perturbation theory and its applications

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1 Second-order order guge-nvrnt perturton theory nd ts pplctons (Short revew of my poster presentton) Some detls cn e seen n my poster Kouj Nkmur (Grd. Unv. Adv. Stud. (NAOJ)) References : K.N. Prog. Theor. Phys., (23), 723. (gr-qc/3339). K.N. Prog. Theor. Phys., 3 (25), 43. (gr-qc/424). K.N. Phys. Rev. D 74 (26), 3R. (gr-qc/657). K.N. Prog. Theor. Phys., 7 (27), 7. (gr-qc/658). K.N. preprnt (rxv:84.384[gr-qc]) (+ α)

2 I. Introducton The second order perturton theory n generl reltvty hs very wde physcl motvton. osmologcl perturton theory Expnson lw of nhomogeneous unverse (ck recton effect, vergng prolem) Non-Gussnty n M (eyond WMAP) lck hole perturtons Rdton recton effects due to the grvttonl wve emsson. lose lmt pproxmton of lck hole - lck hole collson (Gleser, et.l (996)) Perturton of str (Neutron str) Rotton pulston couplng (Kojm 997) There re mny physcl stutons to whch hgher order perturton theory should e ppled.

3 However, generl reltvstc perturton theory requres more delcte tretments of guges. It s worthwhle to formulte the hgher order guge nvrnt perturton theory from generl pont of vew. In ths poster presentton, we show Generl frmework of the second-order gugenvrnt perturton theory. (K.N. PTP, (23), 723; d, 3 (25), 43.) Applctons Second-order cosmologcl perturtons (K.N. PRD, 74 (26), 3R; PTP,7 (27), 7; rxv:7.996[gr-qc]; rxv:84.384[gr-qc]. ) Towrd to the pplcton to the rdton recton. (+α) (Ths prt s not complete.)

4 II. Guge degree of freedom n perturtons (Stewrt nd Wlker, PRSL A34 (974), 49.) Guge degree of freedom n generl reltvstc perturtons rses due to generl covrnce. N Physcl spcetme (PS) p Q M ɛ ɛ In ny perturton theores, we lwys tret two spcetmes : Physcl Spcetme (PS); ckground Spcetme (GS). X ɛ p Q δq Φ ɛ = X ɛ Y ɛ Y ɛ q M ckground spcetme (GS) In perturton theores, we lwys wrte equtons lke Through ths equton, we lwys dentfy the ponts Q( p") = Q (p) + ffq(p) on these two spcetmes. Ths dentfcton s clled guge choce n perturton theory.

5 Q X = X Λ ffl Q = Q+ffl$ uq+ Q Y = YΛ ffl Q = Q+ffl$ vq+ 2 ffl2 $ 2 u Q+O(ffl3 ) 2 ffl2 $ 2 v Q+O(ffl3 ) = (X ffl ff Y ffl ) Λ Q = Q + ffl$ ο Q + Q Q Y Q X = $ ο Q X + $ $ο2 A QX ο Guge trnsformton rules of ech order Expnson of guge choces : We ssume tht ech guge choce s n exponentl mp. Φ Λ ffl > 2 ffl2 $2 Q $ο2 O(ffl ) ο (Sonego nd run, MP, 93 (998), 29.) ο = u v; ο 2 = [u; v] Expnson of the vrle : Q = Q + fflq + ffl2 Q 2 + O(ffl 3 ) 2 Order y order guge trnsformton rules : Inspectng these guge trnsformton rules, we develop second-order guge-nvrnt perturton theory. Q 2Y Q 2X = 2$ ο Q X +

6 III. Guge nvrnt vrles metrc perturton : μg metrc on PS : g, metrc on GS : metrc expnson : μg = g + fflh + Our generl frmework of the second-order ffl2 l + O(ffl 3 ) 2 guge nvrnt perturton theory s sed on sngle ssumpton. lner order (ssumpton) : Suppose tht the lner order h perturton s decomposed s h = H + $ X g so tht the vrle nd H re X the guge nvrnt nd the guge vrnt prts of h, respectvely. These vrles re trnsformed s A under the guge Φ trnsformton := X ffl ff Y. ffl Y H X H = YX X X ο Ths s correct n cosmologcl perturtons (see my poster). ffl

