Towards an analytic solution of rapid roll inflation with a quartic potential

Chinese Journal of Physics 61 (2019) 98–103

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Towards an analytic solution of rapid roll inflation with a quartic potential

T

Chia-Min Lin Fundamental Education Center, National Chin-Yi University of Technology, Taichung 41170, Taiwan

A R T IC LE I N F O

ABS TRA CT

Keywords: Inflation Rapid roll inflation

In this paper, I present an analytic solution to the equation of motion of the inflaton field in a model of rapid roll inflation with a quartic potential by assuming that the Hubble parameter is a constant during inflation. The result is obtained by using Jacobi elliptic functions. The spectral index ns and the running spectral index ns′ is obtained without using rapid roll or extended slow roll approximations. The cosmological consequences of the model is discussed.

1. Introduction Slow roll inflation requires that the effective inflaton mass should be much smaller than the Hubble parameter during inflation, that is m2 ≪ H2. However, when we try to build an inflation model in the framework of string theory or supergravity, one often encounter an Hubble induced inflaton mass, that is m2 ∼ H2 [1,2]. In addition, scalar field ϕ conformally coupled to gravity acquires 1 1 effective mass term 2 ξRϕ2 with ξ = 6 and the effective mass m2 ∼ 12ξH 2 = 2H 2 which violates slow-roll condition. However, inflation can still happen in the form of rapid roll inflation [3]. In order to study rapid roll inflation, rapid roll conditions or extended slow roll conditions are used as an approximation to deal with the problem [3–6]. In this paper, I analyze a specific (but quite representative) case by considering a quartic term in the potential without using these approximation methods. In the following, I use the system of units MP = 2.4 × 1018 GeV = (8πG )−1/2 = 1. 2. Inflation with a quadratic potential As a warm up, I will review a simple inflation model with the potential of the inflaton field ϕ as

V = V0 +

1 2 2 mϕ , 2

(1)

where the constant V0 during inflation can be produced by the mechanism of hybrid inflation [7]. The equation of motion is given by

ϕ¨ + 3Hϕ˙ + m2ϕ = 0.

(2)

Slow roll approximation assumes the first term on the left-hand side of the equation is negligible, therefore the number of e-folds is given by

E-mail address: [email protected]. https://doi.org/10.1016/j.cjph.2019.09.001 Received 1 August 2019; Received in revised form 28 August 2019; Accepted 2 September 2019 Available online 06 September 2019 0577-9073/ © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

Chinese Journal of Physics 61 (2019) 98–103

C.-M. Lin 2

N≡

2

3H 3H dϕ = − 2 ∫ Hdt = ∫ Hϕ˙ dϕ = − ∫ m 2ϕ m

ln ϕ + C = Ht ,

(3)

where C is an integration constant and it is assumed that the Hubble parameter is a constant during inflation. Therefore by defining ϕ (0) = ϕ0 we can write

ϕ = ϕ0 e

2 − m2N 3H

= ϕ0 e

2 − m 2 Ht 3H .

(4)

Actually in this case, we can do better than imposing slow-roll approximation. With the assumption that the Hubble parameter is a constant during inflation, we can solve Eq. (2) by substituting the ansatz ϕ = Be bt into the equation and obtain 4m2 9H2

−3H ± 3H 1 − b=

2

.

(5)

We can neglect the solution with the minus sign because it decays faster, and the result is 2

−3H + 3H 1 − 4m2 9H

ϕ = ϕ0 e

t

2

.

