Two-field constant roll inflation

TL;DR

This paper extends the constant roll inflation framework to two scalar fields, deriving compatible potentials and exploring applications within string theory models.

Contribution

It introduces a two-field constant roll formalism and derives suitable potentials, advancing the theoretical understanding of multi-field inflation in string theory.

Findings

Derived two-field potential compatible with constant roll

Extended constant roll formalism to two scalar fields

Discussed applications to string-models

Abstract

Starting from the idea of realising constant roll inflation in string theory we develop the constant roll formalism for two scalar fields. We derive the two-field potential which is compatible with a constant roll regime and discuss possible applications to string-models.

Equations

$$S_{matter}=\int\sqrt{-g}d^{4}x\Big{(}-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\Big{)}\;,$$

$$d{s}^2=-d{t}^2+a(t{)}^{2}d{\vec{x}}^2,$$

$$3H^2$$

$$-2\dot{H}$$

$$0$$

$$0=3H{\dot{\varphi }}^2+\ddot{\varphi }\dot{\varphi }+\dot{V}=(\ddot{\varphi }+3H\dot{\varphi }+\frac{\delta V}{\delta \varphi })\dot{\varphi }.$$

$$\ddot{\varphi }=-\eta H\dot{\varphi }.$$

$$\ddot{H}=-2\eta H\dot{H},$$

$$H$$

$$a$$

$$V=3H^2+\dot{H},$$

$$V(t)=3h^2+\frac{4k{h}^2}{(k{e}^{h\eta t}-e^{-h\eta t}{)}^2}(3-\eta ).$$

$$\varphi (t)=\pm \sqrt{\frac{2}{\eta }}\ln \frac{\sqrt{k}{e}^{\frac{1}{2}h\eta t}-e^{-\frac{1}{2}h\eta t}}{\sqrt{k}{e}^{\frac{1}{2}h\eta t}+e^{-\frac{1}{2}h\eta t}}+{\varphi }_0,$$

$$V=\frac{h^2}{2}(3+\eta )+\frac{h^2}{2}\cosh (\sqrt{2\eta }(\varphi -{\varphi }_0))(3-\eta ).$$

$$K=-3\log (T+\bar{T}).$$

$$W=e_0+ieT+m{T}^2+i{m}_0{T}^3,$$

$$V=e^K\left(D_{T}W{D}_{\bar{T}}\bar{W}{g}^{T\bar{T}}-3|W{|}^2\right),$$

$$D_T={\partial }_T+{\partial }_{T}K,$$

$$T=\rho +ia,$$

$$V=-\frac{1}{\rho }(e_{0}m+\frac{1}{3}{e}^2)+\frac{a}{\rho }(3e_0{m}_0-\frac{1}{3}em)-\frac{{\rho }^2+a^2}{\rho }(e{m}_0+\frac{1}{3}{m}^2).$$

$$g_{T\bar{T}}{\partial }_{\mu }T{\partial }^{\mu }\bar{T}=\frac{3}{4{\rho }^2}({\partial }_{\mu }\rho {\partial }^{\mu }\rho +{\partial }_{\mu }a{\partial }^{\mu }a).$$

$${\mathcal{L}}_{kin}=-\frac{1}{2}{\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -\frac{3}{4}{e}^{2\sqrt{\frac{2}{3}}\varphi }{\partial }_{\mu }a{\partial }^{\mu }a$$

$$V=$$

$$-e^{\sqrt{\frac{2}{3}}\varphi }(e{m}_0+\frac{1}{3}{m}^2).$$

$$V=e^{-\sqrt{\frac{2}{3}}\varphi }\left[-(e_{0}m+\frac{1}{3}{e}^2)+\frac{(3e_0{m}_0-\frac{1}{3}em{)}^2}{4(e{m}_0+\frac{1}{3}{m}^2)}\right]-e^{\sqrt{\frac{2}{3}}\varphi }(e{m}_0+\frac{1}{3}{m}^2).$$

$$\eta =\frac{1}{3}.$$

$$S_{matter}=\int\sqrt{-g}d^{4}x\Big{(}-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}e^{2b(\phi)}\partial_{\mu}\chi\partial^{\mu}\chi-V(\phi,\chi)\Big{)}\;,$$

