Accelerating Cosmologies in Lovelock Gravity with Dilaton

Transcription

1 37 The Open Astronomy Journal Acceleratng Cosmologes n Lovelock Gravty wth Dlaton Open Access Ilya V Krnos* and Andrey N Makarenko Tomsk State Unversty Tomsk Lenn prosp 36 Russa Tomsk State Pedagogcal Unversty Tomsk Komsomol'sky prosp 75 Russa Abstract: For the descrpton of the Unverse expanson compatble wth observatonal data a model of Lovelock gravty wth dlaton s nvestgated D-dmensonal space wth 3- and (D-4)-dmensonal maxmally symmetrc subspaces s consdered Space wthout matter and space wth perfect flud are under test In varous forms of the theory (thrd order wthout dlaton and second order wth dlaton and wthout t) there are obtaned statonary power-law exponental and exponent-of-exponent form cosmologcal solutons Last two forms nclude solutons whch are clear to descrbe acceleratng expanson of 3-dmensonal subspace Also there s a set of solutons descrbng cosmologcal expanson whch does not tend to sotropzaton n the presence of matter Keywoerds: Cosmology Dlaton Lovelock gravty INTRODUCTION At present tme there are numerous observatonal data known to be ncompatble wth the Standard Cosmologcal Model On the one hand acceleratng expanson observatons from supernovae type Ia [] and gravtatonal lensng [] allow us to calculate metrc tensor On the other hand evaluatng the amount of vsble matter energymomentum tensor can be obtaned However t s mpossble to satsfy Ensten equatons by pluggng n these values Then there are two possbltes: there s a great amount of nvsble matter or Ensten equatons are not true These possbltes pont out two approaches to the problem: to develop theores of dark matter and dark energy or to modfy theory of gravty In the present artcle we deal wth the second approach Modfed gravty has ts begnnng n 90-th The most popular theores are Brans-Dcke theory [3 4] Lovelock gravty [5] and f(r) -gravty (see e g [6]) However for a long tme these theores were not useful for explanaton of expermental data ncompatble wth general relatvty Here we wll nvestgate a scalar-tensor extenson of Lovelock gravty --- Lovelock gravty wth dlaton whch mght have ts orgn n low-energy lmt of strng theory Lovelock gravty wth dlaton contans scalar feld (dlaton) metrc tensor g μ matter felds I and s descrbed (n D -dmensonal space-tme) by Lagrangan where m = D m = (D ) f D s even f D s odd p () V() are arbtrary functons of dlaton μ μ k L L k the generalzed Kronecker delta whch s equal to f L k s even transposton of μ Lμk to f odd one and to zero otherwse; L M ( I g μ ) s the matter Lagrangan We shall call terms L p R L p = L p R LR 4 p p as Lovelock Lagrangans of p -th order s p p () Usually (see e g [7-9]) only -nd order ( e p =0 p >) of Lovelock gravty wthout dlaton (so-called Ensten-Gauss-Bonnet gravty) s nvestgated Solutons for more complcated varants of the theory are not large n number Investgatons of thrd order Lovelock gravty (wthout dlaton) can be found n [0-] studes of the second order wth dlaton can be found n [3-5] Moreover C C Brggs [6 7] obtans explct formulae for m L = p () L p p= L p R R LR 4 p p p p g μ μ V() L M ( I g μ ) () *Address correspondence to ths author at the Tomsk State Unversty Russa Tomsk Lenn prosp 36 Tel: ; Fax: 7 (38) ; E-mal: krnos@malru the 4-th and 5-th Lovelock tensors Also we should draw attenton to researches n f (R L ) -gravty [4 8] and to works n the 3-rd and 4-th orders of theory whch s smlar to Lovelock gravty and obtaned from strng theory lowenergy lmt as well as the latter [9 0] /0 00 Bentham Open

