<figure><img src="image_96.png" /><figcaption>Fig. 93. Double Exponential Inflation potential (DEI) for $\beta = 0.5$. Top left panel: the potential as a function of $\phi/\phi_0$. Top right panel: logarithm of the potential. Bottom left panel: rescaled slow-roll parameter $\epsilon_1\phi_0^2/M_{Pl}^2$. Bottom right panel: rescaled slow-roll parameters $\epsilon_2\phi_0^2/M_{Pl}^2$ (solid line) and $\epsilon_3\phi_0^2/M_{Pl}^2$ (dotted line).</figcaption></figure>

## 6.25. Double Exponential Inflation (DEI)

### 6.25.1. Theoretical justifications

The model was proposed in Ref. [664] as a phenomenological realization of hilltop inflation by means of a single-field potential containing two exponential terms,

$$V(\phi) = \Lambda^4 \left( \alpha_1 e^{\beta_1 \frac{\phi}{M_{Pl}}} + \alpha_2 e^{\beta_2 \frac{\phi}{M_{Pl}}} \right). \qquad (6.418)$$

In this expression, $\alpha_1, \beta_1, \alpha_2$ and $\beta_2$ are dimensionless parameters, and $\Lambda$ sets the overall scale of the potential. Without loss of generality, one can set parameters such that the top of the “hill” (i.e. the local maximum of the potential) corresponds to $\phi = 0$, which amounts to imposing $V(0) > 0$, $V'(0) = 0$ and $V''(0) < 0$. The condition $V''(0) < 0$ implies that $\alpha_1\beta_1^2 + \alpha_2\beta_2^2 < 0$, so $\alpha_1$ and $\alpha_2$ have different signs. Since the ordering of the two exponential terms is arbitrary, one can take $\alpha_1 > 0$ and $\alpha_2 < 0$. The condition $V'(0) = 0$ then leads to $\alpha_1\beta_1 + \alpha_2\beta_2 = 0$, so $\beta_1/\beta_2 = -\alpha_2/\alpha_1 \equiv \beta^2$, which defines the parameter $\beta$ and where we have used that $\alpha_1$ and $\alpha_2$ have different signs. The condition $V(0) > 0$, i.e. $\alpha_1 + \alpha_2 > 0$, implies that $\beta^2 < 1$. Upon introducing $M^4 \equiv \Lambda^4\alpha_1$ and $\phi_0 \equiv M_{Pl}\beta/\beta_1$, the potential reads

$$V = M^4 \left( e^{\beta \frac{\phi}{\phi_0}} - \beta^2 e^{\frac{1}{\beta} \frac{\phi}{\phi_0}} \right). \qquad (6.419)$$

### 6.25.2. Slow-roll analysis

Let us now perform the slow-roll analysis of the potential (6.419). We recall that $\beta^2 < 1$, so $-1 < \beta < 1$. However, since the potential is invariant under the transformation $\beta \to -\beta$ and $\phi \to -\phi$, our considerations can be restricted to the interval $0 < \beta < 1$. The potential is displayed in Fig. 93. It is maximal at $\phi_0$, and possesses two regimes of inflation. When $\phi > 0$, the potential decreases with $\phi$ until it vanishes at

$$\phi_{V=0} = 2\phi_0\beta \frac{\ln\beta}{\beta^2 - 1}, \qquad (6.420)$$

above which it is negative. One can check that, for $0 < \beta < 1$, one has $\phi_{V=0} > 0$. This is why there is a first regime of inflation, at $0 < \phi < \phi_{V=0}$, that we denote DEI1. When $\phi < 0$, the potential decreases as $\phi$ decreases and asymptotes 0 at $\phi \to -\infty$. This second regime of inflation will be denoted DEI2.

Defining

$$x \equiv \frac{\phi}{\phi_0}, \qquad (6.421)$$
the Hubble-flow functions in the slow-roll approximation are given by

$$ \epsilon_1 = \frac{\beta^2 M_{\text{Pl}}^2}{2 \phi_0^2} \frac{(e^{\beta x} - e^{x/\beta})^2}{(e^{\beta x} - \beta^2 e^{x/\beta})^2}, \qquad (6.422) $$

$$ \epsilon_2 = 2 \frac{M_{\text{Pl}}^2}{\phi_0^2} (\beta^2 - 1)^2 \frac{e^{(1+\beta^2)x/\beta}}{(e^{\beta x} - \beta^2 e^{x/\beta})^2}, \qquad (6.423) $$

$$ \epsilon_3 = \frac{M_{\text{Pl}}^2}{\phi_0^2} (\beta^2 - 1) \frac{(e^{\beta x} - e^{x/\beta})(e^{\beta x} + \beta^2 e^{x/\beta})}{(e^{\beta x} - \beta^2 e^{x/\beta})^2}, \qquad (6.424) $$

and are displayed in the lower panels of Fig. 93. Let us describe their behavior in the two regimes of interest.

If $x > 0$, the three Hubble-flow parameters increase with $x$, and diverge as $x$ approaches $x_{V=0}$. Inflation terminates by slow-roll violation when $\epsilon_1(x) = 1$, at a positive field value given by

$$ x_{\text{end}} = x_{\epsilon_1=1}^+ = \frac{\beta}{\beta^2 - 1} \ln \left( \frac{\beta \sqrt{2} \frac{\phi_0}{M_{\text{Pl}}} + 1}{\frac{\sqrt{2}}{\beta} \frac{\phi_0}{M_{\text{Pl}}} + 1} \right). \qquad (6.425) $$

The first and third Hubble-flow parameters vanish at $x = 0$, while the second Hubble-flow parameter is

$$ \epsilon_2^{\min}(x > 0) = 2 \frac{M_{\text{Pl}}^2}{\phi_0^2}. \qquad (6.426) $$

For this reason, slow-roll inflation in DEI1 requires that $\phi_0 \gg 1$.

If $x < 0$, $\epsilon_1$ increases away from 0 as $x$ decreases, and reaches the asymptotic value

$$ \epsilon_1^{\max}(x < 0) = \frac{\beta^2 M_{\text{Pl}}^2}{2 \phi_0^2}, \qquad (6.427) $$

when $x \to -\infty$. Whether or not inflation ends by slow-roll violation in DEI2 thus depends on the values of $\beta$ and $\phi_0$. More precisely, if $\beta > \sqrt{2}\phi_0/M_{\text{Pl}}$, then Eq. (6.427) becomes larger than unity and inflation ends at the field value solution of $\epsilon_1(x) = 1$, in the negative field domain, given by

$$ x_{\epsilon_1=1}^{-} = \frac{\beta}{\beta^2 - 1} \ln \left( \frac{\beta \sqrt{2} \frac{\phi_0}{M_{\text{Pl}}} - 1}{\frac{\sqrt{2}}{\beta} \frac{\phi_0}{M_{\text{Pl}}} - 1} \right). \qquad (6.428) $$

Otherwise, if $\beta < \sqrt{2}\phi_0/M_{\text{Pl}}$, inflation does not stop by violation of the slow-roll conditions and one needs to invoke other mechanisms, which results in the introduction of another free parameter $x_{\text{end}}$. This possibility is not discussed in Ref. [664], and would otherwise corresponds to a PLI regime. For these reasons, we do not consider it either, and impose the condition $\beta > \sqrt{2}\phi_0/M_{\text{Pl}}$. Note that since $\beta < 1$, for this regime to exist, one needs to assume $\phi_0/M_{\text{Pl}} < 1/\sqrt{2}$.

Since $\epsilon_2$ decreases as inflation proceeds in DEI2, its minimum value is obtained by evaluating Eq. (6.423) at $x_{\epsilon_1=1}^{-}$ given by Eq. (6.428). Given that $\phi_0/M_{\text{Pl}} < 1/\sqrt{2}$, the resulting expression can be evaluated in the limit $\phi_0 \ll M_{\text{Pl}}$, and one obtains

$$ \epsilon_2^{\min}(x < 0) \approx 2 \frac{M_{\text{Pl}}^2}{\phi_0^2}, \qquad (6.429) $$

which coincides with $\epsilon_2^{\min}(x > 0)$, i.e. with the value of $\epsilon_2$ at the maximum of the potential. One has therefore $\epsilon_2^{\min} \gg 1$ in this regime, which excludes the possibility to realize slow-roll inflation. The only remaining solution would be to fine tune $\beta$ close to $\sqrt{2}\phi_0/M_{\text{Pl}}$. In that case, however, inflation would end at very large negative values of $x$, where the potential is dominated by its first exponential branch. The predictions of the model become again close to the ones of Power-Law Inflation (PLI, see Section 5.8) in this regime. This is why we will not further consider the regime DEI2, and will focus on DEI1 hereafter.

The slow-roll trajectory can be integrated, and one obtains

$$ N_{\text{end}} - N = \frac{1 + \beta^2}{\beta} \frac{\phi_0^2}{M_{\text{Pl}}^2} (x - x_{\text{end}}) + \frac{\phi_0^2}{M_{\text{Pl}}^2} \ln \left( \frac{e^{x_{\text{end}}/\beta} - e^{\beta x_{\text{end}}}}{e^{x/\beta} - e^{\beta x}} \right). \qquad (6.430) $$

When $x \to 0$, the number of e-folds diverges, which indicates that one can always realize a sufficient number of e-folds by starting close enough to the maximum of the potential. Unfortunately, this trajectory needs to be inverted numerically. Combined with the reheating equation (3.48), this allows us to determine $x_*$, the field value at which the pivot mode crosses out the Hubble radius during inflation. In turn, this determines the mass scale $M$ of the potential from the CMB normalization and one finds

$$ \left(\frac{M}{M_{\text{Pl}}}\right)^4 = 720\pi^2\beta^2 \frac{Q_{\text{rms-P5}}^2}{T^2} \frac{M_{\text{Pl}}^2}{\phi_0^2} \frac{(e^{\beta x_*} - e^{x_*/\beta})^2}{(e^{\beta x_*} - \beta^2 e^{x_*/\beta})^3}. \qquad (6.431) $$

The reheating-consistent slow-roll predictions of DEI1 are displayed in Figs. 283 to 286 for $\phi_0/M_{\text{Pl}} = 10, 20, 50$ and $100$ respectively. In DEI1, as argued above, slow-roll inflation requires $\phi_0 \gg M_{\text{Pl}}$. This is why, in order to gain some analytical insight, it is useful to expand the above expressions in this limit. In this regime (more precisely, under the condition $\phi_0/M_{\text{Pl}} \gg 1/\beta$), one has $x_{\text{end}} \approx 2\beta \ln(\beta)/(\beta^2 - 1)$, and the slow-roll trajectory can be approximated as $x_* \approx x_{\text{end}} - \sqrt{2\Delta N_*} M_{\text{Pl}}/\phi_0$, which gives rise to

$$ \epsilon_{1*} \approx \frac{1}{(1 + 2\Delta N_*)^2}, \qquad \epsilon_{2*} \approx \epsilon_{3*} \approx 4\epsilon_{1*}. \qquad (6.432) $$
<figure><img src="image_97.png" /><figcaption>Fig. 94. S-Dual Inflation (SHI). Top left panel: the potential as a function of $\phi/\mu$. Top right panel: logarithm of the potential. Bottom left panel: rescaled slow-roll parameter $\epsilon_1\mu^2/M_{\text{Pl}}^2$. Notice that $\epsilon_1$ is always smaller than unity for super-Planckian $\mu$, in which case inflation does not gracefully ends. Bottom right panel: rescaled slow-roll parameters $\epsilon_2\mu^2/M_{\text{Pl}}^2$ (solid line) and $-\epsilon_3\mu^2/M_{\text{Pl}}^2$ (red dotted line).</figcaption></figure>

## 6.26. S-Dual Inflation (SDI)

This scenario has been proposed in Ref. [665] and motivated by the wish to have inflation producing a significant amount of tensor modes while having a concave potential. It is loosely motivated by the S-duality in String Theory as the inflaton is considered to be a dilaton field. Because the string coupling constant is given by $g \propto e^{\phi/\mu}$, symmetry under the S-duality transformation $g \to 1/g$ requires the potential to be symmetric under parity $\phi \to -\phi$. Moreover, since a low-energy effective action should be an expansion in the string coupling constant $g$, the potential should be made of exponential terms. From these motivations, Ref. [665] considers a potential of the form

$$V(\phi) = \frac{M^4}{\cosh\left(\frac{\phi}{\mu}\right)}, \qquad (6.433)$$

where $\mu$ is a typical vacuum expectation value for the dilaton field.

