1 Introduction

Inflation in the early Universe has become a well established part of modern cosmology, it naturally generates the density perturbations that become the seeds of large-scale structure and temperature anisotropies of the cosmic microwave background (CMB). The predictions have been confirmed by numerous observations, such as the WMAP [1] and Planck space missions  [2]. Recently, the Planck 2018 data constrain the scalar spectral index and the tensor-to-scalar ratio to be \(n_{\mathcal {R}}=0.9649\pm 0.0042\) at \(68\%\) confidence level and \(r_{0.002}<0.064\) at \(95\%\) confidence level [2], which will narrow down some inflationary models.

In the conventional inflationary models, it is usually assumed that the scalar field rolls down slowly, or more precisely to assumed that the flow parameters satisfy the conditions \(\epsilon _H\ll 1\) and \(\eta _H\ll 1\). This so called slow-roll inflation leads to a nearly scale-invariant spectrum of density perturbations and is consistent with the latest observations. Recently, a new type of inflation called constant-roll inflation has been put forward in  [3], where one of the flow parameters \(\eta _H\) is assumed to be a constant, and not necessarily small. Such class of models can also produce a scale-invariant spectrum and is compatible with the observational constraints without the assumption of slow-roll approximation. The constant-roll condition with small \(\eta _H\) is compatible with the slow-roll inflation, and a special value \(\eta _H=3\) amounts to the ultra-slow-roll inflation  [4, 5], in which the potential is very flat and the inflaton almost stops rolling.The constant-roll inflationary models are discussed in more details in  [6,7,8], and extended to modified theories of gravity [9,10,11,12,13,14] or two fields case [15], and using other constant-roll conditions  [16,17,18]. Other developments are appeared in  [19,20,21,22].

Since the inflation occurs in the early Universe which is believed to be described by quantum gravity, so it is interest to discuss inflation in the framework of quantum gravity theories, such as string theory. It is known that when discuss the effective action in the early universe, the correction terms of higher orders in the curvature coming from superstrings may play a significant role, and the simplest of such correction is the Gauss–Bonnet (GB) term in the low-energy effective action of the heterotic string [23, 24]. There are many works discussing accelerating cosmology with the GB correction in four and higher dimensions [25,26,27,28], and with nonminimal coupled to the inflaton [29,30,31,32,33,34,35,36,37] or to the Higgs [38,39,40,41,42].

In this work, we shall discuss the constant-roll inflation in the model with the inflation field \(\phi \) nonminimal coupled to the GB term with one of the flow parameters, \(\epsilon _1\), \(\epsilon _2\) or \(\eta _H\) is constant. The presence of such nonminimal coupling will generate a new degree of freedom, so here we consider an additional assumption that the first GB flow parameter \(\delta _1\) is also a constant. We derive the analytical expressions of the scalar spectral index \(n_{\mathcal {R}}\) and the tensor to scalar ratio r for each case. Combine with the Planck 2018 observations, we obtain some feasible parameter spaces, and find that in some cases, the chosen of GB flow parameter \(\delta _1>1\) can also produce a nearly scale-invariant scalar power spectrum, which is different from the slow-roll inflation. At last, we also reconstruct the potential in some spectral cases.

The outline of this paper is as follows: in the next section, we briefly review the inflationary model with nonminimal coupling to the Gauss–Bonnet term. In Sections 3, 4 and 5, we focus on the inflationary models with constant \(\epsilon _1\), constant \(\epsilon _2\) or constant \(\eta _H\), respectively, derive the formalism for the scalar spectral index and the tensor-to-scalar ratio of each model and constrain the parameter space with the Planck 2018 data. In Sect. 6, we try to reconstruct the potential in some special cases. And the final section is devoted to summary.

2 Inflation with a Gauss–Bonnet coupling

In this section, we shall review the inflation with the GB coupling, and present the mode equations of scalar and tensor perturbations. Consider the following action with the inflation field \(\phi \) coupled to the GB term

$$\begin{aligned}&S=\int d^{4} x \sqrt{-g}\left[ \frac{1}{2} R-\frac{\omega }{2}(\nabla \phi )^{2}-V(\phi )-\frac{1}{2} \xi (\phi ) R_{\mathrm {GB}}^{2}\right] ,\nonumber \\ \end{aligned}$$
(1)

where \(V(\phi )\) is the potential for the scalar field \(\phi \), \(R_{\mathrm {GB}}^{2}=R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }-4R_{\mu \nu }R^{\mu \nu }+R^2\) is the GB term, and \(\xi (\phi )\) is the coupling functional. The coefficient \(\omega \) in the kinetic term can be chosen the values \(\pm 1\) to ensure conventional inflationary behaviour [29, 30]. We work in Planckian units, \(\hbar =c=8 \pi G=1\).

In the Friedmann–Robertson–Walker homogeneous universe, the background equations can be written as

$$\begin{aligned}&6 H^{2}=\omega \dot{\phi }^{2}+2 V+24 \dot{\xi } H^{3} , \end{aligned}$$
(2)
$$\begin{aligned}&2 \dot{H}=-\omega \dot{\phi }^{2}+4 \ddot{\xi } H^{2}+4 \dot{\xi } H(2 \dot{H}-H^{2}) , \end{aligned}$$
(3)
$$\begin{aligned}&\omega (\ddot{\phi }+3 H \dot{\phi })+V_{, \phi }+12 \xi _{, \phi } H^{2}(\dot{H}+H^{2})=0, \end{aligned}$$
(4)

where a dot represents a derivative with respect to cosmic time and \((...)_{,\phi }\) denotes a derivative with respect to the field \(\phi \).

In standard inflation, it is useful to define a series of flow parameters, such as the Hubble flow parameters

$$\begin{aligned}&\epsilon _{H}=-\frac{\dot{H}}{H^{2}}, \quad \eta _{H}=-\frac{\ddot{H}}{2H\dot{H}}, \end{aligned}$$
(5)

furthermore, we also introduce the horizon flow parameters

$$\begin{aligned}&\epsilon _{1}=-\frac{\dot{H}}{H^{2}}, \quad \epsilon _{i+1}=\frac{\dot{\epsilon _{i}}}{H \epsilon _{i}}, \quad i \ge 1. \end{aligned}$$
(6)

In the presence of the GB coupling, the new degrees of freedom suggest to defining another hierarchy of flow parameters [30]

$$\begin{aligned}&\delta _{1}=4 \dot{\xi } H, \quad \delta _{i+1}=\frac{\dot{\delta _{i}}}{H \delta _{i}}, \quad i \ge 1. \end{aligned}$$
(7)

For the slow-roll inflation, these flow parameters should satisfy the slow-roll conditions \(|\epsilon _i|\ll 1\) and \(|\delta _i|\ll 1\). However, in the constant-roll inflation, such conditions are not necessary to satisfy, we only need \(0<\epsilon _1<1\) to achieve a significative inflation.