2 38 The Open Astronomy Journal 00 Volume 3 Krnos and Makarenko G (3) μ = (g μ R3 g μ RR R 3g μ RR R 6g μ R R R 4g μ R R R 4g μ R R R g μ R R R 8g μ R R R 6R μ R 4RR μ R 4R μ R R 48R μ R R 48R μ R R 6R μ R R 4R μ R R 4RR μ R RR μ R 48R μ R R 48R μ R R 48R μ R R 4R μ R R 4R μ R R 48R μ R R 4R μ R R R μ R R 48R μ R R ) (8) In the present paper a set of solutons n second order wth dlaton and wthout t also n thrd order wthout dlaton are obtaned In the most of solutons extra spatal dmensons are assumed to exst Unobservablty of them s explaned by Kaluza-Klen approach (see e g [ p 86] and references theren) whch s brefly the followng: extra dmensons are compactfed on so small scale that t s mpossble to observe them (by present day devce) In the frst secton seven-dmensonal thrd order Lovelock gravty wthout dlaton s consdered Second secton s devoted to the second order wth dlaton n spaces wth varous number of dmensons THIRD ORDER LOVELOCK GRAVITY WITHOUT DILATON Because t seems mpossble to consder general case of Lovelock gravty n a space wth great number of dmensons let us now dscuss thrd order Lovelock gravty wthout dlaton and wthout cosmologcal constant: felds are g μ and I Lagrangan s L Lovelock 3 = R L 3 L 3 L M ( I g μ ) Here 3 are constants General expressons for - nd and 3-rd Lovelock Lagrangans are: L 3 =R μ R R μ L = R μ R μ 4R μ R μ R (3) μ 8R R R μ s Gauss-Bonnet Lagrangan s thrd Lovelock Lagrangan 4R μ R R μ 3RR μ R μ 4R μ R μ R 6R μ R R μ RR μ R μ R 3 (4) By varaton of acton S = L Lovelock 3 d D x one may obtan G () μ G () μ 3 G (3) μ = 8G T c 4 μ (5) G () μ =(R μ R R μ R R μ R RR μ ) (R R 4R R R )g μ and (3) G μ s wrtten down accordng to [6]: Cosmologcal Equatons Now consder seven-dmensonal flat space and assume metrc tensor to get the form g μ = dag{ a (t) a (t) a (t) b (t) b (t) b (t)} (9) Furthermore let T μ =0 From such a metrc one can obtan nonzero Chrstoffel symbols: 0 = a &a 0 aa = b b & 0 = 0 = &a a c c0 c = 0c = & b b (Latn ndexes from the mddle of alphabet jk run over vsble subspace and Latn ndexes from the begnnng of alphabet abc run over extra subspace; ndex 0 notce the tme coordnate; Greek ndexes run over all the space) Nonzero components of Remann tensor are 0 R 0 R 00 R jj R cc 0 = a(t)&&a(t); R c0c = &&a(t) a(t) ; R c 0c0 = &a c (t) j; R dcd = &a(t) a(t) b(t) & b(t); Now the feld equatons (5) are c R c = b(t) b(t); && b(t) = && b(t) = b & (t) c d; b(t) = & a(t) &a(t) b(t) (7) (0) where G () μ = R μ Rg (6) μ We elected 7-dmensonal space for two reasons Frst just n such a space nonzero thrd order of Lovelock gravty arses Second n such a space we obtan general exact soluton of the present form for the second order theory (see below)

3 Acceleratng Cosmologes n Lovelock Gravty The Open Astronomy Journal 00 Volume 3 39 () Here H (t)= &a(t)/a(t) h(t)= & b(t)/b(t) () are Hubble parameters for vsble and extra dmensons respectvely Note that thrd equaton s consequence of two other equatons A reader s asked to verfy ths correlaton by hmself It seems reasonable to begn consderaton of system () wth allowng = 3 =0 and solvng therefore 7- dmensonal Ensten equatons But all solutons of them are only partcular cases of generalzed Kasner soluton (see [] and [3 7] for 4-dmensonal case) Partcularly n our case (9) there s a soluton a(t)=a (0) (t 0 t) b(t)=b (0) (t 0 t) 3 5 6( 5) 3 5 6( 5) a (0) (t 0 t) 006 b (0) (t 0 t) 0539 (3) where a(t) descrbes accelerated expanson of vsble subspace General