Cosmic String Loop Distribution with a Gravitational Wave Cutoff

Transcription

1 Ouline Inroducion Towards he Cosmological Aracor Cosmic Sring Loop Disribuion wih a Graviaional Wave Cuoff Larissa Lorenz Insiue of Mahemaics and Physics Cenre for Cosmology, Paricle Physics and Phenomenology (CP3) Universie Caholique de Louvain la Neuve work wih C. Ringeval and M. Sakellariadou JCAP 1010:003 [arxiv: ] Conclusions

2 Ouline 1 Inroducion Cosmic Srings Sring Neworks and Loop Formaion 2 Towards he Cosmological Aracor Evoluion Equaion Loop Producion Funcion Aracor in he Radiaion Era Relaxaion Towards Scaling 3 Conclusions

3 Cosmic Srings Ouline 1 Inroducion Cosmic Srings Sring Neworks and Loop Formaion 2 Towards he Cosmological Aracor Evoluion Equaion Loop Producion Funcion Aracor in he Radiaion Era Relaxaion Towards Scaling 3 Conclusions

4 Cosmic Srings Wha are cosmic srings? Cosmic srings are line-like defecs produced in GUT phase ransiions in he early Universe (Kibble 1976). Afer formaion, he cosmic sring nework undergoes sreching, inercommuaion evens and energy loss by graviaional radiaion. Source: J. Rocha (2008) Under hese effecs, he nework rapidly reaches a scaling regime.

5 Cosmic Srings Wha are cosmic srings? Energy densiy ρ 1/d 2 h : srings never dominae he Universe a lae imes. Long cosmic srings are characerized by heir ension U (or dimensionless GU, G m 2 ). Pl Once a candidae for srucure formaion, now consrained by WMAP o conribue < 10%. Source: J. Rocha (2008) for a ν : d h () = 1 ν from CMB: GU <

6 Cosmic Srings Cosmic sring loops Source: J. Rocha (2008) Loops are formed a inersecion or auo-commuaion evens (P = 1 for Nambu Goo). Loops evacuae energy densiy from he nework so ha scaling is reached. Cosmic sring loops are characerized by heir lengh l. We sudy he number densiy disribuion n(l, ) of cosmic sring loops of size l a ime.

7 Sring Neworks and Loop Formaion Ouline 1 Inroducion Cosmic Srings Sring Neworks and Loop Formaion 2 Towards he Cosmological Aracor Evoluion Equaion Loop Producion Funcion Aracor in he Radiaion Era Relaxaion Towards Scaling 3 Conclusions

8 Sring Neworks and Loop Formaion Disribuion of loops A which size are cosmic sring loops ypically formed? one scale model (Kibble 1976): lengh l given by ypical disance beween long srings, i.e. (fracion of) he horizon size d h bu simulaions (Ringeval e al. 2005) indicae ha here is also a loop populaion a smaller l

9 Sring Neworks and Loop Formaion Disribuion of loops A which size are cosmic sring loops ypically formed? one scale model (Kibble 1976): lengh l given by ypical disance beween long srings, i.e. (fracion of) he horizon size d h bu simulaions (Ringeval e al. 2005) indicae ha here is also a loop populaion a smaller l using α = l/d h : dn dα = S(α) αdh 3, S(α) = C α p S(α) is he scaling funcion, where C and p are fixed from simulaions Bu: Simulaions do no include he graviaional wave emission (GW) of loops which dominanes for scales α < α d ΓGU.

10 Sring Neworks and Loop Formaion The Polchinski Rocha model of loop formaion Polchinski & Rocha (2006, 2007): consider sreching he dominan process for long cosmic srings and add loop formaion as a perurbaion loop producion funcion obained from angen vecor correlaors along long srings in scaling wih d N he average loop number and dσ he uni disance along he long sring, i is found ha d N dσ dl d 1 l 3 ( ) 2χ l loop producion (LP) power law shape in good agreemen wih simulaions

11 Sring Neworks and Loop Formaion The Polchinski Rocha model of loop formaion Polchinski & Rocha (2006, 2007): consider sreching he dominan process for long cosmic srings and add loop formaion as a perurbaion loop producion funcion obained from angen vecor correlaors along long srings in scaling wih d N he average loop number and dσ he uni disance along he long sring, i is found ha Rocha (2007): d N dσ dl d 1 l 3 ( ) 2χ l loop producion (LP) power law shape in good agreemen wih simulaions sudy evoluion of loop number densiy wih LP and GW