(6)

When m ≪ H , Eq. (6) reduces to 2

2

2

−3H + 3H ⎛1 − 2m2 ⎞ ⎝ 9H ⎠ t ϕ0 e 2 ⎜

ϕ=

⎟

= ϕ0 e

2 − m 2 Ht 3H ,

(7)

which recovers Eq. (4) and shows the validity of slow-roll approximation. 3. Rapid roll inflation The action of a non-minimally coupled scalar field ϕ is given by

S=

∫ d 4x

R 1 1 −g ⎡ − ∂ μϕ∂μ ϕ − V (ϕ) − ξRϕ2 ⎤. 2 2 ⎣2 ⎦

In a FRW universe with

ds 2

=

−dt 2

+

a (t )2

dx2,

(8)

the equation of motion of the scalar field ϕ is

ϕ¨ + 3Hϕ˙ + 6ξ (H˙ + 2H 2) ϕ + V ′ (ϕ) = 0,

(9)

and the Friedmann equation is

3H 2 =

1 ˙ 1 (ϕ + Hϕ)2 + V (ϕ) ≡ π 2 + V (ϕ). 2 2

(10)

If we assume H and V = V0 are constants during inflation and choosing conformal coupling ξ =

1 , 6

Eq. (9) becomes

ϕ¨ + 3Hϕ˙ + 2H 2ϕ = 0. This is nothing but Eq. (2) with

(11)

m2

=

2H 2

and the solution can be obtained from Eq. (6) as

ϕ = ϕ0 e−Ht .

(12)

If we make a time derivative,

ϕ˙ = −Hϕ,

(13)

and substitute the result into Eq. (10), we obtain

3H 2 = V0,

(14)

which is indeed a constant and implies a de Sitter universe. This also shows the important role played by V0. 4. Rapid roll inflation with a quartic potential Now let us go beyond the simplest model and consider a potential of the form1

V = V0 +

1 2 2 1 M ϕ + λϕ4 . 2 4

(15)

This might be regarded as a “two term approximation” to a more general potential. The potential should be regarded as an effective potential during inflation when V0 is a constant. Inflation ends through the mechanism of hybrid inflation [7], namely the inflaton 1

This potential was considered in the case of hilltop inflation where the parameter space is analyzed in the framework of slow-roll inflation [8]. 99

Chinese Journal of Physics 61 (2019) 98–103

C.-M. Lin

field couples to another scalar field ψ with a Higgs-like symmetry breaking potential. When the field value of ϕ becomes small enough, the effective mass of ψ becomes negative and triggers a phase transition to the end of inflation. 1 By our assumption that Hubble parameter is a constant and conformal coupling ξ = 6 , Eq. (9) becomes

ϕ¨ + 3Hϕ˙ + (2H 2 + M 2) ϕ + λϕ3 = 0.

(16)

I will ignore the quardratic term of the potential from now on by considering M ≪ 2H or gives ϕ a large effective mass.2 Therefore the equation of motion becomes 2

2

M2

= 0 since conformal coupling already

ϕ¨ + 3Hϕ˙ + m2ϕ + λϕ3 = 0,

(17)

and let us keep in mind that (18)

m2 ≡ 2H 2.

Is there an analytic solution to Eq. (17)? This non-linear second-order equation is called a Duffing equation and corresponds to the equation of motion of a damped Duffing oscillator without a driving force. In order to solve the equation, let us consider the ansatz [9]

ϕ (t ) = α (t )cn(ω (t ), k 2).

(19)

Here cn is a Jacobi elliptic function defined as

cn(u, k 2) = cos ψ, where u =

∫0

dθ

ψ

1 − k 2 sin2 θ

.

(20)

By substituting our ansatz into Eq. (17), and use some properties of Jacobi elliptic functions (see the appendix), we could obtain

cn(ω (t ), k 2)[m2α + 3Hα˙ − αω˙ 2 + 2k 2αω˙ 2 + α¨]+cn(ω (t ), k 2)3 [λα3 − 2k 2αω˙ 2] ˙ ˙ + αω¨] = 0. − sn(ω (t ), k 2)dn(ω (t ), k 2)[3Hαω˙ + 2αω

(21)

The above equation holds for all time t if and only if

m2α + 3Hα˙ − αω˙ 2 + 2k 2αω˙ 2 + α¨ = 0

(22)

λα3 − 2k 2αω˙ 2 = 0

(23)

˙ ˙ + αω¨ = 0 3Hαω˙ + 2αω

(24)

