-S634- Journl of the Koren Physicl Society, Vol. 35, August 999 structure with two degrees of freedom. The three types of structures re relted to the

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1 Journl of the Koren Physicl Society, Vol. 35, August 999, pp. S633S637 Conserved Quntities in the Perturbed riedmnn World Model Ji-chn Hwng Deprtment of Astronomy nd Atmospheric Sciences, Kyungpook Ntionl University, Tegu, Kore The evolutions of liner structures in sptilly homogeneous nd isotropic world model re chrcterized by some conserved quntities: the mplitude of grvittionl wve is conserved in the super-horizon scle, the perturbed three-spce curvture in the comoving guge is conserved in the super-sound-horizon scle, nd the ngulr momentum of rottionl perturbtion is generlly conserved. I. SUMMARY We consider sptilly homogeneous nd isotropic world model with the most generl, spcetime dependent, three (sclr, vector, nd tensor) types of liner structures. As the grvity sector we consider both Einstein grvity with hydrodynmic uid or sclr eld, nd clss of generlized versions of grvity which couples the sclr eld with the sclr curvture. The three types of structures decouple from ech other due to the symmetry in the bckground world model nd the linerity of the structures we re ssuming. We identify the generlly conserved quntities existing in ll three types of perturbtions, ignoring the imperfect uid source terms: (A) In ner t bckground the sclr-type (relted to the density condenstion) structure is chrcterized by conserved quntity in the lrge-scle which is the perturbed three-spce curvture in the comoving (or the uniform-eld) guge. This quntity is conserved in the super-sound-horizon scle, nd thus is conserved eectively in ll scles in the mtter dominted er. (B) The vector-type (rottion) structure is chrcterized by the ngulr momentum conservtion. (C) In ner t (vnishing sptil curvture) bckground the mplitude of the tensor-type (grvittionl wve) structure is conserved in the super-horizon scle. In similr sense, the sign of three-spce curvture in the underlying world model, K, cn be lso regrded s conserved quntity. In the uid nd the sclr eld dominted ers the conserved properties remin vlid independently of chnging (i.e., time-vrying) eqution of stte p() nd chnging eld potentil V (), respectively. These conservtion properties lso pply in clss of generlized versions of grvity theories including the Brns-Dicke theory, the non-minimlly coupled sclr eld, the sclrcurvture-squre grvity, the low-energy eective ction of the string theory, etc. As long s the conditions re met the conservtion properties remin vlid independently of chnging grvity theories. The conservtion properties in (B,C) re vlid in the multi-component situtions, wheres (A) pplies to single-component situtions. In the following we will present thebove mentioned results by concretely showing the equtions describing the results. Since this is summry of our previous works, the detils bout derivtions nd explntions re referred to the originl works in the literture. II. GRAVITY THEORIES We consider grvity theories belonging to the following ction S = Z d 4 x p ;gh f( R) ;!() ;V ()+L mi () where nd R re the sclr eld nd the sclr curvture, respectively. L m is the hydrodynmic prt of the Lgrngin with the hydrodynmic energy-momentum tensor T b dened s ( p ;gl m ) p ;gt b gb. III. COSMOLOGICAL PERTURBATIONS We consider spcetime dependent perturbtions in the bckground riedmnn world model ds = ; ( + ) dt ; ( + B ) dtdx (3) + hg (+)+ j +C (j) +C idx dx : (),,, nd indicte the sclr-type structure with four degrees of freedom. The trnsverse B nd C indicte the vector-type structure with four degrees of freedom. The trnsverse-trcefree C indictes the tensor-type -S633-