The potential is even, by construction, so we can restrict our analysis to positive field values only. It is a monotonic decreasing function of the field and inflation proceeds at increasing field values. Defining

$$x \equiv \frac{\phi}{\mu}, \qquad (6.434)$$

the Hubble flow functions in the slow-roll approximation read

$$\epsilon_1 = \frac{M_{\text{Pl}}^2}{2\mu^2} \tanh^2(x), \quad \epsilon_2 = \frac{2M_{\text{Pl}}^2}{\mu^2} \frac{1}{\cosh^2(x)}, \quad \epsilon_3 = -\frac{2M_{\text{Pl}}^2}{\mu^2} \tanh^2(x) = -4\epsilon_1. \qquad (6.435)$$

The potential and the Hubble-flow functions have been presented in Fig. 94. As this figure emphasizes, the first Hubble-flow function asymptotes to

$$\epsilon_1^{\text{max}} = \frac{2M_{\text{Pl}}^2}{\mu^2}, \qquad (6.436)$$

at large field values. As a result, inflation ends naturally only for $\mu < \sqrt{2M_{\text{Pl}}}$ and at a field value given by

$$x_{\epsilon_1=1} = \operatorname{arctanh}\left(\sqrt{2} \frac{\mu}{M_{\text{Pl}}}\right). \qquad (6.437)$$
In this regime, inflation proceeds at increasing field value within the domain $0 < x < x_{\epsilon_1=1}$. However, as can be seen in the bottom-right panel of Fig. 94, $\epsilon_2$ may be larger than unity in this region. More precisely, one has $\epsilon_2(x) = 1$ at the field value

$$x_{\epsilon_2=1} = \operatorname{arccosh} \left( \frac{\sqrt{2} M_{\mathrm{Pl}}}{\mu} \right). \qquad (6.438)$$

For all $\mu < \sqrt{2/5} M_{\mathrm{Pl}}$, one has $x_{\epsilon_2=1} > x_{\epsilon_1=1}$, and since $\epsilon_2$ is a decreasing function of the field value, this implies that slow roll is violated $\epsilon_2 > 1$ over the whole inflating domain. One may want to restrict $\mu$ to the range $\sqrt{2/5} < \mu/M_{\mathrm{Pl}} < 1/\sqrt{2}$, for which $\epsilon_2$ is smaller than one when inflation ends, but one can check that the values of $\epsilon_2$ in the relevant part of the inflationary dynamics are still too large to produce a viable inflationary scenario. For these reasons, we now consider only the super-Planckian values of $\mu > M_{\mathrm{Pl}}/\sqrt{2}$, for which an additional mechanism has to be invoked to end inflation. This could be, for instance, a tachyonic instability triggered by an additional field. We denote the field value at which inflation ends by $x_{\mathrm{end}} = \phi_{\mathrm{end}}/\mu$ making SDI a two-parameter model.

The slow-roll trajectory can be integrated analytically and reads

$$N_{\mathrm{end}} - N = \frac{\mu^2}{M_{\mathrm{Pl}}^2} \ln \left[ \frac{\sinh(x_{\mathrm{end}})}{\sinh(x)} \right], \qquad (6.439)$$

which can be inverted as

$$x = \operatorname{arcsinh} \left[ e^{-\frac{M_{\mathrm{Pl}}^2 (N_{\mathrm{end}} - N)}{\mu^2}} \sinh(x_{\mathrm{end}}) \right]. \qquad (6.440)$$

Combined with the reheating equation (3.48), one can determine uniquely $x_*$, the field value at which the pivot mode crossed the Hubble radius during inflation. The mass scale of the potential is then given by the CMB normalization and one finds

$$\left(\frac{M}{M_{\mathrm{Pl}}}\right)^4 = 720\pi^2 \frac{M_{\mathrm{Pl}}^2}{\mu^2} \sinh(x_*) \tanh(x_*) \frac{Q^2_{\mathrm{rms}-\mathrm{PS}}}{T^2}. \qquad (6.441)$$

The reheating consistent slow-roll predictions for SDI have been plotted in Fig. 287. At small values of $x_{\mathrm{end}}$, the model predictions asymptote a $\mu$-dependent constant spectral index with a very small amount of gravitational waves. This can be immediately understood from Eq. (6.435). The inflationary domain being at $x < x_{\mathrm{end}}$, in the limit of small $x$ one has $\epsilon_1 \to 0$ and $\epsilon_2 \to 2M_{\mathrm{Pl}}^2/\mu^2$, which is typical of a small-field model with non-vanishing mass (see SF12 in Section 6.1). At large values of $x_{\mathrm{end}}$, one can check that, for mildly super-Planckian values of $\mu$, a substantial amount of gravitational waves can be produced (as mentioned above this was one of the original motivations for this model, although it occurs in the convex region of the potential), since $\epsilon_1$ asymptotes a constant at large-field values and the tensor-to-scalar ratio is controlled by $\epsilon_1$ at leading order in slow roll.

## 6.27. Generalized Double Well Inflation (GDWI)

These models are a generalization of Double Well Inflation (DWI) discussed in Section 5.14 and are of the “Mexican-hat” type. The potential is given by

$$V(\phi) = M^4 \left[ \left( \frac{\phi}{\phi_0} \right)^{2p} - 1 \right]^2, \qquad (6.442)$$

where $\phi_0$ is a vev and $p > 1$ is the power index. The case $p = 1$ corresponds to DWI, which is presented in Section 5.14. There, it is shown that DWI has different observable predictions than the quadratic Small Field Inflation (SFI) model of Section 6.1 and, as such, cannot be simply viewed as a large-field regularization of the SFI potential. This is due to the fact that both DWI and quadratic SFI support slow-roll inflation only for $\phi_0 > M_{\mathrm{Pl}}$ for which they significantly differ. Indeed, at the top of the potential, the second Hubble-flow function for these two models is given by $\epsilon_2 = 8M_{\mathrm{Pl}}^2/\phi_0^2$ and it would exceed unity for $\phi_0 < M_{\mathrm{Pl}}$. Such a feature comes from the fact that a quadratic term in the potential implies a non-vanishing effective mass at the top of the potential.

For $p > 1$ the effective mass term vanishes and GDWI can support slow-roll inflation for both $\phi_0 < M_{\mathrm{Pl}}$ and $\phi_0 \ge M_{\mathrm{Pl}}$. As such, for sub-Planckian $\phi_0$, GDWI with $p > 1$ can be considered as a UV completion of the SFI models with the same power index $p$ (see Section 6.1). In particular, the case $p = 2$ has been numerically studied in Ref. [9] and shown to smoothly regularize the quartic SFI model while reducing some of its fine-tuning issues at very small vev $\phi_0 \ll M_{\mathrm{Pl}}$.

The potential of GDWI, and its logarithm, have been represented in Fig. 95 as a function of $\phi/\phi_0$. The negative domain is not represented as the potential is symmetric under the mapping $\phi \to -\phi$. Moreover, we consider only the inflationary domain $\phi < \phi_0$. The potential can also inflate at large field values but, in this region, it behaves exactly as Large Field Inflation (LFI) with a power index of $4p$ (see Section 5.2).

Defining the dimensionless field value

$$x = \frac{\phi}{\phi_0}, \qquad (6.443)$$

the Hubble-flow functions, in the slow-roll approximation, reduce to

$$\epsilon_1 = 8p^2 \left( \frac{M_{\mathrm{Pl}}}{\phi_0} \right)^2 \frac{x^{2(2p-1)}}{(x^{2p}-1)^2}, \qquad \epsilon_2 = 8p \left( \frac{M_{\mathrm{Pl}}}{\phi_0} \right)^2 \frac{x^{2p} (x^{2p} + 2p - 1)}{x^2 (x^{2p}-1)^2}, \qquad (6.444)$$

and

$$\epsilon_3 = 8p \left( \frac{M_{\mathrm{Pl}}}{\phi_0} \right)^2 \frac{x^{2p} \left[ x^{4p} + (2p-1)(p+2)x^{2p} + 2p^2 - 3p + 1 \right]}{x^2 (x^{2p}-1)^2 (x^{2p} + 2p - 1)}. \qquad (6.445)$$
<figure><img src="image_98.png" /><figcaption>Fig. 95. Generalized Double Well Inflation for $p = 2$. Upper panels: the potential and its logarithm as a function of $\phi/\phi_0$. Only the positive domains are represented as the potential is symmetric under $\phi \to -\phi$. Bottom left panel: first Hubble-flow function $\epsilon_1$, divided by $M_{\text{Pl}}^2/\phi_0^2$, as a function of $\phi/\phi_0$. The bottom right panel shows $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line), both divided by $M_{\text{Pl}}^2/\phi_0^2$, as a function of $\phi/\phi_0$.</figcaption></figure>

They have been represented in the lower panels of Fig. 95, and, as can be seen in these plots, inflation proceeds at the top of the hill towards increasing field values. It gracefully ends when $\epsilon_1(x_{\text{end}}) = 1$ and, from Eq. (6.444), $x_{\text{end}}$ is the solution of

$$x_{\text{end}}^{2p} - 2\sqrt{2p} \frac{M_{\text{Pl}}}{\phi_0} x_{\text{end}}^{2p-1} = 1, \qquad (6.446)$$

in the small field domain $0 < x < 1$. This equation cannot be solved analytically for arbitrary values of $p$ and, in general, $x_{\text{end}}$ has to be determined numerically.

The slow-roll trajectory can, however, be determined by quadrature from Eq. (3.11) and reads

$$N_{\text{end}} - N = \frac{\phi_0^2}{8p(p-1)M_{\text{Pl}}^2} \left[ x^2 (p-1 + x^{-2p}) - x_{\text{end}}^2 (p-1 + x_{\text{end}}^{-2p}) \right]. \qquad (6.447)$$

Combined with the reheating equation (3.48), and the numerical solution of Eq. (6.446), one can numerically determine $x_*$, the field value at which the pivot mode crossed the Hubble radius during inflation.

The mass scale $M$ of the potential is then given by the CMB normalization and one finds

$$\left(\frac{M}{M_{\text{Pl}}}\right)^4 = 11520\pi^2 p^2 \frac{M_{\text{Pl}}^2}{\phi_0^2} \frac{x_*^{2(2p-1)}}{(x_*^{2p}-1)^4} \frac{Q_{\text{rms}-\text{PS}}^2}{T^2}. \qquad (6.448)$$

The reheating consistent slow-roll predictions for GDWI are represented in Figs. 290 to 292.

## 6.28. Non-Minimal Large Field Inflation (NMLFI)

### 6.28.1. Theoretical justification

We consider again the conformal model described by Eq. (4.35), except that the potential is now given by a quartic power of the field $\phi$, namely

$$S(g_{\mu\nu}, \chi, \phi) = \frac{M_{\text{Pl}}^2}{2} \int d^4 x \sqrt{-g} \left[ \frac{\chi^2}{6} R + g^{\mu\nu} \partial_{\mu} \chi \partial_{\nu} \chi - \frac{\phi^2}{6} R - g^{\mu\nu} \partial_{\mu} \phi \partial_{\nu} \phi - \frac{\lambda}{2} \phi^4 \right]. \qquad (6.449)$$

Here again, for the sake of clarify, we drop the “bar” over Jordan frame quantities. This model was considered in Ref. [666]. The above action resembles the action of the T-model, see Section 5.30. However, instead of a potential term $\propto F_T(\phi/\chi)(\phi^2 - \chi^2)^2$, we now have a potential which no longer depends on $\chi$ and is simply given by $\phi^4$.
As noticed after Eq. (4.35), this action is invariant under the transformation, $\tilde{g}_{\mu\nu} = e^{-2\sigma}g_{\mu\nu}$, $\tilde{\phi} = e^{\sigma}\phi$ and $\tilde{\chi} = e^{\sigma}\chi$. Notice that the conformal invariance requires the potential to be proportional to $\phi^4$ (if it does not depend on $\chi$). As before, the sign of the kinetic term of $\chi$ (the “conformon”) is the “wrong” one. However, as before, this is not a problem because we can always fix the conformal gauge, for instance by choosing $\chi = \sqrt{6}$. In that case, the action can be re-written as

$$S(g_{\mu\nu}, \phi) = \frac{M_g^2}{2} \int d^4 x \sqrt{-g} \left[ \left(1 - \frac{\phi^2}{6}\right) R - g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - \frac{\lambda}{2} \phi^4 \right]. \quad (6.450)$$

The above action corresponds to a scalar-tensor theory. Using Eqs. (4.72), (4.74) and the equation for the potential in the text below Eq. (4.74), the model can be rewritten in the Einstein frame, with a potential

$$V(\tilde{\phi}) = \frac{M_g^2 \lambda}{4} \frac{\phi^4}{\left(1 - \frac{\phi^2}{6}\right)^2}, \quad (6.451)$$

where $\phi(\tilde{\phi})$ is given in terms of the canonically normalized field $\tilde{\phi}$ in the Einstein frame by

$$\frac{\phi}{\sqrt{6}} = \frac{1 - e^{-\sqrt{\frac{2}{3}} \frac{\tilde{\phi}}{M_g^2}}}{1 + e^{-\sqrt{\frac{2}{3}} \frac{\tilde{\phi}}{M_g^2}}}. \quad (6.452)$$

As noticed in Ref. [666], the coefficient in front of the term $\phi^2 R$ in the action is fixed by the requirement of maintaining the conformal invariance of the model. However, if the model is embedded in conformal supergravity, this restriction can be avoided. We now discuss this case.