$$3H^2$$

$$-2\dot{H}$$

$$0$$

$$\dot{\varphi }\left(\ddot{\varphi }+3H\dot{\varphi }-e^{2b}\frac{\delta b}{\delta \varphi }{\dot{\chi }}^2+\frac{\delta V}{\delta \varphi }\right)+\dot{\chi }\left(\ddot{\chi }+3H\dot{\chi }+2\dot{\chi }\dot{b}+\frac{\delta V}{\delta \chi }\right)=0,$$

$$\dot{X}=-2\eta HX,$$

$$X=\frac{8k{h}^2\eta }{(k{e}^{h\eta t}-e^{-h\eta t}{)}^2}.$$

Full text

Two-field constant roll inflation

Andrei Micu

*Department of Theoretical Physics,

Horia Hulubei National Institute for Physics and Nuclear Engineering

*Str. Reactorului nr. 30, RO–077125, Măgurele, Romania

E-mail: amicu@theory.nipne.ro

Starting from the idea of realising constant roll inflation in string theory we develop the constant roll formalism for two scalar fields. We derive the two-field potential which is compatible with a constant roll regime and discuss possible applications to string-models.

1 Introduction

It is almost out of doubt that inflation has to be part of any cosmological model. While from the GR point of view only the energy content of the universe is relevant, from a particle physics point of view there is a big question of how the conditions for inflation are realised. As string / supergravity models offer most promising perspectives to unify particle physics and gravity a lot of effort has been spent in order to accommodate inflation in such theories. The simplest idea is to generate inflation from a cosmological constant which in turn is given by the vacuum energy. Even if, thanks to supersymmetry, Minkowski and AdS solutions are quite easy to find [1, 2, 3], dS solutions, which should be relevant for inflation, are completely forbidden in many cases [4]. More recently, a series of conjectures regarding the consistency of a theory coupled to quantum gravity, known as the swampland criteria, have been formulated [5, 6, 7]. It was immediately realised that these criteria may be in direct conflict with conditions for inflation [8].111Note that as long as the precise details for inflation are not known many inflationary options in agreement with the swampland criteria can be found [9]. It is therefore important to find models of inflation which are not in the swampland according to these criteria.

In recent years a new type of inflation has been put forward [10] which was dubbed constant roll inflation. The novelty is that imposing a constant roll regime for the inflaton, the exact background solution can be written down explicitly, while the parameter defining the constant roll need not be in any sense small unlike in the slow roll case. This may ensure that the constant roll condition is safe with regard of the swampland criteria and moreover since the constant roll parameter is directly related to the $\eta$ parameter defined in slow roll inflation, this new type of inflation may constitute a simple workaround to the well known supergravity $\eta$-problem [11]. It is therefore natural to analyse whether constant roll inflation can be realised in string theory. One aspect to keep in mind is that for a single inflaton field, the constant roll potential which was derived in [10, 12], turns out to be completely fixed and it is not clear whether concrete string models can accommodate such potentials. A more relevant question to be asked is under what conditions multi-field constant roll inflation can take place, since models derived from string theory usually involve many scalar fields. In this note we shall give an answer to this question for a model with two scalar fields.

The outline of the paper is the following. We begin with a short review of single field constant roll solution in Section 2 and then try to implement the corresponding potential in a simple string model in Section 3. We shall see that the string potential comes quite close to the form required by the constant roll conditions, but nonetheless the identification of the string potential with the constant roll one is still questionable. This will motivate us to study in detail the two-field constant roll regime in Section 4. We derive the explicit solutions for the fields and for the scalar potential and note that the shortcomings of the single field potential may have a resolution in this more general case.

2 Constant roll inflation

In this section we shall briefly review constant roll conditions and the solution for the case of a single scalar field. This was first discussed in [10] and further studied in more details in [13, 14]. Other developments appeared in [15, 16, 18, 17]. A systematic study of constant roll inflation appeared more recently [12] and this reveals more possibilities then originally proposed in [10]. In the following we shall closely follow [12].

The matter action just contains the kinetic term for the inflaton and a scalar potential

[TABLE]

and we use the “mostly plus” convention for the Minkowski metric. In a spatially flat FRW background where the metric is given by

[TABLE]

the equations of motion read

[TABLE]

where $H$ is the Hubble parameter and is given as $H=\frac{\dot{a}}{a}$. It is important to notice that for the case of a single scalar field, once the Einstein equations are fulfilled the equation of motion for the scalar field is identically satisfied. By taking the derivative of the first Einstein equation and replacing $\dot{H}$ from the second one, we find

[TABLE]

Constant roll was defined in [10] by asking that the ratio between the acceleration and the speed of the field $\phi$ is proportional to the Hubble constant

[TABLE]

Here we shall adopt an equivalent formulation in terms of the Hubble parameter only

[TABLE]

and it is easy to see from the Einstein equations that the above definitions are indeed equivalent. This latter definition is more fundamental in the sense that it is enough to determine the inflationary dynamics independent on the field content of the theory. We shall use this property when we generalise constant roll to a matter system of two scalar fields.

