Observational consequences of composite inflation models

Transcription

1 1 Observational consequences of composite inflation models Khamphee Karwan The institute for fundamental study, Naresuan University, Thailand

2 Outlines 1. Composite inflation model 2. Evolution equations 3. Power spectrum of perturbation and bound on the model parameters 4. Conclusions

3 Composite inflation model The inflaton is a composite field made out of the bound state of particles in strongly coupled theory. S = d 4 x g The general action for composite inflation is { M 2 p 2 F(φ)R 1 2 G(φ) νφ µ φ V(φ) wheredis the mass dimension of the composite fieldφand }, (1) F(φ) = 1+ ξ M 2 p φ 2 d and G(φ) = φ 2 2d d.

4 Evolution equations The Friedmann equation and the evolution equation for the background field are respecttively given by 3FH 2 +3 FH = 1 2 G φ 2 +V(φ), (2) G φ+ 1 2 G φ φ 2 +3HG φ+v φ = 3F φ H 2 (2 ǫ), (3) ǫ = Ḣ H 2 = F φ 2HF + G Φ 2 2H 2 M 2 P F + F ΦΦ Φ 2 2H 2 F + F Φ Φ 2H 2 F, (4)

5 During inflation, G φ 2 V (φ). (5) During the time at which the observable perturbations exit the horizon, ǫ 1 constant, F t = F 2FH constant. (6)

6 The power spectrum for the primordial curvature perturbation We suppose that during the horizon exitf t andǫare approximately constant, so that P ζ (1+F t) 1/2( 3Ft 2 +GΦ 2 /2F ) 1/2 H 2 F (ǫ+f t ) 3/2 8π 2 cs k τ =1. (7)

7 The spectrum index for this power spectrum is n s = dlnp ζ dlnk ǫ 2F t + Φ 2 dln [ GΦ 2 / ( 2F +3Ft 2 dφ (8) )] where we have used d/dlnk φ d/d(lna). Using H 2 (1+F t ) V (φ) during inflation,we get A s (1+F t) 1/2( 3F 2 t +GΦ 2 /2F ) 1/2 V 24π 2 F 2 (ǫ+f t ) 3/2 (1+2F t ) cs k τ =1, (9)

8 Constraints the model parameters Techni-Inflation: First, we consider the case where the inflaton is the composite state of techni-quarksin minimal walking technicolor theory. Techni-Inflation. In this case, a V(ϕ) = κ 4 ϕ4. (10) a P. Channuie, J. J. Joergensen and F. Sannino, Minimal Composite Inflation, JCAP 1105, 007 (2011) [arxiv: [hep-ph]].

9 For this model, n s = 1 ( 6ξ 1+ϕ 2 ξ(1+6ξ) + 4+4ξ ) 4+ϕ 2 3[ϕ+4ξϕ+ϕ 3 ξ(1+6ξ)] 2 3ϕ 2 [ (11) and A s = κϕ6( 6ϕ 2 ξ 2 + ( ϕ 2 +4 ) ξ +1 )( ϕ 2 ξ(6ξ +1)+1 ) 768(πϕ 2 ξ +π) 2 (6ϕ 2 ξ 2 +(ϕ 2 4)ξ +1), (12)

10 In the ξ limit: n s 1 4 2N+1 + f(n,φ) ξ 2 +O ( ) 1 ξ 3, A s κn(2n+1) 144π 2 ξ 2 +g(n,φ)o ( ) 1 ξ 3. a n s = 0.960±0.007, 43 N 62, 3.04 ln ( A s 10 10) 3.13, 3.9 < κ ξ < 8.9. a P. A. R. Ade et al. [Planck Collaboration], Planck 2013 results. XXII. Constraints on inflation, arxiv: [astro-ph.co].