At linear order in perturbation theory, the Fourier modes of curvature perturbations satisfy the Mukhanov–Sasaki equation

$$\begin{aligned}&v^{\prime \prime }+\left( c_{\mathcal {R}}^{2} k^{2}-\frac{z_{\mathcal {R}}^{\prime \prime }}{z_{\mathcal {R}}}\right) v=0, \end{aligned}$$
(8)

where a prime represents the derivative with respect to conformal time \(\tau \). \(z_{\mathcal {R}}\) and the sound speed \(c_{\mathcal {R}}\) can be expressed in terms of the horizon and GB flow parameters as

$$\begin{aligned} z_{\mathcal {R}}^{2}= & {} a^{2} \frac{F}{\left( 1-\frac{1}{2} \Delta \right) ^{2}}, \end{aligned}$$
(9)
$$\begin{aligned} c_{\mathcal {R}}^{2}= & {} 1-\Delta ^{2} \frac{2 \epsilon _{1}+\frac{1}{2} \delta _{1}\left( 1-5 \epsilon _{1}-\delta _{2}\right) }{F}, \end{aligned}$$
(10)

with \(\Delta =\delta _{1} /\left( 1-\delta _{1}\right) \) and \(F \equiv 2 \epsilon _{1}-\delta _{1}\left( 1+\epsilon _{1}-\delta _{2}\right) +\frac{3}{2} \Delta \delta _{1}\). And the effective mass term reads

$$\begin{aligned} \frac{z_{\mathcal {R}}^{\prime \prime }}{z_{\mathcal {R}}}= & {} a^{2} H^{2}\left[ 2-\epsilon _{1}+\frac{3}{2} \frac{\dot{F}}{H F}+\frac{3}{2} \frac{\dot{\Delta }}{H\left( 1-\frac{1}{2} \Delta \right) }+\frac{1}{2} \frac{\ddot{F}}{H^{2} F}\right. \nonumber \\&\left. +\frac{1}{2} \frac{\ddot{\Delta }}{H^{2}\left( 1-\frac{1}{2} \Delta \right) } -\frac{1}{4} \frac{\dot{F}^{2}}{H^{2} F^{2}} +\frac{1}{2} \frac{\dot{\Delta }^{2}}{H^{2}\left( 1-\frac{1}{2} \Delta \right) ^{2}}\right. \nonumber \\&\left. +\frac{1}{2} \frac{\dot{\Delta }}{H\left( 1-\frac{1}{2} \Delta \right) } \frac{\dot{F}}{H F}\right] , \end{aligned}$$
(11)

with

$$\begin{aligned} \frac{\dot{F}}{H}= & {} \epsilon _{1} \epsilon _{2}\left( 2-\delta _{1}\right) -\delta _{1} \delta _{2}\left( 1+\epsilon _{1}-\delta _{2}-\delta _{3}\right) \nonumber \\&+\frac{3}{2} \Delta \delta _{2}\left( \Delta +\delta _{1}\right) ,\nonumber \\ \frac{\dot{\Delta }}{H}= & {} \Delta ^{2} \frac{\delta _{2}}{\delta _{1}},\nonumber \\ \ddot{H}= & {} \epsilon _{1} \epsilon _{2}\left( -\epsilon _{1}+\epsilon _{2}+\epsilon _{3}\right) \left( 2-\delta _{1}\right) \nonumber \\&+\epsilon _{1} \delta _{1} \delta _{2}\left( 1+\epsilon _{1}-2 \epsilon _{2}-\delta _{2}-\delta _{3}\right) \nonumber \\&-\delta _{1} \delta _{2}^{2}\left( 1+\epsilon _{1}-\delta _{2}-\delta _{3}\right) \nonumber \\&-\delta _{1} \delta _{2} \delta _{3}\left( 1+\epsilon _{1}-2 \delta _{2}-\delta _{3}-\delta _{4}\right) \nonumber \\&+\frac{3}{2} \Delta \delta _{2}\left( \Delta +\delta _{1}\right) \left( -\epsilon _{1}+\Delta \frac{\delta _{2}}{\delta _{1}}+\delta _{3}\right) \nonumber \\&+\frac{3}{2} \Delta \delta _{2}\left( \Delta ^{2} \frac{\delta _{2}}{\delta _{1}}+\delta _{1} \delta _{2}\right) ,\nonumber \\ \ddot{\Delta }= & {} \Delta ^{2} \frac{\delta _{2}}{\delta _{1}}\left( -\epsilon _{1}+2 \Delta \frac{\delta _{2}}{\delta _{1}}-\delta _{2}+\delta _{3}\right) . \end{aligned}$$
(12)

Similarly, for the tensor perturbations, the Fourier modes satisfy

$$\begin{aligned}&u^{\prime \prime }+\left( c_{T}^{2} k^{2}-\frac{z_{T}^{\prime \prime }}{z_{T}}\right) u=0, \end{aligned}$$
(13)

with

$$\begin{aligned}&z_{T}^{2}=a^{2}\left( 1-\delta _{1}\right) , \end{aligned}$$
(14)
$$\begin{aligned}&c_{T}^{2}=1+\Delta \left( 1-\epsilon _{1}-\delta _{2}\right) . \end{aligned}$$
(15)

And the effective mass term in the tensor mode can be written in terms of the flow parameters

$$\begin{aligned} \frac{z_{T}^{\prime \prime }}{z_{T}}=&a^{2} H^{2}\left[ 2-\epsilon _{1}-\frac{3}{2} \Delta \delta _{2}-\frac{1}{2} \Delta \delta _{2}\left( -\epsilon _{1}+\delta _{2}+\delta _{3}\right) -\frac{1}{4} \Delta ^{2} \delta _{2}^{2}\right] .\nonumber \\ \end{aligned}$$
(16)

In the following sections, we shall focus on the constant-roll inflation in the model with nonminimal coupling to the GB term.

3 Constant-roll inflation with constant \(\epsilon _1\)

We focus on the constant-roll inflation in the model with nonminimal coupling to the GB term. The presence of such coupling will generate a new degree of freedom \(\xi (\phi )\), so in this work we consider an additional assumption that the first GB flow parameter \(\delta _1\) is also a constant, then from the definition of GB flow parameters (7) we have \(\delta _i=0(i\ge 2)\).

In the following, we first discuss the case with \(\epsilon _1=\) constant. From the relations

$$\begin{aligned} \frac{d}{d \tau }\left( \frac{1}{a H}\right) =-1+\epsilon _{1}, \end{aligned}$$
(17)

and assuming that \(\epsilon _1\) is a constant, one can express the factor aH in (11) and (16) as a function of conformal time \(\tau \)

$$\begin{aligned} aH=-\frac{1}{\tau }\left( \frac{1}{1-\epsilon _1}\right) , \end{aligned}$$
(18)

then

$$\begin{aligned} \nu _{\mathcal {R}}^2=\tau ^2\frac{z_{\mathcal {R}}^{\prime \prime }}{z_{\mathcal {R}}}+\frac{1}{4}, \end{aligned}$$
(19)

can be approximated as a constant. Therefore the general solution to the mode equation (8) is a linear combination of Hankel functions of order \(\nu _{\mathcal {R}}\)

$$\begin{aligned} v=\frac{\sqrt{\pi |\tau |}}{2} e^{i\left( 1+2 \nu _{\mathcal {R}}\right) \pi / 4}\left[ c_{1} H_{\nu _{\mathcal {R}}}^{(1)}\left( c_{\mathcal {R}} k|\tau |\right) +c_{2} H_{\nu _{\mathcal {R}}}^{(2)}\left( c_{\mathcal {R}} k|\tau |\right) \right] \end{aligned}$$
(20)