Equation (6) can be integrated analytically and one can obtain the solution for the Hubble constant and for the scale factor as functions of time

[TABLE]

where $h$ and $k$ are constants. This is the form of the solution found in [12] where the constants $k$ and $h$ are allowed to take complex values as long as the solutions for $H$ and $a$ are real. This reality condition imposes constraints on the way these constants can be chosen and the various cases lead to the classes of solutions discussed in [10, 12]. For simplicity we shall use this general form as we are not particularly interested in a certain class of models and bare in mind the fact that this formulation encodes few specific possibilities depending on the reality of the parameters $k$ and $h$. Finally let us stress that under certain conditions for the integration parameters and for the constant roll parameter $\eta$ it was shown that inflation can take place and a sufficient number of e-foldings can be accommodated [10, 12]. Regarding the CMB fingerprints of inflation, like the scalar and tensor perturbations, various studies in the literature seem to favor small-$\eta$, but there is no true consessum on its precise value [13, 12, 19].

So far we displayed the gravitational part of the solution which, up to this point, can be seen to be independent on the matter content of the theory. Now we need to make the link with the matter part. Note that as long as we look for the potential as a function of time only, this is given as

[TABLE]

which, after replacing the solution for the Hubble parameter above, becomes

[TABLE]

Furthermore, the time profile for the scalar field can be obtained using the second Einstein equation and the solution for the Hubble constant (7)

[TABLE]

where $\phi_{0}$ is an additional integration constant. Finally, inverting the equation above and inserting the solution in (9) one finds the general scalar potential which is suitable for constant roll inflation with a single scalar field

[TABLE]

We see that the potential which is compatible with constant roll conditions is completely fixed. This potential is derived from the Einstein equations of motion together with the constant roll condition and therefore is the unique potential which admits such a constant roll regime. Moreover, as we saw before, a solution of the Einstein equations is also a solution of the scalar field equation. Therefore we are dealing with a completely consistent solution of the Einstein and field equations which exhibits constant roll behavior.

3 Constant roll inflation in string theory?

It is natural to ask whether constant roll inflation can be encountered in certain particle models. In particular, we shall ask this question in the context of string/supergravity constructions. We do not attempt to make an extensive survey of possible string potentials, but we shall rather concentrate on a toy model which may be derived from generic string compactifications with fluxes. As we can see from (11), the potential which is needed for single field constant roll inflation is quite rigid and apriori it is not at all clear that some arbitrary string model can accommodate such a potential. We shall nevertheless see that the model we picked up comes quite close to generate suitable conditions for constant the roll regime to take place.

We shall consider a $N=1$ supergravity model coupled to complex field whose real part can be thought of as the radius of the compactification manifold in a would be string model. Therefore we write the Kähler potential as

[TABLE]

The superpotential we consider is inspired from string compactifications with fluxes in a regime of large volume [20, 21]

[TABLE]

where $e_{0}$, $e$, $m$ and $m_{0}$ are constants which parametrize the fluxes.

With these specifications we can compute the scalar potential by the well-known $N=1$ formula

[TABLE]

where $D_{T}$ is the Kähler covariant derivative

[TABLE]

and $g^{T\bar{T}}$ is the inverse Kähler metric. Parametrising the complex scalar field $T$ as

[TABLE]

we find

[TABLE]

Our main interest in this section is to see whether this potential is suitable for constant roll inflation. To check this, first note that the scalar fields are not canonically normalised as the kinetic term takes the form

[TABLE]

Redefining $\sqrt{3/2}\ln{\rho}=\phi$, we find for the kinetic terms

[TABLE]

while the potential becomes

[TABLE]

In order to match this potential with the one which is suitable for a constant roll behavior we need to eliminate the field $a$. Extremizing the potential with respect to $a$ and inserting the solution back into the potential we find

[TABLE]

It is interesting to note that this potential has the right exponentials to combine into a hyperbolic cosine provided the coefficients match in the right way and it is not inconceivable that for a certain choice of flux parameters this actually happens. More problematic is that the constant term which appears in the potential (11) can not be reproduced in this simple model. It may be possible that certain modifications of the initial data of the problem (eg various corrections to the Kähler and/or superpotential) do lead to the presence of a constant term in the potential without spoiling the general structure observed above. This however, has to be checked on a case by case basis and will not concern us here any longer.