11 In the ξ 0 limit: n s ( ) 1 24 ϕ 2 + ( ) 96 ϕ +8 2 ξ +O ( ξ 2), A s κϕ6 768π ξ (κϕ 6 (ϕ 2 8)) 2 768π 2 +O ( ξ 2) 2κ(N+1)3 3π 2 +O(ξ). A s 10 9 requiresκ < From the numerical calculation: κ > for N = 55. (13)

12 Glueball Inflation: We next consider the case where the inflaton is the bound state of gluon, called glueball. The potential for the glueball inflaton is a ( ϕ V(ϕ) = 2ϕ 4 ln Λ), (14) whereλis the confining scale and we have redefined the field such thatϕhas a canonical dimension. a F. Bezrukov, P. Channuie, J. J. Joergensen and F. Sannino, Composite Inflation Setup and Glueball Inflation, Phys. Rev. D 86, (2012) [arxiv: [hep-ph]].

13 In theξ limit : n s 1 f (N)+ g(n,ϕ,λ) ξ + ( ) ξ, (15) A s (ln[ϕ/λ])3 (6ln[ϕ/Λ]+1) π 2 (6ln[ϕ/Λ] 1)ξ 2 +O ( 1/ξ 3). (16) The contributions from the next leading order terms can be neglected if ξ > 10 4 andλ > 0.1. Forξ > 10 4 andλ > 0.1, n s 0.975, 30 N 60

14 In the ξ 0 limit, we get A s 16ϕ6 (ln(ϕ/λ)) 3 +O(ξ) 3π 2 (17) (1+4ln(ϕ/Λ)) 2 Slow-roll evolution requiresln(ϕ/λ) 1, soa s In general,a s increases whenξ decreases. On the next slide, we plot how A s ln ( A s 10 10) andn s depend onxi,λand the number of e-foldingsn.

15 Λ = 10 1 Λ = 10 2 Λ = 10 3 Λ = Λ = 10 1 Λ = 10 2 Λ = 10 3 Λ = As 3 ns N = 30 ξ Λ = 10 1 Λ = 10 2 Λ = 10 3 Λ = N = ξ Λ = 10 1 Λ = 10 2 Λ = 10 3 Λ = As 3 ns N = 60 ξ N = ξ

16 ξ , Λ cannot be constrained. (18)

17 Super Yang-Mills Inflation: We finally consider the case where the inflaton is the composite state of the super partner of glueball called gluino-ball in the super Yang-Mills theory. In this case, a V(ϕ) = 4N 2 cϕ 4 (ln[ϕ/λ ) 2,] (19) whereλis the confining scale andn c is the number of colors. a P. Channuie, J. J. Jorgensen and F. Sannino, Composite Inflation from Super Yang-Mills, Orientifold and One-Flavor QCD, Phys. Rev. D 86, (2012) [arxiv: [hep-ph]].

18 In theξ limit : n s 1+f (N)+ g(n,ϕ,λ) ξ +O ( ) 1 ξ 2, (20) A s Y 4 [3Y +1] 2π 2 N 2 c [3Y 1]ξ 2 +O(1/ξ3 ), (21) where Y = ln(ϕ/λ). If ξ > 10 4,Λ > 0.1, the next leading contributions can be neglected, so that forξ > 10 4 andλ > 0.1, n s 0.975, 37 N 80

19 Similar to the glueball model, the power spectrum amplitude increases whenξ decreases, and the power spectrum amplitude cannot satisfy the observational bound ifξ < 1. For this model, ξ , Λ (22)

20 Λ = Λ = 10 2 Λ = 10 4 Λ = Λ = Λ = 10 2 Λ = 10 4 Λ = 10 5 As 3 ns N = 37 ξ Λ = Λ = 10 1 Λ = N = ξ Λ = Λ = 10 1 Λ = As 3 ns N = 62 ξ N = ξ

21 Conclusions We study how the parameters of various composite inflation model influence the spectral indexn s and the power spectrum amplitudea s of the primordial curvature perturbations, and then constrain these parameters using the observational bound onn s anda s from the Planck data. We find that for the techni-inflation model, the ratioκ/ξ 2 is constrained to be 10 6 for a largeξ. In the case of glueball-inflation, we find that the observational bound forn s anda s can be satisfied at 2-σ if ξ For the Super Yang-Mills Inflation, we find thatξ is constrained to be andλ 10 5