Choosing \(c_1=1\) and \(c_2=0\), the usual Minkowski vacuum state is recovered in the asymptotic past. Then on super-horizon scales, \(c_{\mathcal {R}}k\ll aH\), the power spectrum of the scalar perturbation is

$$\begin{aligned} \mathcal {P}_{\mathcal {R}}= & {} \frac{c_{\mathcal {R}}^{-3}}{F} \frac{H^{2}}{4 \pi ^{2}}\left( \frac{1-\Delta / 2}{a H|\tau |}\right) ^{2} \frac{\Gamma ^{2}\left( \nu _{\mathcal {R}}\right) }{\Gamma ^{2}(3 / 2)}\left( \frac{c_{\mathcal {R}} k|\tau |}{2}\right) ^{3-2 \nu _{\mathcal {R}}} \nonumber \\&\left. \simeq \frac{2^{2 \nu _{\mathcal {R}}-3} c_{\mathcal {R}}^{-3}}{F} \frac{H^{2}}{4 \pi ^{2}} \frac{\Gamma ^{2}\left( \nu _{\mathcal {R}}\right) }{\Gamma ^{2}(3 / 2)}\left( 1-\frac{\Delta }{2}\right) ^2\right. \nonumber \\&\left. \times \left( \frac{1}{1-\epsilon _1}\right) ^{1-2\nu _{\mathcal {R}}}\right| _{c_{\mathcal {R}} k=a H}, \end{aligned}$$
(21)

with the scalar spectral index is

$$\begin{aligned} n_{\mathcal {R}}-1=\frac{d\ln \mathcal {P}_{\mathcal {R}}}{d\ln k}=3-2\nu _{\mathcal {R}}. \end{aligned}$$
(22)

Using the same procedure as in the case of scalar perturbations, we get the power spectrum of tensor perturbations

$$\begin{aligned} \mathcal {P}_{T}= & {} \frac{8 c_{T}^{-3}}{1-\delta _{1}} \frac{H^{2}}{4 \pi ^{2}}\left( \frac{1}{a H|\tau |}\right) ^{2} \frac{\Gamma ^{2}\left( \nu _{T}\right) }{\Gamma ^{2}(3 / 2)}\left( \frac{c_{T} k|\tau |}{2}\right) ^{3-2 \nu _{T}}\nonumber \\&\left. \simeq 2^{2 \nu _{T}} c_{T}^{-3} \frac{H^{2}}{4 \pi ^{2}} \frac{\Gamma ^{2}\left( \nu _{T}\right) }{\Gamma ^{2}(3 / 2)}\left( \frac{1}{1-\delta _{1}}\right) \left( \frac{1}{1-\epsilon _1}\right) ^{1-2\nu _{T}}\right| _{c_{T} k=a H},\nonumber \\ \end{aligned}$$
(23)

with

$$\begin{aligned} \nu _{T}^2=\tau ^2\frac{z_{T}^{\prime \prime }}{z_{T}}+\frac{1}{4}, \end{aligned}$$
(24)

and the tensor spectral index is

$$\begin{aligned} n_{T}=\frac{d\ln \mathcal {P}_{T}}{d\ln k}=3-2\nu _{T}. \end{aligned}$$
(25)

Combing Eqs. (21) and (23), we obtain the tensor to scalar ratio

$$\begin{aligned} r \equiv \frac{\mathcal {P}_{T}}{\mathcal {P}_{\mathcal {R}}} \simeq 2^{3+2 \nu _{T}-2 \nu _{\mathcal {R}}}F \frac{c_{\mathcal {R}}^{3}}{c_{T}^{3}} \frac{\Gamma ^{2}\left( \nu _{T}\right) }{\Gamma ^{2}\left( \nu _{\mathcal {R}}\right) }\frac{\left( 1-\epsilon _1\right) ^{2\nu _T-2\nu _{\mathcal {R}}}}{\left( 1-\Delta / 2\right) ^2 (1-\delta _1)}. \end{aligned}$$
(26)

Below we will assume that the first GB flow parameter \(\delta _1\) is a constant, which contains two cases: \(\delta _1=0\) or \(\delta _1\ne 0\).

3.1 \(\delta _1=0\)

If \(\delta _1=0\), using (22), (26) we get

$$\begin{aligned}&n_{\mathcal {R}}=4-\Big |\frac{3-\epsilon _1}{1-\epsilon _1}\Big |, \end{aligned}$$
(27)
$$\begin{aligned}&r=16 \epsilon _1. \end{aligned}$$
(28)

The \(n_{\mathcal {R}}-r\) region predicted by the model with \(\epsilon _1=\) constant and \(\delta _1=0\) are show in Fig. 1, where the contours are the marginalized joint 68% and 95% confidence level regions for \(n_{\mathcal {R}}\) and r at the pivot scale \(k_*= 0.002\) Mpc\(^{-1}\) from the Planck 2018 TT, TE, EE + lowE + lensing data.

Fig. 1
figure 1

The \(n_{\mathcal {R}}-r\) region predicted by the model with \(\epsilon _1=\) constant and \(\delta _1=0\) . The contours are the marginalized joint 68% and 95% confidence level regions for \(n_{\mathcal {R}}\) and r at the pivot scale \(k_*= 0.002\) Mpc\(^{-1}\) from the Planck 2018 TT, TE, EE + lowE + lensing data [2]

Full size image

We could see that the curve is ruled out by the observations, so we didn’t interest in this case.

3.2 \(\delta _1\ne 0\)

If \(\delta _1=\) constant but not zero, the expression of scalar spectral index is the same as in the previous case

$$\begin{aligned} n_{\mathcal {R}}=4-\Big |\frac{3-\epsilon _1}{1-\epsilon _1}\Big |, \end{aligned}$$
(29)

and the tensor-to-scalar ratio is more complex

$$\begin{aligned} r&=\frac{\left( 16\left( \left( 2 \delta _{1}^{2}-6 \delta _{1}+4\right) \epsilon _{1}+5 \delta _{1}^{2}-2 \delta _{1}\right) \right) }{\left( 2-3 \delta _{1}\right) ^{2}}\nonumber \\&\quad \times \left( \frac{\left( 3 \delta _{1}-2\right) \left( \left( \delta _{1}^{2}+2 \delta _{1}-2\right) \epsilon _{1}-2 \delta _{1}^{2}+\delta _{1}\right) }{\left( 1-\delta _{1} \epsilon _{1}\right) \left( \left( 2 \delta _{1}^{2}-6 \delta _{1}+4\right) \epsilon _{1}+5 \delta _{1}^{2}-2 \delta _{1}\right) }\right) ^{3 / 2} \end{aligned}$$
(30)

Combine these expressions with the observational constraints from Planck 2018, one can obtain the constraints on \(\epsilon _1\) and \(\delta _1\). However, for a significative inflation we have \(\ddot{a}>0\), which is equivalent to \(\epsilon _1<1\), so only the parameter region satisfies \(\epsilon _1<1\) is reasonable. We then obtain two regions of parameter space, and show them in Fig. 2.

Fig. 2
figure 2

The observational constraints on \(\epsilon _1\) and \(\delta _1\). The orange and blue regions correspond to the parameters satisfied \(1\sigma \) and \(2\sigma \) confidence level, respectively

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In Fig. 3, we show the \(n_{\mathcal {R}}-r\) region predicted by the model for the parameters in region I of Fig. 2.

Fig. 3
figure 3

The \(n_{\mathcal {R}}-r\) region predicted by the model for the parameter region I of Fig. 2

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Left panel: The parameter \(\delta _1\) is taken as \(\delta _1=0.04,0.03,0.02\) from left to right, and as \(\epsilon _1\) increase, the \(n_{\mathcal {R}}-r\) dots go along the curves to the left. Right panel: The parameter \(\epsilon _1\) is taken as \(\epsilon _1=0.02,0.015,0.01\) from left to right, and as \(\delta _1\) increase, the \(n_{\mathcal {R}}-r\) dots go along the curves from top to bottom. The \(n_{\mathcal {R}}-r\) region for the parameters in region II of Fig. 2 are shown In Fig. 4.

