String Cosmology. Ed Copeland University of Sussex

Transcription

1 String Cosmology Ed Copland Univrsity of Sussx 1. Dilaton-Moduli Cosmology (Pr Big Bang).. Dilaton-Moduli-Axion Cosmology. 3. Gracful Exit in String Cosmology. 4. Dnsity prturbations. 5. Dilaton-Moduli including moving fiv bran. 6. Inflation in Bran worlds 7. Cyclic Univrs 8. Rolling Tachyons SUSY-, Jun,, DESY, Hamburg

2 Dilaton-Moduli Cosmology (PBB) 4d dim mtric on isotropic torus: ds -dt g ij dx i dx j / d b d ab dx a dx b S Obtain 4 dim NS-NS string ff action: j - b - s - Ú d 4 x g È Í R Î ffctiv dilaton in vol of axion int 1 ( ) ( ) j j - b - ( s ) -j spac 4D [ s] mnl mnlr j H r 1 Not coupling 3.5.3

3 Pr Big Bang Solutions s ( )[ ] Writ: ds h - h W Obtain: a j b a d d 4 ( 1 3 cosx* ) a* h j ( 3 cosx* ) * h b ( 3sin x * ) * h For cos < -1/ x * 3 1. Pol law inflation for. Wak coupling as (Mullr;Vnziano; Vnziano and Gasprini) h < ( j ) h Æ -, Æ

4 String coupling j.. x * p ---- x * inc axion String scal factor

5 Solns: smi infinit propr liftim -v branch: Start from singularity at t for t > v branch: Approach a singularity at t for t < Einstin fram: ~ a ~ a * h ( 1/ ) Both frams lad to inflation: Comoving Hubbl lngth dcrass with tim ~ dh - 1 d(ln a) / h Conncting th two branchs PBB to FRW Gracful Exit Problm in string cosmology

6 Dilaton-Moduli-Axion Cosmologis [ ] [ ] 3 s r ; x x x x x x x x ; x x a a r r r r s r r r 1 r 1 Í Î È - ± s s h h - - -j * b b - j j * - * * * * Why important? { ( ) Æ h Æ j -, EC, Lahiri, Wands No wak coupling limit Axion intrpolats btwn two strong coupling dilatonmoduli vac solutions. Bad for PBB. Similar whn fiv bran movs in Dil-Mod background.

7 In PBB, Fin tuning issus. H &, j& >, h < Initial valus must b vry small implying a cold, mpty, wakly coupld flat vacuum stat. But: Axion intrpolats btwn two strong coupling dilaton-moduli vac solutions. Curavtur tnds to prvnt nough inflation. String coupling must b initially v. small Whr dos initial stat com from? Ensmbl of grav and dilatonic grav wavs? Miln Univrs? j/ -6 s g Turnr,Winbrg; Kalopr, Lind, Bousso; EC, Lahiri, Wands; Vnziano Buonanno, Vnziano, Damour

8 Gracful Exit. Joining th v and v branchs through th singularity h Æ -, j grows, ntring strong coupling rgim. Expct highr ordr corrctions to bcom important. g s a Controls quantum loop corrctions. Invrs string tnsion, controls finit string siz corrctions highr drivativ trms Gnrally rquir combination of both sorts. Gasprini, Maggior,Vnziano; Kalopr, Maddn, Oliv; Brustin,Maddn; Cartir,EC,Maddn

9 Gnral conditions for xit: 1. i.c. of () branch and HS, j& >, rquirs HE < ( / (H )) 1 j - j& HE S. Branch chang from () to (-) has to occur whil H E < 3.Succssful scap and xit compltion rquirs NEC violation, with a bounc in Einstin fram aftr th branch chang has occurrd, nding up with H E > 4. Dilaton stabilisation in radiation ra rquirs potntial or dcay mchanism

10 L c Exampl of xit: a -j ÔÏ ar Ì GB b j( { } (Cartir, EC, Maddn) j) mn mn 4 1 ÔÓ c R - g R mj nj d( Most gnral corrction up to 4 th ordr in drivativs. (Missnr, Kalopr and Missnr) m m j) 4 Constraint: (bc)d -16a --- rproduc string scattring amplitud. Unfortunatly, tru loop corrctions not known, so bst approach to us plausibl trms. L L L c A j L c B j L c

11 H S j& Gnrally difficult to go through th transition in th wak coupling rgim. Typically j.1. 3 final

12 Dnsity prturbations in PBB Comoving Hubbl lngth dcrass in PBB phas, vacuum fluctuations gnratd in flat spac vacuum lav, bcom strtchd and r-ntr today. Einstin fram: { i i j } (1 A)d B d dx ( h )dx dx ds a ( h) - h h d Evolution of mtric prtns about 4D dil-moduli vac solutions. Singl Fourir mod, co-moving k: constraint: A -(B hb) with A j 4h dj b 4h db A ha ,i ij ij k A Brustin, Gasprini, Giovannini, Mukhanov, Vnziano

13 Also: z A 3 Curvatur prtn on uniform nrgy dnsity hyprsurfacs as kh Æ Curvatur prtn on larg scals: Spctral indx: P h d ln Pz n 1 d ln k z 8 ) p lplh (-kh) ln( -k 3 [ ] cmp H-Z: n1 Rcnt claim n (n1 with xponntial potntial inc). Nd to proprly tak account of matching conditions across singularity from collaps to xpansion Durrr and Vrnizzi

14 Similar, grav wav prtns: n T 3 Cmp sc-inv: n T (Gasprini, Vnziano) Axion offrs hop: ds hds k ds -j ds n 3-3 cos s x * Can b sc-inv, but isocurvatur nd to convrt to adiabatic (Durrr,Gasprini,Sakllariadou,Vnziano)