As discussed at the end of Section 4.1.2, conformal supergravity depends on two functions, $N$ and the potential $W$. The Lagrangian density of conformal supergravity was already given in Eq. (4.41) and reads

$$\mathcal{L} = \sqrt{-g} \left[ -\frac{1}{6} \mathcal{N}(X, \bar{X}) R - G_{I\bar{J}} D^\mu X^I D_\mu \bar{X}^J - V(X, \bar{X}) \right], \quad (6.453)$$

with $D_\mu X^I = \partial_\mu X^I - iA_\mu X^I$. The fact that the superfields $X^I$ are now charged is a difference compared with Eq. (4.41). Here, we consider a model where the function $N(X, \bar{X})$ is defined by

$$\mathcal{N}(X^0, X^1) = -|X^0|^2 + |X^1|^2 - 3\Delta |X^0|^2 \left[ \left( \frac{X^1}{X^0} \right)^2 + \left( \frac{\bar{X}^1}{\bar{X}^0} \right)^2 \right], \quad (6.454)$$

where $\Delta$ is a dimensionless parameter, $X^0$ the conformon and $X^1$ the inflaton. Notice that, when $\Delta = 0$, the embedding potential has an enhanced SU(1, 1) symmetry. Compared to the superconformal model of Sections 4.1 and 5.30, we see that only two fields are present, the conformon $X^0$ and the inflaton $X^1$. The Goldstino $S$ is now absent. Straightforward calculations lead to the kinetic terms of those fields and one obtains

$$G_{00} = -1 + 3\Delta \left[ \left( \frac{X^1}{X^0} \right)^2 + \left( \frac{\bar{X}^1}{\bar{X}^0} \right)^2 \right], \quad G_{0\bar{1}} = -6\Delta \frac{\bar{X}^1}{\bar{X}^0}, \quad G_{1\bar{0}} = -6\Delta \frac{X^1}{X^0}, \quad G_{1\bar{1}} = 1. \quad (6.455)$$

We see that the parameter $\Delta$ is proportional to the curvature of the Kähler internal manifold since $\Delta = 0$ implies that $G_{I\bar{I}} = \delta_{I\bar{I}}$.

An important property of the above action is that it is invariant under the following transformation

$$\tilde{g}_{\mu\nu} = e^{-2\sigma} g_{\mu\nu}, \quad \tilde{X}^I = e^{\sigma+i\Lambda} X^I, \quad \tilde{X}^J = e^{-\sigma-i\Lambda} \tilde{X}^J, \quad \tilde{A}_{\mu} = A_{\mu} + \partial_{\mu} \Lambda, \quad (6.456)$$

as we are going to show explicitly. Let us first notice that the transformations (6.456) imply that

$$D_{\mu} X^{I} = e^{-\sigma - i \Lambda} \tilde{D}_{\mu} X^{I} - e^{-\sigma + i \Lambda} \tilde{X}^{I} \partial_{\mu} \sigma. \quad (6.457)$$

Let us split the Lagrangian in two parts $\mathcal{L} = \mathcal{L}_1 + \mathcal{L}_2$ with

$$\mathcal{L}_1 = -\frac{1}{6} \sqrt{-g} \left[ \left( -|X^0|^2 + |X^1|^2 \right) - |X^0|^2 (G_{00} + 1) \right] R, \quad (6.458)$$

$$\mathcal{L}_2 = -\sqrt{-g} \left( G_{I\bar{J}} g^{\mu\nu} D_\mu X^I D_\nu \bar{X}^J + V \right).$$

By the transformation (6.456), $\mathcal{L}_1$ becomes

$$\tilde{\mathcal{L}}_1 = -\frac{1}{6} e^{4\sigma} \sqrt{-\tilde{g}} \left[ e^{-2\sigma} \left( -|\tilde{X}^0|^2 + |\tilde{X}^1|^2 \right) - e^{-2\sigma} |\tilde{X}^0|^2 (\tilde{G}_{00} + 1) \right] \times e^{-2\sigma} \left( \tilde{R} - 6\tilde{g}^{\mu\nu} \tilde{\nabla}_\mu \partial_\nu \sigma - 6\tilde{g}^{\mu\nu} \partial_\mu \sigma \partial_\nu \sigma \right), \quad (6.459)$$

where we have used the fact that the components of $G_{I\bar{I}}$ are invariant under Eq. (6.456). We notice that the exponential terms exactly cancel out. Then, the transformation of the term $\mathcal{L}_2$ can be expressed as

$$\begin{aligned} \tilde{\mathcal{L}}_2 &= -e^{4\sigma} \sqrt{-\tilde{g}} \left\{ \tilde{G}_{I\bar{I}} e^{-2\sigma} \tilde{g}^{\mu\nu} \left[ e^{-2\sigma} \tilde{D}_{\mu} \tilde{X}^I \tilde{D}_{\bar{\nu}} \tilde{X}^{\bar{J}} - e^{-2\sigma} (\tilde{D}_{\mu} \tilde{X}^I) \tilde{X}^{\bar{J}} \partial_{\bar{\nu}} \sigma \right. \right. \\ & \qquad \left. - e^{-2\sigma} \tilde{X}^I (\tilde{D}_{\bar{\nu}} \tilde{X}^{\bar{J}}) \partial_{\mu} \sigma + e^{-2\sigma} \tilde{X}^I \tilde{X}^{\bar{J}} \partial_{\mu} \sigma \partial_{\bar{\nu}} \sigma \right] + V \right\} \\ &= -\sqrt{-\tilde{g}} \tilde{G}_{I\bar{I}} \tilde{g}^{\mu\nu} \left[ \tilde{D}_{\mu} \tilde{X}^I \tilde{D}_{\bar{\nu}} \tilde{X}^{\bar{J}} - (\partial_{\mu} \tilde{X}^I - i \tilde{A}_{\mu} \tilde{X}^I) \tilde{X}^{\bar{J}} \partial_{\bar{\nu}} \sigma - \tilde{X}^I (\partial_{\bar{\nu}} \tilde{X}^{\bar{J}} + i \tilde{A}_{\mu} \tilde{X}^{\bar{J}}) \partial_{\mu} \sigma \right. \\ & \qquad \left. + \tilde{X}^I \tilde{X}^{\bar{J}} \partial_{\mu} \sigma \partial_{\bar{\nu}} \sigma \right] - \sqrt{-\tilde{g}} e^{4\sigma} V. \end{aligned} \quad (6.460)$$
The two terms proportional to the gauge fields $\bar{\lambda}_{\mu}$ cancel out while the second and third terms can be rewritten as a total derivative,<fnref n="12" /> namely

$$ \partial_{\mu} \left[ \sqrt{-g} \bar{g}^{\mu\nu} \tilde{G}_{Ij} \tilde{X}^I \tilde{X}^j (\partial_{\nu} \sigma) \right] = \sqrt{-g} \left[ \bar{\nabla}_{\mu} (\bar{g}^{\mu\nu} \partial_{\nu} \sigma) \tilde{G}_{Ij} \tilde{X}^I \tilde{X}^j + \bar{g}^{\mu\nu} (\partial_{\nu} \sigma) (\bar{\nabla}_{\mu} \tilde{G}_{Ij}) \tilde{X}^I \tilde{X}^j \right. \\ \left. + \bar{g}^{\mu\nu} (\partial_{\nu} \sigma) \tilde{G}_{Ij} \bar{\nabla}_{\mu} (\tilde{X}^I \tilde{X}^j) \right]. \qquad (6.461) $$

Collecting the various terms in $\mathcal{L} = \mathcal{L}_1 + \mathcal{L}_2$, one obtains

$$ \begin{aligned} \bar{\mathcal{L}} = \sqrt{-\bar{g}} & \left[ -\frac{1}{6} \mathcal{N} (\tilde{X}, \tilde{\bar{X}}) \tilde{R} - \tilde{G}_{Ij} \tilde{g}^{\mu\nu} \tilde{D}_{\mu} \tilde{X}^I \tilde{D}_{\nu} \tilde{\bar{X}}^j - e^{4\sigma} V \right] \\ & + \sqrt{-\bar{g}} \left[ \left(-|\tilde{X}^0|^2 + |\tilde{X}^1|^2\right) - |\tilde{X}^0|^2 (\tilde{G}_{00} + 1) - \tilde{G}_{Ij} \tilde{X}^I \tilde{\bar{X}}^j \right] [\bar{g}^{\mu\nu} \partial_{\mu} \sigma \partial_{\nu} \sigma + \bar{g}^{\mu\nu} \tilde{\nabla}_{\mu} (\partial_{\nu} \sigma)] \\ & + \partial_{\mu} \left[ \sqrt{-\bar{g}} \bar{g}^{\mu\nu} \tilde{G}_{Ij} \tilde{X}^I \tilde{\bar{X}}^j (\partial_{\nu} \sigma) \right] - \sqrt{-\bar{g}} \bar{g}^{\mu\nu} (\partial_{\nu} \sigma) (\partial_{\mu} \tilde{G}_{Ij}) \tilde{X}^I \tilde{\bar{X}}^j. \end{aligned} \qquad (6.462) $$

Using the internal metric of Eq. (6.455), one finds that the first term of the second line, the one within square brackets, vanishes. It remains the last term. Using the definition of the internal metric, one has

$$ \partial_{\mu} \tilde{G}_{Ij} = \partial_{\mu} \mathcal{N}_{Ij} = \mathcal{N}_{KIJ} \partial_{\mu} X^{K} + \mathcal{N}_{KIJ} \partial_{\mu} \tilde{X}^{K}, \qquad (6.463) $$

from which

$$ (\partial_{\mu} G_{Ij}) X^I \tilde{X}^j = \mathcal{N}_{KIJ} X^I \tilde{X}^j \partial_{\mu} X^K + \mathcal{N}_{KIJ} X^I \tilde{X}^j \partial_{\mu} \tilde{X}^K = 0. \qquad (6.464) $$

If $V(X, \tilde{X})$ is homogeneous and of second degree in $X$ and $\tilde{X}$ then, as announced, the Lagrangian (6.453) is indeed invariant under the transformation (6.456). An explicit example is given in Ref. [666] with $V = \lambda(X^1 \tilde{X}^1)^2$.

Let us now return to Eq. (6.453) and fix the conformon by assuming $X^0 = \tilde{X}^0 = \sqrt{3}M_g$. Then, the Lagrangian becomes

$$ (\sqrt{-g})^{-1} \mathcal{L} = \left\{ \frac{M_g^2}{2} - \frac{1}{6} |X^1|^2 + \frac{\Delta}{2} \left[ (X^1)^2 + (\tilde{X}^1)^2 \right] \right\} R - \partial_{\mu} X^1 \partial^{\mu} \tilde{X}^1 - \lambda (X^1 \tilde{X}^1)^2. \qquad (6.465) $$

Decomposing $X^1 = (\varphi_1 + i\varphi_2)/\sqrt{2}$, the Lagrangian reads

$$ (\sqrt{-g})^{-1} \mathcal{L} = \left[ \frac{M_g^2}{2} + \frac{1}{2} (\Delta - \frac{1}{6}) \varphi_1^2 - \frac{1}{2} (\Delta + \frac{1}{6}) \varphi_2^2 \right] R - \frac{1}{2} \partial_\mu \varphi_1 \partial^\mu \varphi_1 \\ - \frac{1}{2} \partial_\mu \varphi_2 \partial^\mu \varphi_2 - \frac{\lambda}{4} (\varphi_1^2 + \varphi_2^2)^2. \qquad (6.466) $$

In particular, notice that the model is invariant by changing the sign of $\Delta$ and swapping the fields $\varphi_1$ and $\varphi_2$. If one focuses on the choice $\Delta > 0$, then the ground state of the system is obtained for $\varphi_2 = 0$ and, renaming $\varphi_1 \equiv \phi$, one has

$$ (\sqrt{-g})^{-1} \mathcal{L} = \frac{M_g^2}{2} R - \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} (\frac{1}{6} - \Delta) \phi^2 R - \frac{\lambda}{4} \phi^4. \qquad (6.467) $$

As announced, one therefore obtains a non-minimally coupled large field quartic model but, contrary to Eq. (6.450), there is now an arbitrary coefficient in front of the $\phi^2 R$ term. This Lagrangian describes a scalar-tensor theory in the Jordan frame, from which one obtains the Einstein frame's potential (see Section 4.2.2).

$$ V(\tilde{\phi}) = \frac{\lambda}{8} \frac{\phi^4}{(1 + \xi \phi^2 / M_g^2)^2}, \qquad (6.468) $$

where we have defined $\xi \equiv \Delta - 1/6$ and where the canonically normalized field $\tilde{\phi}$ can be expressed as

$$ \frac{\tilde{\phi}}{M_g} = \sqrt{\frac{1 + 6\xi}{\xi}} \operatorname{arcsinh} \left[ \sqrt{\xi(1 + 6\xi)} \frac{\phi}{M_g} \right] - \sqrt{6} \operatorname{arctanh} \left[ \frac{\xi\sqrt{6\phi}/M_g}{\sqrt{1 + \xi(1 + 6\xi)\phi^2/M_g^2}} \right]. \qquad (6.469) $$

As one may expect, this relation is exactly the same as Eq. (4.82) for Higgs Inflation (with the identification $h = \phi/M_g$ and $\chi = \tilde{\phi}/M_g$). As it is the case for HI, it is not possible to analytically invert this relation to obtain an explicit expression for the potential $V(\phi)$. Let us notice that in the limit where $\Delta \to 0$, one finds

$$ \frac{\tilde{\phi}}{M_g} = \sqrt{6} \operatorname{arctanh} \left( \frac{\phi}{M_g \sqrt{6}} \right) + \mathcal{O}(\Delta), \qquad (6.470) $$

which gives back Eq. (6.452), as expected.