7 L = 2 2 YZ X Z = ο + [ο ; X] Φ ; L = 2 Ψ fl j + 2 χ j Second order : Once we ccept the ove ssumpton for the lner order metrc perturton h, we cn lwys decompose the second order metrc perturtons s follows : l l =: L + 2$ X h 2 A X g ; $Z $ where s guge nvrnt L Z prt nd s guge vrnt prt. Under the guge trnsformton the vector Φ ffl := X feld ffl Z s trnsformed s ff Y ffl omponents of guge nvrnt vrle L n cosmologcl perturtons : ν ; L j = 2 2

8 Q := Q $ X () Q Q = Q + $ X ()Q Q = Q + 2$ X Q + :$ Z $ 2 X Perturtons of n rtrry mtter feld Q : Usng guge vrnt prt of the metrc perturton of ech order, guge nvrnt vrles for n rtrry felds Q other thn metrc re defned y Frst order perturton of Q : Second order perturton of Q : 8 < 9 = Q 2$ X Q := Q ; () Q These mples tht ech order perturton of n rtrry feld s lwys decomposed s 8 < 9 = ; () Q : guge nvrnt prt :$ Z $ 2 X : guge vrnt prt

9 ( μρ + μp)μu g c μu c + μpff = E := P := ρ 2$ X $Z $ 2 $Z $ 2 X (u $Z $ 2 X μu = u + ffl U := (u) + 2 ffl2 (u ) $ X u Energy momentum tensor (perfect flud) μρ = ρ + ffl ρ ffl2 + 2 ρ μt T + ffl T + = ffl2 T, 2 μp = p + ffl p ffl2 + 2 p Frst order guge nvrnt vrles (u) := E ρ $ X ρ,, p P := $ X p Second order guge nvrnt vrles ρ A p 2$ X p A p U := ) 2$ X (u u A

10 = (ρ + p)u T E + μg X +2$ U u 2 U u + E + E ff P G U c U c U c Perturtons of Ensten tensor nd Energy momentum tensor Frst order : G = + $ XG [H], = T E + P ff + A + (ρ + p) g H d u c c u + g U c u A dc u U P X T +$ Second order : μg G [L] + G [H; H] = + ρ $Z $ 2 X H c c + g u H u d L d u c 2H f U c f +g cd A + 2(ρ + p) A + U c c A u H c u c + 2 +(ρ + p) A u P A u u + A P P T + $ 2 Z $ X X +2$ T : guge nvrnt prt : guge vrnt prt

11 G = 8ßG (p) T ; p = ; ; 2: IV. Guge Invrnt Ensten equtons We mpose the Ensten equton of ech order, (p) G Then, the Ensten equton of ech order s necessrly gven n terms of guge nvrnt vrles : lner order : second order : [H] = 8ßG T G, G [H; H] = 8ßG T. + [L] We do not hve to cre out guge degree of freedom t lest n the level where we concentrte only on Ensten equtons.

12 E = Φ= 3 2 ( + Φ j 3K) D D j + ( H@ 2 + 3H@ K + 2@ H + H 2 j j D D Φ 2 D D j j 2 nd order Ensten equtons (cosmologcl, sclr modes) components of perturton of the flud four-velocty V A (dx ), D =, V (d ) + v + U := energy densty perturton U = >: A Φ v V ν A Φ v + V ν A >= >; H@ + + 3K 3H 2 4ßG 2 k k 3 pressure perturton k k 3. 4ßG 2 P = velocty perturton H@ j 3K) D D j + ( 2 v = 2@ D Ψ 2HD Φ +D D k k 8ßG (ffl + p)d. trceless prt of the sptl component of Ensten equton k k 3. Ψ D j j D k k 3.