Fig. 4
figure 4

The \(n_{\mathcal {R}}-r\) region predicted by the model for the parameter region II of Fig. 2

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Left panel: The parameter \(\delta _1\) is taken as \(\delta _1=0.49,0.5,0.51\) from left to right, and as \(\epsilon _1\) increase, the \(n_{\mathcal {R}}-r\) dots go along the curves to the left. Right panel: The parameter \(\epsilon _1\) is taken as \(\epsilon _1=0.02,0.015,0.01\) from left to right. We could see that this panel is almost the same as the one in Fig. 3, that is because the scalar spectral index depends only on \(\epsilon _1\), so for the same chosen of \(\epsilon _1\), the curves look the same. However, different from that panel, in this time the \(n_{\mathcal {R}}-r\) dots go along the curves from bottom to top as \(\delta _1\) increase.

We could see that for the choices of parameter space in Fig. 2, the inflation prediction is in agreement with the observational constraints. However, the constant-roll inflation with \(\epsilon _1=\) constant is no natural end, hence an additional mechanism is required to stop it.

4 Constant-roll inflation with constant \(\epsilon _2\)

In this section, we assume that the second horizon flow parameter \(\epsilon _2\) is a constant. So from the relations (17) and using the definition of flow parameter (6), we obtain that to the first order approximation of \(\epsilon _1\), the relation between aH and \(\tau \) is [16, 18]

$$\begin{aligned}&aH\simeq -\frac{1}{\tau }\left( 1+\frac{\epsilon _1}{1-\epsilon _2}\right) , \end{aligned}$$
(31)

then, we obtain the power spectrum of the scalar perturbation and tensor perturbation

$$\begin{aligned} \mathcal {P}_{\mathcal {R}}&\left. \simeq \frac{2^{2 \nu _{\mathcal {R}}-3} c_{\mathcal {R}}^{-3}}{F} \frac{H^{2}}{4 \pi ^{2}} \frac{\Gamma ^{2}\left( \nu _{\mathcal {R}}\right) }{\Gamma ^{2}(3 /2)}\left( 1-\frac{\Delta }{2}\right) ^2\right. \nonumber \\&\quad \left. \times \left( 1+\frac{\epsilon _1}{1-\epsilon _2}\right) ^{1-2\nu _{\mathcal {R}}}\right| _{c_{\mathcal {R}} k=a H}, \end{aligned}$$
(32)
$$\begin{aligned} \mathcal {P}_{T}&\left. \simeq 2^{2 \nu _{T}} c_{T}^{-3} \frac{H^{2}}{4 \pi ^{2}} \frac{\Gamma ^{2}\left( \nu _{T}\right) }{\Gamma ^{2}(3 / 2)}\left( \frac{1}{1-\delta _{1}}\right) \right. \nonumber \\&\quad \left. \times \left( 1+\frac{\epsilon _1}{1-\epsilon _2}\right) ^{1-2\nu _{T}}\right| _{c_{T} k=a H}, \end{aligned}$$
(33)

with the scalar spectral index and the tensor-to-scalar ratio are

$$\begin{aligned}&n_{\mathcal {R}}-1=\frac{d\ln \mathcal {P}_{\mathcal {R}}}{d\ln k}=3-2\nu _{\mathcal {R}}, \end{aligned}$$
(34)
$$\begin{aligned}&r \equiv \frac{\mathcal {P}_{T}}{\mathcal {P}_{\mathcal {R}}} \simeq 2^{3+2 \nu _{T}-2 \nu _{\mathcal {R}}}F \frac{c_{\mathcal {R}}^{3}}{c_{T}^{3}} \frac{\Gamma ^{2}\left( \nu _{T}\right) }{\Gamma ^{2}\left( \nu _{\mathcal {R}}\right) }\frac{\left( 1+\frac{\epsilon _1}{1-\epsilon _2}\right) ^{2 \nu _{\mathcal {R}}-2 \nu _{T}}}{\left( 1-\Delta / 2\right) ^2 (1-\delta _1)}.\nonumber \\ \end{aligned}$$
(35)

In the following, we still assume that the first GB flow parameter \(\delta _1\) is a constant, which contains two cases: \(\delta _1=0\) or \(\delta \ne 0\).

4.1 \(\delta _1=0\)

If \(\delta _1=0\), the model recovers to the case without GB coupling, and such case have been discussed in other works [16]. Here we list some useful results.

To the first order approximation of \(\epsilon _1\), the scalar spectral index can be approximated as

$$\begin{aligned} n_{\mathcal {R}}\simeq 4-|\epsilon _2+3|+\frac{\left( 2 \epsilon _2^2+7 \epsilon _2+6\right) \epsilon _1}{(\epsilon _2-1)|\epsilon _2+3|}, \end{aligned}$$
(36)

and the tensor-to-scalar ratio

$$\begin{aligned} r \simeq 2^{3-\left| 3+\epsilon _{2}\right| }\left( \frac{\Gamma [3 / 2]}{\Gamma \left[ \left| 3+\epsilon _{2}\right| / 2\right] }\right) ^{2} 16 \epsilon _{1}. \end{aligned}$$
(37)

Since \(\epsilon _{2}\) is a constant, we can integer the definition (6) and get the relation between \(\epsilon _1\) and \(\epsilon _2\)

$$\begin{aligned} \epsilon _{1}(N)=\exp \left( -\epsilon _{2} N\right) , \end{aligned}$$
(38)

where N is the e-folding number before the end of inflation, and we have use the condition \(\epsilon _1 (N=0)=1\) at the end of inflation. The \(n_{\mathcal {R}}-r\) predictions compare with the Planck 2018 data are show in Fig. 5 (blue line) with \(N=60\). We can see that the result is ruled out by the observations. So it is always to assume that the second Hubble flow parameter \(\eta _H\) is a constant in the constant-roll inflationary models, we will discuss this case in the next section.

Fig. 5
figure 5

The \(n_{\mathcal {R}}-r\) region predicted by the model with \(\delta _1=0\) and constant \(\epsilon _1\) (blue line) or constant \(\eta _H\) (orange line)

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4.2 \(\delta _1\ne 0\)

If \(\delta _1=\)constant but not zero, to the first order of \(\epsilon _1\), the expression of scalar spectral index is

$$\begin{aligned} n_{\mathcal {R}}\simeq 1-\frac{2\left( 3+\epsilon _{2}\right) \left( -2 \epsilon _{2}+2 \epsilon _{2}^{2}+\delta _{1}\left( 2+3 \epsilon _{2}-3 \epsilon _{2}^{2}\right) +\delta _{1}^{2}\left( -5-\epsilon _{2}+\epsilon _{2}^{2}\right) \right) }{3\left( 2 \delta _{1}-5 \delta _{1}^{2}\right) \left( 1-\epsilon _{2}\right) } \epsilon _{1}, \end{aligned}$$
(39)

and the tensor-to-scalar ratio

$$\begin{aligned} r\simeq & {} -\frac{16}{3\left( 3 \delta _{1}-2\right) \left( 5 \delta _{1}-2\right) } \sqrt{\frac{2-7\delta _{1}+6 \delta _{1}^2}{2-5 \delta _{1}}}\Big (30 \delta _{1}^{3}-27 \delta _{1}^{2}+6 \delta _{1}\nonumber \\&+\epsilon _{1}\left( 45 \delta _{1}^{4}-69 \delta _{1}^{3}-6 \delta _{1}^{2}+42 \delta _{1}-12\right. \nonumber \\&\left. +\left( -2 \delta _{1}^{3}+7 \delta _{1}^{2}-7 \delta _{1}+2\right) \epsilon _{2}\left( \epsilon _{2}+3\right) \ln 4\right. \nonumber \\&\left. +2\left( 2 \delta _{1}^{3}-7 \delta _{1}^{2}+7 \delta _{1}-2\right) \epsilon _{2}\left( \epsilon _{2}+3\right) (\gamma -2+\ln 4)\right) \Big ),\nonumber \\ \end{aligned}$$
(40)

with \(\gamma \approx 0.577216\) is the Euler–Mascheroni constant. Combine these expressions with the observational constraints and adopting the relation \(\epsilon _{1}(N)=\exp \left( -\epsilon _{2} N\right) \), one can obtain the constraints on \(\epsilon _1\) and \(\delta _1\). By setting the e-folding number \(N=60\), we find three regions of parameter space, and show them in Fig. 6.