### 6.28.2. Slow-roll analysis

From the previous theoretical motivations, we consider the class of Non-Minimal Large Field Inflation (NMLFI) models defined as having a potential in the Jordan frame identical to the LFI models of Section 5.2, i.e., $U(\phi) = \bar{\lambda} M_g^2 (\tilde{\phi}/M_g)^p$ [667,668], where we now denote by $\tilde{\phi}$ the Jordan frame real scalar field. Here $M_g$ is the gravitational mass scale in the Jordan frame and $\bar{\lambda}$ a dimensionless coupling constant. As explained

<footnote n="12">¹² For any vector $V^\mu$, $\sqrt{-g} \nabla_\mu V^\mu = \partial_\mu (\sqrt{-g} V^\mu)$.</footnote>
<figure><img src="image_99.png" /><figcaption>Fig. 96. Non-Minimal Large Field Inflation (NMLFI) for $p < 4$, the potential develops a maximum at intermediate field values. The top left panel shows the potential as a function of $\chi = \bar{\phi}/M_{\text{pl}}$ for the two sets of parameter values reported in the figure. Top right panel: logarithm of the potential. For $\xi = 0.1$ and $p = 0.1$ (blue curves), both the NMLFI1 (left of the maximum) and NMLFI2 (right of the maximum) inflationary regimes end when the first Hubble flow function (lower left panel) exceeds unity. The slow-roll parameters $\epsilon_2$ (solid curve) and $|\epsilon_3|$ (dotted curve) are represented in the bottom right panel. For $\xi = 1$ and $p = 2$ (red curves), NMLFI1 still gracefully ends whereas inflation on the right of the potential maximum, NMLFI3, needs an additional mechanism to end inflation. The potential for $p > 4$ is represented in Fig. 97.</figcaption></figure>

in Section 4.2.2, only if the vacuum state of the theory after inflation is at $\bar{\phi}/M_g \approx 0$, one can identify the numerical value of $M_g \approx M_{\text{Pl}}$, see Eq. (4.79). The non-minimal coupling functions appearing in the scalar-tensor action of Eq. (4.72) are of the form

$$F(h) = 1 + \xi \left( \frac{\bar{\phi}}{M_g} \right)^2, \quad Z(h) = 1, \qquad (6.471)$$

with $\xi \ge 0$. As for Higgs Inflation in Section 4.2, we introduce the two dimensionless fields

$$\bar{h} \equiv \sqrt{\xi} \frac{\bar{\phi}}{M_g}, \quad \chi \equiv \frac{\bar{\phi}}{M_g}, \qquad (6.472)$$

where $\bar{\phi}$ is the canonically normalized scalar field in the Einstein frame. The potential of NMLFI in the Einstein frame can only be given in the parametric way and reads

$$V(\bar{\phi}) = M^4 \frac{\bar{h}^p}{(1 + \bar{h}^2)^2}, \qquad (6.473)$$

where $M$ is the inflationary mass scale in the Einstein frame and verifies $M^4 = M_g^4 \bar{\lambda}/\xi^{p/2}$, see Eq. (4.89). The function $\bar{h}(\chi)$ is the solution of Eq. (4.85), namely

$$\chi = \sqrt{6 + \frac{1}{\xi}} \ln \left[ \sqrt{1 + (1 + 6\xi)\bar{h}^2} + \sqrt{(1 + 6\xi)\bar{h}^2} \right] + \sqrt{6} \ln \left[ \frac{\sqrt{1 + \bar{h}^2}}{\sqrt{1 + (1 + 6\xi)\bar{h}^2} + \sqrt{6\xi\bar{h}^2}} \right], \qquad (6.474)$$

which cannot be inverted explicitly. This is not an issue as the Hubble flow functions can nevertheless be determined in a parametric form. Following what has been done for HI in Section 4.2.4, from Eq. (4.97), one gets

$$\epsilon_1(\bar{h}) = \frac{\xi}{2\bar{h}^2} \frac{[p + (p - 4)\bar{h}^2]^2}{1 + (1 + 6\xi)\bar{h}^2}, \quad \epsilon_2(\bar{h}) = 2\xi \frac{1 + \bar{h}^2}{\bar{h}^2} \frac{p + (p + 4 + 12p\xi)\bar{h}^2}{[1 + (1 + 6\xi)\bar{h}^2]^2}, \qquad (6.475)$$

and

$$\epsilon_3(\bar{h}) = 2\xi \frac{p + (p - 4)\bar{h}^2}{\bar{h}^2} \times \frac{p + 3p(1 + 6\xi)\bar{h}^2 + [4 + 3p + 48\xi + 36p\xi(1 + 4\xi)]\bar{h}^4 + (1 + 6\xi)(4 + p + 12p\xi)\bar{h}^6}{[1 + (1 + 6\xi)\bar{h}^2]^2 [p + (4 + p + 12p\xi)\bar{h}^2]} \qquad (6.476)$$
<figure><img src="image_100.png" /><figcaption>Fig. 97. Non-Minimal Large Field Inflation (NMLFI) for $p > 4$, the potential has no maximum and only the NMLFI regime exists. For $p = 4$, the potential has a plateau and one recovers the potential of Higgs Inflation for any value of $\xi$, see Fig. 7. The top left panel shows the potential as a function of $\chi = \phi/M_{pl}$. Top right panel: logarithm of the potential. Bottom left panel: the first slow-roll parameter $\epsilon_1$. At large field value $\chi$, $\epsilon_1$ is constant. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid curve) and $\epsilon_3$ (dotted curve). The NMLFI potential for $p < 4$ is represented in Fig. 96.</figcaption></figure>

The potential and the Hubble flow functions have been plotted in Figs. 96 and 97, for various values of $\xi$ and $p$, in terms of the dimensionless canonically normalized field $\chi$.

As can be seen on this figure, the potential admits a local maximum for all values of $p < 4$. Solving for $\epsilon_1 = 0$, the maximum occurs at the parameter value $\tilde{h}_{Vmax}$ given by

$$ \tilde{h}_{Vmax} = \sqrt{\frac{p}{4-p}}. \qquad (6.477) $$

The corresponding value of the canonically normalized field $\chi_{Vmax}$ can be obtained by plugging Eq. (6.477) into Eq. (6.474). For $p < 4$, inflation can then occur in two regions. Either at decreasing parametric field values, for $\tilde{h} < \tilde{h}_{Vmax}$, in a regime that will be referred to as NMLFI1, or at increasing parametric field values for $\tilde{h} > \tilde{h}_{Vmax}$. Since $\epsilon_1(\tilde{h})$ increases when $\tilde{h}$ decreases, NMLFI1 always gracefully ends at a parametric field value close to zero. In the other domain, at large $\tilde{h}$ values, Eq. (6.475) implies

$$ \lim_{\tilde{h} \to \infty} \epsilon_1 = \frac{(p-4)^2}{2} \frac{\xi}{1+6\xi}. \qquad (6.478) $$

We immediately see that if $\xi$ is too small, the asymptotic limit of $\epsilon_1 < 1$ and inflation never ends. More specifically, let us define

$$ \xi_0(p) \equiv \frac{2}{p(p-8)+4} = \frac{2}{(p-p_*)(p-p_+)'} \qquad (6.479) $$

where $p_\pm$ are the two roots of the quadratic denominator:

$$ p_- \equiv 2(2-\sqrt{3}) \approx 0.54, \qquad p_+ \equiv 2(2+\sqrt{3}) \approx 7.46. \qquad (6.480) $$

From Eq. (6.478), one has $\lim_{\tilde{h} \to \infty} \epsilon_1 > 1$ provided two conditions are satisfied: $p < p_-$ and $\xi > \xi_0(p)$. Under these conditions, in the domain $\tilde{h} > \tilde{h}_{Vmax}$ inflation stops at the field value where $\epsilon_1$ reaches unity. This regime will be refereed to as NMLFI2.

Still in the domain $\tilde{h} > \tilde{h}_{Vmax}$, for $p < p_-$ and $\xi < \xi_0(p)$, but also for $p_- < p < 4$, the asymptotic limit of $\epsilon_1 < 1$ and inflation never ends. One therefore needs an additional mechanism to stop inflation, as for instance a tachyonic instability triggered by an extra field. This inflationary regime is then a model with three parameters, $p$, $\xi$ and $\chi_{end}$ (or $\tilde{h}_{end}$), that we refer to as NMLFI3.

For $p > 4$ the potential has no maximum, inflation can only proceed at decreasing field values and this regime will be also referred to as NMLFI1. The limiting case $p = 4$ is exactly Higgs Inflation (HI) with $v = 0$, a vanishing vev, and unconstrained values for $\xi$, see Section 4.2.4.

Let us notice that since both NMLFI2 and NMLFI3 can explore some part of the large field region, one should pay attention to the actual value of $\tilde{h}_{end}$, the parametric field value after inflation, in order to determine how much the numerical value of $M_g$ differs from $M_{pl}$. From Eq. (4.79),
one indeed has, at the end of inflation

$$M_g^2 = \frac{M_{Pl}^2}{1 + \bar{h}_{\text{end}}^2} \frac{1 + \bar{h}_{\text{end}}^2 + 8\xi\bar{h}_{\text{end}}^2}{1 + \bar{h}_{\text{end}}^2 + 6\xi\bar{h}_{\text{end}}^2} \quad (6.481)$$

For large enough $\bar{h}_{\text{end}}$, and provided $\bar{h}$ remains frozen after inflation, one has $M_g < M_{Pl}$ and these models can potentially address the Planck mass hierarchy problem [669].

For both NMLF1 and NMLF2, the parametric field value at which inflation stops is solution of $\epsilon_1 = 1$. From Eq. (6.475), this equation admits, a priori, two solutions

$$\bar{h}_{\pm}^2 = \frac{p(p-4)\xi - 1 \pm \sqrt{(1+2p\xi)(1+6p\xi)}}{2 - \xi[p(p-8)+4]} \quad (6.482)$$

For $p_- < p < p_+$ the denominator is always positive. Therefore, one has $\bar{h}_+^2 > 0$ whereas $\bar{h}_-^2 < 0$ and the end of inflation for NMLF1 occurs at the parametric field value $\bar{h}_{\text{end}} = \bar{h}_+$.

For $p > p_+$, one always have $\bar{h}_-^2 < 0$ whereas $\bar{h}_+^2 > 0$ under the additional condition that $\xi < \xi_0(p)$. Let us mention that $\xi_0(p)$ is also a root of the numerator in Eq. (6.482) such that determining its sign requires some attention. For these cases, NMLF1 ends again at $\bar{h}_{\text{end}} = \bar{h}_+$. If $\xi > \xi_0(p)$ (still for $p > p_+$), one has $\epsilon_1 > 1$ for all the values of $\bar{h}$ and the potential ends up being too steep to support inflation at all.

At last, for $p < p_-$ one always have $\bar{h}_+^2 > 0$ whereas $\bar{h}_-^2 > 0$ only under the additional condition of having $\xi > \xi_0(p)$. As a result, for $p < p_-$, NMLF1, which proceeds at $\bar{h} < \bar{h}_{V\text{max}}$, ends once more at $\bar{h}_{\text{end}} = \bar{h}_+$ whereas NMLF2, which proceeds at $\bar{h} > \bar{h}_{V\text{max}}$, ends at $\bar{h}_{\text{end}} = \bar{h}_-$ but only if $\xi > \xi_0(p)$. For $\xi < \xi_0(p)$, as already discussed, inflation does not end by itself and only NMLF3 exists in the domain $\bar{h} > \bar{h}_{V\text{max}}$.

