Prepared for submission to JCAP
arXiv:1307.2880v3 [hep-ph] 3 Oct 2013
Observational Constraints on Composite Inflationary Models Phongpichit Channuie† and Khamphee Karwan‡ † School
of Science, Walailak University, 222 Thaiburi, Thasala District, Nakhon Si Thammarat, 80161, Thailand ‡ The Institute for Fundamental Study, Naresuan University, Phitsanulok 65000, Thailand and Thailand Center of Excellence in Physics, Ministry of Education, Bangkok 10400, Thailand E-mail:
[email protected],
[email protected]
Abstract. In the light of Planck data, we examine observational constraints on single-field inflation in which the inflaton is a composite field stemming from a four-dimensional strongly interacting field theory. We derive the power spectra of the primordial perturbations associated to the composite inflationary models based on the non-minimally coupled composite field to gravity. In the case of a composite model inspired by the minimal walking technicolor theory, we find for a large non-minimal coupling, ξ, limit that the spectral index for the curvature perturbations ns and the tensor-to-scalar ratio r depends solely on the number of e-foldings N , and a ratio of the inflaton self-coupling κ to ξ 2 has to lie within the range 2.3 × 10−10 < κ/ξ 2 < 1.3 × 10−9 to satisfy the 95% CL observational bound on the amplitude of the curvature perturbation As . For small ξ, we find that the 95% CL observational bound for As and r can be satisfied if κ > 10−12 for N = 60. In the case of a glueball model, the confining scale Λ is weakly constrained and the coupling ξ is flavored to be greater than 4.5 × 104 in order to satisfy the 95% CL observational bound for the As , r and ns . For a super Yang-Mills, we discover that the confining scale Λ is also weakly constrained and the coupling ξ is constrained to be not less than 8.0 × 104 in order to satisfy the 95% CL observational bound of the As , r and ns .
Keywords: inflation, particle physics - cosmology connection, physics of the early universe ArXiv ePrint: 1307.2880
Contents 1 Introduction
1
2 Composite Formulations and Background Evolutions
2
3 Power Spectra and Spectral Index
4
4 Contact with Observations 4.1 Techni-Inflation (TI) 4.2 Dilatonic/Glueball Inflation (GI) 4.3 Super Yang-Mills Inflation (SYMI)
7 8 11 14
5 Conclusions
18
1
Introduction
It was widely excepted that there was a period of accelerating expansion in the very early universe. Such period is traditionally known as inflation. The inflationary paradigm [1–5] tends to solve important issues, e.g. the magnetic monopoles, the flatness, and the horizon problems, plagued the standard big bang theory and successfully describes the generation and evolution of the observed large-scale structures of the universe. The inflationary scenario is formulated so far by the introduction of (elementary) scalar fields (called inflaton) with a nearly flat potential (see, e.g. [6–11]). However, the theories featuring elementary scalar fields are unnatural meaning that quantum corrections generate unprotected quadratic divergences which must be fine-tuned away if the models must be true till the Planck energy scale. Therefore, it would be of great interest to imagine natural models underlying the cosmic inflation. In general, however, the inflaton need not be an elementary degree of freedom. Recent investigations show that it is possible to construct models in which the inflaton emerges as a composite state of a four-dimensional strongly coupled theory [12–14]. These types of models have already been stamped to be composite inflation. There were other models of super or holographic composite inflation[26–33]. Practically, all speculative ideas concerning physics of very early universe can be falsified by using the observation of the large scale structure. In particular, the temperature fluctuations observed in the Cosmic Microwave Background (CMB) is basically regarded as providing a clear window to probe the inflationary cosmology. To testify all inflationary models, we need observables including: (i) the scalar spectral index ns , (ii) the amplitude of the power spectrum for the curvature perturbations, (iii) tensor-to-scalar ratio r, (iv) the non-gaussianity parameter fNL . Yet, other relevant parameters are the running of scalar spectral index α ≡ dns /d ln k and the spectral index for tensor perturbations nT . Most recently, the Planck satellite data showed that the spectral index ns of curvature perturbations is constrained to be ns = 0.9603±0.0073 (68% CL) and ruled out the exact scale-invariance (ns = 1) at more than 5σ confident level (CL), whilst the amplitude of the power spectrum for the curvature per 2 10 . turbations |ζ|2 is bounded to be As = 3.089+0.024 −0.027 (68% CL) [15] with As ≡ ln |ζ| × 10
–1–
Having constrained by Planck, the tensor-to-scalar ratio r is bounded to be r < 0.11 (95% CL). In this work, we compute the power spectrum for the primordial curvature perturbations and the tensor-to scalar ratio for various composite inflationary models. Then we constrain these quantities using the observational bound for ns , r and As from the Planck data. The paper is organized as follows: In section (2), we first spell out the setup for generic model of composite paradigm. We then derive equations of motion and useful expressions. In section (3), we calculate the power spectrum of the primordial curvature perturbations and the tensor-to-scalar ratio by supposing that the potential energy of the inflaton dominates the kinetic energy and the slow-roll parameter is nearly constant. In section (4), the spectral index, tensor-to-scalar ratio, and the amplitude of the curvature perturbation for various composite inflation models are examined, and the range of the model parameters in which these quantities satisfy the observational bound is estimated. Finally, the conclusions are given in section (5).
2
Composite Formulations and Background Evolutions
Recently, it has already been shown that cosmic inflation can be driven by four-dimensional strongly interacting theories non-minimally coupled to gravity [12–14]. The general action for composite inflation in the Jordan frame takes the form for scalar-tensor theory of gravity as 1 2 Z MP 1 4 √ µν SCI,J = d x −g F (Φ)R − G(Φ)g ∂µ Φ∂ν Φ − V (Φ) . (2.1) 2 2 The functions F (Φ) and G(Φ) in this action are defined as F (Φ) = 1 +
2 ξ Φd MP2
and
G(Φ) = Φ
2−2d d
,
(2.2)
where d is the mass dimension of the composite field Φ. The non-minimal coupling to gravity is signified by the dimensionless coupling ξ. Here, we write the general action for the composite inflation in the form of scalar-tensor theory of gravity in which the inflaton non-minimally couples to gravity. At the moment, the non-minimal term ξΦ2/d R/MP2 has purely phenomenological origin. The reason resides from the fact that one want to relax the unacceptable large amplitude of primordial power spectrum generated if one takes ξ = 0 or small. Let us next derive the equations of motion describing the background evolution. According to the above action, the Friedmann equation and the evolution equations for the background field are respectively given by 1 3MP2 F H 2 + 3MP2 F˙ H = 3MP2 H 2 F (1 + 2Ft ) = GΦ˙ 2 + V (Φ) , 2 1 2 2 2 ˙ 2 2 ¨ ˙ 3MP F H + 2MP F H + 2MP F H + 2MP F = − GΦ˙ 2 + V (Φ) , 2 1 ¨ + 3HGΦ˙ + GΦ Φ ˙ 2 + VΦ = 3MP2 FΦ H˙ + 2H 2 , GΦ 2
(2.3) (2.4) (2.5)
where Ft = F˙ /(2HF ), H is the Hubble parameter, subscripts “Φ” denote derivative with respect to Φ, and the dot represents derivative with respect to time, t. In order to derive 1
We used the signature of the matrix as (−, +, +, +) throughout the paper.