Fig. 6
figure 6

The observational constraints on \(\epsilon _2\) and \(\delta _1\) with the e-folding number \(N=60\). The orange and blue regions correspond to the parameters satisfied \(1\sigma \) and \(2\sigma \) confidence level, respectively

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The observational constraint on \(n_{\mathcal {R}}-r\) for the parameters in region I of Fig. 6 are show in Fig. 7. Left panel: The parameter \(\delta _1\) is taken as \(\delta _1=-0.01,-0.006,-0.002\) from top to bottom, and as \(\epsilon _1\) increase, the \(n_{\mathcal {R}}-r\) dots go along the curves from left to right. Right panel: The parameter \(\epsilon _1\) is taken as \(\epsilon _1=0.12,0.16,0.2\) from left to right, and as \(\delta _1\) increase, the \(n_{\mathcal {R}}-r\) dots go along the curves to the left.

Fig. 7
figure 7

The \(n_{\mathcal {R}}-r\) region predicted by the model for the parameter region I of Fig. 6

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Similarly, in Fig. 8, we show the \(n_{\mathcal {R}}-r\) region predicted for the parameters in region II of Fig. 6. Left panel: The parameter \(\delta _1\) is taken as \(\delta _1=0.1,0.09,0.08\) from left to right, and as \(\epsilon _1\) increase, the \(n_{\mathcal {R}}-r\) dots go along the curves from top to bottom. Right panel: \(\epsilon _1\) is taken as \(\epsilon _1=0.05,0.052,0.054\) from left to right, and as \(\delta _1\) increase, the dots go along the curves from top to bottom.

Fig. 8
figure 8

The \(n_{\mathcal {R}}-r\) region predicted by the model for the parameter region II of Fig. 6

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Finally, the \(n_{\mathcal {R}}-r\) region for the parameters in region III of Fig. 6 are show in Fig. 9. With \(\delta _1\) is taken as \(\delta _1=0.505,0.503,0.501\) from left to right in the left panel, and as \(\epsilon _1\) increase, the \(n_{\mathcal {R}}-r\) dots go along the curves to the right. In the right panel, \(\epsilon _1\) is taken as \(\epsilon _1=0.065,0.07,0.075\) from left to right, and the dots go along the curves from bottom to top as \(\delta _1\) increase.

Fig. 9
figure 9

The \(n_{\mathcal {R}}-r\) region predicted by the model for the parameter region III of Fig. 6

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5 Constant-roll inflation with constant \(\eta _H\)

In this section, the second Hubble flow parameter \(\eta _{H}=-\ddot{H}/(2H\dot{H})\) is assumed to be a constant. Combine with the Eq.(17), the relation between aH and \(\tau \) to the first order approximation of \(\epsilon _1\) can be obtained as [18]

$$\begin{aligned} a H \approx -\frac{1}{\tau }\left( 1+\frac{\epsilon _{1}}{1+2 \eta _{H}}\right) . \end{aligned}$$
(41)

Substitute this relation to the expressions of power spectrum, we get

$$\begin{aligned} \mathcal {P}_{\mathcal {R}}&\left. \simeq \frac{2^{2 \nu _{\mathcal {R}}-3} c_{\mathcal {R}}^{-3}}{F} \frac{H^{2}}{4 \pi ^{2}} \frac{\Gamma ^{2}\left( \nu _{\mathcal {R}}\right) }{\Gamma ^{2}(3 / 2)}\left( 1-\frac{\Delta }{2}\right) ^2\right. \nonumber \\&\left. \quad \times \left( 1+\frac{\epsilon _{1}}{1+2 \eta _{H}}\right) ^{1-2\nu _{\mathcal {R}}}\right| _{c_{\mathcal {R}} k=a H}, \end{aligned}$$
(42)
$$\begin{aligned} \mathcal {P}_{T}&\left. \simeq 2^{2 \nu _{T}} c_{T}^{-3} \frac{H^{2}}{4 \pi ^{2}} \frac{\Gamma ^{2}\left( \nu _{T}\right) }{\Gamma ^{2}(3 / 2)}\left( \frac{1}{1-\delta _{1}}\right) \right. \nonumber \\&\left. \quad \times \left( 1+\frac{\epsilon _{1}}{1+2 \eta _{H}}\right) ^{1-2\nu _{T}}\right| _{c_{T} k=a H}, \end{aligned}$$
(43)

with the scalar spectral index and the tensor-to-scalar ratio are

$$\begin{aligned}&n_{\mathcal {R}}-1=3-2\nu _{\mathcal {R}}, \end{aligned}$$
(44)
$$\begin{aligned}&r \equiv \frac{\mathcal {P}_{T}}{\mathcal {P}_{\mathcal {R}}} \simeq 2^{3+2 \nu _{T}-2 \nu _{\mathcal {R}}}F \frac{c_{\mathcal {R}}^{3}}{c_{T}^{3}} \frac{\Gamma ^{2}\left( \nu _{T}\right) }{\Gamma ^{2}\left( \nu _{\mathcal {R}}\right) }\frac{\left( 1+\frac{\epsilon _{1}}{1+2 \eta _{H}}\right) ^{2 \nu _{\mathcal {R}}-2 \nu _{T}}}{\left( 1-\Delta / 2\right) ^2 (1-\delta _1)}.\nonumber \\ \end{aligned}$$
(45)

Similarly as in the previous section, we set the parameter \(\delta _1\) to be a constant and discuss two cases: \(\delta _1=0\) or \(\delta \ne 0\).

5.1 \(\delta _1=0\)

If \(\delta _1=0\) and \(\eta _H=constant\), which is the original constant-roll model without GB coupling [3]. In this case, to the first order approximation of \(\epsilon _1\), the scalar spectral index can be approximated as

$$\begin{aligned} n_{\mathcal {R}}\simeq 4-\mid 2 \eta _H -3\mid +\frac{2 \left( 4 \eta _H^2+5 \eta _H -6\right) \epsilon _1}{\mid 2 \eta _H -3\mid (2 \eta _H +1)}, \end{aligned}$$
(46)

and the tensor-to-scalar ratio

$$\begin{aligned} r \simeq 2^{3-\left| 3-2\eta _{H}\right| }\left( \frac{\Gamma [3 / 2]}{\Gamma \left[ \left| 3-2\eta _{H}\right| / 2\right] }\right) ^{2} 16 \epsilon _{1}. \end{aligned}$$
(47)

Since \(\eta _H\) is a constant, from the definition of flow parameters (5) and (6), and the condition \(\epsilon _{1}(N=0)=1\), we obtain the relation [18]

$$\begin{aligned} \epsilon _{1}(N)=\frac{\eta _H \exp \left( 2 \eta _H N\right) }{\exp \left( 2 \eta _H N\right) +\eta _H-1}. \end{aligned}$$
(48)

Then we get the \(n_{\mathcal {R}}-r\) predictions with \(N=60\) and show them in Fig. 5 (orange line). For a constant \(\eta _H\), the result is consistent with the observations at \(2\sigma \) confidence level. And we updated the constraint to \(\eta _H\) with the Planck 2018 data, which is \(-0.0163<\eta _H<-0.0006\).