The parametric slow-roll trajectory can be determined as done for Higgs Inflation in Eq. (4.103), and, can be analytically integrated. The case $p=4$ requires special attention and one gets

$$\begin{aligned} \Delta N &= \frac{2 + 3\xi p}{4\xi(p-4)} \ln \left[ \frac{p + (p-4)\bar{h}^2}{p + (p-4)\bar{h}_{\text{end}}^2} \right] - \frac{3}{4} \ln \left( \frac{1 + \bar{h}^2}{1 + \bar{h}_{\text{end}}^2} \right), & \text{for } p \neq 4, \\ \Delta N &= \frac{1 + 6\xi}{8\xi} (\bar{h}^2 - \bar{h}_{\text{end}}^2) - \frac{3}{4} \ln \left( \frac{1 + \bar{h}^2}{1 + \bar{h}_{\text{end}}^2} \right), & \text{for } p = 4, \end{aligned} \quad (6.483)$$

where $\Delta N = N_{\text{end}} - N$. For $p \neq 4$, the trajectory cannot be inverted analytically and one has to solve Eq. (6.483) numerically to determine $\bar{h}(\Delta N)$, and hence $\chi(\Delta N)$ from Eq. (6.474). The special case $p=4$ can nevertheless be inverted in terms of a Lambert function as

$$\bar{h}^2(\Delta N) = -1 - \frac{6\xi}{1+6\xi} W_1 \left\{ -\frac{(1+6\xi)(1+\bar{h}_{\text{end}}^2)}{6\xi} e^{-\frac{1}{6\xi}[1+6\xi(1+\bar{h}_{\text{end}}^2)+8\xi\Delta N]} \right\}, \quad \text{for } p=4. \quad (6.484)$$

From Eq. (6.483), one can check that, for $p < 4$, $\Delta N \to \infty$ for $\bar{h} \to \bar{h}_{V\text{max}}$ and an infinite number of e-folds can be realized at the top of the potential. However, the divergence is only logarithmic and this implies that NMLF2, which inflates in the domain $[\bar{h}_{V\text{max}}, \bar{h}_{\text{end}}]$ is a very fine-tuned, if not ruled-out, model. Indeed, it exists only for large enough values of $\xi > \xi_0(p)$ and this implies that $\Delta N$ can be made larger than, say, 60 e-folds only if the initial field value is exponentially fine-tuned to the top of the potential. Then, Eq. (6.475) implies that

$$\epsilon_2(\bar{h}_{V\text{max}}) = \frac{8\xi(4-p)}{2+3p\xi} > \frac{8}{1-p}, \quad (6.485)$$

where the last inequality comes from $\xi > \xi_0(p)$. Because NMLF2 requires $p < p_-$, one gets $\epsilon_2(\bar{h}_{V\text{max}}) > 8$ and since all of the e-folds of inflation are done at the top of the potential, NMLF2 violates slow-roll and is hardly compatible with cosmological observations. For these reasons, it is no longer considered in the following.

For NMLF1 and NMLF3, the previous equations allow us to determine the parameter $\bar{h}_*$ at which the pivot scale crosses the Hubble radius during inflation by solving the Einstein frame's reheating equation of Section 4.1.4. Notice that, contrary to HI, the value of the coupling $\bar{\lambda}$ is not set by the underlying model and the non-minimal coupling $\xi$ is a model parameter not fixed by the amplitude of the CMB anisotropies. As such, one can forget about $\bar{\lambda}$ and trade it for the Einstein frame mass scale $M$. Once $\bar{h}_*$ is determined, $M$ is simply given by the normalization of the potential

$$\frac{M^4}{M_g^4} = 720\pi^2\xi \frac{(1 + \bar{h}_*^2)^2 [p + (p-4)\bar{h}_*^2]^2 Q_{\text{rms-PS}}^2}{\bar{h}_*^{p+2} [1 + (1 + 6\xi)\bar{h}_*^2]} \quad (6.486)$$

Let us notice that in a situation where $\bar{\lambda}$ would be fixed by the underlying theory, then, one should solve instead the coupled reheating equations as explained in Section 4.2.4.

The reheating-consistent slow-roll prediction for NMLF1 have been represented in Figs. 293 to 298 for various values of $p$ and $\xi$, in the two regimes $p < 4$ where it is a small field model and for $p > 4$ where it becomes large field-like. Predictions for NMLF3 can be found in Figs. 299 to 316 for various values of the three parameters $p, \xi$ and $\chi_{\text{end}}$. Here as well, we have split the parameter domains into a “small” region, for $p < p_-$ with $\xi < \xi_0(p)$ and a “large” region for $p_- < p < 4$. Let us notice that, for NMLF3, the values of $\chi_{\text{end}}$ reported on these plots imply that the numerical value of $M_{\text{pl}}$ could be up to three orders of magnitude larger than the numerical value of $M_g$, after inflation. Another remark is that the case $p=4$ (HI) is unique in the sense that only for a quartic power index $p$ the potential in the Einstein frame is of the plateau-type. For any other values of $p$, one ends up with potentials radically different than the plateau-type.
## 6.29. Superconformal $\alpha$-Attactor B Inflation (SABI)

### 6.29.1. Theoretical justifications

In this section, we consider a generalization of the “T-models” (TMI, see Section 5.30), which leads to another family of $\alpha$-attractor models. The idea is to introduce a new embedding Kähler potential and a new superconformal superpotential given by

$$ \mathcal{N}(X^I, \bar{X}^I) = -|X^0|^2 \left[ 1 - \frac{|X^1|^2 + |S|^2}{|X^0|^2} + 3\zeta \frac{|S|^4}{|X^0|^2 (|X^0|^2 - |X^1|^2)} \right]^\alpha \quad (6.487) $$

$$ \mathcal{W} = S(X^0)^2 f\left(\frac{X^1}{X^0}\right) \left[1 - \left(\frac{(X^1)^2}{(X^0)^2}\right)^{\frac{3\alpha-1}{2}}\right]. \quad (6.488) $$

The field content of the model is the same as for the TMI model, namely a conformon $X^0$, the inflaton $X^1 = \Phi$ and a Goldstino $X^3 = S$. As before, the term proportional to the parameter $\zeta$ is introduced to make sure that the inflationary trajectory is stable.

The main new aspect of the model is the presence of the parameter $\alpha$. For $\alpha = 1$, one recovers the embedding Kähler potential (5.472) and the superconformal superpotential (5.473). Then, the conformal symmetry is fixed by choosing $X^0 = \bar{X}^0 = \sqrt{3} M_g$ and, using Eq. (4.42), one obtains the Kähler and superpotential, namely

$$ K = -3\alpha M_g^2 \ln(1+k), \quad (6.489) $$

$$ W = 3M_g^2 S f \left( \frac{\Phi}{\sqrt{3}M_g} \right) \left( 1 - \frac{\Phi^2}{3M_g^2} \right)^{\frac{3\alpha-1}{2}}, \quad (6.490) $$

where the function $k$ has already been defined in Eq. (5.475). Compared to the Kähler potential of TMI, Section 5.30, we see that it is still given by a logarithm of the function $1 + k$ but, now, multiplied by the parameter $\alpha$.

From the above expressions of $K$ and $W$, one can then calculate the kinetic term and the potential of the inflaton field. As done in Section 5.30, this calculation is carried out along the inflationary trajectory $S = 0$. One obtains

$$ G_{A\bar{B}} = - \frac{3\alpha}{1+k} \frac{\partial^2 k}{\partial X^A \partial \bar{X}^B} + \frac{3\alpha}{(1+k)^2} \frac{\partial k}{\partial X^A} \frac{\partial k}{\partial \bar{X}^B}, \quad (6.491) $$

that is to say

$$ G_{A\bar{B}} = \frac{\alpha}{M_g^2(1+k)^2} \begin{pmatrix} 1 & 0 \\ 0 & 1+k \end{pmatrix}. \quad (6.492) $$

These two expressions are the generalization of Eqs. (5.481) and (5.482). One immediately deduces that the canonically normalized inflaton field $\varphi$ can be expressed in terms of $\Phi$ in the following way

$$ \Phi = \sqrt{6} M_g \tanh \left( \frac{\varphi}{\sqrt{6\alpha} M_g} \right), \quad (6.493) $$

and this expression should be compared to its TMI’s counterpart in Eq. (5.483). Finally, by using Eq. (5.480), the potential of the canonically normalized field $\varphi$ can be derived and one obtains

$$ V(\varphi) = 9M_g^4 \left| f \left[ \tanh \left( \frac{\varphi}{\sqrt{6\alpha} M_g} \right) \right] \right|^2. \quad (6.494) $$

As a consequence, one obtains a potential which ressembles a lot the potential of the T-models, the difference being that the argument of the function $f$ now involves $\varphi/(\sqrt{6\alpha} M_g)$ instead of $\varphi/(\sqrt{6} M_g)$. We have therefore obtained a mixed model, combining aspects of the T-model with aspects of the $\alpha$-attractor.

In order to have an explicit potential, one must choose the function $f$, which is equivalent to the choice of the function $F_T(.)$ in Section 5.30. As noticed in Sections 4.1.2 and 5.30, the choice $F_T(\phi/\chi) = (\phi/\chi)^2/(1+\phi/\chi)^2$ leads to the Starobinsky model for $\alpha = 1$. If $\alpha \neq 1$, this leads to an alternative derivation of the $\alpha$-attractor model, as mentioned at the end of Section 5.29. One can also choose the more general form

$$ F_T\left(\frac{\phi}{\chi}\right) = \left(\frac{\phi/\chi}{1+\phi/\chi}\right)^{2n}, \quad (6.495) $$

where $n$ is a new free index. This leads to the potential

$$ V(\phi) = 9M^4 \left[ \frac{\tanh\left(\frac{\phi}{M_g\sqrt{6\alpha}}\right)}{1 + \tanh\left(\frac{\phi}{M_g\sqrt{6\alpha}}\right)} \right]^{2n}, \quad (6.496) $$

Clearly, this potential is a direct generalization of the $\alpha$-attractor potential of Section 5.29.
<figure><img src="image_101.png" /><figcaption>Fig. 98. Superconformal α-attractor B Inflation (SABI). Top left panel: the potential as a function of φ/M_g. Top right panel: logarithm of the potential. Bottom left panel: the first slow-roll parameter ϵ_1. Bottom right panel: slow-roll parameters ϵ_2 (solid line) and ϵ_3 (dotted line).</figcaption></figure>

### 6.29.2. Slow-roll analysis

As explained in the previous section, the potential of the SABI model depend on two parameter, α and n, and can also be written as (we have redefined the scale M)

$$V(\phi) = M^4 \left(1 - e^{-\sqrt{\frac{2}{3\alpha}}x}\right)^{2n}, \qquad (6.497)$$

where x = φ/M_g. For n = 1, one recovers the α-attractor model, see Section 5.29, and for n = 1 and α = 1, one has the Starobinsky model, see Section 4.1. These models may also be referred to as α-attractor E models in the literature. The potential (6.497) is represented in Fig. 98 for different values of α and n. Since after inflation the field settles at the minimum of the potential where it vanishes, the numerical value of M_g ≈ M_{Pl}.
The three Hubble flow functions are given by

$$\epsilon_1 = \frac{4n^2}{3\alpha \left(e^{\sqrt{\frac{2}{3\alpha}}x} - 1\right)^2}, \quad \epsilon_2 = \frac{2n}{3\alpha \left[\sinh\left(\frac{x}{\sqrt{6\alpha}}\right)\right]^2}, \quad \epsilon_3 = \frac{4n}{3\alpha \tanh\left(\frac{x}{\sqrt{6\alpha}}\right) \left(e^{\sqrt{\frac{2}{3\alpha}}x} - 1\right)}. \qquad (6.498)$$

Evidently, when α = 1 and n = 1, these expressions reduce to Eqs. (4.48) while if n = 1 (and α unspecified), one recovers the expressions of the Hubble flow functions given in Section 5.29. The Hubble flow functions have been represented in the lower panels of Fig. 98.
In this scenario inflation gracefully ends when ϵ_1 = 1, at a field value x_{end} given by

$$x_{\text{end}} = \sqrt{\frac{3\alpha}{2}} \ln \left( 1 + \frac{2n}{\sqrt{3\alpha}} \right). \qquad (6.499)$$

However, as it was the case for Higgs inflation in Section 4.2, and also for the α-attractor scenario (see Section 5.29), violation of the slow-roll conditions can occur before. The value of the field for which ϵ_2 = 1 can be expressed as

$$x_{\epsilon_2=1} = \sqrt{6\alpha} \operatorname{arcsinh} \left( \sqrt{\frac{2n}{3\alpha}} \right), \qquad (6.500)$$

and the field value for which ϵ_3 = 1 is

$$x_{\epsilon_3=1} = \sqrt{6\alpha} \operatorname{arctanh} \left( \frac{2}{1 + \sqrt{1 + 6\alpha/n}} \right). \qquad (6.501)$$