–2–
the slow-roll parameter ǫ, we consider the combination (eq.(2.4) − eq.(2.3)) /MP2 H 2 F which yields −
˙2 ¨ ˙ ˙2 F˙ FΦ Φ FΦ Φ GΦ H˙ FΦΦ Φ + = ǫ , Ft = = . = −F + + t 2 2 2 2 2 H 2H F 2H F 2HF 2HF 2H MP F
(2.6)
For convenience, we will set MP2 = 1 in the following calculations. During inflation, we suppose that the inflaton field Φ is slowly evolving such that V (Φ) ≫ G(Φ)Φ˙ 2 /2. Consequently, we can write the Friedmann equation and the evolution equation for the field Φ as 3H 2 F (1 + 2Ft ) = V (Φ) , VΦ FΦ (2 − ǫ) − , Φ′ = G 3GH 2
(2.7) (2.8)
where a prime denote a derivative with respect to N = ln a and a is the cosmic scale factor. ¨ ≪ H|Φ| ˙ during inflation, so that we can neglect Φ ¨ in eq. (2.6) Eq. (2.8) suggests that |Φ| and hence we get GΦ˙ 2 FΦΦ Φ˙ 2 ǫ = −Ft + + . (2.9) 2H 2 F 2H 2 F Substituting ǫ from this equation in to eq. (2.8) and using eq. (2.7), we get q 2 2 + −2YF F + F FΦ2 − 2F (YFΦ + G) − 8F FΦ (FΦΦ + G) (YF − 2FΦ ) − 2F G Φ Φ ′ , Φ = 2FΦ (FΦΦ + G) (2.10) ′ where Y = VΦ /V . Substituting Φ back into eq. (2.9), we get ǫ = − −2YF GFΦ − 3GFΦ2 − 2Y 2 F FΦ2 + YFΦ3 + 2YF FΦ FΦΦ − 4FΦ2 FΦΦ − 2F G2 + q 2 FΦ2 − 2F (YFΦ + G) − 8F FΦ (FΦΦ + G) (YF − 2FΦ )+ YFΦ q −1 2 2 . G FΦ − 2F (YFΦ + G) − 8F FΦ (FΦΦ + G) (YF − 2FΦ ) 2FΦ2 (FΦΦ + G) (2.11)
To compute the number of e-foldings of inflation, the priority is to determine the expression for the field Φ at the end of inflation. Without any approximation, we can use eqs. (2.6) and (2.5) to write ǫ in terms of Φ and Φ′ as −FΦ 6FΦ (YΦ′ − 2) + 6YF + (Φ′ )2 GΦ + GΦ′ (2Φ′ FΦΦ + FΦ (YΦ′ − 8)) + 2G2 (Φ′ )2 ǫ= . 6FΦ2 + 4F G (2.12) To obtain the above equation, we have used the Friedmann equation and the evolution equation for Φ of the form GΦ′2 2 , (2.13) V (Φ) = 3H F 1 + 2Ft − 2F VΦ FΦ GΦ Φ′2 − +3 (2 − ǫ) , (2.14) Φ′′ = (ǫ − 3) Φ′ − 2 2G GH G FΦΦ Φ′2 + FΦ Φ′′ − ǫFΦ Φ′ GΦ′2 ǫ = −Ft + + . (2.15) 2F 2F
–3–
We can determine the relation between Φ and Φ′ when inflation ends, by setting ǫ in eq. (2.12) to be unity, consequently we get q 2 2 + 8 F (3YF + 2G) − 3F 2 (−F G + G (2F 2 8GF + 6YF Φ Φ Φ Φ ΦΦ + YFΦ ) + 2G ) Φ Φ Φ′ = 2FΦ GΦ − 2G (2FΦΦ + YFΦ ) − 4G2 4GFΦ + 3YFΦ2 + . (2.16) −FΦ GΦ + G (2FΦΦ + YFΦ ) + 2G2 This relation should satisfy the evolution equation (2.14), so that we substitute this relation into eq. (2.14) by using Φ′′ = Φ′ dΦ′ /dΦ and setting ǫ = 1. As a result, we get an algebraic equation for Φ at the end of inflation. In general, it is complicated to solve this equation for Φ analytically. However, for our convenience, we will perform our calculation using the field variable ϕ with canonical dimension for which F = 1 + f ξϕ2 and G = g0 , where f and g0 are constant. In terms of field ϕ, the algebraic equation for the field can be solved analytically if one considers simple potentials such as V (ϕ) ∝ ϕ4 , etc. In the case of a complicated form of the potentials in which the equation for the field at the end of inflation cannot be solved analytically, we estimate, instead, the expression for the field by making an expansion for large and small ξ respectively as ϕ X 2 − 8X + 4 0= 2X 2 g0 ϕ2 X 4 − 22X 3 + 96X 2 − 88X + 16 + 12(X − 2)X 3 + O(1/ξ 2 ) , (2.17) + 24f ϕX 4 ξ and
√ 2 2 2X −√ 0= g0 ϕ g0 √ √ √ 3/2 2g0 ϕ3 − 2g0 ϕ2 (X + 2) + 2 2 g0 ϕ(2X + 1) + 2(X − 1)X fξ 2 − + O ξ , (2.18) g02 ϕ
where X = ϕY is either constant or non-polynomic function of ϕ, e. g., X = 4 for V ∝ ϕ4 . Here, however, the equation for the field at the end of inflation is cumbersome to write down explicitly. Having computed the field Φ at the end of inflation, one can determine the number of e-foldings via Z Φe Z Φe Z ae Z te 1 ˜ H ˜ 1 ae dΦ = dΦ , (2.19) Hdt˜ = = d˜ a= N (Φ) = ln ˜′ a a ˜ ˜˙ Φ Φ Φ a t Φ where the subscript “e” denotes the evaluation at the end of inflation and Φ′ is given by eq. (2.10). At the observable perturbation exits the horizon, we can evaluate the field Φ once the number of e-foldings N is known. Determining the value of Φ and Φ′ when the perturbations exit the horizon allows us to compute the spectral index and power spectrum amplitude for the curvature perturbations in the next sections.