5.2 \(\delta _1\ne 0\)

If \(\delta _1=\)constant but not zero, the scalar spectral index and the tensor-to-scalar ratio to the first order of \(\epsilon _1\) is obtained as

$$\begin{aligned}&n_{\mathcal {R}}\simeq 1-\frac{2\left( 2 \eta _{H}-3\right) \left( 4 \eta _{H}+8 \eta _{H}^{2}+\delta _{1}\left( 2-6 \eta _{H}-12 \eta _{H}^{2}\right) +\delta _{1}^{2}\left( -5+2 \eta _{H}+4 \eta _{H}^{2}\right) \right) }{3 \delta _{1}\left( 5 \delta _{1}-2\right) \left( 1+2 \eta _{H}\right) } \epsilon _{1}, \end{aligned}$$
(49)

and

Fig. 10
figure 10

The observational constraints on \(\eta _H\) and \(\delta _1\) with the e-folding number \(N=60\). The orange and blue regions correspond to the parameters satisfied \(1\sigma \) and \(2\sigma \) confidence level, respectively

Full size image
$$\begin{aligned} r\simeq & {} -\frac{16}{3\left( 3 \delta _{1}-2\right) \left( 5 \delta _{1}-2\right) } \sqrt{\frac{2-7\delta _{1}+6 \delta _{1}^2}{2-5 \delta _{1}}}\Big (30 \delta _{1}^{3}-27 \delta _{1}^{2}\nonumber \\&+6 \delta _{1}+ \epsilon _{1}\left( 45 \delta _{1}^{4}-69 \delta _{1}^{3}-6 \delta _{1}^{2}+42 \delta _{1}-12\right. \nonumber \\&\left. +2\left( -2 \delta _{1}^{3}+7 \delta _{1}^{2}-7 \delta _{1}+2\right) \eta _H\left( 2\eta _H-3\right) \ln 4\right. \nonumber \\&\left. +4\left( 2 \delta _{1}^{3}-7 \delta _{1}^{2}+7 \delta _{1}-2\right) \eta _H\left( 2\eta _H-3\right) (\gamma -2+\ln 4)\right) \Big ).\nonumber \\ \end{aligned}$$
(50)

Using the relation (48) and setting the e-folds \(N=60\), we find four reasonable regions of parameter space, which are show in Fig. 10. We can see that in this case of constant-roll inflation, the absolute of GB flow parameter \(\delta _1\) can be larger then one(region IV), which is different from the conventional slow-roll inflation with GB coupling.

The tensor-to-scalar ratio r versus the spectral index \(n_{\mathcal {R}}\) for the parameters region I of Fig. 10 are show in Fig. 11. \(\delta _1\) is taken as \(\delta _1=0.502,0.506,0.51\) from left to right in the left panel, and the \(n_{\mathcal {R}}-r\) dots go along the curves from right to left as \(\eta _H\) increase. The parameters \(\eta _H\) taken as \(\eta _H=0.03,0.02,0.01\) are show in the right panel, and this time the \(n_{\mathcal {R}}-r\) dots go from bottom to top as \(\delta _1\) increase.

Fig. 11
figure 11

The \(n_{\mathcal {R}}-r\) region predicted by the model for the parameter region I of Fig. 10

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In Fig. 12, we show \(n_{\mathcal {R}}\) versus r for the parameters in region II of Fig. 10. Left panel: The parameter \(\delta _1\) is taken as \(\delta _1=0.025,0.015,0.005\) from left to right, and as \(\eta _H\) increase, the \(n_{\mathcal {R}}-r\) dots go along the curves from right to left. Right panel: The parameters \(\eta _H\) is taken as \(\eta _H=0.009,0.005,0.001\) from left to right, and as \(\delta _1\) increase, the \(n_{\mathcal {R}}-r\) dots go along the curves from left to right.Left panel: The parameter \(\delta _1\) is taken as \(\delta _1=0.025,0.015,0.005\) from left to right, and as \(\eta _H\) increase, the \(n_{\mathcal {R}}-r\) dots go along the curves from right to left. Right panel: The parameters \(\eta _H\) is taken as \(\eta _H=0.009,0.005,0.001\) from left to right, and as \(\delta _1\) increase, the \(n_{\mathcal {R}}-r\) dots go along the curves from left to right.

Fig. 12
figure 12

The \(n_{\mathcal {R}}-r\) region predicted by the model for the parameter region II of Fig. 10

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Similarly, the \(n_{\mathcal {R}}\) versus r for the parameters in region III of Fig. 10 are show in Fig. 13. In the left panel The parameter is taken as \(\delta _1=-0.002,-0.005,-0.008\) from left to right, and the dots go along the curves to the top as \(\eta _H\) increase. The parameter \(\eta _H\) is taken as \(\eta _H=-0.01,-0.03,-0.05\) from left to right in the right panel. In this time the \(n_{\mathcal {R}}-r\) dots go from left to right as \(\delta _1\) increase.

Fig. 13
figure 13

The \(n_{\mathcal {R}}-r\) region predicted by the model for the parameter region III of Fig. 10

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Finally, the \(n_{\mathcal {R}}-r\) constraint for the parameters in region IV of Fig. 10 are show in Fig. 14 with the parameters taken as \(\delta _1=-1.8,-1.78,-1.76\) from left to right in the left, where the \(n_{\mathcal {R}}-r\) dots go from bottom to top as \(\eta _H\) increase. The parameter in the right panels \(\eta _H=0.43,0.44,0.45\) from left to right. In this time as \(\delta _1\) increase, the dots go along the curves from bottom to top.

Fig. 14
figure 14

The \(n_{\mathcal {R}}-r\) region predicted by the model for the parameter region IV of Fig. 10

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6 Reconstruct the potential

In this section, we shall find the corresponding scalar potential \(V(\phi )\) and the GB coupling \(\xi (\phi )\) for the constant-roll inflation. If the GB flow parameter \(\delta _1\) is a constant, from the definition of flow parameters, we have \(\delta _2=0\), then the background equations (2) and (3) can be written as

$$\begin{aligned}&\omega \dot{\phi }^2=(2\epsilon _1-\delta _1-\delta _1 \epsilon _1){H^2} , \end{aligned}$$
(51)
$$\begin{aligned}&6 (1-\delta _1) H^2 =\omega \dot{\phi }^2+2V . \end{aligned}$$
(52)

In the following, we will discuss the models with \(\epsilon _1=\)constant, \(\epsilon _2=\)constant and \(\eta _H=\)constant, respectively.

6.1 \(\epsilon _1=\) constant

In the model with \(\epsilon _1=\) constant, the case with \(\delta _1=0\) is ruled out by the observations, so we only interest in the case with \(\delta _1=\)constant but not zero. Substitute (51) into the definition of the flow parameter \(\epsilon _1\)

$$\begin{aligned} \epsilon _{1}=-\frac{\dot{H}}{H^{2}}=-\frac{\dot{\phi }H_{,\phi }}{H^{2}}, \end{aligned}$$
(53)

we obtain a first-order differential equation of the Hubble parameter and get two solutions

$$\begin{aligned} H(\phi )= c_1 e^{\pm \frac{i\epsilon _1\sqrt{\omega }\; \phi }{\sqrt{\delta _1-2 \epsilon _1+\delta _1 \epsilon _1 }}}, \end{aligned}$$
(54)

with \(c_1\) is a integration constant. Substitute the Eqs. (54) and (51) into the Hamilton–Jacobi equation (52), we get the potential of the scalar field \(\phi \)

$$\begin{aligned} V(\phi )=V_0(6-5\delta _1-2\epsilon _1 +\delta _1 \epsilon _1)e^{\pm \frac{2 i\epsilon _1\sqrt{\omega }\; \phi }{\sqrt{\delta _1-2 \epsilon _1+\delta _1 \epsilon _1 }}} \end{aligned}$$
(55)

where the overall factor \(V_0=c_1^2/2\) can be restricted by the amplitude of the primordial curvature perturbations.