13 V. Towrd the pplcton to the rdton recton prolem v lck hole perturtons The cpture of solr-mss compct ojects y mssve lck holes (glctc centers 6 M sun ) one of the promsng sources of grvttonl wves for LISA. ( GW H H The lck hole perturton (mss M) Perturton prmeter μ=m ο 6 Energy momentum tensor T μν Z ff(4) z(f )) p dz (x μ df g dz ν = μ df df

14 Perturtons n rdton recton prolem To dscuss the rdton recton effect y lck hole perturtons we consder the followng order countng: Energy momentum tensor : (ckground) (Pont prtcle, geodesc) (geodesc + ts devton) = ffl ffl T + ffl2 T 2 T T GW H GW metrc perturtons : (ckground metrc) g (ckground + GW emsson) g + fflh

15 + H X r T +$ H c In terms of guge nvrnt vrles Energy momentum tensor nd equton of moton of ech order s gven s follows : Frst order : Guge nvrnt energy momentum tensor : (energy momentum tensor for pont prtcle) := T $ X T = T T Eq. of moton n guge nvrnt form :»» c r μ T μ r T = H H c c T T r T = = Ths equton gves the geodesc equton round the lck hole.

16 μ μ T r h d 2H c 2H +2$ X h H d H c H μ μ T r dc H c H c + 2H c T h h h $ X $2 X c T (r T h Second order : Guge nvrnt energy momentum tensor : := T 2$ X T T ρ 2 ff $ X Y $ T (Ths descres devtons from geodesc motons.) Eq. of moton n guge nvrnt form : 2$ X T T = r T = h cd 2H h T c 2)H + H H T c H H n o + ) r T = 2H c c + 2H c T c = T H These terms correspond to the self-force. Eq. for the devton from geodesc should e gven n guge nvrnt form, lthough we do not consder the regulrzton, yet. H

17 However, to evlute ths self-force force completely, there re mny prolems whch should e clrfed. Guge nvrnt tretments of perturtons Schwrzschld cse... Prolems n the tretments of l=, modes. No guge nvrnt vrles n l=, modes. (n mny ref.) <----> We hve defned guge-nvrnt vrles for the perturtons wth FRW ckground (specl cse of spherclly symmetrc spcetmes) nd these lso nclude sphercl modes. How should we understnd these consstently. Kerr cse...??? (Newmn-Penrose formulton) Tretments of pont prtcle <---> regulrzton We should clrfy the guge nvrnt tretments of pont prtcle or the regulrzton (or extrcton of tl prt) of the metrc n the guge nvrnt mnner. A systemtc hgher order perturtve expnson lke the post- Newtonn expnson s possle??? (It mght e drem)

18 VI. Summry We hve shown the frmework of the generl reltvstc second order perturtons from generl pont of vew, whch s developed n [K.N., PTP (23), 723; d, 3 (25), 43.]. We hve verfed the frmework n the ove references s pplcle to cosmologcl perturtons. We hve derved the second order Ensten equtons n terms of guge nvrnt vrles defned long ths generl frmework. [K.N., PRD74 (27), 3. gr-qc/658] In ths frmework, we do not specfy nythng out the ckground spcetme nor the physcl menng of the nfntesml prmeter for perturtons. (I hope) Ths frmework wll e pplcle to ny theory n whch generl covrnce s mposed. Ths frmework wll hve very mny pplctons.

19 Lst of pplcton cnddtes Second-order cosmologcl perturton theory (n progress) Ignorng the frst order vector- nd tensor-modes Sngle perfect flud system. (OK) Sngle sclr feld system. (OK) Extenson of our formulton to nclude the frst order vectornd tensor-modes. Sngle perfect flud system (OK) Sngle sclr feld system (OK) Extensons to mperfect flud system (n progress) Extensons to the mult-felds system Extensons to the Ensten-oltzmnn system Nonlner effects n M physcs Rdton recton Prolem sed on the lck hole perturton theory (Just plnnng).