3
Power Spectra and Spectral Index
In order to compute the power spectrum for the curvature perturbations, we suppose that Ft and ǫ are approximately constant during the time at which the perturbations exit the
–4–
horizon. Supposing that ǫ is approximately constant, one can show that [17] a=−
1 , Hτ (1 − ǫ)
d2 a 2−ǫ = , 2 2 a dτ τ (1 − ǫ)2
(3.1)
R where τ = dt/a is the conformal time. When Ft and ǫ are nearly constant, the second order action for the curvature perturbation in the scalar-tensor theory [18, 19] takes the form ! ( ) Z dζk (τ ) 2 GΦ˙ 2 3Ft2 3 2 2 Ft + ǫ 2 + S2 = dτ d ka F ζ (τ ) , (3.2) +k dτ 1 + Ft k (1 + Ft )2 2H 2 F (1 + Ft )2 where ζk (τ ) is the Fourier mode of the curvature perturbation. In terms of the variable vk (τ ) ≡ zζk (τ ), where s GΦ˙ 2 Ft2 + , (3.3) z ≡ a 3F (1 + Ft )2 2H 2 (1 + Ft )2 the action (3.2) becomes ) ( 2 Z 2z d dv (τ ) k + c2s k2 vk2 (τ ) + z −1 2 vk2 (τ ) , S2 = dτ d3 k dτ dτ ( ) Z dvk (τ ) 2 1 1 3 2 2 2 2 2 = dτ d k + cs k vk (τ ) + 2 µ − vk (τ ) . dτ τ 4
(3.4)
Here, #−1 ˙2 G Φ , c2s = (Ft + ǫ) (1 + Ft ) 3Ft2 + 2H 2 F r 3 9 + 3ǫ + 3Ft + 10ǫ2 + 5ǫFt + Ft2 + O (ǫ3 ) ≃ + ǫ + Ft + O ǫ2 . µ= 4 2 "
(3.5) (3.6)
Since Ft and ǫ are approximated to be constant in our calculation, cs and µ are presumably constant. The evolution equation from the action (3.4) is d2 vk 1 1 2 2 2 + cs k − 2 µ − vk = 0 . (3.7) dτ 2 τ 4 The solutions of this equation can be written as p vk (τ ) = cs k|τ | c1 Hµ(1) (cs k|τ |) + c2 Hµ(2) (cs k|τ |) , (1)
(3.8)
(2)
where c1 and c2 are the integration constant, and Hµ and Hµ are the Hankal functions of the first and second kind respectively. Using the asymptotic expression for the Hankal function, r r 2 i(x− π µ− π ) 2 −i(x− π µ− π ) (2) (1) 2 4 2 4 e , Hµ (x ≫ 1) ∼ e , (3.9) Hµ (x ≫ 1) ∼ πx πx the solution on the sub-horizon scale (cs k|τ | ≫ 1) becomes r 2 i(cs k|τ |− π µ− π ) 2 4 , e vk = c1 π
–5–
(3.10)
where we have neglected e−ics k|τ | . On sub-horizon scales, the first term in the square bracket of eq. (3.7) is much larger than the second one, so that the solution on the sub-horizon scale takes the plane-wave form [17, 18]: eics k|τ | . (3.11) vk ≃ √ 2cs k Matching this plane-wave solution to eq. (3.10), we get r π −i( π µ+ π ) c1 = e 2 4 . 4cs k
(3.12)
Using the asymptotic behavior for the Hankal function, r 2 −i π µ− 3 Γ(µ) −µ Hµ(1) (x ≪ 1) ∼ e 22 2 x , π Γ(3/2) and the above expression for c1 , we can write the solution on superhorizon scale as Γ(µ) 2 1 2µ−3 2 2 (cs k|τ |)−2µ+1 . |vk | = 2cs k Γ(3/2)
(3.13)
(3.14)
Using this result, the power spectrum for the curvature perturbation is obtained as Pζ =
k3 k3 |vk |2 H 2 (1 + Ft )2 2 (cs k|τ |)−2µ+3 . |ζ| = ≃ 2π 2 2π 2 z 2 8π 2 F c3s 3Ft2 + GΦ′2 /2F
At the horizon exit cs k|τ | = 1, the expression for the power spectrum becomes 1/2 (1 + Ft )1/2 3Ft2 + GΦ′2 /2F H 2 . Pζ ≃ 8π 2 F (ǫ + Ft )3/2 cs k|τ |=1
(3.15)
(3.16)
The spectrum index for this power spectrum can be computed via ns =
d ln Pζ 1 d ln P +1≃ + 1 ≃ 1 − 2ǫ − 2Ft + δΦ , d ln k H dt
(3.17)
where Φ′ d ln GΦ′2 / 2F + 3Ft2 3Φ′ d ln [ǫ + Ft ] δΦ ≡ + . 2 dΦ 2 dΦ
(3.18)
The amplitude of the curvature perturbation can be directly read from the power spectrum as 1/2 2 + GΦ′2 /2F 1/2 V (1 + F ) 3F t t , (3.19) |ζ|2 ≃ 3/2 2 2 24π F (ǫ + Ft ) (1 + 2Ft ) cs k|τ |=1
where we have used eq. (2.7) to write H 2 in terms of V . For the primordial tensor perturbation, the power spectrum can be written as [19] 8 H 2 (3.20) PT = F 4π 2 k|τ |=1 –6–
Although the power spectrum for the curvature perturbations is evaluated at cs k|τ | = 1 and the power spectrum for the tensor perturbations is evaluated at k|τ | = 1, up to the lowest order in slow-roll parameter, one can compute the ratio of tensor to scalar amplitudes for the power spectra as 16 (ǫ + Ft )3/2 PT = r= . (3.21) 1/2 1/2 Pζ (1 + F ) 3F 2 + GΦ′2 /2F t
t
k|τ |=1
This is because c2s ≃ 1 + O (ǫ) when Ft and ǫ are nearly constant. One also can verify this approximation by substituting eq.(3.22) and eq.(3.23) into eq.(3.5). When Ft is approximately constant, and Φ is slowly rolling, the expression for ǫ can be obtained by differentiating eq. (2.7) with respect to time. The result is ǫ = Ft −
VΦ F Ft . V FΦ
(3.22)
The evolution equation (2.10) will be simplified if we add the approximations Ft ∼ constant and ǫ ∼ constant. Imposing the conditions GΦ˙ 2 /2 ≪ V , Ft ∼ constant and ǫ ∼ constant and using ǫ from eq. (3.22), eq. (2.5) becomes −1 F2 FΦ VΦ F VΦ FΦ VΦ FΦ Φ′ = 2 . (3.23) − + Φ − 1+ G V G V G 2F G V 2G In the next section, we will examine ns , |ζ|2 and r both analytically and numerically. For the analytical calculation, we write ns , |ζ|2 and r in terms of Φ at the horizon exit using eqs. (3.22) and (3.23). It is worthy noting that the power spectra are obtained by supposing that Ft and ǫ are constant. Here it is more calculable for our investigations to impose Φ′ given in eq.(3.22) and ǫ in eq(3.23) since the obtained expressions, ns , |ζ|2 and r, takes simple form, however, reflecting reliable features. Note that for our numerical analysis we will use the expressions Φ′ given in eq(2.10) for better approximations. In order to determine ns , |ζ|2 and r numerically, we estimate the value of the field Φ around the horizon exit from eq. (2.19) by suitably choosing the number of e-foldings N before inflation ends. Using this value of Φ with eq. (2.10), we determine Φ′ and Ft at the horizon exit. From this Ft , we compute ǫ from eq. (3.22), and then we compute ns , |ζ|2 and r using eqs. (3.17), (3.19) and (3.21) respectively.
4
Contact with Observations
As it is well known in particle physics, the elementary scalar sector, like the Higgs field in the SM, is plagued by the so-called hierarchy problem. Therefore, a natural hope for employing the composite fields is to solve such problem. In this section, we examine observational constraints on single-field inflation in which the inflaton is a composite field stemming from a four-dimensional strongly interacting field theory non-minimally coupled to gravity using the Planck data. This non-minimal coupling of scalar fields to gravity was pioneered in several earlier [34–39] and recent works [40] where a similar phenomenological large value of ξ was needed. However, it is very interesting to further study a potential origin of such a large coupling. It was potentially shown that the models of composite inflation nicely respect tree-level unitary for the scattering of the inflation field during inflation all the way to the Planck scale [12–14].