2 -S634- Journl of the Koren Physicl Society, Vol. 35, August 999 structure with two degrees of freedom. The three types of structures re relted to the density condenstion, the rottion, nd the grvittionl wve, respectively. Since these three types of structures evolve independently, we will hndle them seprtely. We lso consider the generl perturbtions in the hydrodynmic energy-momentum tensor nd the sclr eld: T b (x t)= Tb (t)+t b (x t) (x t)= (t)+(x t): (3) IV. GAUGE ISSUE We hve ten degrees of freedom in the metric perturbtion. Two (one temporl nd one sptil) degrees of freedom in the sclr-type perturbtion, nd two (both sptil) degrees of freedom in the vector-type perturbtion re due to the spcetime coordinte (guge) trnsformtion. In order to hndle these ctitious degrees of freedom we hve the right to impose four conditions on the respective perturbed metric or energy-momentum content. Due to the sptil symmetry (homogeneity) the three sptil degrees of freedom cn be uniquely xed, thus mking the remining vribles sptilly gugeinvrint [] for exmple, ( + ) is sptilly guge-invrint combintion. After removing the sptil guge degrees of freedom, only the sclr-type perturbed vribles re ected by the temporl guge condition. When we x the temporl guge condition, we hve severl meningful choices. Since usully we do not know which guge condition will turn out to be most useful H = 3 +! ; f + R +V;3H for the problem priori,itisdvntgeous to strt from equtions without xing the temporl guge condition, thus in guge-redy form, [,]. Using the guge-redy pproch wehve investigted the cosmologicl perturbtion bsed on the hydrodynmic uid, sclr eld, nd clss of generlized grvity, [{6]. rom these studies we nd the best conserved quntity is the perturbed three-spce curvture in the comoving guge (v ) or in the uniform-eld guge ( ) in the generlized grvity theory the uniform-eld guge is the better guge condition, nd the uniform-eld guge coincides with the comoving guge in the minimlly coupled sclr eld. Both guge conditions completely x the guge trnsformtion nd ech vrible evluted in the guge is the sme s the corresponding unique guge invrint combintion of the vrible nd vrible used in the guge condition. Since we hve mny dierent guge conditions we proposed to write the guge-invrint combintions in the following wy, []: v ; H k v ; H ; H (4) where k is comoving wvenumber nd H =. v is the sme s in the comoving guge condition which tkes v, etc v is velocity relted perturbed vrible, [7]. V. EVOLUTION EQUATIONS The equtions for bckground re: ; K (5) +3H +! ; f +V! = (6) +3H ( + p) = (7) K is the sign of the bckground sptil curvture. It cn be considered s n integrtion constnt, nd thus is conserved quntity for the given world model. In the following we summrize the evolution equtions for three types of perturbtions. We consider ner t bckground, thus neglect K term. (A) In hndling the sclr-type structure the proper c s H 3 ( + p) H H ( + p) c s H v! ; = choice of the guge condition simplies the nlyses nd the resulting equtions. A guge invrint combintion of the perturbed curvture vrible bsed on the comoving (or uniform-eld) guge is found to be the best conserved quntity under vrious chnges. The equtions describing the evolutions of the hydrodynmic [7], the minimlly coupled sclr eld [8], nd the generlized grvity theories [9], respecively, re the following: ; c s v = stresses (8) (9)

3 Conserved Quntities in the Perturbed riedmnn World Model { Ji-chn Hwng 3! ! S635- ; = () where is Lplcin opertor bsed on g (3) nd c s p=. or pressureless idel uid, insted of Eq. (8) we hve v =. (B) We introduce B by (v) nd C cy (v) where Y (v) is (trnsverse) vector hrmonic function, [,]. v! v (v) ; b is guge invrint combintion relted to the mplitude of hydrodynmic vorticity! s v! /!, []. The rottionl structure is described by 4 ( + p) v! = stress: () Thus, neither the generlized nture of the grvity theory nor the presence of sclr eld ects the rottionl perturbtion in the hydrodynmic prt. We note tht Eq. () is vlid even in generlized grvity with the Ricci-curvture-squre term in the ction, []. (C) The grvittionl wve is described by [,] 3 3 C ; C = stress: () VI. SOLUTIONS Equtions (8-) immeditely led to generl solutions in certin limiting situtions. (A) rom Eqs. (8-) we hve the lrge-scle solution, for the hydrodynmic (ignoring the stresses), the sclr eld, nd the generlized grvity: v = C(x) ; D(x) Z t ~ c s H dt (3) 3 ( + p) = C(x) ; D(x)Z t = C(x) ; D(x)Z t H dt (4) 3 3! dt (5) + 3 where C nd D (or ~ D) re integrtion constnts indicting coecients of reltively growing nd decying solutions, respectively. Compred with the solutions in the other guge conditions, the decying modes in these solutions re higher order in the lrge-scle expnsion, [6,7]. Thus, ignoring the trnsient (nd lso subdominting in the lrge-scle) modes we hve the conserved quntity = C(x) = v : (6) rom Eqs. (8-) we notice tht Eq. (3) is vlid in the super-sound horizon scle, wheres Eqs. (4,5) re vlid in the super-horizon scle. In the lrge-scle limit in mny dierent guge conditions shows the conserved behvior: in the cse of n idel uid see Eqs. (4,73) in [3], nd Eq. (34,35) in [4] in the cse of the sclr eld see Eqs. (9) in [5] nd in the cse of the generlized grvity see Sec. VI in [6]. The often discussed conserved vrible introduced in [3] is which is in the uniform-density guge. In [7] we mde rguments why we regrd v s the best conserved quntity. Conservtion properties re further discussed in [4]. (B) or vnishing nisotropic stress, Eq. () for the rottion mode hs solution [5,] 3 ( + p) v! L(x): (7) Thus, the rottion mode of the hydrodynmic prt is chrcterized by conservtion of the ngulr momentum L. (C) In the lrge-scle limit (super-horizon scle), ignoring the nisotropic stress, from Eq. () we hve generl solution for the grvittionl wve [6] C (x t)=c (x) ; d(x)z t dt (8) 3 where c nd d re integrtion constnts indicting coecients of reltively growing nd decying solutions, respectively. Ignoring the trnsient mode, the mplitude of grvittionl wve inthe super-horizon scle is temporlly conserved. VII. UNIIED ORMS The equtions nd the lrge-scle solutions for the sclr- nd tensor-type structures cn be written in uni- ed forms s: Q ) ; c 3 Q (3 A = (9) =C(x) ; D(x)Z t ( 3 Q) ; dt () where, for the sclr-type uid, the generlized grvity, nd the tensor-type structures, respectively, we hve: = v = Q = + p c s H c A! c s () Q =! +3 = = c A! () =C Q = c A! : (3)