Soluton n the Second Order Assume 3 =0 n () Then we have From the frst equaton there are 3 possbltes: 3 5 H = H = h h H = 3 5 Second and thrd possbltes are satsfed n all cases ( H >0 f h <0) frst one --- under >0 Consder them h one by one H = Pluggng nto the second equaton we h have Then 78 &h 3 = h 6 ( h 44 h 4 ) h = 6 3 x where x obeys equaton x 5 3x 4 5x 3 5x 3x =0 wth H = 3 5 Then h obeys equaton =tan 3(t t 0 ) h Pluggng nto the second equaton we have 8 arctanh ( h 5 h ) h -6(tt 0 )h 5=0 3 H = 3 5 Pluggng nto the second equaton we have (4)

4 40 The Open Astronomy Journal 00 Volume 3 Krnos and Makarenko Then h obeys equaton 8 arctanh ( h 5 h ) In all cases above there s only parametrc dependence H(t) and h(t) Some explct solutons n Ensten-Gauss- Bonnet gravty for 7 or other dmensons wll be obtaned n the second secton 3 Exponental Soluton n the Thrd Order Now consder equatons of the thrd Lovelock gravty wth constant Hubble parameters: H=0 h=0 Then the system () takes a form H = Therefore (because of ()) h = where C C are arbtrary postve constants (9) It s clear that &a(t)>0 &&a(t)>0 e the abovementoned soluton descrbes accelerated expanson of vsble dmensons At that extra dmensons shrnk Then t s possble that vsble and extra dmensons were equvalent and the Unverse look as 4-dmensonal one only after expanson of one subspace and contracton of another one H 3Hh h H 3 h 36 H h Hh H 3 h 3 =0 3H 6Hh 6h 7 H h 7 Hh 3 4 H 3 h h H h H 3 h 3 =0 3h 6Hh 6H 7 H h 7 H 3 h 4 Hh 3 H H 4 h 88 3 H 3 h 3 =0 (5) Subtractng the second equaton from the thrd one we have Then h= -H s a soluton of ths equaton Pluggng ths equalty nto the second equaton of (5) one can obtan 4 varous solutons: H = ± ± 3 (6) 4 3 Because of the expanson of vsble dmensons let us take only H >0 Then we have H = ± 3 h = ± 3 (7) Pluggng obtan h = H nto the frst equaton of (5) we H = ± ± (0 / 3) 3 (8) 40 3 Comparng that wth (6) we have Then 3 = Soluton (7) seems to be useful for the descrpton of nflaton Fnally f T μ =0 >0 and 3 = then system (5) has soluton (9) where functons a(t) and b(t) are expressed by (9) wth arbtrary postve constants C and C EINSTEIN-GAUSS-BONNET GRAVITY WITH DILATON Such a theory s the followng: felds g μ I ; Lagrangan L EGBd = R g μ μ V()()L L M ( I g μ ) (0) Here () V() are functons of dlaton L s a Gauss-Bonnet Lagrangan (3) Theory under consderaton s dfferent from generalzed Brans-Dcke theory (see e g []) even n 4-dmensonal space that s why we nvestgate both (3)-dmensonal space and spaces wth extra dmensons (where Ensten-Gauss-Bonnet gravty wthout dlaton s sensble) Varatng the acton wth Lagrangan (0) we get feld equatons:

5 Acceleratng Cosmologes n Lovelock Gravty The Open Astronomy Journal 00 Volume 3 4 () () Here G μ s the second Lovelock tensor (7) Cosmologcal Equatons Consder space of D = p q dmensons wth two maxmally symmetrc subspaces: p -dmensonal and q - dmensonal Square nterval n such a space s u ( t) u ( t) ( t) 0 ds = e dt e ds e ds () where ds p and ds q are square ntervals n p - and q - dmensonal subspaces respectvely u 0 (t) u (t) u (t) are arbtrary functons of tme t If metrc s () then Chrstoffel symbols are (as above Latn ndexes from the mddle of alphabet jk run over vsble p -subspace and Latn ndexes from the begnnng of alphabet abc run over extra q -subspace; ndex 0 notce the tme coordnate; Greek ndexes run over all the space) jk ~ = 0 00 jk = &u 0 0 j = e u 0 0 g j ab 0 = 0 a = bc p a ~ = bc u q = e u 0 g ab a 0a a = a0 = Remann tensor Rcc tensor and