–7–
4.1
Techni-Inflation (TI)
The authors of [12] recently demonstrated that it is possible to obtain a successful inflation in which the inflaton is a composite field stemming from a four-dimensional strongly interacting field theory. In this work, they engaged the simplest models of technicolor passing precision tests well known as the minimal walking technicolor (MWT) theory [20–23] with the standard (slow-roll) inflationary paradigm as a template for composite inflation. The inflaton identified with the lightest composite state, i.e. Φ ≡ ϕ, in which we couple non-minimally to gravity. The resulting action in the Jordan frame is given by [12]: Z 1 + ξϕ2 1 µν 4 √ R − g ∂µ ϕ∂ν ϕ − VTI (ϕ) , (4.1) STI = d x −g 2 2 where VTI (ϕ) = −
m2 2 κ 4 ϕ + ϕ , 2 4
(4.2)
in which κ is a self coupling and the inflaton mass is m2TI = 2m2 . Since mTI is order of the GeV energy scale, κ should be order of unity and Φ during inflation is order of Planck mass, we neglect m2TI term in our calculation. For this model, we have F (ϕ) = 1 + ξϕ2
and G = 1 .
(4.3)
For this form of the potential, eqs. (3.17) and (3.19) yield 4 + 4ξ 4 + ϕ2 76 + 30ξϕ2 6ξ + , − ns = 1 − 1 + ϕ2 ξ (1 + 6ξ) 3 [ϕ + 4ξϕ + ϕ3 ξ (1 + 6ξ)]2 3ϕ2 [1 + ξ (4 + ϕ2 (1 + 6ξ))] (4.4) and κϕ6 6ϕ2 ξ 2 + ϕ2 + 4 ξ + 1 ϕ2 ξ(6ξ + 1) + 1
2
|ζ| =
768 (πϕ2 ξ + π)2 (6ϕ2 ξ 2 + (ϕ2 − 4) ξ + 1)
,
(4.5)
where ϕ is evaluated at the horizon exit. From eq. (4.4), we see that ns explicitly depends on the non-minimal coupling ξ but not on the inflaton-self coupling κ. When ξ → 0, we have 96 24 + 8 ξ + O ξ2 , (4.6) ns ≃ 1 − 2 + 2 ϕ ϕ
and when ξ → ∞, we get
4 −7 + ϕ2 8 ns ≃ 1 − 2 + + O 1/ξ 3 . 4 2 3ϕ ξ 9ϕ ξ
(4.7)
The value of the field ϕ at the horizon exit has to lie within a suitable range, such that the slow-roll evolution of the field ϕ or equivalently inflation lasts sufficiently long. Hence, possible ranges of ϕ can be estimated from eq. (2.10). When ξ → 0 and ξ → ∞, eq. (2.10) respectively yields ϕ′ 4 16 ≃ − 2 + 2 ξ + O ξ2 , ϕ ϕ ϕ
and
–8–
ϕ′ 2 ≃ − 2 + O 1/ξ 2 . ϕ 3ϕ ξ
(4.8)
From these equations, we see that when ξ is large, the slow-roll evolution of the field requires 2/(3ϕ2 ξ) ≪ 1. Hence, when the inflaton slowly evolves, eq. (4.7) approximately yields ns ∼ 1 − 4 (ϕ′ /ϕ), implying that ns can significantly deviate from unity if ϕ′ /ϕ & 0.001. In this case, the dominant contributions to N yield 4N ≃ 3ϕ2 ξ − 3ϕ2e ξ with ϕ2e ≃ 2/ (3ξ). Hence we find that ϕ2 ξ ∼ 4N /3 + 2/3. Using eq. (4.7), one gets ns ≃ 1 − 4/ (2N + 1). Apparently, the dominant contributions to ns depend only on the number of e-foldings. For this model of inflation, one can show that when ξ ≫ 1, r≃
32 64 64 − 4 3 + O ξ −4 ≃ 3 (1 − ns )2 + 4 3 + O ξ −4 , 4 2 3ϕ ξ 9ϕ ξ 9ϕ ξ
(4.9)
so that r can satisfy the observational bound, i.e. r < 0.11, when ns lies within the 95% (CL) boundary from observation, i.e. 0.945 . ns . 0.974. Using the observational bound on the amplitude of the curvature perturbation, we can put a constraint on the ratio κ/ξ 2 . This is so since for large ξ the amplitude for the curvature perturbation takes the form κϕ2 ϕ2 − 4 κϕ4 2 + + O 1/ξ 2 |ζ| ≃ 2 2 128π 768π ξ κN (2N + 1) 3 ∼ + O 1/ξ . (4.10) 144π 2 ξ 2
The above approximation show that at leading order, the amplitude of the curvature perturbation depends on the number of e-foldings and the ratio κ/ξ 2 . From our numerical calculation, we find that ns can lie within the 95% CL observational bound if the number of e-foldings is within the range 36 . N . 80. For such a range of N and the observational bound on As , the ratio κ/ξ 2 is constrained to be 2.3 × 10−10 < κ/ξ 2 < 1.3 × 10−9 . We evidently see from the figure (1) that ns does not depend on ξ and κ when ξ is large. This feature is in agreement with the analytical one given above. From figure (2), κ is possibly constrained to be order of unity required by the MWT theory if and only if ξ ∼ 104 − 105 . Apparently, our numerical result fits well to that of [12]. 0.99 0.98
κ =−6 1 κ = 10−12 κ = 10−13 κ = 10
ns
0.97 0.96 0.95 0.94 0.93 −6 10
10−4
10−2
10+0 ξ
10+2
10+4
10+6
Figure 1. The spectral index as a function of κ and ξ for N ≃ 60. The green band corresponds to the 95% CL observational bound.
–9–
Nevertheless, the above investigations yield only for large ξ. Now we turn to a small ξ case, and we will see the situations change as follows. In such case, eq. (4.8) suggests that 2 ≪ ϕ when the field is slowly rolling. It follows from eq. (4.6) and eq. (4.8) that both ϕ′ /ϕ 6 5
0.2 κ = 10−12 0.15 κ = 10
κ=1 0.1
3 2
r
As
4
−6
κ = 10−13 0.05
1 0 −6 10
10−4
10−2
10+0 ξ
10+2
10+4
0 10+6
Figure 2. The amplitude of the curvature perturbation As (the red curves) and tensor-to-scalar ratio r (the blue-dashed curves) for various values of κ and ξ for N ≃ 60. The yellow band corresponds to the observational bound for As at 2σ (CL). The grey band represents to the 95% CL observational bound on r.
