1. Introduction
Inflation, characterized by an exponential expansion of the Universe during its early stages, was proposed to address issues in standard Big Bang cosmology, including the horizon, flatness, and monopole problems [1–5]. This expansion is typically driven by a scalar field known as the inflaton, where the simplest model is the canonical inflaton field with a flat potential. Utilizing the Higgs field, the only detected scalar field in the Standard Model of particle physics, as the inflaton leads to a conflict with observations of the cosmic microwave background (CMB) radiation due to the large prediction of the tensor-to-scalar ratio r. To overcome it, inflation models with the field non-minimally coupled to gravity have been proposed [6–8]. One notable model is the new Higgs inflation model with the kinetic term non-minimally derivative coupling to the Einstein tensor, represented by Gμν∂μφ∂νφ/M2 [7, 9]. Under the general relativity (GR) limit F = H2/M2 = 0, this model becomes the canonical situation conflicting with observations, while under the high friction limit F ≫ 1, the tensor-to-scalar ratio is 1/3 of that in the canonical Higgs inflation, aligning with the observational data [9, 10]. Furthermore, the effective self-coupling λ of the Higgs boson in this model can be lowered to the order of unity without introducing a new degree of freedom [7, 9]. For more works on the inflation model with non-minimally derivative coupling, see [11–25].
Apart from the slow-roll inflation scenario, the constant-roll inflation scenario, where one slow-roll parameter remains constant and need not be small, can also yield predictions consistent with observational data [26–28]. The characteristics of primordial perturbations in constant-roll inflation can differ significantly from those in slow-roll inflation. For instance, in the canonical constant-roll inflation with a constant ${\eta }_{\phi }=-\ddot{\phi }/(H\dot{\phi })$ and ηφ > 3/2, scalar perturbations outside the horizon continue to evolve [27]. Additionally, the power spectrum of scalar perturbations can exhibit sharp enhancements at small scales in ultra-slow-roll inflation with ηφ ≈ 3, potentially leading to the formation of primordial black holes and inducing secondary gravitational waves [29–41]. Despite the differences in the evolution of perturbations between ultra-slow-roll inflation ηφ = 3 − α and slow-roll inflation ηφ = α in the canonical inflation model [42], if we neglect the contribution from εH to the scalar spectral tilt ns, the predictions for the scalar spectral tilt ns and tensor-to-scalar ratio r are the same [43, 44]. This indicates the existence of a duality in constant-roll inflation models with large and small ηφ. For more about the constant-roll inflation, please refer to [45–66].
In this paper, we research the constant-roll inflation model with non-minimally derivative coupling. We calculate the power spectrum of scalar and tensor fluctuations using the Bessel function method and derive expressions for the scalar spectral tilt ns, the tensor spectral tilt nT, and the tensor-to-scalar ratio r. Additionally, we explore the dual between the ultra-slow-roll inflation and slow-roll inflation. The organization of this paper is as follows: section 2 presents the background evolution and defines the slow-roll parameters. Section 3 calculates the power spectrum for scalar and tensor perturbations using the Bessel function method within the constant-roll condition. Section 4 explores the dual behavior between ultra-slow-roll inflation and slow-roll inflation. Finally, section 5 draws the conclusions.
2. Inflation with non-minimally derivative coupling
In this section, we derive the background evolution, define the slow-roll parameters, and discuss the general formulae for the the scalar and tensor spectral for the inflationary models with non-minimally derivative coupling.
2.1. The background
The action for the inflationary model with the non-minimal derivative coupling is