12 Sring Neworks and Loop Formaion Exending he Polchinski Rocha model The emission of graviaional waves (for α < α d = ΓGU) renders he cosmic sring loops smooher and smooher on shor scales. No (or much less) loops should be formed a all below he scale of graviaional backreacion (BR) given by α c = Υ (GU) 1+2χ. here: Γ, Υ coefficiens wih Γ 10 2, Υ 10 Our approach: Use Polchinski Rocha loop producion funcion wih coefficiens adjused o numerical simulaions in he region α > α d. Accoun for graviaional wave emission. Change loop producion funcion below α c o phenomenologically include graviaional backreacion. Do no neglec ransien soluions. Wha is he full scaling regime?

13 Evoluion Equaion Ouline 1 Inroducion Cosmic Srings Sring Neworks and Loop Formaion 2 Towards he Cosmological Aracor Evoluion Equaion Loop Producion Funcion Aracor in he Radiaion Era Relaxaion Towards Scaling 3 Conclusions

14 Evoluion Equaion Loop number densiy disribuion n(l, ) LPF loop producion funcion P(l, ) «d a 3 dn = a 3 P(l, ) d dl n(l, ) no. densiy disribuion of cosmic sring loops of size l a ime GW graviaional wave emission γ d BR graviaional back reacion γ c

15 Evoluion Equaion Loop number densiy disribuion n(l, ) GW akes over LP LPF loop producion funcion P(l, ) n(l, ) no. densiy disribuion of cosmic sring loops of size l a ime «d a 3 dn = a 3 P(l, ) d dl shrinking by GW emission ««a 3 dn γ d a 3 dn dl l dl = a 3 P(l, ) GW graviaional wave emission γ d BR graviaional back reacion γ c

16 Evoluion Equaion Loop number densiy disribuion n(l, ) GW akes over LP LPF loop producion funcion P(l, ) n(l, ) no. densiy disribuion of cosmic sring loops of size l a ime GW graviaional wave emission γ d γ d γ c BR cus off LP BR graviaional back reacion γ c «d a 3 dn = a 3 P(l, ) d dl shrinking by GW emission: ««a 3 dn γ d a 3 dn dl l dl = a 3 P(l, ) new: smoohing by BR ««a 3 dn γ d a 3 dn dl l dl = a 3 P(l, )

17 Evoluion Equaion Loop number densiy disribuion n(l, ) GW akes over LP LPF loop producion funcion P(l, ) n(l, ) no. densiy disribuion of cosmic sring loops of size l a ime GW graviaional wave emission γ d γ d γ c BR cus off LP BR graviaional back reacion γ c «d a 3 dn = a 3 P(l, ) d dl shrinking by GW emission: ««a 3 dn γ d a 3 dn dl l dl = a 3 P(l, ) new: smoohing by BR ««a 3 dn γ d a 3 dn dl l dl change of variables: = a 3 P(l, ) (l, ) (γ, ) wih γ l F(γ, ) dn/dl

18 Evoluion Equaion Evoluion of F(γ, ) (a3 F) (γ + γ d ) (a3 F) γ = a 3 P(γ, ) γ d = ΓGU: GW emission scale, γ c = Υ (GU) 1+2χ : graviaional BR scale P(γ, ) piecewise, bu coninuous for γ > γ c and γ < γ c variable ranges for γ and : 0 < γ < γ max (horizon-sized loops) ini < < 0 Wha is a phenomenological form for P?