and ξ can give the contributions to ns if their magnitudes are of the same order. It can also be shown that at leading order the ratio ϕ′ /ϕ is related to the number of e-foldings between the horizon exit and the end of inflation as N ≃ ϕ2 /8 − ϕ2e /8 ≃ ϕ2 /8 − 1 with ϕ2e ∼ 8. Hence, from eq. (4.6), we see that, for a given the number of e-foldings, ns decreases when ξ decreases. In the small ξ limit, the tensor-to-scalar ratio for this model takes the form r≃
16 128 256 − 128ξ + O ξ 2 ≃ (1 − ns ) − ξ + O ξ2 , 2 ϕ 3 3
(4.11)
which may not satisfy the observational bound when ns gets closer to the lower bound from observation. When ξ is small, we can expand |ζ|2 as 2κ (N + 1)3 ξ κϕ6 ϕ2 − 8 κϕ6 2 2 |ζ| ≃ − + O ξ + O (ξ) . (4.12) ≃ 768π 2 768π 2 3π 2
This relation shows that the observational bound on the power spectrum amplitude can be used to constrain κ for sufficiently small ξ. Nevertheless, since ϕ2 ≫ 2 during inflation, the amplitude of the curvature perturbation may be flavored to satisfy the observational bound if κ ≪ 10−7 , which is much smaller than unity expected from the theory at κ ∼ O(1). Let us consider for instance for N ∼ 60. We clearly see from figure (2) that the observational bound on As can be satisfied for lower ξ when κ decreases, and r tends to increase when ξ decreases. One can check that this features are valid for a wide range of N , so that they imply the lower bound on κ for a given N . For N = 60, figure (2) shows that if κ . 10−12 , As and r cannot be simultaneously satisfied the 95% CL bound constrained by Planck. From figures (1) and (2), we can also conclude that ns , r and As can fit observational bound for a wide range of κ when suitable ξ’s is chosen.
– 10 –
4.2
Dilatonic/Glueball Inflation (GI)
The simplest examples of strongly coupled theories are pure Yang-Mills theories featuring only gluonic-type fields. It would be of great interest to investigate inflation using these theories. The authors of [13] demonstrated that it is possible to achieve successful inflation where the inflaton emerges as the lightest glueball field associated to a pure Yang-Mills theory. In this case, the inflaton is the interpolating field describing the lightest glueball. The effective Lagrangian for the lightest glueball state, constrained by the Yang-Mills trace anomaly, non-minimally coupled to gravity in the Jordan frame reads " # Z 1 + ξΦ1/2 4 √ −3/2 µν SGI = d x −g R−Φ g ∂µ Φ∂ν Φ − VGI (Φ) , (4.13) 2 where VGI (Φ) =
Φ ln 2
Φ Λ4
,
(4.14)
The effective potential given above is known in particle physics. It is the generating functional for the trace anomaly of a generic purely gluonic Yang-Mills theory such that Φ is a composite operator. It is convenient to introduce the field ϕ possessing unity canonical dimension and related to Φ as follows: Φ = ϕ4
(4.15)
From the above assignment, the action then becomes Z 1 + ξϕ2 µν 4 √ R − 16g ∂µ ϕ∂ν ϕ − VGI (ϕ) , SGI = d x −g 2
(4.16)
where VGI (ϕ) = 2ϕ4 ln which yields F (ϕ) = 1 + ξ ϕ2
ϕ Λ
,
and G = 32 .
(4.17)
(4.18)
Then we define new parameters such that Y ≡ ln[ϕ/Λ] , A ≡ 1 + ξϕ2 , B ≡ ξϕ2 (16 + 3ξ) ,
(4.19)
C ≡ 832ξ + 54ξ 2 + B and D± ≡ 32 + 2 (±2ξ + B) .
(4.20)
For this model of composite inflation, we find h ns ≃ 1− 5ϕA3 (16 + B) + 2Y A2 (1536 + ξ(320 + ϕ2 (3072 + C)(80 + 21ξ))) + 4Y A(1280 + ξ(256 + ϕ2 (2560 + C(80 + 21ξ)))) + 4(768 + ξ(96+ i ϕ2 (2048 + 480ξ + 21ξ 2 + B(112 + 15ξ + B)))Y × h 2 i−1 2ϕ2 Y (16 + B) ξA + (32 + 4ξ + B)Y , (4.21)
– 11 –
and |ζ|2 =
ϕ6 Y 3 (16 + B) (ξA + D+ Y )
3π 2 A2 (1 + ξϕ2 + 4Y )2 (−ξA + D− Y )
.
(4.22)
One can see from eq. (4.21) that ns depends on both ξ and Λ. In the above expressions, ϕ > Λ. In the small ξ and large ξ limits, eq. (4.21) respectively becomes ns ≃ 1 −
and
h1 3 5 3 3 − − + + 2 2 2 2 4ϕ 32ϕ (ln[ϕ/Λ]) 16ϕ ln[ϕ/Λ] 4 32ϕ2 7 3 + − 2048ϕ2 (ln[ϕ/Λ])3 32(ln[ϕ/Λ])2 i 1 3 2 ξ + O ξ , + + 128ϕ2 (ln[ϕ/Λ])2 16ϕ2 ln[ϕ/Λ]
(4.23)
1 3 + ln [ϕ/Λ] (1 + 6 ln [ϕ/Λ]) 2 ln [ϕ/Λ] (1 + 6 ln [ϕ/Λ])2 h i−1 h + 2ϕ2 ln[ϕ/Λ](1 + 6 ln[ϕ/Λ])3 ξ − 5 − 92 ln[ϕ/Λ] + 128ϕ2 ln[ϕ/Λ]
ns ≃1 −
− 564(ln[ϕ/Λ])2 + 1152ϕ2 (ln[ϕ/Λ])2 i − 1200(ln[ϕ/Λ])3 − 1152(ln[ϕ/Λ])4 .