In the case of the α-attractor model, regardless of the value of α, the field reaches first the value x_{ε_2=1}, then x_{ε_3=1} and, finally, x_{end}. It is interesting to notice that, here, this hierarchy does no longer exist: when the parameter n is changed, the field value can reach x_{ε_2=1} before inflation stops, or not.
<figure><img src="image_102.png" /><figcaption>Fig. 99. Superconformal α-Attactor T Inflation (SATI). Top left panel: the potential as a function of $\phi/M_g$. Top right panel: logarithm of the potential. Bottom left panel: the first slow-roll parameter $\epsilon_1$. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line).</figcaption></figure>

The slow-roll trajectory can be analytically derived and reads

$$N_{\text{end}} - N = \frac{1}{2n} \sqrt{\frac{3\alpha}{2}} (x_{\text{end}} - x) + \frac{3\alpha}{4n} \left( e^{\sqrt{\frac{2}{3\alpha}}x} - e^{\sqrt{\frac{2}{3\alpha}}x_{\text{end}}} \right). \qquad (6.502)$$

As it was the case for the α-attractor model, this trajectory can be inverted and expressed in term of the “-1-branch” of the Lambert function $W_{-1}$. One finds

$$x = \sqrt{\frac{3\alpha}{2}} \left\{ -\frac{4n}{3\alpha}\Delta N + \sqrt{\frac{2}{3\alpha}}x_{\text{end}} - e^{\sqrt{\frac{2}{3\alpha}}x_{\text{end}}} - W_{-1} \left[ -\exp\left(-\frac{4n}{3\alpha}\Delta N + \sqrt{\frac{2}{3\alpha}}x_{\text{end}} - e^{\sqrt{\frac{2}{3\alpha}}x_{\text{end}}}\right) \right] \right\}, \qquad (6.503)$$

where, as usual, $\Delta N = N_{\text{end}} - N$. The reason that inflation proceeds along the -1 branch of the Lambert function has already been explained in Section 5.29.

Finally, the value of $x_*$, at which the pivot mode crossed out the Hubble radius during inflation can be expressed as

$$x_* = \sqrt{\frac{3\alpha}{2}} \left[ -\frac{4n}{3\alpha}\Delta N_* + \ln\left(1 + \frac{2n}{\sqrt{3\alpha}}\right) - \left(1 + \frac{2n}{\sqrt{3\alpha}}\right) - \sqrt{\frac{3\alpha}{2}} W_{-1} \left\{ -\exp\left[-\frac{4n}{3\alpha}\Delta N_* + \ln\left(1 + \frac{2n}{\sqrt{3\alpha}}\right) - \left(1 + \frac{2n}{\sqrt{3\alpha}}\right) \right] \right\} \right], \qquad (6.504)$$

where, in this expression, we have used the value of $x_{\text{end}}$ derived above. From the knowledge of $x_*$, the energy scale $M$ of the potential can be inferred and one obtains

$$\frac{M^4}{M_g^4} = \frac{1920\pi^2 n^2}{\alpha} \left(1 - e^{-\sqrt{\frac{2}{3\alpha}}x_*}\right)^{-2(n+1)} e^{-2\sqrt{\frac{2}{3\alpha}}x_*} \frac{Q_{\text{rms}-\text{PS}}^2}{T^2}. \qquad (6.505)$$

The reheating consistent slow-roll prediction for Superconformal α-attractor B Inflation have been represented in Figs. 317 to 319.

## 6.30. Superconformal α-Attactor T Inflation (SATI)

These models have been discussed in Ref. [541]. In Section 6.29, we have seen how to generate a class of potentials that depend on $\tanh[\phi/(M_g\sqrt{6\alpha})]$, see Eq. (6.494). The precise shape of the potential then depends on an arbitrary function $f \sim F_T(\cdot)$. A specific choice was
made in Section 6.29 for this function and, here, we study another choice. In fact, this choice of $F_T(.)$ was already considered in Eq. (5.471) and is just a power-law. This directly leads to the SATI potential

$$V(\phi) = M^4 \left[ \tanh \left( \frac{\phi}{\sqrt{6\alpha} M_g} \right) \right]^{2n}, \qquad (6.506)$$

which represents a generalization of the TMI model discussed in Section 5.30. It describes a two parameters model with $\alpha$ and $n$, matching TMI for $\alpha = 1$.

Defining $x \equiv \phi/M_g$, one obtains the Hubble flow functions

$$\epsilon_1 = \frac{4n^2}{3\alpha} \sinh^{-2} \left( \frac{2x}{\sqrt{6\alpha}} \right), \quad \epsilon_2 = \frac{8n}{3\alpha} \frac{\cosh \left( \frac{2x}{\sqrt{6\alpha}} \right)}{\sinh^2 \left( \frac{2x}{\sqrt{6\alpha}} \right)}, \quad \epsilon_3 = \frac{3 + \cosh \left( \frac{4x}{\sqrt{6\alpha}} \right)}{\sinh^2 \left( \frac{2x}{\sqrt{6\alpha}} \right) \cosh \left( \frac{2x}{\sqrt{6\alpha}} \right)}. \qquad (6.507)$$

For $\alpha = 1$, one can check that the Hubble flow function of the TMI model of Section 5.30 are recovered. The potential, its logarithm and the Hubbel flow functions have been plotted in Fig. 99 as a function of $x$. The field value after inflation is expected to be at the minimum of the potential and vanishing. As a result, the numerical value of $M_g \approx M_{pl}$.

In this scenario, as shown in Fig. 99 (bottom left panel), the first slow-roll parameter increases as the vacuum expectation of the field decreases and this implies that inflation stops by violation of the slow-roll conditions, when $\epsilon_1 = 1$. The corresponding vacuum expectation value of the field reads

$$x_{\text{end}} = \frac{\sqrt{6\alpha}}{2} \arcsinh \left( \frac{2n}{\sqrt{3\alpha}} \right). \qquad (6.508)$$

The slow-roll trajectory can be integrated exactly and one obtains

$$N_{\text{end}} - N = \frac{3\alpha}{4n} \left[ \cosh \left( \frac{2x}{\sqrt{6\alpha}} \right) - \cosh \left( \frac{2x_{\text{end}}}{\sqrt{6\alpha}} \right) \right]. \qquad (6.509)$$

This formula can be inverted analytically and, as a consequence, $\phi/M_g$ during slow-roll inflation can be expressed as

$$x = \frac{\sqrt{6\alpha}}{2} \operatorname{arccosh} \left( \sqrt{1 + \frac{4n^2}{3\alpha}} + \frac{4n}{3\alpha} \Delta N \right), \qquad (6.510)$$

where $\Delta N = N_{\text{end}} - N$. The value of $x_*$ is just given by the above expression with $\Delta N = \Delta N_*$. Again, one verifies that the above formulas are equivalent to those presented in Section 5.30 when $\alpha = 1$.

Finally, the mass scale $M$ that normalizes the potential can be expressed as

$$\frac{M^4}{M_g^4} = \frac{1920\pi^2 n^2}{\alpha \sinh^2\left(\frac{2x_*}{\sqrt{6\alpha}}\right) \left[\tanh\left(\frac{x_*}{\sqrt{6\alpha}}\right)\right]^{2n}} \frac{Q_{\text{rms-PS}}^2}{T^2}. \qquad (6.511)$$

The reheating consistent observable predictions for SATI have been represented in Figs. 320 to 322 for various values of $n$ and $\alpha$. As before, one notices that the dependence of the spectral index and tensor-to-scalar ratio with respect to $n$ are very small. Indeed, if $n\Delta N$ dominates in Eq. (6.510), one obtains

$$x_* \approx \frac{\sqrt{6\alpha}}{2} \operatorname{arccosh} \left( \frac{4n}{3\alpha} \Delta N_* \right). \qquad (6.512)$$

Plugging this approximation into Eqs. (6.507) gives

$$\epsilon_{1*} \approx \frac{3\alpha}{4\Delta N_*^2}, \quad \epsilon_{2*} \approx \frac{2}{\Delta N_*}, \quad \epsilon_{3*} \approx \frac{1}{\Delta N_*}, \qquad (6.513)$$

and the Hubble-flow functions are independent of $n$ in the large $\Delta N_*$ limit. However, one notices that $\epsilon_{1*}$ retains a dependence in $\alpha$ while the two other Hubble flow parameters remains unaffected.

The reheating consistent slow-roll prediction for Superconformal $\alpha$-attractor T Inflation have been represented in Figs. 320 to 322.

# 7. Three parameters models

## 7.1. Running-mass Inflation (RMI)

### 7.1.1. Theoretical justifications

This model has been derived and studied in Refs. [418,670–678]. Following Ref. [673], let us briefly discuss its physical origin. At tree level, a potential can always be expanded as $V(\phi) \approx M^4 + m^2\phi^2/2 + \lambda\phi^4/4 + \dots$. Since the potential must be flat to support inflation, quantum corrections may play an important role. Typically, they modify the potential with a term of the form $(c_1 + c_2\phi^2 + c_4\phi^4) \ln(\phi/\mu)$, where $\mu$ is the renormalization scale. In a non-supersymmetric framework, the quartic term dominates and one is led to models similar to RCMI, RCQI or CWI, see Section 5.4, 5.5 and 5.11. On the other hand, in a supersymmetric context, at least if supersymmetry is spontaneously broken, the quadratic and the quartic terms cancel and one is left with a model similar to LI, see Section 5.12. If, however, supersymmetry is explicitly broken by the presence of soft terms, then the most important term will be the quadratic one.
Concretely, the above reasoning leads to a specific shape for the inflaton potential. We start from a flat direction in supersymmetry. Then, we assume that supersymmetry is explicitly broken and, as a consequence, that the potential receives corrections $\propto m^2\phi^2$, where m is a soft mass. Higher order terms are supposed to be negligible since we assume $\phi/M_{Pl} \ll 1$. We thus have

$$V = V_0 + \frac{1}{2}m^2\phi^2 + \dots, \qquad (7.1)$$

The one loop corrections to this tree potential will typically induce a logarithmic dependence of the soft mass through the renormalization group equation

$$\frac{dm^2}{d \ln \phi} = \beta_{\text{mat}}, \qquad (7.2)$$

where $\beta_{\text{mat}}$ is proportional to the inflaton couplings with the other fields present in the theory. Therefore, by Taylor expanding the solution of the previous equation aroused $\phi = \bar{\phi}$, we can write

$$m^2 = m^2(\bar{\phi}) + \beta_{\text{mat}} \ln \left( \frac{\phi}{\bar{\phi}} \right) + \dots. \qquad (7.3)$$

As a consequence, the potential (7.1) can be re-expressed as

$$V(\phi) = V_0 + \frac{1}{2}m^2(\bar{\phi})\phi^2 + \frac{1}{2}\beta_{\text{mat}}\phi^2 \ln\left(\frac{\phi}{\bar{\phi}}\right). \qquad (7.4)$$

As noticed in Refs. [673,676,678], the beta function can typically be expressed as

$$\beta_{\text{mat}} = \frac{-2C}{\pi} \alpha \bar{m}^2 + \frac{D}{16\pi^2} |\lambda|^2 m_{\text{loop}}^2, \qquad (7.5)$$

if we assume that the inflaton interacts with gauge bosons and fermions. The quantity $\alpha$ is the coupling constant between $\phi$ and the gauge boson, $\lambda$ is a Yukawa coefficient, $\bar{m}$ is the gaugino mass, $m$ the fermionic mass and $C$ and $D$ are dimensionless numbers of order one.

In the next section, we explore the cosmological consequences of this type of potential. In particular, we will see that it can lead to four different kind of inflationary scenarios.

### 7.1.2. Slow-roll analysis

We now perform the slow-roll analysis of the potential previously derived. In order to carry out this task, it is more convenient to re-write the potential as follows

$$V(\phi) = M^4 \left[ 1 - \frac{c}{2} \left( -\frac{1}{2} + \ln \frac{\phi}{\phi_0} \right) \frac{\phi^2}{M_{\text{Pl}}^2} \right], \qquad (7.6)$$

where we have defined the two parameters $c$ and $\phi_0$ by

$$c = -\frac{M_{\text{Pl}}^2 \beta_{\text{mat}}}{2V_0}, \qquad m^2(\bar{\phi}) = -\beta_{\text{mat}} \left[ \frac{1}{2} + \ln \left( \frac{\phi_0}{\bar{\phi}} \right) \right]. \qquad (7.7)$$

In this expression, $M$, $c$ and $\phi_0$ are free parameters. The dimensionless parameter $c$ can be positive or negative. With the form of the beta function given in Eq. (7.5), the coefficient $c$ is given by $\alpha m^2 M_{\text{Pl}}^2 / V_0$. If one assumes that the soft masses are of order $m \approx H \approx V_0^{1/2} / M_{\text{Pl}}^2$, then $c \approx \alpha \approx 10^{-2}$ to $10^{-1}$ or may be smaller depending on the assumption on the couplings. This also mean that, in order for Eq. (7.3) to be valid, one has $|\ln(\phi/\phi_0)| \ll 1$. Also, the model is commonly worked out in the vacuum dominated regime (otherwise it is equivalent to a large field model, LFI, see Section 5.2), which means that $c\phi_0^2/M_{\text{Pl}}^2 \ll 1$. The location $\phi = \phi_0$ is an extremum of $V(\phi)$, a maximum if $c > 0$ and a minimum if $c < 0$. The potential and its logarithm are represented in Fig. 100.