We only note that provided we succeed to identify the constant roll potential along the lines mentioned above, the constant roll parameter $\eta$ is going to be fixed at a value

[TABLE]

A first restriction for this parameter in order for inflation to occur is that $\eta\leq\frac{1}{2}$ [12], which the above value satisfies. However, as mentioned in the previous section, smaller values for $\eta$ seem to be favored by observations [19] while clearly such a regime is not accessible in the simple string model considered so far.

The weak point in the discussion above is that the string potential generally depends on more scalar fields (in our case on two fields), while the constant roll potential was derived for a single inflaton field. Above, we reduced the string model to one with a single field by extremizing the potential along one direction in order to decide whether the model is suitable for constant roll behavior. However, a more meaningful approach would be to allow for the possibility that all fields participate in the constant roll regime and let the theory itself decide whether some of the fields should be spectators (ie are going to be fixed at the extremal values of the potential). This will be the purpose of the next section where we shall study the constant roll behavior for a system comprising two scalar fields.

4 Two field constant roll inflation

In this section we want to analyse how the constant roll setup of [10, 12], can be generalised to a system of two real scalar fields. Having more fields one can choose the system to be not minimally coupled and allow a non-trivial metric on the scalar field space. Nevertheless, we shall not consider a completely general metric, but based on the simple model discussed in the previous section we write the matter action

[TABLE]

where for the moment $b(\phi)$ is some arbitrary function of $\phi$, but later on we shall specialise to the specific form which we used in the example before.

For a FRW background, the Einstein and (scalar) field equations read

[TABLE]

Compared to the single field case, the scalar field equations of motion are no longer a consequence of the Einstein equations. Following the same strategy by taking the time derivative of the first Einstein equation and replacing $\dot{H}$ from the second one we find

[TABLE]

and we see that in the brackets we obtain precisely the field equations. Therefore, if we find a solution of the Einstein equations which satisfies one of the field equations, the other one is implied by the relation above.

As in the previous case, we are interested to see under what conditions the above system admits constant roll solutions i.e. $\ddot{H}/(H\dot{H})=2\eta=$ constant. Fixing this relation for $H$ ensures that the solutions for the Hubble parameter and the scale factor are going to be the same as in the single field case (7) and therefore, the conditions for inflation are the same as the ones found in [12]. However our interest is to find the field profiles and the scalar potential which gives rise to such constant roll inflation. We therefore need to find a suitable Ansatz for the scalar fields such that this condition is satisfied. It is easy to see that the equations of motion above require that

[TABLE]

where $X=\dot{\phi}^{2}+e^{2b}\dot{\chi}^{2}$. This condition is the two-field equivalent of (5) and it reduces to the single field condition if we set one of the fields to be constant.222 The fact that the kinetic term $X$ has to satisfy the relation above is similar to multi-field slow roll inflation where precisely the same combination has to satisfy the slow roll conditions.

Having the solution for $H$ from the single-field analysis (7), we can find $X$ immediately from the second Einstein equation as

[TABLE]

A solution for $X$ in this case does not immediately lead to a solution for $\dot{\phi}$ and $\dot{\chi}$ as we saw in the single field case. To proceed further we shall make the following Ansatz

[TABLE]

where $\theta$ is a free parameter. These are definitely sufficient conditions for (27) to be satisfied and there may be room for generalizations which however will not be our concern in this communication. Note that the parameter $\theta$ controls how much the fields $\phi$ and $\chi$ participate in the constant roll regime. Indeed, when either $\sin\theta$, or $\cos\theta$ vanishes one field will be spectator and only the other one will participate in order to ensure that the constant roll condition (26) is satisfied.

For a more intuitive understanding of the Ansatz above note that this is actually equivalent to require that both fields $\phi$ and $\chi$ obey the corresponding one-field constant roll condition (5), namely

[TABLE]

where in the second equation we took into account the effect of the non-trivial metric in the field space for $\chi$. In fact, once we impose that the field $\phi$ obeys the constant roll condition (5), the condition on $\chi$ follows automatically and then (28) is the unique solution for $\dot{\phi}$ and $\dot{\chi}$.