(4.24)
To estimate the magnitude of ϕ during inflation, we write eq. (2.10) in the small ξ and large ξ limits respectively as 4 ln (ϕ/Λ) + 1 1 − 32ϕ2 ln (ϕ/Λ) + 16 ln2 (ϕ/Λ) + 8 ln (ϕ/Λ) ϕ′ 2 ≃− + ξ + O ξ ,(4.25) 2 2 ϕ 32ϕ ln (ϕ/Λ) 1024ϕ2 ln (ϕ/Λ) q ′ 9 ln2 (ϕ/Λ) + 4 ln (ϕ/Λ) + 1 − 3 ln (ϕ/Λ) − 1 ϕ ≃ + O (1/ξ) . (4.26) ϕ 2 ln (ϕ/Λ) In the large ξ limit, eq. (4.26) suggests that the field can slowly evolve when the condition ln (ϕ/Λ) ≫ 1 holds. Using the leading contributions to ϕ′ for the slowly evolving field and a large ξ limit, we get N = N (ϕ) − N (ϕe ) ,
(4.27)
where p 54N (ϕ) ≡81(ln[ϕ/Λ])2 + 3 (9 ln[ϕ/Λ] + 2) 9(ln[ϕ/Λ])2 + 4 ln[ϕ/Λ] + 1 −1 9 ln[ϕ/Λ] + 2 √ + 54 ln[ϕ/Λ] + 5 sinh . 5
(4.28)
The expression for the field at the end of inflation ϕe can be computed √ from eq. (2.17) by setting X = 1 + 1/ ln (ϕ/Λ). At the leading order, we get ln (ϕe /Λ) ∼ 2/ 3 − 1 ∼ 0.15, which depends neither on ξ nor Λ. Applying these results to eq. (4.24), one can see that at leading
– 12 –
order, ns mainly depends only on the number of e-foldings if ξ is significantly large. For this model of inflation, the tensor-to-scalar ratio in the large ξ limit is given by 8 8 12 ln[ϕ/Λ] − 8ϕ2 + 3 4 r≃ + + O ξ −2 ≃ (1 − ns ) + O ξ −1 , (4.29) 2 2 2 3(ln[ϕ/Λ]) 9ϕ ξ(ln[ϕ/Λ]) 3
where in the second equality we use ln[ϕ/Λ] ≫ 1 during the slow-roll evolution of ϕ. This expression shows that r may be exceeded the upper bound constrained by observation if ns gets closed to its observational lower bound. When ξ is large, |ζ|2 can be expanded as |ζ|2 ≃
(ln[ϕ/Λ])3 (6 ln[ϕ/Λ] + 1) + O 1/ξ 3 . 2 2 π (6 ln[ϕ/Λ] − 1)ξ
(4.30)
This relation shows that the power spectrum amplitude depends on the number of e-foldings, and becomes small when ξ is getting large. Moreover, one can see that the parameter Λ is weakly constrained by observational bound on the amplitude of the curvature perturbation. Only the coupling ξ can be effectively constrained for this case. From our numerical calculation presented in figure (3), r tends to decrease while ns increases with increasing the number of e-foldings. Since ns and r mainly depend on N if ξ is large and Λ & 0.1, we first consider this case and discover that the 95% CL observational bound of ns and r can be simultaneously satisfied if the number of e-foldings lies between 40 and 60. The numerical results in figure (3) further show that ns and As get higher, while r becomes smaller when Λ decreases. However, if we bring down the values of Λ below 10−5 , ns will be getting smaller. The decreasing of As for increasing ξ raises our analytical treatment to be matched with the numerical one. From the numerical calculation, As does not significantly depend on Λ when ξ > 104 and Λ & 0.1. Another salient conclusion is that, if Λ < 0.1, the range of the number of e-foldings, in which the 95% CL observational bound on ns and r can be simultaneously satisfied, gets smaller and wider. From figure (3), by varying Λ and N such that ns and r still satisfy their 95% CL observational bound, we show that ξ which makes As satisfy the 95% CL observational bound is smallest when Λ & 0.1 and N = 40. Again from the figure we discover the lower bound for ξ is constrained to be larger than 5.0 × 104 in order to satisfy the 95% CL observational constraint of ns , As and r. Now we turn to the case when ξ is small, eq. (4.25) shows that the slow-roll evolution of the field requires ϕ2 ≫ 1/8 and ϕ2 ln (ϕ/Λ) ≫ 1/32. It follows from eq. (4.23) that this conditions can be satisfied when ns ∼ 0.96, e.g., ϕ2 ∼ 3/0.16, and Λ = 10−3 . We next check whether the power spectrum amplitude can satisfy the observational bound when ξ is small. For small ξ, we can expand |ζ|2 as |ζ|2 ≃
h ϕ6 (ln[ϕ/Λ])2 16ϕ6 (ln[ϕ/Λ])3 − − 1 − 8 ln[ϕ/Λ] − 16(ln[ϕ/Λ])2 3π 2 (1 + 4 ln[ϕ/Λ])2 3π 2 (1 + 4 ln[ϕ/Λ])3 i (4.31) + 64ϕ2 (ln[ϕ/Λ])2 ξ + O ξ 2 .
Since during the slow evolution of the inflaton, ϕ2 ≫ 1/8 and ϕ2 ln (ϕ/Λ) ≫ 1/32, the first term in the above relation gives |ζ|2 > 10−4 . This amount of |ζ|2 is much larger than |ζ|2 ∼ 10−9 required by observations. It can be seen that this large contribution to |ζ|2 is difficult to be canceled by the second terms in the above equation. This implies that the power spectrum from the glueball model of composite inflation cannot satisfy the observational data for a small ξ.
– 13 –
4
0.98 0.975
0.2 10−2
0.97
10−6
3.5 10−1
10−3
0.965
10−5
0.15
3
10−1
2.5
0.96
r ns
As
10−7
10−2
0.1
0.95 0.945
10−3
0.05 10−5
0.955
0.94
N =40
10−6
2 +4 10
0
10+5
N = 40
0.935
10−7
0.93 +0 10
10+1
10+2
ξ 4
10+3 ξ
Λ= Λ= Λ= Λ= Λ= Λ= 10+4
10−1 10−2 10−3 10−5 10−6 10−7 10+5
10+6
1 0.2
0.99
10−2 10−6
3.5
0.98 10−1
10−3
10−5 10−7
0.15
r ns
As
0.97 3 0.1
0.96
10−1
2.5
0.95
10−2
0.05
10−3
2 +4 10
0.94
N =60
10−5 10−6 10−7
10+5
0.93 +0 10
0
ξ
N = 60 10+1
10+2
10+3 ξ
Λ= Λ= Λ= Λ= Λ= Λ= 10+4
10−1 10−2 10−3 10−5 10−6 10−7 10+5
10+6
Figure 3. For the right panels, we plot the spectral index ns for various values of Λ and ξ for N ≃ 40 (top-right) and for N ≃ 60 (bottom-right). The green band corresponds to the 2σ observational bound. For the left ones, we present the amplitude of the curvature perturbation As (the dashed-blue curves) and tensor-to-scalar ratio r (the red curves) for various values of κ and ξ for N ≃ 40 (top-left) and for N ≃ 60 (bottom-left). Each curve is labeled by the value of Λ. The yellow band corresponds to the observational bound for As at 2σ (CL), while the grey band represents r < 0.11 (95% CL).
In general, the potential arises in this model of composite inflation is quite subtle, because it becomes negative when ϕ < Λ and its minimum is also negative. However, the authors of [43] investigated cosmological evolution in models in which the effective potential become negative at some values of the inflaton field. They also discovered several qualitatively new features as compared to those of the positive one. In the next section, we will consider the another compelling paradigm for composite inflation model that leads to a new form of the potential for inflation. 4.3
Super Yang-Mills Inflation (SYMI)
This model has been explored in [14] in the context of four-dimensional strongly interacting field theories non-minimally coupled to gravity. The authors showed that it is viable to achieve successful inflation driven by orientifold field theories. When the number of colors Nc is large, such theories feature super Yang-Mills properties. In this investigation, we assign the inflaton as the gluino-ball state in SYM theory. The effective Lagrangian in supersymmetric gluodynamics was constructed in 1982 by Veneziano and Yankielowicz (VY) [24]. The component bosonic form of the VY Lagrangian was summarized in [25]. As always investigated in standard fashion, we take the scalar component part of the super-glueball action and coupled it non-minimally to gravity. Focusing only on the modulus of the inflaton field and introducing the field ϕ possessing unity canonical dimension and
– 14 –
related to Φ as Φ = ϕ3 , the action of the theory for our investigation is given by Z 1 + Nc2 ξϕ2 9Nc2 µν 4 √ SSYMI = d x −g R− g ∂µ ϕ∂ν ϕ − VSYMI (ϕ) , 2 α
(4.32)
where VSYMI (ϕ) = 4αNc2 ϕ4 (ln[ϕ/Λ])2 ,
(4.33)
with Nc a number of colors. With the action given above, we find F (ϕ) = 1 + Nc2 ξ ϕ2
and G =
18Nc2 . α
(4.34)
It is convenient to introduce a set of parameters: X ≡ Nc2 ξϕ2 , Z ≡ 1 + X , W ≡ (3 + αξ) X , Σ± ≡ αξ (±2 + 3X) ,
(4.35)
U1 ≡ α2 ξ 2 X (3 + 2X) , V1 ≡ αξZ (17 + 24X) , Ψ ≡ 1 + X + 2Y ,
(4.36)
U2 ≡ α2 ξ 2 X (6 + 7X) , V2 ≡ αξZ (4 + 9X) , Γ ≡ 3 + (3 + αξ) X ,
(4.37)
T1 ≡ αξZ (1 + 2Z (2 + X)) , T2 ≡ α2 ξ 2 X (1 + 3X (5 + 2X)) ,
(4.38)
Θ ≡ 2N 2 ϕ2 (3 + W ) Y , Ξ± ≡ ±αξZ + (9 + 9X + Σ± ) Y .