$\begin{eqnarray}\begin{array}{rcl}S & = & \displaystyle \frac{1}{2}\displaystyle \int {{\rm{d}}}^{4}x\sqrt{-g}\left[{M}_{{pl}}^{2}R-{g}^{\mu \nu }{\partial }_{\mu }\phi {\partial }_{\nu }\phi \right.\\ & & \left.+\displaystyle \frac{1}{{M}^{2}}{G}^{\mu \nu }{\partial }_{\mu }\phi {\partial }_{\nu }\phi -2V(\phi )\right],\end{array}\end{eqnarray}$
where ${M}_{{pl}}=1/\sqrt{8\pi G}$ is the reduced Planck mass, R is the Ricci scalar, φ is the scalar field, Gμν is the Einstein tensor, V(φ) is the potential, and M2 is the coupling parameter with the dimension of energy. For the homogeneous and isotropic background with the flat Friedmann–Robertson–Walker (FRW) metric, the background equations are [10] $\begin{eqnarray}{H}^{2}={\left(\displaystyle \frac{\dot{a}}{a}\right)}^{2}=\displaystyle \frac{1}{3{M}_{{pl}}^{2}}\left[\displaystyle \frac{{\dot{\phi }}^{2}}{2}(1+9F)+V(\phi )\right],\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}\left[{a}^{3}\dot{\phi }\left(1+3F)\right)\right]=-{a}^{3}\displaystyle \frac{{\rm{d}}V}{{\rm{d}}\phi },\end{eqnarray}$
$\begin{eqnarray}\dot{H}-\displaystyle \frac{\dot{\phi }\ddot{\phi }}{{M}^{2}{M}_{{pl}}^{2}}H=\displaystyle \frac{1}{2}\displaystyle \frac{{\dot{\phi }}^{2}}{{M}^{2}{M}_{{pl}}^{2}}\dot{H}-\displaystyle \frac{3}{2}\displaystyle \frac{{\dot{\phi }}^{2}}{{M}^{2}{M}_{{pl}}^{2}}{H}^{2}-\displaystyle \frac{1}{2}\displaystyle \frac{{\dot{\phi }}^{2}}{{M}_{{pl}}^{2}},\end{eqnarray}$
where F = H2/M2 and an over-dot means the derivative with respect to the cosmic time t.In this paper, we define the slow-roll parameters as5 ), the background equation (4 ) can be rewritten as6 ), we get
$\begin{eqnarray}{\epsilon }_{H}=-\displaystyle \frac{\dot{H}}{{H}^{2}},\quad {\eta }_{H}=-\displaystyle \frac{\ddot{H}}{2H\dot{H}},\quad {\eta }_{\phi }=-\displaystyle \frac{\ddot{\phi }}{H\dot{\phi }}.\end{eqnarray}$
In the canonical inflation models, the slow-roll parameters satisfy ηH = ηφ, a condition not met in the inflation models with non-minimal derivative coupling. By using the slow-roll parameters ( $\begin{eqnarray}-{\epsilon }_{H}+2{\eta }_{\phi }Q=-{\epsilon }_{H}Q-3Q-Q/F,\end{eqnarray}$
where the definition of Q is $\begin{eqnarray}Q=\displaystyle \frac{{\dot{\phi }}^{2}}{2{M}^{2}{M}_{{pl}}^{2}}.\end{eqnarray}$
Solving the background equation ( $\begin{eqnarray}Q=\displaystyle \frac{F{\epsilon }_{H}}{b+F{\epsilon }_{H}},\end{eqnarray}$
where b = 1 + 3F + 2ηφF.For later use, we also give the following relation,
$\begin{eqnarray}{\dot{\epsilon }}_{H}=2H({\epsilon }_{H}-{\eta }_{H}){\epsilon }_{H},\end{eqnarray}$
$\begin{eqnarray}\dot{F}=-2{HF}{\epsilon }_{H},\end{eqnarray}$
and $\begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}\tau }\left(\displaystyle \frac{1}{{aH}}\right)=-1+{\epsilon }_{H},\end{eqnarray}$
where the conformal time τ is related to the cosmic time t by dt = adτ.2.2. The perturbations
In the uniform field gauge δφ(x, t) = 0, the action for the primordial scalar perturbation ζ is [10, 11, 67]5 ) and equation (6 ), we have
$\begin{eqnarray}{S}_{{\zeta }^{2}}=\int {\rm{d}}t{{\rm{d}}}^{3}{{xM}}_{{pl}}^{2}{a}^{3}\left\{\displaystyle \frac{{\rm{\Sigma }}}{{\bar{H}}^{2}}{\dot{\zeta }}^{2}-\displaystyle \frac{{\theta }_{s}}{{a}^{2}}{\left({\partial }_{i}\zeta \right)}^{2}\right\},\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}\bar{H}=\displaystyle \frac{H(1-3Q)}{1-Q},\quad {\rm{\Sigma }}={{QM}}^{2}\left[1+\displaystyle \frac{3F(1+3Q)}{(1-Q)}\right],\end{array}\end{eqnarray}$
and $\begin{eqnarray}{\theta }_{s}=\displaystyle \frac{1}{a}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}\left[\displaystyle \frac{a}{\bar{H}}(1-Q)\right]-1-Q.\end{eqnarray}$
By the variation principle, we can derive the equation of motion governing the primordial perturbation ζ; and in the Fourier space, we get the Mukhanov–Sasaki equation $\begin{eqnarray}{v}_{k}^{{\prime\prime} }+\left({c}_{s}^{2}{k}^{2}-\displaystyle \frac{{z}^{{\prime\prime} }}{z}\right){v}_{k}=0,\end{eqnarray}$