19 Loop Producion Funcion Ouline 1 Inroducion Cosmic Srings Sring Neworks and Loop Formaion 2 Towards he Cosmological Aracor Evoluion Equaion Loop Producion Funcion Aracor in he Radiaion Era Relaxaion Towards Scaling 3 Conclusions

20 Loop Producion Funcion Shape of he loop producion funcion P(γ, ) Polchinski Rocha: 5 P(γ, ) = c γ 2χ 3 for γ > γ d, fix parameers c and χ from simulaions (his includes loops from loops ec.) assumpion for γ < γ c : 5 P(γ, ) = c c γ 2χc 3 c impose coninuiy a γ = γ c, i.e. c c = cγ 2(χ χc) c

21 Loop Producion Funcion Shape of he loop producion funcion P(γ, ) 5 P Polchinski Rocha: 5 P(γ, ) = c γ 2χ 3 for γ > γ d, fix parameers c and χ from simulaions (his includes loops from loops ec.) assumpion for γ < γ c : 5 P(γ, ) = c c γ 2χc 3 c impose coninuiy a γ = γ c, i.e. c c = cγ 2(χ χc) c χ c =1 χ c =3 χ c =10 1 χ c < : deails of smoohing solve evoluion equaion in he wo domains and choose iniial condiions γ c γ γ

22 Loop Producion Funcion Solving he evoluion equaion (a3 F) (γ + γ d ) (a3 F) γ { = a 3 cγ 2χ 3, γ > γ c c c γ 2χc 3, γ < γ c

23 Loop Producion Funcion Solving he evoluion equaion (a3 F) (γ + γ d ) (a3 F) γ The soluions o his equaion wih a ν are: { = a 3 cγ 2χ 3, γ > γ c c c γ 2χc 3, γ < γ c F(γ, ) = c c (γ + γ d ) 2χc 3 ( µ c 4 2F 1 3 2χ c, µ c ; µ c + 1; γ d γ + γ d ) + I c(γ + γ d ) a 3 µ c 3ν 2χ c 1 I c (x) are wo unknown funcions relaed by coninuiy: I c(x) = I(x) + K (γ d + γ c) 4» «x 3, a where K is a consan x 4 γ c + γ d

24 Loop Producion Funcion Solving he evoluion equaion (a3 F) (γ + γ d ) (a3 F) γ The soluions o his equaion wih a ν are: { = a 3 cγ 2χ 3, γ > γ c c c γ 2χc 3, γ < γ c F(γ, ) = c c (γ + γ d ) 2χc 3 ( µ c 4 2F 1 3 2χ c, µ c ; µ c + 1; γ d γ + γ d ) + I c(γ + γ d ) a 3 µ c 3ν 2χ c 1 I c (x) are wo unknown funcions relaed by coninuiy: I c(x) = I(x) + K (γ d + γ c) 4» «x 3, a where K is a consan x 4 γ c + γ d The unknown funcions can be fixed from he iniial loop number densiy disribuion N ini (l) a ini.

25 Loop Producion Funcion Asympoic expansions Wih I(x) and I c (x) fixed, he final soluion soon has wo pars: 4 4 aini «3 4 F(γ γc, ) = ini N ini j»γ + γ d 1 «ff ini + C(γ + γ d ) 2χ 3 γ d f ini a γ + γ d C(γ + γ d ) 2χ 3 2χ+1 aini «3 γ d ini f ini a γ + γ d 4 4 aini «3 4 F(0 γ < γc, ) = ini N ini j»γ + γ d 1 «ff ini + Cc(γ + γ d ) 2χc 3 γ d fc ini a γ + γ d 2 γ + γ 33 d C(γ + γ d ) 2χ 3 2χ+1 aini «3 γ d ini γc + γ 4 a d f + K γc + γ d 6 7 ini a γ + γ d γ + γ d 4 a() 5

26 Loop Producion Funcion Asympoic expansions Wih I(x) and I c (x) fixed, he final soluion soon has wo pars: 4 4 aini «3 4 F(γ γc, ) = ini N ini j»γ + γ d 1 «ff ini + C(γ + γ d ) 2χ 3 γ d f ini a γ + γ d C(γ + γ d ) 2χ 3 2χ+1 aini «3 γ d ini f ini a γ + γ d 4 4 aini «3 4 F(0 γ < γc, ) = ini N ini j»γ + γ d 1 «ff ini + Cc(γ + γ d ) 2χc 3 γ d fc ini a γ + γ d 2 γ + γ 33 d C(γ + γ d ) 2χ 3 2χ+1 aini «3 γ d ini γc + γ 4 a d f + K γc + γ d 6 7 ini a γ + γ d γ + γ d 4 a() 5 Afer races of N ini have vanished: γ γ c γ d γ c < γ γ d 4 F Cµ γ 2χ 2 c 2 2χ γ d independen of χ c 4 F Cµ γ 2χ 2 2 2χ γ d smooh maching γ γ d 4 F C γ 2χ 3 C, χ from simulaions