With these assumptions we are in position to find the solutions for $\phi$ and $\chi$. Up to the constant $\sin{\theta}$, the differential equation for $\phi$ is the same as in the single field case and therefore, up to this constant, the solution will be the same as in Section 2

[TABLE]

As long as $\chi$ is concerned, the non-trivial metric on the field space becomes important. Without the factor $e^{-b}$ in the expression for $\dot{\chi}$, the solution for this field would have been the same as for $\phi$. The presence of this additional factor changes the solution completely. At this stage we need a precise definition for the function $b(\phi)$ and, as anticipated, we shall choose the form used in the previous section when analysing the string model

[TABLE]

Replacing this back into (28) we find

[TABLE]

and this can be integrated to obtain

[TABLE]

where again, $\chi_{0}$ is an integration constant.

At this stage we have obtained the time profile of the solution which gives the constant roll behavior (6) under the assumption (29). Following the single field case, we need to determine the scalar potential as a function of $\phi$ and $\chi$. As mentioned before, when looking for the potential as a function of time, the solution is given again by (8), independent of the matter content. While in the single field case, the field dependence of the potential was obtained simply by inverting the function $\phi(t)$ and inserting it in (9), when dealing with more scalar fields it is not clear how to find the function $t(\phi,\chi)$ which should be inserted in the potential. As we shall see, this ambiguity will remain until the end and will constitute the main part for the flexibility of the two-field potential.

Before that we have to make sure that the form of the potential is consistent with the field equations of motion. In the single field case this was automatic, but for two fields we need to explicitly impose one of the field equations, the other one being implied by the Einstein equations. As mentioned before, the solution in the two-field case depends on the parameter $\theta$ which quantifies the amount by which each of the fields contributes to the constant roll regime. In the case $\theta=n\pi/2$ for some integer $n$, one of the fields is a spectator and we are effectively dealing with a single field regime. In these particular cases it is natural to think that the individual potentials $V(\phi)$ and $V(\chi)$, obtained by inverting the functions $\phi(t)$ and $\chi(t)$ and introducing them into (9), are the relevant scalar potentials. After some straightforward algebra these potentials are found to be

[TABLE]

The potential $V(\phi)$ has the same form as the one found in the single field case (11) the only difference being the the angle $\theta$ which enters in the argument of the hyperbolic cosine. For $\sin\theta=1$, the potential $V(\phi)$ is precisely the one found in the single field case and since $\cos\theta=0$ equations (28) imply that $\chi$ is constant. Moreover, the equations of motion tell us that for $\dot{\chi}=0$ the potential is independent on $\chi$ and, as anticipated, the potential in this case is simply given by $V(\phi)$. In other words, for $\cos\theta=0$ and $\chi=$constant, the field profile (30) is a solution of the equations of motion only if the full potential reduces to $V(\phi)$ in (4).

Likewise, for $\sin\theta=0$, the potential should reduce to $V(\chi)$.333To take the limit $\sin\theta\to 0$ we should first note that in this case $\dot{\phi}=0$. This means that the warp factor $e^{\phi}$ is constant and so the differential equation for $\chi$ reduces to the one in the single field case. Therefore the solution in this case should be again the same with the solution in the single field case. This means that the appearance of the fields $\phi$ and $\chi$ in the potential strongly depends on the angle $\theta$ and the full potential should should interpolate between $V(\phi)$ for $\sin\theta=1$ and $V(\chi)$ for $\cos\theta=1$. We therefore write the following Ansatz for the full potential

[TABLE]

where $f(t)$ is some arbitrary function of time which must assume the value $f(t)=0$ for $\cos\theta=0$ and $f(t)=1$ for $\sin\theta=0$. Note that when replacing $\phi$ and $\chi$ by the corresponding solutions in (30) respectively (33), $V(\phi)\to V(t)$ and $V(\chi)\to V(t)$ ensuring that the full potential reduces to $V(t)$. The appearance of the function $f(t)$ might be surprising, but as we will shortly see, we need such a function so that we can solve the equations of motion. In fact, the equations of motion will determine the form of this function and then we shall still be left with the freedom to choose $t$ as a function of either $\phi$ or $\chi$ or a combination thereof. One may legitimately wonder whether for the potential (35) equation (25) is still valid as in the derivation it was mutually assumed that $V$ does not explicitly depend on time. Note however, that since the function $f(t)$ multiplies the term $V(\phi)-V(\chi)$ which identically vanishes once we introduce the the solutions for the fields (30) and (33), term like $\dot{f}$ will not be present when taking the time derivative of $V$ in (35). For the same reason, even if we replace $t$ by a certain combination of $\phi$ and $\chi$, in the equations of motion no derivatives of the function $f$ will appear.