where the prime denotes the derivative with respect to the conformal time τ, the effective sound speed is ${c}_{s}^{2}={\bar{H}}^{2}{\theta }_{s}/{\rm{\Sigma }}$, the canonically normalized field is v = zζ, and $\begin{eqnarray}z={{aM}}_{{pl}}\displaystyle \frac{\sqrt{2{\rm{\Sigma }}}}{\bar{H}}.\end{eqnarray}$
By using the slow-roll parameters ( $\begin{eqnarray}\begin{array}{rcl}{c}_{s}^{2} & = & 1-\displaystyle \frac{2{\epsilon }_{H}F(1+7F+6F{\eta }_{\phi })}{{\left(1+3F\right)}^{2}+2F(1+3F){\eta }_{\phi }+12{\epsilon }_{H}{F}^{2}}\\ & & -\displaystyle \frac{8{\epsilon }_{H}^{2}{F}^{3}}{b[b+3F(b+4F{\epsilon }_{H})]},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{z^{\prime\prime} }{z}={a}^{2}{H}^{2}\left[2+3{\eta }_{H}-4{\epsilon }_{H}-6{\eta }_{\phi }-{\eta }_{\phi }(2{\epsilon }_{H}-2{\eta }_{H}+{\eta }_{\phi })\right.\\ \left.+\displaystyle \frac{{\eta }_{H}-{\eta }_{\phi }}{F}\right]+\displaystyle \frac{{a}^{2}{H}^{2}b{\epsilon }_{H}(3-2{\eta }_{\phi }+3{\epsilon }_{H}-2{\eta }_{H})}{b+3F(b+4F{\epsilon }_{H})}\\ +\displaystyle \frac{8{a}^{2}{H}^{2}F{\epsilon }_{H}m{\left(b+F{\epsilon }_{H}\right)}^{2}}{{b}^{2}\left(b-2F{\epsilon }_{H}\right)\left(b+4F{\epsilon }_{H}\right)}\\ +\displaystyle \frac{48{a}^{2}{H}^{2}{F}^{2}{\epsilon }_{H}^{2}{\eta }_{\phi }^{2}(b+F{\epsilon }_{H})}{b{\left(b-2F{\epsilon }_{H}\right)}^{2}}\\ +\displaystyle \frac{48{a}^{2}{H}^{2}{F}^{3}{\eta }_{\phi }^{2}{\epsilon }_{H}^{3}(b+F{\epsilon }_{H})}{{b}^{2}{\left(b+4F{\epsilon }_{H}\right)}^{2}}\\ -\displaystyle \frac{{a}^{2}{H}^{2}{\epsilon }_{H}^{2}{\left[4F{\eta }_{\phi }(b+F{\epsilon }_{H})+b(b+4F{\epsilon }_{H})\right]}^{2}}{{\left(b+4F{\epsilon }_{H}\right)}^{2}{\left[b+3F(b+4F{\epsilon }_{H})\right]}^{2}}\\ -\displaystyle \frac{4{a}^{2}{H}^{2}F{\epsilon }_{H}(b+F{\epsilon }_{H})}{b+3F(b+4F{\epsilon }_{H})}\left[\displaystyle \frac{2{\eta }_{\phi }{\epsilon }_{H}}{b-2F{\epsilon }_{H}}+\displaystyle \frac{{\eta }_{H}-{\eta }_{\phi }}{F(b+4F{\epsilon }_{H})}\right.\\ +\displaystyle \frac{3({\eta }_{H}-{\epsilon }_{H}-2{\eta }_{\phi })-2{\eta }_{\phi }(2{\epsilon }_{H}-{\eta }_{\phi }-{\eta }_{H})}{b+4F{\epsilon }_{H}}\\ -\displaystyle \frac{4F{\epsilon }_{H}{\eta }_{\phi }^{2}(b-2F{\epsilon }_{H})}{b{\left(b+4F{\epsilon }_{H}\right)}^{2}}\\ \left.+\displaystyle \frac{8F{\eta }_{\phi }^{2}{\epsilon }_{H}(b+F{\epsilon }_{H})}{b(b-2F{\epsilon }_{H})(b+4F{\epsilon }_{H})}\right],\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}m=2{\eta }_{\phi }({\eta }_{\phi }+{\epsilon }_{H})+4F{\eta }_{\phi }^{2}({\eta }_{H}+{\eta }_{\phi })+(1+3F)\\ \times \left[3({\eta }_{H}-{\epsilon }_{H}-2{\eta }_{\phi })\right.\\ \left.-2{\eta }_{\phi }(2{\epsilon }_{H}+{\eta }_{\phi }-2{\eta }_{H})+\displaystyle \frac{{\eta }_{H}-{\eta }_{\phi }}{F}\right].\end{array}\end{eqnarray}$
The definition of the power spectrum of the scalar perturbation is $\begin{eqnarray}{{ \mathcal P }}_{\zeta }=\displaystyle \frac{{k}^{3}}{2{\pi }^{2}}{\left|{\zeta }_{k}\right|}^{2},\end{eqnarray}$
and the scalar spectral tilt ns is defined by [68] $\begin{eqnarray}{n}_{s}-1={\left.\displaystyle \frac{\mathrm{dln}{{ \mathcal P }}_{\zeta }}{\mathrm{dln}k}\right|}_{{aH}=\mathrm{const}}.\end{eqnarray}$
For the tensor perturbation, the quadratic action is [10, 67]23 ) is5 ), equation (6 ), and the relation (8 ), we obtain
$\begin{eqnarray}S=\int {{\rm{d}}}^{3}x{\rm{d}}t\displaystyle \frac{{M}_{{pl}}^{2}{a}^{3}}{8}\left[(1-Q){\dot{\gamma }}_{{ij}}^{2}-\displaystyle \frac{1}{{a}^{2}}(1+Q){\left({\partial }_{l}{\gamma }_{{ij}}\right)}^{2}\right].\end{eqnarray}$
The Mukhanov–Sasaki equation for tensor perturbation is $\begin{eqnarray}{u}_{k}^{s^{\prime\prime} }+\left({c}_{t}^{2}{k}^{2}-\displaystyle \frac{{z}_{t}^{\prime\prime} }{{z}_{t}}\right){u}_{k}^{s}=0.\end{eqnarray}$
Here, us = ztγs, the label s = {+, ×}, denotes the polarization of the tensor perturbations, and $\begin{eqnarray}{\gamma }_{{ij}}=\displaystyle \sum _{s=+,\times }{{\rm{e}}}_{{ij}}^{s}{\gamma }^{s},\end{eqnarray}$