scalar curvature are R 0 0 j = e u 0 Xg j R 0 a0b = e u 0 Yg ab R jkl = e u 0 ( k g jl l g jk ) R ajb = e u 0 j g ab R a bcd = e u 0 ( a c g bd a d g bc ) R 00 = px qy (3) Gauss Bounnet Lagrangan (3) Here we ntroduce the followng notatons: p e (u 0 u ) q e (u 0 u ) X u&& &u 0 Y u&& &u 0 (p m) n (p m)( p m )(p m )K(p n) (q m) n (q m)(q m )(q m )K(q n) p R ~ p p(p ) q (4) (5) R ~ q q(q ) (6) where R ~ p and R ~ q are nternal curvatures of p- and q- dmensonal subspaces respectvely ~ jk nternal Chrstoffel symbols a and ~ bc are Now feld equatons are These equatons are equvalent to those n [5] f we substtute g μ μ by gμ μ n Lagrangan and p A p q A pq &u &u q & 0 eu V() 0 eu (){p 3 A p p q q 3 A q 4 (p q pq ) 4 p q } e u 0 () &{ (p q qp ) (pq q ) (p q pq )} = 8G T c 4 00 (7)

6 4 The Open Astronomy Journal 00 Volume 3 Krnos and Makarenko e u 0 {( p)x qy (p ) q (p )q }g j eu 0 & g j V()g j ()e4u 0 g j {(p ) 4 4(p ) q 4(p ) 3 X 4(p ) 3 q 8(p )q Y 4(p )q 4(p )q X 8(p ) q X 4(p ) q Y (p ) q q 3 A q 4q Y } e 4u 0 g j {( & && &u 0 &)[( p ) q (p )q ] &[(p ) X (p )qy (p ) 3 (p )q (p ) q ] &[(p )qx q Y (p ) q q (p )q ]} = 8G T c 4 j (8) e u 0 {( q)y px (q ) p (q )p }g ab eu 0 & g ab V()g ab ()e4u 0 g ab {(q ) 4 4(q ) p 4(q ) 3 Y 4(q ) 3 p 8(q )p X 4(q )p 4(q )p Y 8(q ) p Y 4(q ) p X (q ) p p 3 A p 4 p X} e 4u 0 g ab {( & && &u 0 &)[(q ) p (q )p ] &[(q ) Y (q )px (q ) 3 (q )p (q ) p ] &[(q )py p X (q ) p p (q )p ]} = 8G T c 4 ab [ && ( &u 0 p q ) &] V () ()e u 0 p 3 A p p q q 3 A q 4 (p q pq ) 4 p q 4 px[( p ) q (p )q ] 4qY[ p (q ) p(q ) ]} =0 { (9) (30) put ()= e V()=0 T μ =0 Henceforth we may put u 0 =0 (for smplfcaton) p = 3 (to dentfy p-subspace wth vsble space) Statonary Solutons Let us now turn to fnd solutons of (7)-(30) under p =3 u 0 =0 The smplest solutons are statonary ones Hence put u = const u = const = const Then (4) are = p e u = q e u X = Y =0 Consder space wth homogeneous dust e T 00 0 (other T μ =0) After that system (7)-(30) get the form of algebrac equatons 3 q V() (){q q 3 }= 8G c 4 T 00 ; (3) q A q V() (){4q A A q A q p 3 q }=0; (3) (q ) 3 V() () {(q ) 4 (q ) }=0; (33) V () (){q q 3 A q }=0 (34) From (34) and (3) we have 3 q = V() () V () () 8G T c 4 00 (35) Try to fnd lnear combnaton of (3)-(33) n order to cancel terms wth ( ) Let are coeffcents for (3) (33) and (3) n such a combnaton Then we need q 3 (q ) 4 = q 3 4q (q ) =q Therefore

7 Acceleratng Cosmologes n Lovelock Gravty The Open Astronomy Journal 00 Volume 3 43 Now put = q Then = 3 = q - Now multplyng (3) by 3/ (q) and puttng them together we have (takng (34) nto account) (36) From ths and (35) one can get q V() q () V () 8G T () c 4 00 =0 (37) Put now V()=ae ()=be Then (37) get the form Then takng (39) nto account we obtan Fnally and s (38) = 6 ( q) (q 3) q 6 (4) u = ln p u = ln q (43) It should be noted that the followng constrants was appled: T 00 0 a 0 b q 0 q q 4 p 0 q 0 It s easy to see: = (q ) 8G ln T a( ) c 4 00 (38) Pluggng ths nto (35) we have: 3 q = 4G c 4 Pluggng ( q) (q 3) T 00 (39) A p derved from that nto (34) we can get q {(q )(q 3) q(q )} ( q) (q 3) q = a (q ) b a( ) where where 8G T c 4 00 That s quadratc equaton on whch solutons are ( q) (q 3) q ± D = (40) (q 3 q ) It s easy to derve soluton for (3)-dmensonal space wth perfect flud of arbtrary equaton of state parameter At that we should not specfy V() and () because of () do not partcpate n equatons and V() s specfed from those Dlaton also do not contrbute n equatons therefore we should solve just Ensten equatons