With these in mind, we shall now impose the $\phi$ equation of motion for the potential (35). Replacing $\ddot{\phi}$ from the constant roll condition (29) and using the potential (35), the $\phi$ equation of motion becomes

[TABLE]

In this equation everything apart from the function $f$ is known and replacing the solutions for $H$, $\dot{\phi}$, $\dot{\chi}$ and $V(\phi)$ we find

[TABLE]

It is clear that this function satisfies our basic conditions that it should take the value 1 for $\sin\theta=0$ and vanishes for $\cos\theta=0$. As we anticipated the function has a non-trivial $t$-dependence which is essential for the field equations to be satisfied. Replacing back into the potential we find

[TABLE]

This is the two-field potential which is compatible with the constant roll conditions (29) in the sense that the solution of these constraints (30) and (33) are also solutions of the equations of motion corresponding to this potential. The time parameter still enters this potential and should be replaced by some function of $\phi$ and $\chi$. Imposing the equations of motion has fixed the freedom of choosing the form of the function $f(t)$ in (35) but we are still left with the arbitrariness of choosing $t$ as a specific function of $\phi$ and $\chi$. Therefore the scalar potential in the two-field case is no longer fixed and it may be easier to tune a certain model to obtain a potential which ensures the constant roll behavior. A more detailed analysis of the potential (38) will be left for future work since an immediate prescription of how to replace $t$ as a function of $\phi$ and $\chi$ is not clear. We shall nevertheless comment on the form of the component potentials $V(\phi)$ and $V(\chi)$.

As also mentioned before, $V(\phi)$ is pretty much the same as the single field potential (11). The main difference consists of the fact that another free parameter, in the form of $\sin\theta$ enters the argument of the hyperbolic cosine. Recall that in the simplistic string model we discussed before in the context of single field constant roll, the constant roll parameter was fixed at the value $1/3$ from the exponents obtained in the potential (20). When comparing to the two-field case, the constant roll parameter becomes

[TABLE]

and therefore, an arbitrarily low value for $\eta$ can be obtained by an appropriate choice of $\theta$ so that agreement with values favored by observations can be easily reached.

In what concerns $V(\chi)$, note that it depends on $\chi$ power-like. This is encouraging as in the simple string model discussed, the field $a$ also appears power-like in the potential. With the identification of $\eta$ above, $V(\chi)$ becomes

[TABLE]

Finally, in this case the expressions for $\phi(t)$ and $\chi(t)$ can be simplified

[TABLE]

From this point it seems more tractable to find a replacement of $t$ in (38) in terms of $\phi$ and $\chi$, but we shall not insist on this any further since it is meaningful only for a specific model.

5 Conclusions

In this note we studied the conditions under which the constant roll regime may appear in models with two scalar fields. Our analysis was motivated in a first place by the fact that many realistic models describe more than one scalar field. Moreover, the one-field constant roll potential derived in [10, 12] is completely fixed making it more difficult to match this potential with one obtained from a specific model.

Imposing a constant roll condition for each of the fields, (28), we were able to derive the time profile of the fields and write the general form for the potential. As expected, the two-field potential has a higher degree of complexity and is no longer fixed, which makes it easier to use in specific models. The main source for this flexibility is the fact that the time parameter left in (38) has to be replaced by some function of $\phi$ and $\chi$. The precise combination is quite arbitrary as long as when replacing back the solutions (30) and (33) the time parameter $t$ is recovered.

The second degree of flexibility, as compared to the single field case, is represented by the parameter $\theta$ which controls how much each of the fields $\phi$ and $\chi$ participate in the constant roll regime. In a more indirect way, the same parameter $\theta$ dictates the amount by which each of the individual potentials $V(\phi)$ and $V(\chi)$ enter the full potential (38). For the string model presented, this parameter is tied to the constant roll factor $\eta$ by the relation (39). This implies that, in the two-field case, by suitably choosing $\theta$, the constant roll parameter can be made small enough to match the preferred observational values.

Finally it should be mentioned that the solution discussed in the two field case is still a particular one as the Ansatz (28) is not the unique solution for which relation (27) holds.

Acknowledgments: The author would like to thank D. Ghilencea for enlightening discussions at the beginning of this project. This work was supported by project “Nucleu” PN-19060101/2019.