For the GB coupling \(\xi (\phi )\), we substitute the result of Hubble parameter (54) into the definition of GB flow parameter \(\delta _{1}=4 \dot{\xi } H \) and get a differential equation of \(\xi (\phi )\), the solutions are

$$\begin{aligned} \xi (\phi )=\frac{\delta _1}{16\epsilon _1 V_0}e^{\mp \frac{2 i\epsilon _1\sqrt{\omega }\; \phi }{\sqrt{\delta _1-2 \epsilon _1+\delta _1 \epsilon _1 }}}+\xi _0 \end{aligned}$$
(56)

where \(\xi _0\) is an integration constant represents the GB term with out coupling to \(\phi \), which is no contribution to the result, and is always set to zero. In addition, the coefficient \(\omega \) in the expressions can be chosen the values \(\pm 1\) to ensure the results \(H(\phi )\),\(V(\phi )\),\(\xi (\phi )\) are real functions and generate reasonable inflation. For instance, if \(\delta _1-2 \epsilon _1+\delta _1 \epsilon _1>0\), \(\omega \) should be chosen \(-1\).

6.2 \(\epsilon _2=\) constant

If the flow parameter \(\epsilon _2\) is a constant, as discussed in Sect. 3, the case with \(\delta _1=0\) is also ruled out by the observations. For the case with \(\delta _1\ne 0\), the differential equation to the Hubble parameter is complicated and no analytic solutions. However, for some special cases, we can get the approximate expressions. In the following, we focus on two special cases: \(\epsilon _1\ll \delta _1\) and \(\delta _1\ll \epsilon _1<1\).

6.2.1 \(\epsilon _1\ll \delta _1\)

We first discuss the relation \(\epsilon _1\ll \delta _1\), which can be satisfied near the beginning of inflation. In this case, the background equation (51) is approximated as

$$\begin{aligned} \omega \dot{\phi }^2=-\delta _1{H^2}. \end{aligned}$$
(57)

Substitute it into the definition of the flow parameter \(\epsilon _2\)

$$\begin{aligned} ~~\epsilon _{2}\equiv \frac{\dot{\epsilon }_{1}}{H \epsilon _{1}}=\frac{\dot{\phi }^{2} H_{, \phi \phi }+\ddot{\phi } H_{, \phi }}{\dot{\phi } H H_{\phi }}-\frac{2 \dot{\phi } H_{, \phi }}{H^{2}}, \end{aligned}$$
(58)

and solve the second-order differential equation of H, we get the Hubble parameter as a function of \(\phi \)

$$\begin{aligned} H(\phi )=c_2 e^{\pm \frac{i\;c_1 \sqrt{\delta _1} \text {exp}[\mp {\frac{i\;\epsilon _2 \sqrt{\omega }\; \phi }{\sqrt{\delta _1}}}]}{\epsilon _2 \sqrt{\omega } }}, \end{aligned}$$
(59)

with \(c_1\) and \(c_2\) are integration constants. Then combine the solution (59) with the background equations (51) and (52), we get the potential of the scalar field \(\phi \)

$$\begin{aligned} V(\phi )=V_0 (6-5 \delta _1) e^{\pm \frac{2i\;c_1 \sqrt{\delta _1} \text {exp}[\mp {\frac{i\;\epsilon _2 \sqrt{\omega }\; \phi }{\sqrt{\delta _1}}}]}{\epsilon _2 \sqrt{\omega } }}, \end{aligned}$$
(60)

where \(V_0=c_2^2/2\) is the overall factor.

Combine of the definition \(\delta _{1}=4 \dot{\xi } H \) and the Hubble parameter (59), we get a differential equation the GB coupling \(\xi (\phi )\), and the solutions are

$$\begin{aligned} \xi (\phi )=\frac{\delta 1 }{8 \epsilon _2 V_0}\text {Ei}\left( \mp \frac{2i\;c_1 \sqrt{\delta _1} \text {exp}[\mp {\frac{i\;\epsilon _2 \sqrt{\omega }\; \phi }{\sqrt{\delta _1}}}]}{\epsilon _2 \sqrt{\omega } }\right) +\xi _0, \end{aligned}$$
(61)

where \(Ei(x)\equiv -\int _{-x}^{\infty } \frac{e^{-t}}{t} \, dt\) is the exponential integral function, and \(\xi _0\) is a integration constant represent the GB term with out coupling to \(\phi \).

6.2.2 \(\delta _1\ll \epsilon _1\)

The relation \(\delta _1\ll \epsilon _1<1\) is likely to happen near the end of inflation, then the background equation (51) is approximated as

$$\begin{aligned} \omega \dot{\phi }^2=2\epsilon _1{H^2}, \end{aligned}$$
(62)

combine with the definition of \(\epsilon _1\), we get the relation

$$\begin{aligned} \omega \dot{\phi }=-2 H_{,\phi }, \end{aligned}$$
(63)

which is the same as the standard slow-roll inflation without GB coupling. Substitute (63) into the definition of the flow parameter \(\epsilon _2\), Eq. (58), we obtain the differential equation of the Hubble parameter and the solution is

$$\begin{aligned} H(\phi )=c_{2} e^{c_{1} \phi -\frac{1}{8} \epsilon _{2} \omega \phi ^{2}}, \end{aligned}$$
(64)

with \(c_1\) and \(c_2\) are integration constants. The potential of the scalar field can be obtained by substitute the Eqs. (63) and (64) into (52), which can be written as

$$\begin{aligned} V(\phi )=V_0 \left( 6-6 \delta _1-\frac{4 c_1^2}{\omega }+2c_1\epsilon _2\phi -\frac{1}{4}\epsilon _2^2\;\omega \phi ^2\right) e^{2 c_{1} \phi -\frac{1}{4} \epsilon _{2}\omega \phi ^{2}}, \end{aligned}$$
(65)

with the overall factor \(V_0=c_{2}^{2}/2\) .