where ${{\rm{e}}}_{{ij}}^{s}$ is the polarization tensor and satisfies ${\sum }_{i}{{\rm{e}}}_{{ii}}^{s}=0$, ${\sum }_{i,j}{{\rm{e}}}_{{ij}}^{s}{{\rm{e}}}_{{ij}}^{{s}^{{\prime} }}=2{\delta }_{{{ss}}^{{\prime} }}$. The sound speed in equation ( $\begin{eqnarray}{c}_{t}^{2}=\displaystyle \frac{1+Q}{1-Q},\end{eqnarray}$
and the normalized parameter is $\begin{eqnarray}{z}_{t}=\displaystyle \frac{\sqrt{2}}{2}{{aM}}_{{pl}}\sqrt{1-Q}.\end{eqnarray}$
By using the slow-roll parameters ( $\begin{eqnarray}{c}_{t}^{2}=1+\displaystyle \frac{2F{\epsilon }_{H}}{1+(3+2{\eta }_{\phi })F},\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{z}_{t}^{{\prime\prime} }}{{z}_{t}} & = & {a}^{2}{H}^{2}\left[2-{\epsilon }_{H}-\displaystyle \frac{F{\epsilon }_{H}}{1+(3+2{\eta }_{\phi })F}\left(\displaystyle \frac{{\eta }_{H}-{\eta }_{\phi }}{F}\right.\right.\\ & & \left.+3({\eta }_{H}-2{\eta }_{\phi }-{\epsilon }_{H})-2{\eta }_{\phi }({\epsilon }_{H}-{\eta }_{H})\right)\\ & & -\left.\displaystyle \frac{{F}^{2}{\epsilon }_{H}^{2}{\eta }_{\phi }^{2}}{{\left[1+(3+2{\eta }_{\phi })F\right]}^{2}}\right].\end{array}\end{eqnarray}$
The definition of the power spectrum for the tensor perturbations is $\begin{eqnarray}{{ \mathcal P }}_{T}=\displaystyle \frac{{k}^{3}}{2{\pi }^{2}}{\left|{\gamma }_{{ij}}\right|}^{2}=\displaystyle \frac{{k}^{3}}{{\pi }^{2}}\displaystyle \sum _{s=+,x}{\left|\displaystyle \frac{{u}_{k}^{s}}{{z}_{t}}\right|}^{2}.\end{eqnarray}$
3. The constant-roll inflation
In addition to the slow-roll inflation scenario, where all the slow-roll parameters are small, the constant-roll inflation scenario, where one of the slow-roll parameters is held constant and need not be small, can also produce predictions consistent with observational data from CMB. In this section, we study the constant-roll scenario where the slow-roll parameter ηφ is a constant.
Combining equations (7 ) and (8 ) and calculating $\dot{Q}$, we get9 ) becomes,
$\begin{eqnarray}\begin{array}{rcl}\dot{{\eta }_{\phi }} & = & 3H({\eta }_{\phi }-{\eta }_{H}+{\epsilon }_{H})+{\eta }_{\phi }H(3{\epsilon }_{H}-2{\eta }_{H}+2{\eta }_{\phi })\\ & & -\displaystyle \frac{({\eta }_{H}-{\eta }_{\phi })H}{F}.\end{array}\end{eqnarray}$
By using the condition that the slow-roll parameter ηφ is a constant, we obtain $\begin{eqnarray}{\eta }_{H}={\eta }_{\phi }+\displaystyle \frac{3F{\epsilon }_{H}({\eta }_{\phi }+1)}{1+(3+2{\eta }_{\phi })F},\end{eqnarray}$
and relation ( $\begin{eqnarray}{\dot{\epsilon }}_{H}=2H{\epsilon }_{H}\left({\epsilon }_{H}-{\eta }_{\phi }-\displaystyle \frac{3F{\epsilon }_{H}({\eta }_{\phi }+1)}{1+(3+2{\eta }_{\phi })F}\right).\end{eqnarray}$
Because ηφ is a constant and it may not be small, so ${\dot{\epsilon }}_{H}$ can still be the first order in perturbation.In the GR limit, F = 0, we recover the standard result ηH = ηφ. In the high friction limit, F ≫ 1,
$\begin{eqnarray}{\eta }_{H}={\eta }_{\phi }+\displaystyle \frac{3{\epsilon }_{H}({\eta }_{\phi }+1)}{3+2{\eta }_{\phi }}.\end{eqnarray}$
Because the slow-roll parameter ηφ may not be small in the constant-roll inflation scenario, the slow-roll parameter ηH may not be small in general either.For the constant-roll inflation with constant ηφ, from equation (11 ) and to the first order approximation of εH, we get the relation of the conformal Hubble parameter aH and the conformal time τ [28],
$\begin{eqnarray}\displaystyle \frac{1}{{aH}}\approx \left(-1+\displaystyle \frac{{\epsilon }_{H}}{1+2{\eta }_{\phi }}\right)\tau .\end{eqnarray}$
3.1. Scalar perturbations
To the first order of εH, the relation (18 ) becomes15 ), we write34 ) and (35 ), we get15 ) becomes the standard Bessel equation, and the solution can be expressed as the Hankel function of order ν [69], 40 ) and the asymptotic behavior of the Hankel function, we obtain the mode function outside the horizon16 ) and the solution (41 ), from the definition of the power spectrum for the scalar perturbation (20 ), we obtain