20 Lst of pplcton cnddtes The correspondence etween oservles n experments (oservton) nd guge nvrnt vrles defned here. Ex. The relton etween guge nvrnt vrles nd phse dfference n the lser nterferometer for GW detecton. Post-Mnkowsk expnson lterntve to post Newtonn expnson (post-mnkowsk descrpton of nry system). The second-order perturton of the Ensten tensor s lredy gven!!! ut we hve to specfy the energy momentum tensor of nry system. In prtculr, we hve to tret two-pont prtcle system nd some regulrzton procedures re necessry to tret ths system. etc. There re mny pplctons to whch our formulton should e ppled. I wnt to clrfy these prolems step y step.

21 おわり (END)

22 II. Guge n generl reltvty (R.K. Schs (964).) There re two knds of guge n generl reltvty. The concepts of these two guge re closely relted to the generl covrnce. Generl covrnce : There s no preferred coordnte system n nture. The frst knd guge s coordnte system on sngle spcetme mnfold. The second knd guge ppers n the perturton theory. Ths s pont dentfcton etween the physcl spcetme nd the ckground spcetme. To expln ths second knd guge, we hve to remnd wht we re dong n perturton theory.

23 X ffl Y ffl The guge choce s not unque y vrtue of generl covrnce. Generl covrnce : There s no preferred coordntes n nture (ntutvely). N Physcl spcetme (PS) p Q M ɛ ɛ Guge trnsformton : The chnge of the pont dentfcton mp. X ɛ p Q δq Φ ɛ = X ɛ Y ɛ Y ɛ q M ckground spcetme (GS) Dfferent guge choce :, Representton of physcl vrle : Guge trnsformton : Q X := X Λ ffl Q, Q Y := Y Λ ffl Q ffl := X ffl Φ, Q Y = Φ Λ fflq X X! Y ff Y ffl

24 + + 2 D k χ lk V + 2 D k χ lk 8 D k ν k Source terms n the 2 nd -order Ensten eq. (for exm...) 2 (ffl + p) D v D v 3D k +8ßG := Φ D k 2 2 2H 2 2 Φ 8 @ 3 Φ Φ A 2K Φ A A Φ +HD V 2HD k A 4 Φ Φ V D (k l ν ν l) +3H 2 k ν k ν +8ßG 2 (ffl + p) +D l D k Φ 2HD k χ l ν Dk kl 2 l χ lk χ χ +H kl χ D k lm χ χ ml D [l lm χ lm K) χ lm ( χ Mode couplng : χ k]m 2 : sclr-sclr : sclr-vector : sclr-tensor : vector-vector : vector-tensor : tensor-tensor

25 h = h (d ) (d ) + 2h (d ) ( (dx ) ) + h j (dx ) (dx j D D j 3 fl j h(tl) + 2D A h (TV)j) + h (TT)j ; ( fl j osmologcl perturtons ckground metrc (d ) + (dx ) (dx j ) A (d ) j g = 2 ( ) metrc perturton fl j : metrc on mxmlly symmetrc 3-spce μg g = + + fflh ffl2 l + O(ffl 3 ) 2 decomposton of lner perturton D h (V L) + h (V ) ; D h (V ) = ; h = = 2 h (L) fl j + 2 h (T )j ; h (T ) j h := flj h (T )j ; h (T )j = (TV) ; D h (TT)j = : D h = Unqueness of ths decomposton ---> Exstence of ( + 2K) Green ( + 3K) functons,, : curvture constnt ssocted wth the metrc K

26 X := 2 (T V ) + 2 D h (T H! X := 2 ν := h D X 2H! X H j := 2 2 Ψ + 2 χ j := h j 2D ( X j ) + 2Hfl j X Guge vrnt nd nvrnt vrles of lner order metrc perturton. guge vrnt vrles : X := X (d ) + X (dx ) X := h (V L) 2 h (T L) ; A ; where. A guge nvrnt vrles : YX X X ο ψ H := 2 2 Φ := h 2 j D ν = ; fl χ j = = D χ j (J. rdeen (98)) where : = H X H Y H =

27 おまけ