Let us substitute the ansatz α = Ae βt and ω = Ωe βt into Eq. (24), and obtain β = −H . Then after substituting this into Eq. (22), we can see that it can be satisfied if k 2 = 1/2 and m2 = 2H 2 . Interestingly, the latter condition is exactly Eq. (18)3 Finally from Eq. (23), we obtain Ω2 = λA2 / H 2, therefore Ω = ± λ A/ H . It does not matter whether we take the plus or minus sign for Ω because cn is an even function and we will take the plus sign. Actually there will be an integration constant C for ω since it appears with at least one time derivative in the equations, so eventaully we have our analytical expression

λ A −Ht 1 ϕ (t ) = Ae−Ht cn ⎛⎜ e + C , ⎞⎟. H 2⎠ ⎝

(25)

The time derivative of ϕ is given by using the equations in the appendix as

ϕ˙ = −Hϕ +

λ 2

A4 e−4Ht − ϕ4 .

(26)

From the above two equations, the constants A and C can be determined by the initial condition ϕ (0) = ϕ0 and ϕ˙ (0) = ϕ˙ 0 as 2

A4 =

λϕ04 + 2ϕ˙ 0 + 4Hϕ0 ϕ˙ 0 + 2H 2ϕ02 (27)

λ

ϕ 1 C = cn−1 ⎛ 0 , ⎞ − ⎝ A 2⎠ ⎜

⎟

λA . H

(28)

Now let us see what we can learn from the above solution. From Eq. (26), we can obtain

π ≡ ϕ˙ + Hϕ =

λ 2

A4 e−4Ht − ϕ4 .

(29)

Therefore For example, if M ∼ 0.1H, we have M2 ∼ 0.01H2. Here this condition comes to us so that the term of coefficient m2 cancels out with the term of coefficient 2H2 so that our ansatz can work. It may be interesting to investigate whether this “miraculous” cancellation is just a coincidence or there is some deep mathematical reason. 2 3

100

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C.-M. Lin

−

π 2 + V ′ϕ/2 H˙ = = 2 H π 2/2 + V

λ 4 −4Ht Ae 2 λ 4 −4Ht A e + 4

V0

, (30)

where Eqs. (9) and (10) is used. Our approximation that the Hubble expansion rate H is a constant is valid if |H˙ / H 2| ≪ 1. From Eqs. (29) and (25), we can obtain

λ 2 −2Ht λ A −Ht 1 Ae 1 − cn4 ⎛⎜ e + C , ⎞⎟ . 2 H 2⎠ ⎝

π=

(31)

This means for any (reasonable4) choices of the initial condition ϕ0 and ϕ˙ 0 , which determines A and C, π goes exponentially fast towards the attractor π ∼ 0, namely ϕ˙ ∼ −Hϕ, because the Jacobi ellipic function cn is bounded. 5. Primordial curvature perturbation Since in rapid roll inflation, the inflaton field is rolling rapidly, primordial density perturbation cannot come from the fluctuation of the inflaton field. We can consider a curvaton scenario [10–12], modulated reheating scenatio [13,14], or even a curvaton with modulated decay [15–17]. The following conclusion are basically the same. As an illustration, let us consider a modulated reheating scenario, where the primordial curvature perturbation ζ is given by the modulated decay width Γ through a modulated coupling constant Λ from a fluctuating scalar field χ as

ζ∼

δχ δΓ δΛ ∝ ∝ , Γ Λ χ

(32)

where Γ ∼ Λ M. The idea is motivated because in string theory, there is “no free parameters” in the sense that the “coupling constants” are determined dynamically by scalar fields. Therefore the quantum fluctuations of a light scalar field resulted in the fluctuation of the coupling constant, hence the decay width. At horizon crossing, k = aH = a0 He ∫ Hdt . Since δχ ∼ H, if we assume the effective mass of χ is much smaller than the Hubble parameter, the tilt of the spectrum can be expressed as 2

ns − 1 =

−1 d ln H 2 d ln H 2 dt H˙ H˙ = = 2 2 ⎛1 + 2 ⎞ , Hdt ∫ d ln k dt d ln(a 0 He H ⎝ H ⎠ ) ⎜