(4.39)
For this model, eqs. (3.17) and (3.19) can be expressed in terms of the above parameters as follows: h 1 ns = 1− 7α2 ξZ 3 (3 + W ) + 6Z 2 (36Z 2 + 2U1 + V1 )Y Θ + Ξ2+ i + 12Z(45Z 2 + U2 + 4V2 )Y 2 + 8(27Z 2 (3 + 2X) + 18T1 + T2 )Y 3 , (4.40) and
2
|ζ| =
Nc4 ϕ5 ΓY 4
q
Nc2 (αϕξ (1 + X) + ϕ (Ξ+ − αξZ))2 2π 2 Z 2 Ξ− Ψ
,
(4.41)
where the field ϕ is evaluated at the horizon exit. In this analysis, we deduce ϕ > Λ during inflation. Using ξ-expansion, we find for ξ → 0: 2α 2 + 5Y + 6Y 2 ns ≃1 − 9N 2 ϕ2 Y 2 2 9α + 38α2 − 72αN 2 ϕ2 Y + 64α2 Y + 48α2 − 72αN 2 ϕ2 Y 3 + O ξ 2 , (4.42) + 2 2 3 162N ϕ Y
– 15 –
and for ξ → ∞: 7 + 24Y ns ≃1 + 2Y (1 + 3Y )2 7α + 71α − 54N 2 ϕ2 Y + 240α − 216N 2 ϕ2 Y 2 + 300αY 3 + 144αY 4 1 . +O − 3 2 2 ξ2 2αN ϕ Y (1 + 3Y ) ξ (4.43) In order to determine the magnitude of the field ϕ during inflation, we write eq.(2.10) for this model in the small ξ and large ξ respectively as ϕ′ α 1 ≃− 2+ ϕ 9Nc2 ϕ2 ln[ϕ/Λ] i h αξ 2 2 2 α + 4α − 9N ϕ ln[ϕ/Λ] + 4α(ln[ϕ/Λ]) + c 81Nc2 ϕ2 (ln[ϕ/Λ])2 + O(ξ 2 ) ,
(4.44)
and and i 2 1 p ϕ′ 1 h (4 + 8 ln[ϕ/Λ] + 9(ln[ϕ/Λ])2 ) ≃ −3− + ϕ 2 ln[ϕ/Λ] ln[ϕ/Λ] h 1 2Nc2 9Nc4 ϕ2 2 − + 4N + c 2Nc4 ξϕ2 α ln[ϕ/Λ] 6 2 Nc ϕ 4α + 10α ln[ϕ/Λ] + 8α + 27Nc2 ϕ2 (ln[ϕ/Λ])2 9 p − + α ln[ϕ/Λ] α ln[ϕ/Λ] Nc8 ϕ4 (4 + 8 ln[ϕ/Λ] + 9(ln[ϕ/Λ])2 ) ii h p −3Nc4 ϕ2 ln[ϕ/Λ] − 2Nc4 ϕ2 + Nc8 ϕ2 (4 + 8 ln[ϕ/Λ] + 9(ln[ϕ/Λ])2 ) + O(1/ξ 2 ) .
(4.45)
For a large coupling and slowly evolving ϕ, eq.(4.45) yields the slow-roll conditions ln [ϕ/Λ] ≫ 1. It follows from eq. (4.43) that if ξ is significantly large, ns will mainly depend on ln (ϕ/Λ) only. Using the leading contributions to ratio ϕ′ /ϕ during the slow evolution of the field, one can write the number of e-foldings in terms of ln (ϕ/Λ) via N ≃ N (ϕ) − N (ϕe ) ,
(4.46)
where p 108N (ϕ) ≡81(ln[ϕ/Λ])2 + (12 + 27 ln[ϕ/Λ]) 9(ln[ϕ/Λ])2 + 8 ln[ϕ/Λ] + 4 −1 9 ln[ϕ/Λ] + 4 √ . + 108 ln[ϕ/Λ] + 20 sinh 2 5
(4.47)
The expression for ϕe in this case can be estimated from eq. (2.17) by setting X = 1 + 2/ ln (ϕe /Λ). At leading order, we get ln (ϕe /Λ) ∼ 0.3 which is independent of ξ and Λ. This implies that for significantly large ξ, ns depends only on the number of e-foldings. The tensor-to-scalar ratio for this model in the limit ξ ≫ 1 can be written as 16 4 ln[ϕ/Λ] − 3Nc2 ϕ2 + 2 16 −2 −1 + + O ξ ≃ 2 (1 − n ) + O ξ . (4.48) r≃ s 3(ln[ϕ/Λ])2 3Nc2 ϕ2 ξ(ln[ϕ/Λ])2
– 16 –
4
0.99 0.2 10−3 10−1 10−4
10−7
0.98
10−13
3.5 10−1
0.97
0.15
3
r ns
As
10−3
0.96
0.1
10−4
0.95 N =55
2.5
0.05 0.94 10−7
2 +4 10
10−13
0.93 +0 10
0 10+6
10+5 ξ
4
N = 55 10+1
10+2
10+3 ξ
Λ = 10−1 Λ = 10−3 Λ = 10−4 Λ = 10−7 Λ = 10−13 10+4
10+5
10+6
1 0.2
0.99
10−3
3.5
0.98 10−1
10−4
10−7
10−13
0.15
r ns
As
0.97 3
10−1
0.1
10−3
0.96
10−4
0.95
N =80
2.5
0.05 10−7
0.94 N = 80
10−13
2 +4 10
0.93 +0 10
0 10+6
10+5 ξ
10+1
10+2
10+3 ξ
Λ = 10−1 Λ = 10−2 Λ = 10−3 Λ = 10−4 Λ = 10−7 Λ = 10−13 10+4
10+5
10+6
Figure 4. For the right panels, we plot the spectral index ns for various values of Λ and ξ for N ≃ 55 (top-right) and for N ≃ 80 (bottom-right). The green band corresponds to the 95% CL observational bound. For the left ones, we present the amplitude of the curvature perturbation As (the dashed-blue curves) and tensor-to-scalar ratio r (the red curves) for various values of κ and ξ for N ≃ 55 (top-left) and for N ≃ 80 (bottom-left). Each curve is labeled by the value of Λ. The yellow band corresponds to the 95% CL observational bound, whilst the grey band represents r < 0.11 (95% CL).