Running mass inflation can be realized in four different ways [673], denoted as RMI1, RMI2, RMI3 and RMI4 in what follows. RMI1 corresponds to the case where $c > 0$ and $\phi < \phi_0$, see Fig. 100 (top panels). In this case, $\phi$ decreases during inflation which proceeds from the right to the left. RMI2 also corresponds to $c > 0$ but with $\phi > \phi_0$ and $\phi$ increases during inflation which now proceeds from the left to the right. RMI3 refers to the situation where $c < 0$ and $\phi < \phi_0$ all the time. In this case, $\phi$ increases during inflation which proceeds from the left to the right. Finally, RMI4 has $c < 0$ and $\phi > \phi_0$ decreases as inflation proceeds from the right to the left.

Using the potential (7.6), one can calculate the three slow-roll parameters $\epsilon_1$, $\epsilon_2$ and $\epsilon_3$. Defining $x \equiv \phi/\phi_0$, one obtains the following expressions

$$\epsilon_1 = \frac{c^2}{2} \left[ \frac{\frac{\phi_0}{M_{\text{Pl}}} x \ln x}{1 - \frac{c}{2} \frac{\phi_0^2}{M_{\text{Pl}}^2} \left( -\frac{1}{2} + \ln x \right) x^2} \right]^2, \qquad (7.8)$$

$$\epsilon_2 = 2c \frac{1 + \frac{c}{4} \frac{\phi_0^2}{M_{\text{Pl}}^2} x^2 + \left(1 - \frac{c}{4} \frac{\phi_0^2}{M_{\text{Pl}}^2} x^2\right) \ln x + \frac{c}{2} \frac{\phi_0^2}{M_{\text{Pl}}^2} x^2 \ln^2 x}{\left[1 - \frac{c}{2} \frac{\phi_0^2}{M_{\text{Pl}}^2} \left(-\frac{1}{2} + \ln x\right) x^2\right]^2}, \qquad (7.9)$$
<figure><img src="image_103.png" /><figcaption>Fig. 100. Top left panel: running mass potential for $c = 0.8$ (blue line) or $c = -0.8$ (green line) and $\phi_0 = 0.5M_p$. Top right panel: logarithm of the potentials for the same values of $c$ and $\phi_0$. Bottom left panel: slow-roll parameter $\epsilon_1$ for a potential with $c = \pm 0.8$ and $\phi_0 = 0.5M_p$. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line) for $c = \pm 0.8$ and $\phi_0 = 0.5M_p$. The value $c = \pm 0.8$ may not be physical and was chosen only in order to produce a clear plot.</figcaption></figure>

and

$$ \epsilon_3 = \frac{c \ln x}{\left[1 - \frac{c}{2} \frac{\phi_0^2}{M_{\text{Pl}}^2} \left(-\frac{1}{2} + \ln x\right) x^2\right]^2} \left[1 + \frac{c}{4} \frac{\phi_0^2}{M_{\text{Pl}}^2} x^2 + \left(1 - \frac{c}{4} \frac{\phi_0^2}{M_{\text{Pl}}^2} x^2\right) \ln x + \frac{c}{2} \frac{\phi_0^2}{M_{\text{Pl}}^2} x^2 \ln^2 x\right]^{-1} \times \left[1 + \frac{c}{2} \frac{\phi_0^2}{M_{\text{Pl}}^2} x^2 + \frac{c^2}{16} \frac{\phi_0^4}{M_{\text{Pl}}^4} x^4 + c \left(2 \frac{\phi_0^2}{M_{\text{Pl}}^2} x^2 + \frac{c}{2} \frac{\phi_0^4}{M_{\text{Pl}}^4} x^4\right) \ln x \right. \\ \left. + c \left(3 \frac{\phi_0^2}{M_{\text{Pl}}^2} x^2 - \frac{c}{2} \frac{\phi_0^4}{M_{\text{Pl}}^4} x^4\right) \ln^2 x + \frac{c^2}{2} \frac{\phi_0^4}{M_{\text{Pl}}^4} x^4 \ln^3 x\right]. \quad (7.10) $$

The slow-roll parameters are represented in the bottom panels in Fig. 100.

Let us now examine how inflation ends in this model. The slow-roll parameter $\epsilon_1$ has a maximum in the $x < 1$ region and a maximum in the $x > 1$ region, see Fig. 100. If these maxima were larger than one, inflation could in principle stop by violation of the slow-roll conditions. In the vacuum dominated approximation, however, we see from Eq. (7.8), that $\epsilon_1 \simeq (c^2/2)(\phi_0^2/M_{\text{Pl}}^2)x^2 \ln^2 x$. This means that the vev $x_{\text{end}}$ satisfies $x_{\text{end}} \ln x_{\text{end}} = \pm(\sqrt{2}/c)(M_{\text{Pl}}/\phi_0)$. But we have established previously that the vacuum dominated condition precisely implies that $cM_{\text{Pl}}/\phi_0 \gg 1$ and one would have $\ln x_{\text{end}} \gg 1$. But for the model to be valid, we have already mentioned that the condition $|\ln x| \ll 1$ should be enforced. We conclude that the value of $x_{\text{end}}$ obtained above lies outside the regime of validity of the potential. The end of inflation either occurs by violation of slow-roll but in a regime where additional unknown corrections arise and modify the shape of $V(\phi)$, or by tachyonic instability. In this last case, inflation stops in a regime where our calculations are valid. This also means that we must consider an additional parameter in the model, namely $x_{\text{end}}$. In this article, this is the assumption made which implies that RMI is indeed a three parameters model.

We now turn to the calculation of the observable predictions. The first step is to obtain the slow-roll trajectory. One obtains

$$ N - N_{\text{end}} = \frac{1}{c} (\ln |\ln x| - \ln |\ln x_{\text{end}}|) - \frac{1}{4} \frac{\phi_0^2}{M_{\text{Pl}}^2} (x^2 - x_{\text{end}}^2) \\ + \frac{1}{4} \left( \frac{\phi_0}{M_{\text{Pl}}^2} \right)^2 [\text{Ei}(2 \ln x) - \text{Ei}(2 \ln x_{\text{end}})], \quad (7.11) $$

where the exponential integral function $\text{Ei}(x)$ is defined by $\text{Ei}(x) = -\int_{-x}^{+\infty} dt e^{-t} / t$ [281,282]. This expression cannot be inverted analytically. However, in the limit $(c\phi_0/M_{\text{Pl}})x \ll 1$ (the vacuum dominated regime), the above expression can be approximated by

$$ N - N_{\text{end}} \approx \frac{1}{c} (\ln |\ln x| - \ln |\ln x_{\text{end}}|), \quad (7.12) $$
from which it follows that

$$x(N) = \exp \left[ e^{c(N-N_{\text{end}})} \ln x_{\text{end}} \right]. \qquad (7.13)$$

The slow-roll predictions of the four models, RMI1, RMI2, RMI3 and RMI4 are presented in Figs. 323, 327, 331 and 335 for $|c| = 10^{-2}$, $\phi_0/M_{\text{Pl}} < 1/\sqrt{|c|}$, and $1/e < x_{\text{end}} < e$, respectively. In order to interpret them, it is interesting to use some approximations. From the trajectory (7.13), it is straightforward to calculate $x_*$. Recalling that inflation is supposed to stop at $x_{\text{end}}$, one obtains $x_* = \exp(e^{-c\Delta N_*} \ln x_{\text{end}})$. Then, using Eqs. (7.8), (7.9) and (7.10) in the vacuum dominated limit, we find that

$$\epsilon_{1*} \approx \frac{c^2}{2} \left( \frac{\phi_0}{M_{\text{Pl}}} \right)^2 \exp \left( 2e^{-c\Delta N_*} \ln x_{\text{end}} \right) e^{-2c\Delta N_*} \ln^2 x_{\text{end}}, \qquad (7.14)$$

$$\epsilon_{2*} \approx 2c \left(1 + e^{-c\Delta N_*} \ln x_{\text{end}}\right). \qquad (7.15)$$

In fact, in order to compare with the existing literature, it turns out to be convenient to define the following quantity

$$s \equiv c \ln x_* = -c e^{-c\Delta N_*} \ln x_{\text{end}}. \qquad (7.16)$$

For RMI1 and RMI4, $s > 0$ while for RMI2 and RMI3 one has $s < 0$. In terms of s Eqs. (7.14) and (7.15) can be re-written as

$$\epsilon_{1*} \approx \frac{s^2}{2} \left( \frac{\phi_0}{M_{\text{Pl}}} \right)^2 e^{-2s/c}, \qquad \epsilon_{2*} \approx 2c \left(1 - \frac{s}{c}\right). \qquad (7.17)$$

These equations imply that the locus of the model predictions in the plane $(\epsilon_1, \epsilon_2)$ are given by $\epsilon_2 \approx 2(c-s) + 4\epsilon_1 M_{\text{Pl}}^2/\phi_0^2$. If we neglect $\epsilon_{1*}$ (with respect to $\epsilon_{2*}$) one recovers the formula derived in Refs. [673,676,678], namely $n_s - 1 \approx 2(s-c)$. The same route for the third slow-roll parameter gives $\epsilon_2\epsilon_3 \approx -2cs$ and neglecting again $\epsilon_1$ gives the scalar running $\alpha_s \approx 2sc$. The above analytic estimates agree well with the complete slow-roll predictions represented in Figs. 323, 327, 331 and 335.

From the CMB normalization, we obtain the following expression for the mass scale

$$\frac{M_{\text{Pl}}^4}{M_{\text{Pl}}^4} = 720 \pi^2 c^2 \frac{Q_{\text{rms-PS}}^2}{T^2} \frac{\phi_0^2}{M_{\text{Pl}}^2} \left\{ 1 - \frac{c}{2} \frac{\phi_0^2}{M_{\text{Pl}}^2} \left[ -\frac{1}{2} + \ln(x_*) \right] x_*^2 \right\}^3. \qquad (7.18)$$

In the vacuum dominated regime, this expression can be approximated by

$$\frac{M_{\text{Pl}}^4}{M_{\text{Pl}}^4} \approx 720 \pi^2 s^2 \frac{Q_{\text{rms-PS}}^2}{T^2} \frac{\phi_0^2}{M_{\text{Pl}}^2} e^{s/c}. \qquad (7.19)$$

One can then easily deduce the mass scale M for a given value of c, $\phi_0$ and $x_{\text{end}}$, the three parameters of the model.