⎟

(33)

and the running spectral index is given by

ns′ ≡

−3 · 1 ⎛ H˙ ⎞ dns H˙ = 2 ⎛1 + 2 ⎞ . 2 d ln k H ⎠ H ⎝H ⎠ ⎝ ⎜

⎟

⎜

⎟

(34)

From Eq. (30), we can obtain

ns − 1 = −

λA4 e−4Ht λ

V0 − 4 A4 e−4Ht

, (35)

and

ns′ =

λA4 e−4Ht λ

V0 − 4 A4 e−4Ht

4V0 ·

(

λ 4 −4Ht Ae 4

(V − 0

+ V0

λ 4 −4Ht 2 Ae 4

)

) ≃ −4(n

s

− 1). (36)

λA4 e−4Ht / V0

and the result can be used The approximation used above is not indispensable, because given ns we can calculate the ratio to obtain ns′. However, this result is inconsistant with the latest Planck data [18] which gives ns − 1 ≃ −0.04 and |ns′ | ≲ 0.01. Therefore our assumption to ignore the mass of χ may be wrong. If the effective mass of χ is non-negligible, there is a correction to the spectral tilt as [11]

Δns =

2 V ″ (χ ) . 3 H2

(37)

We can fit Planck data |ns′ | ≲ 0.01 if

λA4 e−4Ht ≲ 0.0025 V0

(38)

at horizon exit and obtain ns − 1 ≃ −0.04 if V ″ (χ ) ∼ −0.06H 2 . This implies the effective mass of χ is about the same order of magnitude as the Hubble parameter. Since in this case the potential of χ has a hilltop form, we may call χ a “hilltop modulon”. The same result can also be achieved by considering a hilltop curvaton [19]. Instead of considering a hilltop modulon or hilltop curvaton, another possibility to evade Planck constraint is to have ns ∼ 1. This 4

For example, we do not choose the initial condition at the vaccum so that inflation does not happen. 101

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might be achieved if some part of primordial density perturbation comes from cosmic string contribution [20,21]. Since V0 presumably comes from the mechanism of hybrid inflation, the production of cosmic string is quite generic [22]. However, without further simulations by using the latest observational results, it is not clear whether ns ∼ 1 can still be achieved by considering cosmic string contribution. 6. Conclusion and discussion In this paper, I have presented an analytic expression of solutions for a rapid roll inflation with a quartic term in the potential. Although nowadays numerical solution is readily available, it is still satisfying if an analytic expression can be found even for simpler cases, and it could shed some light to improve our understanding of rapid roll inflation. In our model, there are free parameters λ and V0. We may realize this model in the framework of supersymmetric particle physics theories along the line of modified supernature inflation [23–25], where V0 is determined by SUSY breaking scale and the dimensionless parameter λ is given by the ratio between a soft SUSY breaking parameter and the Planck scale, therefore it could be naturally small. I will leave the investigation to future work. Declaration of Competing Interest None. Acknowledgement This work is supported by the Ministry of Science and Technology (MOST) of Taiwan under grant number MOST 106-2112-M167-001. Appendix A. Jacobi elliptic functions The Jacobi elliptic function cn is defined as

cn(u, k 2) = cos ψ, where u =

∫0

ψ

dθ 1 − k 2 sin2 θ

.

(A1)

There are also

sn(u, k 2) = sin ψ,

(A2)

dn(u, k 2) =

(A3)

and

1 − k 2 sin2 ψ .

Some useful identities are the following:

sn2 (u, k 2) + cn2 (u, k 2) = 1

(A4)

dn2 (u,

(A5)

k 2)

=1−

k 2sn2 (u,

k 2)

d cn(u, k 2) = −sn(u, k 2)dn(u, k 2). du

(A6)

Roughly speaking, as we can generalize a circle to a ellipse, Jacobi elliptic functions are a generalization of trigonometric functions. When k2 → 0, we have

sn(u, k 2) = sin u,

(A7)

cn(u,

k 2)

= cos u,

(A8)

dn(u,

k 2)

= 1.

(A9)

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