where we have used ln[ϕ/Λ] ≫ 1 during the slow-roll evolution of ϕ to express the terms on the right-handed side. This expression shows that r may be slightly exceeded the upper bound constrained by observation if ns gets closed to its observational lower bound. When ξ is large, eq. (4.41) can be expanded as |ζ|2 ≃
Y 4 [3Y + 1] + O(1/ξ 3 ) . 2π 2 Nc2 [3Y − 1] ξ 2
(4.49)
It follows from the above relation that the observational bound on power spectrum amplitude can be used to constrain the coupling ξ, and tends to decrease when ξ is getting large. From our numerical calculation presented in figure (4), r tends to decrease while ns increases with increasing the number of e-foldings. Since ns and r mainly depend on N if ξ is large and Λ & 0.1, we first consider this case and discover that the 95% CL observational bound of ns and r can be simultaneously satisfied if the number of e-foldings lies between 79 and 80. Like the glueball paradigm, the numerical results in figure (4) imply that ns and As tends to increase, while r becomes smaller when Λ decreases. However, if we bring down the values of Λ below 10−7 , ns tends to decrease. The decreasing of As for increasing ξ raises our analytical treatment to be matched with the numerical one. From the numerical calculation, As does not significantly depend on Λ when ξ > 104 and Λ & 0.1. Another feature is that, if Λ < 0.1, the range of the number of e-foldings, in which the observational bound on ns and r can be simultaneously satisfied, gets smaller and broader.
– 17 –
From figure (4), by varying Λ and N such that ns and r still satisfy their 95% CL observational bound, we show that ξ which makes As satisfy the observational bound is smallest when Λ & 0.1 and N ∼ 79. Again from the figure we discover the lower bound for ξ is constrained to be larger than 104 in order to satisfy the 95% CL observational bound on ns , As and r. Next we consider a small coupling ξ case, during the slow evolution of the field ϕ, eq.(4.44) yields the constraint, ϕ2 ≫ 2α/9Nc2 and ϕ2 ln [ϕ/Λ] ≫ α/9Nc2 . At leading order, ′ eq. (4.42) that ns can satisfy the observational bound if |ϕ′ /ϕ| ∼ yields ns ∼ 1 − |ϕ /ϕ|, so −3 2 O 10 . In the small ξ limit, |ζ| from eq. (4.41) can be expanded as |ζ|2 ≃
3Nc4 ϕ6 Y 4 Nc4 ϕ6 Y 3 h − 2α − 8αY + 27Nc2 ϕ2 Y − 8αY 2 − 2 3 2 2 2π (1 + 2Y ) 6π (1 + 2Y ) i + 18Nc2 ϕ2 Y 2 ξ + O(ξ 2 ) .
(4.50)
From this equation, we see that |ζ|2 will be of the order of 10−9 as required by observation if Nc4 ϕ6 Y 4 ≪ (1 + Y )2 . Nevertheless, during inflation, the conditions ϕ2 ≫ 2α/9Nc2 and ϕ2 ln [ϕ/Λ] ≫ α/9Nc2 for the slow evolution of inflaton is required. Hence, |ζ|2 can be order of 10−9 if Y 2 /(Nc2 (1 + 2Y )2 ) is significantly small. This ratio can be small if ϕ & Λ or Nc2 ≫ 1. Since ϕ = Λ at the minimum of the potential, during inflation it is difficult for ϕ to get close to Λ such that Y ≪ 1. In order to satisfy the observational bound on the amplitude of the curvature perturbation, Nc2 has to be unacceptable large if Λ < 1. In the same manner as the previous section for glueball-inspired paradigm, for a small coupling, it is difficult for |ζ|2 to be fitted in the observational bound |ζ|2 ∼ O(10−9 ).
5
Conclusions
In this work, we constrain the model parameters of various composite inflationary models using the observational bound for the scalar spectrum index ns , the tensor-to-scalar ratio r and amplitude of power spectrum of the primordial curvature perturbations As from Planck data. In order to satisfy the observational constraints, the general action for the composite inflation has to be in the form of scalar-tensor theory, such that the inflaton is non-minimally coupled to gravity. We compute the power spectra for the curvature perturbations by supposing that the slow-roll parameter ǫ and the ratio Ft = FΦ ϕ′ /(2F ) are approximately constant during the early stage of inflation in which the interesting perturbations modes exit the horizon. Both the analytic expressions and numerical value for ns , r and the amplitude for the curvature perturbation As are calculated. For each inflationary scenario studied in this work we summarize the main results as follows. ◮ According to our analytical analysis, we can show that ns , r and As for the TechniInflation can satisfy the observational data for a wide range of the inflaton self-coupling κ when an appropriate value of ξ is chosen. When ξ is large, ns and r solely depend on the number of e-foldings. Moreover, the amplitude of the curvature perturbations depends on the ratio κ/ξ 2 , so that we can constrain κ for any appropriate value of the coupling ξ matching with recent investigation for single field model non-minimally coupled to gravity [16]. However, we find, for a small coupling ξ, one can constrain the self-coupling κ independent of ξ. From the numerical calculation, the value of ns , r and |ζ|2 will lie within the 95% CL observational bound if the number of e-foldings lie between 36 and 80, and 2.3 × 10−10 < κ/ξ 2 < 1.3 × 10−9 for a large coupling. Moreover, we discover that in order to
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satisfy the 95% CL observational bound for r and As , κ is required to be larger than 10−12 for N ≃ 60. ◮ For composite inflationary models from pure and super Yang-Mills theories, we show analytically that ns , r and |ζ|2 can satisfy the observational bound if ξ is significantly large. Similar to the Techni-Inflation, ns and r depends only on N when ξ is large and Λ is not too small. At leading order for a large ξ limit, As depends only on N and ξ. The quantities ns , r and As are weakly sensitive to the scale Λ if and only if Λ & 0.1. • From the numerical results for the glueball model, we discover that for Λ & 0.1 and ξ > 104 , the number of e-foldings has to lie within the range 40 . N . 60 to obtain 0.943 . ns . 0.975 and r < 0.11. this viable range of N will be shifted to the smaller value if Λ gets smaller. The 95% CL observational bound for ns , r and As can constrain ξ to be larger than 104 . • For the super Yang-Mills model, the numerical investigation yields 0.943 . ns . 0.975 and r > 0.11 if 79 . N . 80 when ξ > 104 and Λ & 0.1. The lower limit for N can be smaller if Λ is smaller. The quantities ns , r and As can potentially satisfy the 95%CL observational bound if the coupling ξ > 104 . For these two models, the effects of Λ on ns , r and As can be significantly compensated by varying the number of e-folding, so that the scale Λ is weakly constrained. Another crucial consequence for the model of inflation is the (p)reheating mechanism. We anticipate to investigate this mechanism by following closely references [41, 42]. Acknowledgement P.C. has just left the Department of Physics at the Faculty of Liberal Arts and Science (FLAS), Kasetsart University, Kamphaeng Saen Campus. There is no way to acknowledge all FLAS’s stuff members, or even any of them properly. P.C. was truly and deeply indebted to Ms. Wilai Jangboon, Dr. Suntree Sangjan and Mr. Wadchara Thongsamer for their incredible hospitality during his time in Nakhon Pathom. K.K. is supported by Thailand Research Fund (TRF) through grant RSA5480009.
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