wth cosmologcal constant Soluton s V =( 3w) 8G T c 4 00 u = w 8G ln T p c 4 00 Here w s equaton of state parameter (p = w p s pressure T 00 s energy densty) artcular case (when = 0) was derved by A Ensten n 97 [4] 3 Exponental Solutons For the dynamcal solutons we need to do further smplfcaton of (7)-(30) Therefore n addton to p = 3 and u 0 = 0 put p = q = 0 e subspaces are flat In addton to smplcty such a condton s caused by Cosmc Mcrowave Background observatons [5 6] ndcate the flatness of vsble subspace For extra subspace q = 0 s only a smplfcaton After that equatons (7)-(30) are (4) (44)

8 44 The Open Astronomy Journal 00 Volume 3 Krnos and Makarenko {&& u q&& u 3 q(q ) q }g j & g j V()g j ()g j {8&& u (q q ) 4&& u (4q q q ) (q )q 4 4q(5q 3) 8qq &u 3 6q 3 } g j {( & )[ && q 4q ] &[4&& u 4q&& u 4 (q )q 4q ] &[4q&& u q u&& 6q qq 4q ]} = 8G T c 4 j (45) {3&& u (q )&& u 6 q 3(q ) }g ab & g ab V()g ab ()g {&& u ( ab (q ) 4(q ) ) 4&& u (6(q ) (q ) 3 6(q ) ) 4 &u 4 q 3 &u 4 4(q )(q 3) 7(q ) 3 (q )(q ) &u 3 } g ab {( & )[6 && (q ) 6(q ) ] &[6(q )&& u (q ) u&& (q ) (q )(q ) 6(q ) ] &[&& u 6(q )&& u 8 3q (q ) ]} = 8G T c 4 ab && (3 q ) & V () (){&& u [ q 4q ] 4q&& u [6 (q ) 6(q ) ] 4 &u 4 (q )q &u 4 4q(q ) 7q 3 qq &u 3 }=0 (46) (47) Fnd at frst solutons wthout dlaton wthout matter and wth constant Hubble parameters: =0 V()=0 T μ =0 = const = const (48) Then system (44)-(46) wll be a system of algebrac equatons for whch we have found two analytcal solutons for arbtrary q and negatve : u u& = ± & (49) = q( q ) Also partcular cases from q = to q = (4=6 s requred for the bosonc strngs and n the case of q =0 Lovelock gravty s just Ensten gravty) have been studed but solutons dfferent from (49) have been obtaned only for q =3 ( e for 7-dmensonal space just as n secton ): = 3 5 and &u = 3 = 5 3 = (50) (5) where constants and take values of and ndependently from each other and >0 Therefore scale factors are a(t) e u = a 0 e t b(t) e u = b 0 e t It s clear that solutons (49) are not useful for us by the followng cause: when vsble subspace expands extra subspace expands too then extra subspace must be vsble n ths case But solutons (50) for = and (5) for = satsfy our purpose Now let us try to obtan exponental solutons n the presence of perfect flud For that substtute condtons (48) by =0 V()=0 T 00 = T j = wg j (5) T ab = wg ab = const = const After pluggng those nto (44)--(46) and subtractng factors g j and g ab we see that left-hand sdes of equatons are ndependent of tme Hence the rght-hand sdes also must be constant From 00-component of local conservaton law for energy-momentum tensor ( μ T μ0 =0) one can obtan (takng (5) nto account) = 0 exp[( w)(3 q )t] Therefore = const under at least one of a two condtons (here H h ): w = ; h = 3 q H In the frst case matter can be descrbed by cosmologcal constant n the second one comovng bulk s constant In the

9 Acceleratng Cosmologes n Lovelock Gravty The Open Astronomy Journal 00 Volume 3 45 latter case equatons (44)-(46) as equatons on H w have two solutons: H =0 =0 w s arbtrary e flat space wth Lorenz metrc >0 h < sgn n expresson for q(q ) H s arbtrary <0 sgn s " " 3 <0 h < sgn s " " (q )(q ) H s arbtrary = 3c4 H (q 3 3q 3H q 3 54H q 8H q 6H ) 6Gq 3 w = H q q 5H q 8H q 3H q 45H q 54H (53) e one can obtan any value for H by matchng energy densty and EoS parameter w It s clear that h <0 f H >0 that's why ths soluton satsfes all requrements Fnally such a soluton descrbes ansotropc expanson of the Unverse wth matter whch