Substitute the result of Hubble parameter (64) into the definition of GB flow parameter \(\delta _{1}=4 \dot{\xi } H \), then we get the solution of GB coupling

$$\begin{aligned} \xi (\phi )=\frac{\delta _{1}}{8 \epsilon _{2} V_0} e^{-\frac{4 c_1^{2}}{\epsilon _{2}\omega }} \mathrm {Ei}\left[ \frac{\left( \epsilon _{2}\omega \phi -4 c_{1}\right) ^{2}}{4 \epsilon _{2}\omega }\right] +\xi _0. \end{aligned}$$
(66)

6.3 \(\eta _H=\) constant

In this section, we reconstruct the potential for the constant-roll inflation with \(\eta _H\) and \(\delta _1\) are constants. If \(\delta _1=0\), the model recover to the case without GB coupling, which is discussed in other reference [3]. Form the background equation, we get the relation

$$\begin{aligned} \omega \dot{\phi }=-2 H_{,\phi } \end{aligned}$$
(67)

Using the same method as in the previous subsections and combine the definition of \(\eta _H\)

$$\begin{aligned} \eta _H \equiv -\frac{\ddot{H}}{2H\dot{H}}=-\frac{\dot{\phi }^{2} H_{, \phi \phi }+\ddot{\phi } H_{, \phi }}{2\dot{\phi } H H_{\phi }}, \end{aligned}$$
(68)

we get the Hubble parameter and the scalar potential as functions of \(\phi \)

$$\begin{aligned}&H(\phi )= c_1 e^{\frac{\sqrt{\eta _H} \sqrt{\omega } \phi }{\sqrt{2}}}+c_2 e^{-\frac{\sqrt{\eta _H} \sqrt{\omega } \phi }{\sqrt{2}}}, \end{aligned}$$
(69)
$$\begin{aligned}&V(\phi )= 2 c_1 c_2 (3+\eta _H)+(3-\eta _H)(c_1^2 e^{\sqrt{2\eta _H} \sqrt{ \omega } \phi }+c_2^2 e^{-\sqrt{2\eta _H} \sqrt{ \omega } \phi }),\nonumber \\ \end{aligned}$$
(70)

with \(c_1\) and \(c_2\) are integration constants. The results are consistent with other reference [3].

Similarly, for the case with \(\delta _1\ne 0\), the differential equation to the Hubble parameter is no analytic solutions. And we discuss the approximate expressions in two special cases: \(\epsilon _1\ll \delta _1\) and \(\delta _1\ll \epsilon _1<1\).

6.3.1 \(\epsilon _1\ll \delta _1\)

If \(\epsilon _1\ll \delta _1\), combine the background equations (57) and (52) with the definition of parameter \(\eta _H\) (68), we can get two solutions of the Hubble parameter \(H(\phi )\) and the corresponding scalar potential \(V(\phi )\)

$$\begin{aligned}&H(\phi )= c_{2} \left( e^{\pm \frac{2 i c_{1} \eta _H \sqrt{\omega }}{\sqrt{\delta _{1}}}}-e^{\pm \frac{2 i \eta _H \sqrt{\omega } \phi }{\sqrt{\delta _{1}}}}\right) ^{\frac{1}{2}} , \end{aligned}$$
(71)
$$\begin{aligned}&V(\phi )=V_0\left( 6-5 \delta _{1}\right) \left( e^{\pm \frac{2 i c_1 \eta _{H} \sqrt{\omega }}{\sqrt{\delta _{1}}}}-e^{\pm \frac{2 i \eta _{H} \sqrt{\omega } \phi }{\sqrt{\delta _{1}}}}\right) , \end{aligned}$$
(72)

with \(c_1\) and \(c_2\) are integration constants and \(V_0=c_{2}^{2}/2\). Substitute the Hubble parameter (71) into the definition of GB flow parameter \(\delta _{1}=4 \dot{\xi } H \), then the solution of GB coupling can be written as

$$\begin{aligned} \xi (\phi )=\xi _{0}+\frac{\delta _{1}}{16 V_0 \eta _{H}} e^{\frac{2 i c_{1} \eta _{H} \sqrt{\omega }}{\sqrt{\delta _{1}}}} \ln \left[ e^{\frac{2 i c_{1} \eta _{H} \sqrt{\omega }}{\sqrt{\delta _{1}}}}-e^{\frac{2 i \eta _{H} \sqrt{\omega } \phi }{\sqrt{\delta _{1}}}}\right] , \end{aligned}$$
(73)

for the \(+\) case of (71), and for the − case is

$$\begin{aligned} \xi (\phi )&=\xi _{0}+\frac{\delta _{1}}{16 V_0 \eta _{H}} e^{-\frac{2 i c_{1} \eta _{H} \sqrt{\omega }}{\sqrt{\delta _{1}}}}\left( \ln \left[ e^{\frac{2 i c_{1} \eta _{H} \sqrt{\omega }}{\sqrt{\delta _{1}}}}-e^{\frac{2 i \eta _{H} \sqrt{\omega } \phi }{\sqrt{\delta _{1}}}}\right] \right. \nonumber \\&\quad \left. -\frac{2 i \eta _{H} \sqrt{\omega } \phi }{\sqrt{\delta _{1}}}\right) . \end{aligned}$$
(74)

6.3.2 \(\delta _1\ll \epsilon _1\)

If \(\delta _1\ll \epsilon _1<1\), using the same method, combine the relation (63) with the definition of parameter \(\eta _H\) (68), we can get the solutions of the Hubble parameter \(H(\phi )\)

$$\begin{aligned} H(\phi )= c_1 e^{\frac{\sqrt{\eta _H} \sqrt{ \omega } \phi }{\sqrt{2}}}+c_2 e^{-\frac{\sqrt{\eta _H} \sqrt{ \omega } \phi }{\sqrt{2}}}, \end{aligned}$$
(75)

then substitute it into the background equation (52), we get the scalar potential \(V(\phi )\)

$$\begin{aligned} V(\phi )= & {} 2 c_1 c_2 (3+\eta _H-3 \delta _1)+(3-\eta _H-3 \delta _1)\nonumber \\&\times \left( c_1^2 e^{\sqrt{2\eta _H} \sqrt{ \omega } \phi }+c_2^2 e^{-\sqrt{2\eta _H} \sqrt{ \omega } \phi }\right) , \end{aligned}$$
(76)

and the GB coupling \(\xi (\phi )\) can be obtained by integer the definition of \(\delta _1\)

$$\begin{aligned} \xi (\phi )=\frac{\delta _{1} }{8 c_{1} c_{2} \eta _{H}}\tanh ^{-1}\left[ \frac{c_{1} \mathrm {e}^{\sqrt{2 \eta _{H}} \sqrt{ \omega }\phi }}{c_{2}}\right] +\xi _{0}. \end{aligned}$$
(77)

7 Summary

In this paper, we discuss the constant-roll inflation in the model with the inflaton \(\phi \) nonminimal coupled to the GB term. Since the presence of a new degree of freedom from GB coupling \(\xi (\phi )\), so we discuss the constant-roll inflation with constant \(\epsilon _1\), constant \(\epsilon _2\) and constant \(\eta _H\), with the additional assumption that the first GB flow parameter \(\delta _1\) is a constant, respectively. Using the Bessel function approximation, we present the mode equations of scalar and tensor perturbations and get the analytical expressions for the scalar and tensor power spectrum to the first order of \(\epsilon _1\). We derive the scalar spectral index and the tensor to scalar ratio and constraint the parameter space by using the Planck 2018 results. First, we assume that \(\epsilon _1\) is a constant, the case with \(\delta _1=0\) is ruled out by the observations, and in the case with \(\delta _1=\) constant but not zero, we obtain two regions of parameter space and show the results on the \(n_{\mathcal {R}}-r\) region. Second, in the assumption with \(\epsilon _2=\) constant, \(\delta _1=0\) is just the constant-roll inflation without GB coupling, which is also ruled out by the observations. If \(\delta _1=\)constant but not zero, we obtain three regions of parameter space. Finally, we assume that \(\eta _H=\) constant, the case with \(\delta _1=0\) is the original constant-roll inflation discussed in [3]. Here we update the constraint from Planck 2018 data, which is \(-0.0163<\eta _H<-0.0006\). In the case with \(\delta _1=\) constant but not zero, we obtain four feasible regions of parameter space, and find that in some region, the chosen of GB flow parameter \(\delta _1>1\) can also produce a nearly scale-invariant scalar power spectrum, which is different from the conventional slow-roll inflation with GB coupling. In addition, we also find the analytical expressions for corresponding scalar potential of the constant-roll inflation and the nonminimum coupling \(\xi (\phi )\) in some cases.