4 -S636- Journl of the Koren Physicl Society, Vol. 35, August 999 VIII. APPLICATIONS We hve shown conserved quntities in ll three types of structures in the riedmnn world model. Even in the super-horizon scle, not every quntity is conserved, nd rther there exist some specil (guge-invrint) vribles which re conserved independently of chnging eqution of stte p(), chnging sclr eld potentil V (), nd chnging grvity theories f( R). Using the conserved quntities, s long s the linerity ssumption is vlid, we cn esily trce the evolution of structures from the recent er to the erly stge. Let us consider scenrio where generlized grvity (or Einstein grvity with sclr eld) domintes the erly evolution stge of our observble ptch ofthe universe, nd t some point Einstein grvity tkes over the dominnce till the present er. If we further ssume tht the generlized grvity er provides n ccelerted expnsion (intion) stge, the observtionlly relevnt scles my exit the horizon to become superhorizon scles during the er. Under some conditions we cn derive the generted quntum uctutions bsed on the vcuum expecttion vlue, nd s the scle becomes the super-horizon it cn be interpreted s the clssicl uctutions bsed on the sptil verge. As long s the relevnt scles remin in the super-horizon stge during the trnsit epoch ofthegrvities (or chnging potentil, or chnging eqution of stte) there exist conserved quntities: C nd v (or ) re the conserved quntities. Now, let us explin more concretely how the conserved quntities provide connections between the hydrodynmic perturbtions in Newtonin regime nd the quntum uctutions in the erly universe. We consider the sclr-type structures for tensor-type structures, see []. The quntum genertion process is most esily hndled using perturbed sclr eld () eqution in the uniform-curvture guge ( ), i.e., using. The nlytic forms of power-spectrum of re vilble (in generl scles) in vriety of intion stges bsed on the sclr eld nd the generlized grvity, [7]. Using Eq. (4) the power-spectrum of is directly relted to the power-spectrum of, nd the ltter quntity is conserved during super-horizon scle evolution which my be the cse for the observtionlly relevnt lrge-scle structures in post intionry er. As long s the scle remins in the super-horizon it is conserved independently of the reheting process, possible grvity chnge (e.g., from the generlized grvity to Einstein one), nd the chnge from the sclr eld dominted to uid dominted stges i.e., = C(x) = v. Lter on, in the hydrodynmic stge, from v we cn derive the rest of perturbtion vribles fterll, since we re considering the liner perturbtion, vrible is liner combintion of the other vribles. It is known tht ( in the zero-sher guge which tkes ), v ( = in the comoving guge), nd v (v in the zero-sher guge) closely correspond to the Newtonin potentil uctution, density contrst, nd velocity uctutions, respectively, [,7]. We hve [7]: Z = C ; H t dt + H d (4) v = k (5) v = k t ;CZ dt + d (6) which revlid for generl p(), but ssuming K = nd vnishing stresses. The observed nisotropy ofthe cosmic microwve bckground rdition in the lrge ngulr scle is lso relted to C t the lst scttering epoch s T=T = ; C, where we ssumed mtter dominted er 5 nd ignored the decying mode. As mentioned, D ~ terms in Eq. (3) is (k=h) higher order compred with d(x) term in these solutions. Thus, in order to determine the Newtonin quntities (, v, nd v which provide the initil conditions for lter nonliner evolution stge) wht we need is the sptil structures encoded in C(x). rom Eqs. (4,6) the power-spectrum of sptil distribution of C, P C, cn be directly relted to the power-spectrum of in the in- tionry stge s P = C = P = v = P = = H j j P = : (7) The evlution of quntum uctutions of bsed on the vcuum expecttion vlue leds to P for exmple, in ner-exponentil intion bsed on sclr eld, in the lrge-scle nd in simplest vcuum choice, we hve P = = H=, [8,7]. In this prdigm of lrgescle structures generted from quntum uctutions, we cn probe the physics in intion stge using the observed lrge-scle structures nd the nisotropy in the cosmic microwve bckground rdition. Recent endevors for reconstructing nd constrining the intion physics from observtion cn be found in [9]. ACKNOWLEDGEMENTS We thnk Dr. H. Noh for useful discussions. This work ws supported by the Kore Science nd Engineering oundtion, Grnt No nd through the SRC progrm of SNU-CTP. REERENCES [] J. M. Brdeen in Prticle Physics nd Cosmology, edited by L. ng nd A. Zee (London, Gordon nd Brech, 988),. [] J. Hwng, Astrophys. J. 375, 443 (99). [3] J. Hwng, Astrophys. J. 45, 486 (993). [4] J. Hwng, Astrophys. J. 47, 533 (994).

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