not tends to sotropzaton In Ensten gravty t s possble only for maxmally stff flud: w = Now turn to the cosmologcal constant case: w = Then t s possble to consder (44)-(46) as equatons on H h and These have the followng solutons: H = h and s arbtrary = c4 h (6 q 5q q 4 h 6q 3 h h q 6h q) ; (54) 6G Here we also should emphasze an exstence of solutons wth matter whch do not tend to sotropzaton 3 Exponent-of-Exponent form Solutons Try to obtan solutons wth a dynamcal dlaton For that purpose consder equatons (44)--(47) and notce that functons u ( t) and u ( t) make contrbuton only through dervatves u& u& u& & u& & but (t) partcpate explctly To fnd solutons wth constant dervatves let's elmnate (t) by ntroducng new tme varable At frst we put ) = e V ( ) = e T = 0 (56) ( μ Now turn from tme t to new varable : / = e t (55) It s evdent that the frst soluton s unsatsfactory The second one s adequate under h <0 H >0 Such condtons are fulflled n three cases: Dervatves wth respect to wll be denoted by the prme After such a substtuton and puttng u " = u" = " = 0 one can obtan (57) (58)

10 46 The Open Astronomy Journal 00 Volume 3 Krnos and Makarenko (59) (60) Now assume that we have obtaned some quanttes u' u' ' whch satsfy these equatons What should they be to descrbe acceleratng expanson of vsble subspace and contracton of extra one? It s easy to see that the scale factor of vsble subspace would be a(t) e u ' = a 0 exp u 'c 0 e t/ (6) where a 0 c 0 are arbtrary postve constants Then ts frst and second dervatves wth respect to tme t would be Fnally feld equatons n the case of flat subspaces wthout matter and wth V() and () n the form of (56) have exponent-of-exponent form solutons (6) (6) wth abovementoned parameters 5 Power-Law Solutons Now consder space wth dust-lke matter: T 00 0 other T μ =0 And try to obtan solutons of system (44)--(47) wth scale factors of power-law form: &a(t)=a 0 u ' c 0 e t/ exp u ' c 0 e t/ &&a(t)= a u ' c 0 0e t/ exp u ' c 0 e t/ a u ' 0 c 0 e t exp u ' c 0 e t/ It s clear that both dervatves would be postve (as need for acceleratng expanson) f u ' >0 >0 (the latter s not necessary but s suffcent) By the same manner we obtan the scale factor for extra dmensons: b(t) e u ' = b 0 exp u' c 0 e t/ ( b 0 s postve constant) and ts frst dervatve: &b(t)=b 0 u' c 0 e t/ exp u' c 0 e t/ (6) whch would be negatve (as need for contracton) under u' <0 Therefore t s necessary to fnd solutons of (57)-(60) satsfed condtons u' >0 u' <0 >0 Numercal calculatons gve us solutons for dfferent dmensons from q = to q =0 For example (63) Then all Ensten terms n left-hand sdes of equatons (44)-(46) are proportonal to /t and Gauss-Bonnet terms 4 are proportonal to /t Hence solutons wth scale factors (63) are possble f T 00 t (64) & t (65) () t (66) V() t (67) Consder these condtons one by one It s possble to derve tme dependence of energy densty from conservaton law of energy-momentum tensor: T 00 = const t 3nqm q = = = = = 0383 u' = 0378 u' = 3 q = = 00 = 00 = 00 = u' = 0075 u' = q =3 = 000 = 00 = 000 = u' = u' = 00788