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{z^{\prime\prime} }{z}\approx {a}^{2}{H}^{2}\left[2-3{\eta }_{\phi }+{\eta }_{\phi }^{2}+\displaystyle \frac{2-3F}{1+3F}{\epsilon }_{H}\right.\\ \,\left.-{\eta }_{\phi }{\epsilon }_{H}\left(\displaystyle \frac{3(1-F)}{1+3F}+\displaystyle \frac{4F(1+6F)(3-4{\eta }_{\phi })}{(1+3F)[1+(3+2{\eta }_{\phi })F]}\right)\right].\end{array}\end{eqnarray}$
To solve the Mukhanov–Sasaki equation ( $\begin{eqnarray}\displaystyle \frac{z^{\prime\prime} }{z}=\displaystyle \frac{{\nu }^{2}-1/4}{{\tau }^{2}}.\end{eqnarray}$
To the first order of εH, combining equations ( $\begin{eqnarray}\begin{array}{rcl}\nu & = & \left|\displaystyle \frac{3}{2}-{\eta }_{\phi }\right|+\displaystyle \frac{\bar{\nu }{\epsilon }_{H}}{\left|3/2-{\eta }_{\phi }\right|},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}\bar{\nu } & = & \displaystyle \frac{2-3{\eta }_{\phi }+{\eta }_{\phi }^{2}}{1+2{\eta }_{\phi }}+\displaystyle \frac{2-3F}{2(1+3F)}\\ & & -{\eta }_{\phi }\left[\displaystyle \frac{3(1-F)}{2(1+3F)}+\displaystyle \frac{2F(1+6F)(3-4{\eta }_{\phi })}{(1+3F)[1+(3+2{\eta }_{\phi })F]}\right].\end{array}\end{eqnarray}$
Assuming that both ν and cs are constants, then the Mukhanov–Sasaki equation ( $\begin{eqnarray}{v}_{k}(\tau )=\sqrt{-\tau }\left[{c}_{1}{H}_{\nu }^{(1)}(-{c}_{s}k\tau )+{c}_{2}{H}_{\nu }^{(2)}(-{c}_{s}k\tau )\right].\end{eqnarray}$
Choosing the Bunch–Davies vacuum, at small scales, −cskτ ≫ 1, we have the initial condition $\begin{eqnarray}{v}_{k}\to \displaystyle \frac{1}{\sqrt{2{c}_{s}k}}{{\rm{e}}}^{-{\rm{i}}{c}_{s}k\tau }.\end{eqnarray}$
Using the initial condition ( $\begin{eqnarray}{v}_{k}={{\rm{e}}}^{-{\rm{i}}(\nu -1/2)\tfrac{\pi }{2}}{2}^{\nu -\tfrac{3}{2}}\displaystyle \frac{{\rm{\Gamma }}(\nu )}{{\rm{\Gamma }}(3/2)}\displaystyle \frac{1}{\sqrt{2{c}_{s}k}}{\left(-{c}_{s}k\tau \right)}^{\tfrac{1}{2}-\nu }.\end{eqnarray}$
Combining equation ( $\begin{eqnarray}\begin{array}{l}{{ \mathcal P }}_{\zeta }\approx {2}^{2\nu -3}{\left[\displaystyle \frac{{\rm{\Gamma }}(\nu )}{{\rm{\Gamma }}(3/2)}\right]}^{2}\displaystyle \frac{{H}^{2}}{8{\pi }^{2}{M}_{{pl}}^{2}{c}_{s}{\theta }_{s}}\\ \qquad \times {\left(1-\displaystyle \frac{{\epsilon }_{H}}{1+2{\eta }_{\phi }}\right)}^{2\nu -1}{\left(\displaystyle \frac{{c}_{s}k}{{aH}}\right)}^{3-2\nu }.\end{array}\end{eqnarray}$
Substituting equation (42 ) into the definition of the scalar spectral tilt ns, equation (21 ), we obtain
$\begin{eqnarray}{n}_{s}-1=3-2\nu .\end{eqnarray}$
If ∣ηφ∣ ≪ 1, to the first order of perturbation, we get [10, 70] $\begin{eqnarray}{n}_{s}-1\approx 2{\eta }_{\phi }-\displaystyle \frac{2(2+3F)}{1+3F}{\epsilon }_{H}.\end{eqnarray}$
In the GR limit, we have
$\begin{eqnarray}\nu =\left|\displaystyle \frac{3}{2}-{\eta }_{\phi }\right|+\displaystyle \frac{(6-5{\eta }_{\phi }-4{\eta }_{\phi }^{2}){\epsilon }_{H}}{\left|3-2{\eta }_{\phi }\right|(1+2{\eta }_{\phi })},\end{eqnarray}$
and $\begin{eqnarray}{n}_{s}-1=3-\left|3-2{\eta }_{\phi }\right|-\displaystyle \frac{2(6-5{\eta }_{\phi }-4{\eta }_{\phi }^{2}){\epsilon }_{H}}{\left|3-2{\eta }_{\phi }\right|(1+2{\eta }_{\phi })}.\end{eqnarray}$
In the high friction limit, we have
$\begin{eqnarray}\nu =\left|\displaystyle \frac{3}{2}-{\eta }_{\phi }\right|+\displaystyle \frac{3(3-4{\eta }_{\phi })(1-3{\eta }_{\phi }-6{\eta }_{\phi }^{2}){\epsilon }_{H}}{\left|3-2{\eta }_{\phi }\right|(1+2{\eta }_{\phi })(3+2{\eta }_{\phi })},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{n}_{s}-1 & = & 3-\left|3-2{\eta }_{\phi }\right|\\ & & -\displaystyle \frac{6(3-4{\eta }_{\phi })(1-3{\eta }_{\phi }-6{\eta }_{\phi }^{2}){\epsilon }_{H}}{\left|3-2{\eta }_{\phi }\right|(1+2{\eta }_{\phi })(3+2{\eta }_{\phi })}.\end{array}\end{eqnarray}$
As discussed above, ${\dot{\epsilon }}_{H}$ is still the first order, so we will get a different expression for the scalar spectral tilt if we take the derivative of the amplitude of the scalar power spectrum at the horizon crossing csk = aH with respect to the scale k. To show this point, we give the derivation below.