15 Bran cosmology Rapidly dvloping fild, chck hp-th! Diffrnt approachs involv looking for inflation from colliding brans; no inflation from colliding brans; inflation from scalar filds living on bran, with gravity affctd in bulk; pr big bang typ modls, cyclic modls providing cosmology. Exciting fild

16 Dilaton-Moduli Cosmology including moving fiv bran. 4D ff htrotic M-thory with moving fiv bran. Inducs transition btwn asym rolling radii solns. Always drivs solns to strong coupling. Can nd in bran collision with small instanton transition. Bran voln dpnds on dilaton and moduli. EC,Gray,Lukas

17 S E - 1 k p Ú d 4 x g È1 Í Î R q ( ) ( ) 5 -j j b ( z) b j - ffctiv dilaton in 4D b - siz of orbifold z - posn of fiv bran; z Œ q - 5 fiv bran ch arg [,1] Not coupling Mtric: ds - -n dt -a dx j j( t), a a( t), b b( t), z z( t), n

18 solutions: a 1 3 ln t-t T a j b z p p j,i b,i ln ln dá Ê 1 Ë t-t T t-t T T t-t Ï-, t () branch t Œ Ì Ó t, (-) branch 4 3p b, n p j,n 3 for n i,f Êp Á Ë ˆ Êp Á Ë ˆ 1 -d Ê1 (p (p ˆ -1 j,f b,f - z b,f b,i q5d Á P Á, P j -b ln 3 p3.5.3 j,f pj,i Ë p p j,i b,i t t )lná Ê - -d T 1 ˆ Ë t t )lná Ê - -d T 1 ˆ Ë - 1 d - 1 d j b Constraints on intgration const. d > ( ) branch; d < (-) branch 1 ˆ d p b, i - pj,i ( )

19 An xampl: 1. Ngativ tim branch.. p p b,i j,i - Æ 3 p b,f Æ p - j,f b - uppr; j - lowr z - volution. Strong coupling xpansion b-j givn by Bcom larg asymp. Boundaris at z,1. Fiv bran collids with z boundary at 1 t T ª

20 5D Bran scnario (Bintruy t al (); Shiromizu t al ()) k Ê r ˆ L4 H r Á1 4 3 Ë l 3 a ; && f 3Hf & V L 4 4D Cosm const assum Contrib of bulk gravitons on bran l 3 Ê M ˆ : M 4p Á Ë l 5 4 Á M5 Bran tnsion Not: Modifid Fridmann qn incrasd Hubbl paramtr incrasd friction 3.5.3

21 Exampl: Inflation with stp potntials that nds without nd for minima and rhats du to gravitational particl production. V( f) V -kaf Standard inflation rquirs a < Hr can hav a >> Inf whn r / l >> 1 Ends naturally whn r / l ª1 Prtns gnratd, COBE È1 V 1/ GV r / 1/ nd l ª1 Í 3/ a Îa Offrs possibility of using moduli filds, dilaton and also Quintssntial inflation GV Spctral indx: n 1-4/(N1).9 for N5 R.4: ratio tnsor/scalar amplitud obsrvabl with Planck. Rhat with grav particl production.

22 Cyclic Univrs Stinhardt and Turok No Big Bang squncs of bangs and crunchs. Exampl bran collisions, radion fild potntial vital. Dscribd by ffctiv 4D fild thory involving radion and non-trivial coupling to radiation. Claim: scal invariant dnsity prturbations gnratd during contraction phas. At crunch, tmp and dnsity rmain finit and small bcaus of spcial coupling. Ngligibl production of gravitational wavs

23 1. Quintssnc (trillion yars). Dclratd xpansion (billion yrs) 3. H, contraction bgins. 4. Dnsity flucns on obsrvd scals. 5. KE dom 6. Bounc and rvrsal. 7. End of KE dom 8. RD bgins (1-5 s) 9. MD bgins (1 1 s) 1. Pot dominats, fild turns round, back to (1) 4 4 È1 1 4 S Ú d x a Í R ( f ) - V( f) b ( f) rr Î Rquir: ab Æ const, as a Æ for finit nrgy dnsity at crunch. Rquir xponntial potntial for scalinv fluctuations during contraction

24 V L m -a 3 Tachyon cosmology V 1- T T Sn; Gibbons; Kutsaov t al; Alxandr; Mazumdar t al; Fairbairn and Tytgat Ida: Opn string vacuum unstabl (T), condnsation of closd strings stabl (T1). p -V 1- T - r(t) (1 - T ) Condnsat smoothly intrpolats btwn: Possibl rol in p -r w -1 for T ; inflation at arly tims and as nw dark mattr p w for T 1 at lat tims?

25 Summary 1. Low nrgy string action admits inflating rolling radii solns.. PBB on such xampl -- wak coupling rgim in infinit past, ntrs strong coupling poch as approachs singularity issus ovr fin tuning. 3. Gracful xit to FRW phas rquirs highr ordr loop corrctions gnrally arising in strong coupling rgim. 4. Dnsity fluctuations gnratd too stp for structur formation (dbatabl). Isocurvatur axion fluctuations can b scal inv. 5. Including moving bulk brans possibl full solution rquirs all filds and drivs th systm into strong coupling asymptotically. 6. Cyclic univrs, strong claims, rsults basd on ffctiv fild thoris. Could hav sam problms as PBB modls

26 Not mntiond many aras (string statistical mchanics, multi-bran modls, warpd gomtris ) In th words of Th Spontanous Harmony Brakrs: Whn doing string cosmology you nd to Think with lots of fantasy Think a nw world