m m v2 + 7 m v4 + 7 m v6 + (13) 2 2 12 36 6 To nd out the riti al points we set right-hand side (RHS) of Eqs. (11) and (12) to zero and we obtain two xed points u r = 0; v r = v0 and v r = 0; u r = +1=2 where v0 is a solution of the algebrai equation 7  v6 + 7  v4 + v03  = 0 (14) m 9 m0 6 m0 6 2 0.006 Plugging these in ( 11 ) and ( 12 ) and keeping only linear terms we get Æv0 = u r Æv (15) Æu0 = 2u r Æu (16) with the solutions 1=2 Æv / e   and 1=2 Æu / e 2  whi h learly demonstrate the stability of the riti al point under small perturbations. As the system evolves, 4 the point moves in the (u, v) plane along the urve given by the onstraint equation ( 13 ) on whi h lies the ritial point. Stability guarantees that the motion will take pla e towards the riti al pont. We have evolved the system numeri ally for di erent values of the parameter m , we nd that traje tories starting anywhere in the phase spa e, onsistent with the onstraint, end up at the stable xed point as shown in Fig. 2. Therefore the a elerated expansion is a late time attra tor of nite range gravity. 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