For the constant-roll inflation with large ηφ, the scalar perturbations may continue evolving outside the horizon [27]. If scalar perturbations are constant after the horizon crossing, aH ≥ csk, to the first order of εH, the amplitude of the power spectrum for scalar perturbations outside the horizon is20 ), if scalar perturbations are constants after the horizon crossing, the scalar spectral tilt can also be derived by using the the amplitude of the power spectrum for the scalar perturbation,49 ) and (52 ) into equation (51 ), to the first order of εH, we get43 ) if ηφ is not small. If ∣ηφ∣ ≪ 1, then the result (53 ) reduces to the result (44 ). In the following discussion, we use the familiar result (43 ).
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal A }}_{\zeta } & = & {{ \mathcal P }}_{\zeta }{| }_{{c}_{s}k={aH}}\\ & \approx & {2}^{| 3-2{\eta }_{\phi }| -3}{\left[\displaystyle \frac{{\rm{\Gamma }}(| 3/2-{\eta }_{\phi }| )}{{\rm{\Gamma }}(3/2)}\right]}^{2}\displaystyle \frac{{H}^{2}}{8{\pi }^{2}{M}_{{pl}}^{2}{\epsilon }_{H}}\\ & & \times \displaystyle \frac{1+3F+2F{\eta }_{\phi }}{1+3F}(1+D{\epsilon }_{H}),\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}D & = & \displaystyle \frac{4\bar{\nu }}{| 3-2{\eta }_{\phi }| }\left(\mathrm{ln}2+\displaystyle \frac{{\rm{\Gamma }}^{\prime} (| 3/2-{\eta }_{\phi }| )}{{\rm{\Gamma }}(| 3/2-{\eta }_{\phi }| )}\right)-\displaystyle \frac{| 3-2{\eta }_{\phi }| -1}{1+2{\eta }_{\phi }}\\ & & \times \displaystyle \frac{18{F}^{2}{\eta }_{\phi }}{(1+3F)[1+(3+2{\eta }_{\phi })F]}.\end{array}\end{eqnarray}$
Besides the definition ( $\begin{eqnarray}{n}_{s}-1={\left.\displaystyle \frac{\mathrm{dln}{{ \mathcal A }}_{\zeta }}{\mathrm{dln}k}\right|}_{{c}_{s}k={aH}}.\end{eqnarray}$
To the first order of approximation of εH, $\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\mathrm{dln}k}{H{\rm{d}}t} & = & (1-{\epsilon }_{H})-{H}^{-1}\displaystyle \frac{\dot{{c}_{s}}}{{c}_{s}}\\ & \approx & 1-\left(\displaystyle \frac{2F{\eta }_{\phi }(1+7F+6F{\eta }_{\phi })}{(1+3F)[1+(3+2{\eta }_{\phi })F]}+1\right){\epsilon }_{H}.\end{array}\end{eqnarray}$
Substituting equations ( $\begin{aligned} n_{s} & -1 \approx 2 \eta_{\phi}-\frac{2(2+3 F) \epsilon_{H}}{1+3 F} \\ & +2 \eta_{\phi} \epsilon_{H}\left[\frac{\left|3-2 \eta_{\phi}\right|-1}{1+2 \eta_{\phi}}\right. \\ & -\frac{4 \bar{\nu}}{\left|3-2 \eta_{\phi}\right|}\left(\ln 2+\frac{\Gamma^{\prime}\left(\left|3 / 2-\eta_{\phi}\right|\right)}{\Gamma\left(\left|3 / 2-\eta_{\phi}\right|\right)}\right) \\ & \left.+\frac{(1+3 F)(1+4 F)+2 F \eta_{\phi}\left(2+F+6 F \eta_{\phi}\right)}{(1+3 F)\left[1+\left(3+2 \eta_{\phi}\right) F\right]}\right] \end{aligned}$
It is obvious that this result is different from equation (3.2. Tensor perturbation
For tensor perturbations, to solve equation (23 ), we use the same method as that solving the scalar perturbation. To the first order of εH, we have
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{z}_{t}^{{\prime\prime} }}{{z}_{t}} & \approx & {a}^{2}{H}^{2}\left[2-{\epsilon }_{H}+\displaystyle \frac{F{\epsilon }_{H}{\eta }_{\phi }(3-2{\eta }_{\phi })}{1+(3+2{\eta }_{\phi })F}\right]\\ & = & \displaystyle \frac{{\mu }^{2}-1/4}{{\tau }^{2}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\mu \approx \displaystyle \frac{3}{2}+\bar{\mu }{\epsilon }_{H},\end{eqnarray}$
and $\begin{eqnarray}\bar{\mu }=\displaystyle \frac{(3-2{\eta }_{\phi })[1+3F+F{\eta }_{\phi }(3+2{\eta }_{\phi })]}{3(1+2{\eta }_{\phi })[1+(3+2{\eta }_{\phi })F]}.\end{eqnarray}$
Like scalar perturbations, assuming that both ct and μ are constants, to the first order of εH, we obtain the power spectrum for the tensor perturbation,