27 Aracor in he Radiaion Era Ouline 1 Inroducion Cosmic Srings Sring Neworks and Loop Formaion 2 Towards he Cosmological Aracor Evoluion Equaion Loop Producion Funcion Aracor in he Radiaion Era Relaxaion Towards Scaling 3 Conclusions

28 Aracor in he Radiaion Era Scaling soluion in he radiaion era 4 F(γ,) γ 2χ 2 ν = 1/2 2χ 2 γ /γd c χ c = 2 γ 2χ 3 γ d = 10 4 γ c = 10 6 C, χ from Ringeval e al. (2005) 10 0 γ c γ d γ

29 Relaxaion Towards Scaling Ouline 1 Inroducion Cosmic Srings Sring Neworks and Loop Formaion 2 Towards he Cosmological Aracor Evoluion Equaion Loop Producion Funcion Aracor in he Radiaion Era Relaxaion Towards Scaling 3 Conclusions

30 Relaxaion Towards Scaling Relaxaions and processes The loop number densiy disribuion F(γ, ) undergoes wo relaxaions: 1 from iniial disribuion N ini (l) o radiaion scaling soluion F (γ, ) 2 from radiaion scaling soluion F (γ, ) o maer scaling soluion

31 Relaxaion Towards Scaling Relaxaions and processes The loop number densiy disribuion F(γ, ) undergoes wo relaxaions: 1 from iniial disribuion N ini (l) o radiaion scaling soluion F (γ, ) 2 from radiaion scaling soluion F (γ, ) o maer scaling soluion Model cosmological background in erms of z using (H 0, Ω r0,ω m0 ): 1 1 rad (z) 2H 0 Ωr0 (1 + z) ma (z) 3H 0 Ωm0 (1 + z) 3/2 good approximaion apar from ransiion and for z > 2 insananeous ransiion from radiaion o maer a z = 9 Ω m0 1 = 9 (1 + zeq) 1 16 Ω r0 16 h = 0.72, Ω m0 h 2 = 0.13, Ω r0 h 2 =

32 Relaxaion Towards Scaling Relaxaions and processes The loop number densiy disribuion F(γ, ) undergoes wo relaxaions: 1 from iniial disribuion N ini (l) o radiaion scaling soluion F (γ, ) 2 from radiaion scaling soluion F (γ, ) o maer scaling soluion There are wo relaxaion processes: 1 sweeping lefwards : N ini [l > d h ( ini )] = 0 2 damping of erms wih N ini 4 4 aini «3 4 F(γ γc, ) = ini N ini j»γ + γ d 1 «ff ini + C(γ + γ d ) 2χ 3 γ d f ini a γ + γ d C(γ + γ d ) 2χ 3 2χ+1 aini «3 γ d ini f ini a γ + γ d 4 4 aini «3 4 F(0 γ < γc, ) = ini N ini j»γ + γ d 1 «ff ini + Cc(γ + γ d ) 2χc 3 γ d fc ini a γ + γ d 2 γ + γ 33 d C(γ + γ d ) 2χ 3 2χ+1 aini «3 γ d ini γc + γ 4 a d f + K γc + γ d 6 7 ini a γ + γ d γ + γ d 4 a() 5

33 Relaxaion Towards Scaling Iniial disribuion and radiaion era relaxaion 4 F(γ,) z = z = z = z = z = z = z = Vachaspai & Vilenkin (1984): random walk model ( inin 4 ini ) 5/2 ini (l) = C i. l where 1 C i (GU) 3/ γ c γ no damping (cancellaions) sweeping complee a γ d 1 + z h (γ) γ + γd = 1 + z ini 2 + γ d

34 Relaxaion Towards Scaling Relaxaion from radiaion o maer Radiaion scaling soluion F (γ, ) is he maer era s iniial disribuion. For ν = 2/3, he 2 F 1 funcion becomes a polynomial expression: 1 f(x) = 1 x «(1 x) 2 2χ 2 2χ damping of N complee by (γ γ d ) 1 + z d (γ) 1 + z = ( CM sweeping less efficien C R ) 1/(3/2 3χR ) γ 2(χ M χ R )/(3/2 3χ R ) for large γ, z d 300, bu inermediae scales γ c < γ γ d relax las