11 Acceleratng Cosmologes n Lovelock Gravty The Open Astronomy Journal 00 Volume 3 47 From comparson ths expresson wth (64) we have 3n m = q Note that under ths condton extra subspace contracts ( m <0) f vsble subspace expanses acceleratve ( n >) (but we haven't obtan such a soluton see below) For (3)-dmensonal space condton (64) leads to n =/3 e to Fredmann soluton From condton (65) t s easy to obtan (t)=ln(t / t 3 ) where t 3 are arbtrary constants ( t 3 >0) In order to avod unnecessary complcaton put = Therefore (t)=ln t t 3 From comparson ths expresson wth (66) (67) we see: where are constants Pluggng all those nto equatons (44)-(47) and puttng one can obtan a system of algebrac equatons on n wth parameters where T 00 (t 0 ) s energy densty at some tme moment t 0 Such a system of equatons has solutons not at all values of parameters Consderng dmensons from q = to q = we have obtaned a set of solutons wth arbtrary X Here and are functons of X and 0 <n = m< e vsble and extra subspaces expands wth deceleraton (and wth the same velocty) In another set of solutons X possesses fxed values Here n two cases n m and n another cases n These are solutons for q = 6 9 : Note that n the last two cases = 0 therefore these solutons are solutons n Brans-Dcke theory ( e theory wth Lagrangan (0) wthout Gauss-Bonnet term) In all obtaned power-law solutons 0 <n< 0 <m< that s why such solutons don t descrbe accelerated expanson of vsble space or contracton of extra dmensons However solutons wth n m are nterestng for another cause In Ensten theory there s no ansotropc power-law soluton n the presence of dust However n Ensten-Gauss- Bonnet theory wth dlaton that s possble CONCLUSION Derent varants of Lovelock gravty wth dlaton were consdered n D-dmensonal space wth two maxmally symmetrc subspaces: 3-dmensonal and (D4)- dmensonal Absence of matter and exstence of perfect flud were nvestgated We have several types of obtaned solutons: Statonary Power-law Exponental Exponent-of-exponent form solutons Among the last two forms solutons whch descrbe acceleratng expanson of 3-dmensonal subspace and contracton of (D 4)-dmensonal one were elected Unobservablty of the latter subspace was justfed on the bass of Kaluza- Klen approach Also a set of ansotropc solutons whch do not tend to sotropzaton n the presence of matter n contrast to Ensten gravty have been obtaned Such a possblty s of mportance because t allows us to assume that extra dmensons become small durng the Unverse evoluton Ths ssue we are gong to nvestgate n more detal n another work Moreover t would be nterestng to extend the results of ths work for account of thrd-order Lovelock terms Ths wll be done elsewhere Studyng of future sngulartes n such models would also be mportant For 4-dmensonal modfed gravtes ths problem was consdered n [7 8] Unfortunately most of solutons descrbe only flat maxmally symmetrc subspaces For curved subspaces there are

12 48 The Open Astronomy Journal 00 Volume 3 Krnos and Makarenko only statonary solutons Those are of nterest as exact solutons of very complcated equatons and as possble bass for numercal dynamcal solutons n the case of curved subspaces ACKNOWLEDGEMENTS Ths work s partally supported by RFBR grant and by RF Presdental grant for LSS IVK s supported by grants of Tomsk State Unversty academc councl and of Dynasty foundaton n the frameworks of Internatonal Center for Fundamental Physcs n Moscow The authors are grateful to S D Odntsov K E Osetrn and A V Toporensky for useful dscussons REFERENCES [] Kowalsk M Rubn D Alderng G et al Improved cosmologcal constrants from new old and combned supernova datasets 4 July 008: Avalable from: [] Bernsten GM Comprehensve two-pont analyses of weak gravtatonal lensng surveys February 009: Avalable from: [3] Brans CH Dcke RH Mach s prncple and a relatvstc theory of gravtaton Phys Rev 96; 4: [4] Dcke RH Mach s prncple and nvarance under transformaton of unts Phys Rev 96; 5: [5] Lovelock D The ensten tensor and ts generalzatons J Math Phys 97; : [6] Nojr S Odntsov SD Introducton to modfed gravty and gravtatonal alternatve for dark energy Int J Geometrc Methods Phys 007; 4: 5-46 [7] Nojr S Odntsov SD Sasak M Gauss-Bonnet dark energy 7 May 008; Avalable from: [8] Nojr S Odntsov SD Modfed Gauss-Bonnet theory as gravtatonal alternatve for dark energy 5 May 008; Avalable from: [9] Elzalde E Makarenko AN Obukhov VV Osetrn KE Flppov AE Statonary vs sngular ponts n an acceleratng FRW cosmology derved from sx-dmensonal Ensten Gauss-Bonnet gravty 3 August 008; Avalable from: [0] Dehghan MH Shamrzae M Thermodynamcs of asmptotcally flat charged black holes n thrd order lovelock gravty 3 March 008; Avalable from: [] Dehghan MH Mann RB Thermodynamcs of rotatng charged black branes n thrd order lovelock gravty and the counterterm method 3 March 008; Avalable from: hep-th/06043 [] Dehghan MN Bostan N Spacetmes wth longtudnal and angular magnetc felds n thrd order lovelock gravty 3 March 008; Avalable from: [3] Krnos IV Makarenko AN Osetrn KE Cosmologcal solutons n the Lovelock theory and the Ensten-Gauss-Bonnet theory wth a dlaton Grav Cosmol 009; 9: 59-6 [4] Cognola G Elzalde E Nojr S Odntsov SD Zerbn S Strngnspred Gauss-Bonnet gravty reconstructed from the unverse expanson hstory and yeldng the transton from matter domnance to dark energy 7 August 008; Avalable from: [5] Bamba K Guo ZK Ohta N Acceleratng cosmologes n the ensten-gauss-bonnet theory wth dlaton Prog Theor Phys 007; 8: [6] Brggs CC A general expresson for the quartc lovelock tensor Aprl 008; Avalable from: [7] Brggs CC A general expresson for the quntc lovelock tensor Aprl 008; Avalable from: [8] Cognola G Elzalde E Nojr S Odntsov SD Zerbn S Dark energy n modfed Gauss-Bonnet gravty: late-tme acceleraton and the herarchy problem 5 October 008; Avalable from: [9] Nojr S Odntsov SD Sam M Dark energy cosmology from hgher-order strng-nspred gravty and ts reconstructon 5 October 008; Avalable from: [0] Elzalde E Jhngan S Nojr S Odntsov SD Sam M Thongkool I Dark energy generated from a (super)strng eectve acton wth hgher order curvature correctons and a dynamcal dlaton 3 August 008; Avalable from: [] Carroll SM Spacetme and geometry: an ntroducton to general relatvty Addson Wesley: San Francsco 004 [] Kasner E Geometrcal theorems on Ensten s cosmologcal equatons Am J Math 9; 43: 7 [3] Landau LD Lfshtz EM The classcal theory of felds Oxford Pergamon Press: 00 [4] Ensten A Kosmologsche Betrachtungen zur allgemenen Relatvtdtstheore Stzungsber preuss Akad Wss 97; : 4-5 [5] Jae AH Ade PAR Balb A et al Cosmology from MAXIMA- BOOMERANG and COBE/DMR cosmc mcrowave background observatons Phys Rev Lett 00; 86: [6] Spergel DN Verde L Pers HV et al Frst year wlknson mcrowave ansotropy probe (WMAP) observatons: determnaton of cosmologcal parameters Astrophys J Suppl 003; 48: 75 [7] Nojr S Odntsov SD The future evoluton and fnte-tme sngulartes n f(r) gravty unfyng the nflaton and cosmc acceleraton 4 November 008; Avalable from: hep-th/ [8] Bamba K Nojr S Odntsov SD The unverse future n modfed gravty theores: approachng the fnte-tme future sngularty 4 November 008; Avalable from: Receved: August Revsed: September Accepted: September Krnos and Makarenko; Lcensee Bentham Open Ths s an open access artcle lcensed under the terms of the Creatve Commons Attrbuton Non-Commercal Lcense ( whch permts unrestrcted non-commercal use dstrbuton and reproducton n any medum provded the work s properly cted