$\begin{eqnarray}\begin{array}{l}{{ \mathcal P }}_{T}\approx \displaystyle \frac{{2}^{2\mu }}{{M}_{{pl}}^{2}}{\left(\displaystyle \frac{H}{2\pi }\right)}^{2}{\left[\displaystyle \frac{{\rm{\Gamma }}(\mu )}{{\rm{\Gamma }}(3/2)}\right]}^{2}{\left(1-\displaystyle \frac{{\epsilon }_{H}}{1+2{\eta }_{\phi }}\right)}^{2\mu -1}\\ \quad \times \left(\displaystyle \frac{1+3F+2F{\eta }_{\phi }-2F{\epsilon }_{H}}{1+(3+2{\eta }_{\phi })F}\right){\left(\displaystyle \frac{{c}_{t}k}{{aH}}\right)}^{(3-2\mu )}.\end{array}\end{eqnarray}$
Therefore, the tensor spectral tilt is $\begin{eqnarray}\begin{array}{rcl}{n}_{T} & = & {\left.\displaystyle \frac{\mathrm{dln}{{ \mathcal P }}_{T}}{\mathrm{dln}k}\right|}_{{aH}=\mathrm{const}}\\ & = & -\displaystyle \frac{2(3-2{\eta }_{\phi })[1+3F+F{\eta }_{\phi }(3+2{\eta }_{\phi })]}{3(1+2{\eta }_{\phi })[1+(3+2{\eta }_{\phi })F]}{\epsilon }_{H}.\end{array}\end{eqnarray}$
If ∣ηφ∣ ≪ 1, then we get nT ≈ −2εH.In the GR limit, we get
$\begin{eqnarray}{n}_{T}=-\displaystyle \frac{2(3-2{\eta }_{\phi })}{3(1+2{\eta }_{\phi })}{\epsilon }_{H}.\end{eqnarray}$
In the high friction limit, we get $\begin{eqnarray}{n}_{T}=-\displaystyle \frac{2(3-2{\eta }_{\phi })[3+{\eta }_{\phi }(3+2{\eta }_{\phi })]}{3(1+2{\eta }_{\phi })(3+2{\eta }_{\phi })}{\epsilon }_{H}.\end{eqnarray}$
To the first order of εH, the amplitude of the power spectrum for tensor perturbations becomes
$\begin{eqnarray}\begin{array}{l}{{ \mathcal P }}_{T}\left|{}_{{c}_{t}k={aH}}\approx \displaystyle \frac{8}{{M}_{{pl}}^{2}}{\left(\displaystyle \frac{H}{2\pi }\right)}^{2}\left[1-(A-B){\epsilon }_{H}\right]\right.,\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}A & = & \displaystyle \frac{2[3+C(3-2{\eta }_{\phi })]}{3(1+2{\eta }_{\phi })},\\ B & = & -\displaystyle \frac{2F[C{\eta }_{\phi }(3-2{\eta }_{\phi })+3]}{3[1+(3+2{\eta }_{\phi })F]},\end{array}\end{eqnarray}$
and the constant $C=-2+\gamma +\mathrm{ln}2\approx -0.73$.Combining the scalar power spectrum (49 ) and the tensor power spectrum (61 ), we obtain the tensor-to-scalar ratio
$\begin{eqnarray}\begin{array}{rcl}r & = & \displaystyle \frac{{{ \mathcal P }}_{T}{| }_{{c}_{t}k={aH}}}{{{ \mathcal P }}_{\zeta }{| }_{{c}_{s}k={aH}}}\\ & = & \displaystyle \frac{{2}^{7-| 3-2{\eta }_{\phi }| }{\left[{\rm{\Gamma }}(3/2)\right]}^{2}(1+3F)}{{\left[{\rm{\Gamma }}(| 3/2-{\eta }_{\phi }| )\right]}^{2}[1+(3+2{\eta }_{\phi })F]}{\epsilon }_{H}.\end{array}\end{eqnarray}$
It is apparent that the consistency relation r = − 8nT does not hold for the constant-roll inflation if ∣ηφ∣ is not small. If ∣ηφ∣ ≪ 1, then $\begin{eqnarray}r\approx 16{\epsilon }_{H},\end{eqnarray}$
the consistency relation still holds.In the GR limit, we get
$\begin{eqnarray}r=\displaystyle \frac{{2}^{7-| 3-2{\eta }_{\phi }| }{\left[{\rm{\Gamma }}(3/2)\right]}^{2}}{{\left[{\rm{\Gamma }}(| 3/2-{\eta }_{\phi }| )\right]}^{2}}{\epsilon }_{H}.\end{eqnarray}$
In the high friction limit, we get $\begin{eqnarray}r=\displaystyle \frac{3{\left[{\rm{\Gamma }}(3/2)\right]}^{2}{2}^{7-| 3-2{\eta }_{\phi }| }}{{\left[{\rm{\Gamma }}(| 3/2-{\eta }_{\phi }| )\right]}^{2}(3+2{\eta }_{\phi })}{\epsilon }_{H}.\end{eqnarray}$
4. The duality in constant-roll inflation
In the GR case, from equation (46 ), we see that if we neglect the contribution from εH, then we have the same scalar spectral tilt ns − 1 ≈ 2α when we replace ηφ = α by ηφ = 3 − α with ∣α∣ < 3/2. From equation (65 ), we see that the duality between ηφ = α and ηφ = 3 − α also exists for the tensor-to-scalar ratio r. The duality is also referred to as the duality between constant slow-roll and ultra-slow-roll inflation and explained as the two branches of the solution of the inflation field from the same potential [44], although the behaviors of background and perturbations are different for the constant slow-roll and ultra-slow-roll inflationary models if εH is not negligible [42]. In this section, we discuss this duality behavior, aiming to understand the connection between the slow-roll and ultra-slow-roll scenarios within the framework of inflationary models with non-minimally derivative coupling.