35 Relaxaion Towards Scaling Relaxaion from radiaion o maer z = 3059 z = 2000 z = 1000 z = 100 z = 50 z = F(γ,) γ c ma γ c rad γ d γ γ d = 10 4, γ c(rad) = 10 6, γ d (ma) = 10 7 producion resumed

36 Conclusions We sudied he evoluion of he cosmic sring loop number densiy disribuion dn/dl(l, ) using he Polchinski Rocha model for loop producion wih numerically adjused coefficiens, including emission of graviaional waves and including smoohing by graviaional backreacion. dn/dl(l, ) rapidly assumes a a universal form on all lengh scales, insensiive o he deails of he backreacion. This form only depends on he backreacion scale. Our sudy may be exended by recalculaing he GW consrains using he new loop disribuion, using he parameers o describe oher energy loss mechanisms or considering he case P 1 (cosmic supersrings).

37 Wih I(x) and I c (x) fixed, he final soluion has hree pars: 4 4 aini «3 4 F(γ γc, ) = ini N ini j»γ + γ d 1 «ff ini + C(γ + γ d ) 2χ 3 f ini a C(γ + γ d ) 2χ 3 2χ+1 aini «3 γ d ini f ini a γ + γ d γ d γ + γ d

38 Wih I(x) and I c (x) fixed, he final soluion has hree pars: 4 4 aini «3 4 F(γ γc, ) = ini N ini j»γ + γ d 1 «ff ini + C(γ + γ d ) 2χ 3 f ini a C(γ + γ d ) 2χ 3 2χ+1 aini «3 γ d ini f ini a γ + γ d γ d γ + γ d 4 F(γτ γ < γc, ) = ini 4 aini a C(γ + γ d ) 2χ 3 «3 4 ini N ini j»γ + γ d 1 «ff ini + Cc(γ + γ d ) 2χc 3 fc ini 2χ+1 aini a «3 γ d ini f + K γ + γ d 2 γc + γ 4 a d 6 γ + γ d 4 γ d γ + γ d γ + γ d γc + γ d a()

39 Wih I(x) and I c (x) fixed, he final soluion has hree pars: 4 4 aini «3 4 F(γ γc, ) = ini N ini j»γ + γ d 1 «ff ini + C(γ + γ d ) 2χ 3 f ini a C(γ + γ d ) 2χ 3 2χ+1 aini «3 γ d ini f ini a γ + γ d γ d γ + γ d 4 F(γτ γ < γc, ) = ini 4 aini a C(γ + γ d ) 2χ 3 «3 4 ini N ini j»γ + γ d 1 «ff ini + Cc(γ + γ d ) 2χc 3 fc ini 2χ+1 aini a «3 γ d ini f + K γ + γ d 2 γc + γ 4 a d 6 γ + γ d 4 γ d γ + γ d γ + γ d γc + γ d a() aini «3 4 F(γ < γτ, ) = ini N ini j»γ + γ d 1 «ff ini + Cc(γ + γ d ) 2χc 3 fc ini a Cc(γ + γ d ) 2χc 3 2χc+1 aini «3 γ d ini fc ini a γ + γ d where C c/µ, f(x) 2 F 1 (3 2χ, µ; µ + 1; x) o simplify noaions γ d γ + γ d γ τ () (γ c + γ d ) ini γ d

40 BR GW LP loops long srings γ 0 γ τ γ c γ d γ max

41 1 γ γ c BR GW LP loops long srings 2 γ τ γ < γ: on-sie producion vs. produced and shrunk 3 γ < γ τ : non-conaminaed γ 0 γ τ γ c γ d γ max

42 1 γ γ c BR GW LP loops long srings 2 γ τ γ < γ: on-sie producion vs. produced and shrunk 3 γ < γ τ : non-conaminaed γ Soluion 3 dies ou when γ τ ( τ ) = 0, ha is a 0 γ τ γ c γ d γ max τ ini ini = γ c γ d.