Combining equations (37 ) and (43 ), and neglecting the contribution from εH, we get63 ), we get69 ) and (70 ), we find that they are the same only if F = 0, i.e., the duality exists only in the GR case, and there is no such duality in inflationary models with non-minimally derivative coupling.
$\begin{eqnarray}{n}_{s}-1=3-| 3-2{\eta }_{\phi }| .\end{eqnarray}$
It is obvious that the duality between ηφ = α and ηφ = 3 − α holds, and $\begin{eqnarray}{n}_{s}-1\approx 2\alpha ,\end{eqnarray}$
if ∣α∣ < 3/2. Now we discuss whether the duality holds for the tensor-to-scalar ratio r. For the constant slow-roll inflation with ηφ = α, from equation ( $\begin{eqnarray}r=\displaystyle \frac{{2}^{4+2\alpha }{\left[{\rm{\Gamma }}(3/2)\right]}^{2}(1+3F)}{{\left[{\rm{\Gamma }}(3/2-\alpha )\right]}^{2}[1+(3+2\alpha )F]}{\epsilon }_{H}.\end{eqnarray}$
For the ultra-slow-roll inflation with ηφ = 3 − α, the tensor-to-scalar ratio becomes $\begin{eqnarray}r=\displaystyle \frac{{2}^{4+2\alpha }{\left[{\rm{\Gamma }}(3/2)\right]}^{2}(1+3F)}{{\left[{\rm{\Gamma }}(3/2-\alpha )\right]}^{2}[1+(9-2\alpha )F]}{\epsilon }_{H}.\end{eqnarray}$
Comparing equations (When the slow-roll parameter εH is extremely small and can be neglected, the scalar spectral tilts obtained from both the slow-roll inflation and the ultra-slow-roll inflation coincide. In the limit where F = 0 (GR limit), the tensor-to-scalar ratio (69 ) derived from slow-roll inflation matches equation (70 ) obtained from ultra-slow-roll inflation, thus the duality recovers in the GR limit. However, in a more general case, the tensor-to-scalar ratios r from slow-roll inflation and ultra-slow-roll inflation are different. Consequently, even for the scalar spectral tilt ns and the tensor-to-scalar ratio r, the duality behavior between slow-roll and ultra-slow-roll inflation models with non-minimally derivative coupling is not observed. Therefore, the duality of ns and r is not a universal feature in the constant-roll inflation models.
5. Conclusion
We study the constant-roll inflation model with the kinetic term non-minimally derivative coupling to the Einstein tensor. For the constant-roll condition, we choose the slow-roll parameter ${\eta }_{\phi }=-\ddot{\phi }/(H\dot{\phi })$ to be a constant. With this constant-roll condition, we calculate the power spectra for the scalar and tensor perturbations and subsequently derive the scalar spectral tilt ns, tensor spectral tilt nT, and tensor-to-scalar ratio r. Due to the effect of large ηφ, ${\dot{\epsilon }}_{H}$ remains to be the first order, the expressions for ns are different with different ordering of taking the derivative of the scalar power spectrum with respect to the scale k and the horizon crossing condition csk = aH in the constant-roll inflation, and the consistency relation between the tensor-to-scalar ratio and tensor spectral tilt, r = − 8nT, does not hold. Between the slow-roll inflation with ηφ = α being small and the ultra-slow-roll inflation with ηφ = 3 − α, although the duality for the scalar spectral tilt still holds if we neglect the contribution of εH to ns, but the results for the tensor-to-scalar ratio r are different. Thus, the duality of ns and r is not a universal feature in inflation with non-minimally derivative coupling.
The inflation model with non-minimally derivative coupling is important because the new Higgs inflation model saves the Higgs inflation from unitarity bound violations. This investigation into constant-roll inflation presents a different perspective from the typical slow-roll inflation scenario, thereby enhancing our understanding of inflation models with non-minimally derivative coupling. Scalar-induced gravitational waves usually originate from inflation models featuring a transitional ultra-slow-roll phase. Therefore, this investigation into the ultra-slow-roll condition is also helpful for the issue of scalar-induced gravitational waves in the inflation model with non-minimally derivative coupling.