## 7.2. Valley Hybrid Inflation (VHI)

### 7.2.1. Theoretical justifications

Hybrid inflation is a two-fields model with the potential given by the following expression [221,318,418,679–682]

$$V(\phi, \psi) = \frac{1}{2} m^2 \phi^2 + \frac{\lambda'}{4} (\psi^2 - \Delta^2)^2 + \frac{\lambda}{2} \phi^2 \psi^2, \qquad (7.20)$$

where $\phi$ is the inflaton, $\psi$ the waterfall field, $\lambda$ and $\lambda'$ are two coupling constants and $\Delta$ a constant of dimension one. A priori, given the above potential, inflation can occur in different regimes. However, the standard lore is that inflation can proceed along the valley given by $\psi = 0$ and, in this case, the potential reduces to an effective single field potential that can be written as

$$V(\phi) = M^4 \left[ 1 + \left( \frac{\phi}{\mu} \right)^p \right], \qquad (7.21)$$

with $p = 2$ and where one has used the following parameter redefinition

$$M = \frac{\lambda'^{1/4} \Delta}{\sqrt{2}}, \qquad \mu = \sqrt{\frac{\lambda'}{2} \frac{\Delta^2}{m}}. \qquad (7.22)$$

Inflation along the valley has been shown to be a dynamical attractor of the two-field dynamics in Refs. [683,684]. However, as recently shown in Ref. [685], the hybrid potential can also support an inflationary phase along a mixed valley-waterfall trajectory, which is genuinely a two-fields dynamics. As we use a single field description here, those effects cannot be described by the potential of Eq. (7.21). For this reason, we will refer to the single field approximation as the “valley hybrid regime”. Let us stress that, if the waterfall inflationary regime occurs, then it will erase any observable effects coming the valley hybrid regime. As a result, Eq. (7.21) is a good description of hybrid inflation only if the model parameters are such that the waterfall regime remains sub-dominant. According to Ref. [685,686], this is the case provided

$$\sqrt{\lambda'} \frac{\Delta^3}{m} \ll M_{\text{Pl}}^2. \qquad (7.23)$$

a condition that will be assumed in the following. The effective potential (7.21) was also obtained in Ref. [687] in the context of supergravity brane inflation, and in Ref. [617] in the context of hilltop supernatural inflation. It depends on three parameters, namely M, $\mu$ and $p$. In fact, as mentioned before, $p = 2$ for the two-field model given in Eq. (7.20) but we will consider the most general situation with $p > 0$ unspecified. Let us stress again that all multifield effects such as the generation of isocurvature modes or cosmic strings cannot be accounted within the single field dynamics [205,688–690].
It is also worth mentioning that the potential (7.21) with $p = 2$ can also be obtained in the supergravity context [691–694]. The main idea is to consider a supergravity model which is not R-symmetry invariant and described by the following Kähler and super-potentials:

$$K = XX^{\dagger} + \frac{b}{6M^2} (XX^{\dagger})^2 - \frac{c}{9M^2} XX^{\dagger} [X^2 + (X^{\dagger})^2], \qquad (7.24)$$

$$W = fX, \qquad (7.25)$$

Here $X$ is a superfield, $M < M_{Pl}$ a mass scale and $b, c$ two dimensionless constants, a priori of order one. The quantity $f$ is a constant of dimension two that can be viewed as the supersymmetry breaking scale. From these expressions, the scalar potential reads

$$V = f^2 \left[ 1 - \frac{2b}{3M^2} XX^{\dagger} + \frac{c}{3M^2} (X^2 + X^{\dagger 2}) + \mathcal{O}\left(\frac{1}{M^4}\right) \right], \qquad (7.26)$$

or, re-writing $X = \alpha + i\beta$, it reads

$$V \approx f^2 \left[ 1 + \frac{2}{3M^2}(b-c)\alpha^2 - \frac{2}{3M^2}(b+c)\beta^2 \right]. \qquad (7.27)$$

For a field evolution along the $\alpha$ direction, we recover a potential of the VHI type with $p = 2$ ($b-c$ must be positive). In this setup, $\alpha/M \ll 1$ is required in order for the field $\alpha$ to be approximately canonically normalized, the Kähler potential being not minimal. It is also interesting to comment on the $\eta$-problem in this model since this is a generic issue in supergravity. If one calculates the slow-roll parameter $\eta \equiv M_{Pl}^2 V_{aa}/V$, one finds that

$$\eta = \frac{4M_{Pl}^2}{3M^2}(b-c). \qquad (7.28)$$

Therefore, one must take $M \lesssim M_{Pl}$ and fine-tune the difference $b-c$ to a small number.

### 7.2.2. Slow-roll analysis

We now turn to the slow-roll analysis of the VHI scenario. Recall that we consider the following potential

$$V(\phi) = M^4 \left[ 1 + \left( \frac{\phi}{\mu} \right)^p \right], \qquad (7.29)$$

where the parameter $M$ and $\mu$ have been expressed in terms of the parameters of the two-field model in Eq. (7.22). The first three Hubble flow functions in the slow-roll approximation can be derived from Eq. (7.29) in a straightforward fashion. Defining the quantity $x$ by the following expression

$$x \equiv \frac{\phi}{\mu}, \qquad (7.30)$$

they read

$$\epsilon_1 = \frac{p^2}{2} \left( \frac{M_{Pl}}{\mu} \right)^2 \frac{x^{2p-2}}{(1+x^p)^2}, \qquad \epsilon_2 = 2p \left( \frac{M_{Pl}}{\mu} \right)^2 \frac{x^{p-2}x^p - p+1}{(1+x^p)^2}, \qquad (7.31)$$

and

$$\epsilon_3 = p \left( \frac{M_{Pl}}{\mu} \right)^2 x^{p-2} \frac{2x^{2p} - (p-1)(p+4)x^p + (p-1)(p-2)}{(1+x^p)^2 (x^p - p + 1)}. \qquad (7.32)$$

A specific feature of hybrid inflation in comparison to large and small field models is that $\epsilon_2$ and $\epsilon_3$ can be negative (see Fig. 101). In particular

$$\epsilon_2 \underset{p \to 0}{\approx} -2p(p-1) \left( \frac{M_{Pl}}{\mu} \right)^2 x^{p-2}, \qquad (7.33)$$

and $\epsilon_3$ blows up in the limit $x^p \to p-1$. Together with the potential, the three Hubble flow functions have been represented in Fig. 101.

The slow-roll trajectory is obtained by integrating Eq. (3.11) with the valley hybrid potential and reads

$$N - N_{\text{end}} = \frac{\mu^2}{2p M_{\text{Pl}}^2} \left[ -x^2 + x_{\text{end}}^2 + \frac{2}{2-p} (x_{\text{end}}^{2-p} - x^{2-p}) \right], \qquad (7.34)$$

which is, up to a sign, the same as for the SFI models [see Eq. (6.5)]. The case $p=2$ requires special attention, but as for SFI, is recovered as the limit $p \to 2$ in the previous equation. One obtains

$$N - N_{\text{end}} = \frac{\mu^2}{4 M_{\text{Pl}}^2} \left[ -x^2 + x_{\text{end}}^2 - 2 \ln \left( \frac{x}{x_{\text{end}}} \right) \right], \qquad (7.35)$$

which is again very similar to SFI, up to a sign. The trajectory (7.34) cannot be inverted analytically in the general case. It is however possible to perform this inversion for many integer values of $p$, but those expressions will be omitted for the sake of clarity. We simply give an approximate solution valid only in the limit $x \ll 1$ and $p > 2$

$$x \approx \left[ x_{\text{end}}^{2-p} + p(p-2) \frac{M_{\text{Pl}}^2}{\mu^2} (N - N_{\text{end}}) \right]^{1/(2-p)}. \qquad (7.36)$$

If the waterfall inflation does not take place, i.e. under the condition (7.23), valley hybrid inflation ends by a tachyonic instability in the small field regime $x < 1$, also referred to as “the vacuum dominated regime”. From the two-fields potential (7.20), one sees that the transverse direction becomes tachyonic at the inflaton value

$$\phi_{\text{end}} = \sqrt{\frac{\lambda'}{\lambda}} \Delta. \qquad (7.37)$$
<figure><img src="image_104.png" /><figcaption>Fig. 101. Valley Hybrid inflation (VHI) for $p = 1/2$ (red line) and $p = 2$ (blue line). Upper panels: the potential and its logarithm for $\mu = 0.6M_{\text{pl}}$. Bottom left panel: slow-roll parameter $\epsilon_1$ for $p = 1/2$, $\mu = 0.6M_{\text{pl}}$ (red line), $p = 2$, $\mu = 0.6M_{\text{pl}}$ (blue line) and $p = 2$, $\mu = 0.9M_{\text{pl}}$ (green line). For small values of $\mu$ and $p > 1$, the inflationary regions are separated into a large field one and the vacuum dominated one. The latter may not exist due to slow-roll violations if the field first rolls down the potential in the large field domain (see the text for a detailed discussion). The shaded area indicates the regions in which acceleration cannot occur. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line) for $\mu = 0.6M_{\text{pl}}$.</figcaption></figure>

In the single field approach, $x_{\text{end}}$ is therefore an extra-parameter and VHI is a three parameters model according to our classification. However, as can be seen in Fig. 101, one should pay attention to the various domains in which inflation can take place. They are given by the behavior of $\epsilon_1(x)$.

If $p > 1$, the slow-roll parameter $\epsilon_1$ vanishes when the field goes to zero and at infinity while it reaches a maximum for

$$x_{\epsilon_1}^{\text{max}} = (p-1)^{1/p}, \qquad (7.38)$$

equals to

$$\epsilon_1^{\text{max}} = \frac{1}{2} \left( \frac{M_{\text{Pl}}}{\mu} \right)^2 (p-1)^{\frac{2p-2}{p}}. \qquad (7.39)$$

Defining

$$\mu_{\epsilon} \equiv \frac{M_{\text{Pl}}}{\sqrt{2}} (p-1)^{1-1/p}, \qquad (7.40)$$

for all $\mu > \mu_{\epsilon}$, one has $\epsilon_1(x) < 1$ and inflation can proceed all over the domain $x > 0$. On the contrary, if $\mu < \mu_{\epsilon}$, then inflation can, a priori, proceed in two disconnected domains. Either $0 < x < x_{\epsilon_1}^{-}$ or $x > x_{\epsilon_1}^{+}$ where $x_{\epsilon_1}^{\pm}$ are the two roots of $\epsilon_1 = 1$, i.e. the solutions of

$$x^{2p} + 2x^p - \frac{p^2}{2} \left( \frac{M_{\text{Pl}}}{\mu} \right)^2 x^{2p-2} + 1 = 0. \qquad (7.41)$$

This equation cannot be solved explicitly in the general case but, as for the trajectory, there are explicit analytic expressions for many integer values of $p$. For instance, for $p = 2$, one gets

$$x_{\epsilon_1=1}^{\pm(p=2)} = \frac{1}{\sqrt{2}} \frac{M_{\text{Pl}}}{\mu} \left( 1 \pm \sqrt{1 - 2 \frac{\mu^2}{M_{\text{Pl}}^2}} \right). \qquad (7.42)$$

The positive sign corresponds to the largest root while the minus one to the smallest (see Fig. 101). In the limit $\mu \ll M_{\text{Pl}}$, one has $x_{\epsilon_1=1}^{+} \simeq pM_{\text{Pl}}/(\sqrt{2}\mu)$ which is also the expression of $x_{\text{end}}$ for the large field model LFI (see Section 5.2). This does not come as a surprise since in that situation Eq. (7.29) is indeed dominated by the monomial term. In fact, the two above-mentioned domains precisely corresponds to a large field one for $x > x_{\epsilon_1=1}^{+}$ and a vacuum dominated one for $x < x_{\epsilon_1=1}^{-}$. It is a common mistake to assume that the large field domain remains unobservable due to the existence of the vacuum dominated one. In fact, as shown in Ref. [683], the large field regime becomes observable provided $\mu \ll \mu_{\epsilon}$. In that situation, after having crossed $x_{\epsilon_1=1}^{+}$, the field fast-rolls in the region $\epsilon_1(x) > 1$. Then, it enters the domain $x < x_{\epsilon_1=1}^{-}$ with a strong initial velocity and, as a consequence,
crosses the whole vacuum dominated region, still in fast-roll, to reach $x_{end}$. All observable predictions in such a situation are therefore similar to that obtained in the LFI models. Let us notice that, if there exists a mechanism that can gently put the field without a strong initial velocity inside the $x < x_{\epsilon_1}^{-1}$ domain, then inflation can still occur in the vacuum dominated region, even though $\mu < \mu_{\epsilon}$. But if the field is coming from the region $x > x_{\epsilon_1}^{+}$, then this regime does not exist anymore.

For $p = 1$, $\epsilon_1(x)$ is a decreasing function of the field and takes a finite value $M_{Pl}^2/(2\mu^2)$ for $x \to 0$. The behavior is similar to the case $p > 1$ and if $\mu > M_{Pl}/\sqrt{2}$ inflation can take place all over $x > x_{end}$. However, if $\mu < M_{Pl}/\sqrt{2}$ then the vacuum dominated region does not exist anymore and $x_{\epsilon_1}^{+} = x_{\epsilon_1}^{-} = M_{Pl}/(\sqrt{2}\mu) - 1$ One should also notice that if $p = 1$ the relation $\epsilon_2 = 4\epsilon_1$ applies.

Finally, for $p < 1$, $\epsilon_1(x)$ is a decreasing function of the field but it blows up when $x \to 0$. In that situation, inflation stops at $x = \max(x_{\epsilon_1}^{-1}, x_{end})$ but the field will still fast-roll till the tachyonic instability develops at $x_{end}$. As a result, even if for some cases $x_{\epsilon_1}^{-} > x_{end}$, the observable predictions remain mostly the same.

According to the previous discussion, for $p > 1$, the VHI effective potential is therefore adequate to describe the vacuum dominated regime only, i.e. for $x_{end} < x < x_{\epsilon_1}^{-}$, where $x_{end}$ is the instability point given by Eq. (7.37). In that situation, solving Eq. (3.48) together with the trajectory (7.34) gives the observable field value $x_{*}$ at which the pivot mode crossed the Hubble radius during inflation. The potential parameter $M$ is fixed from the amplitude of the CMB anisotropies

$$ \frac{M^4}{M_{Pl}^4} = 720\pi^2 p^2 \frac{M_{Pl}^2}{\mu^2} \frac{x_*^{2p-2}}{(1+x_*^p)^3} \frac{Q_{rms-PS}^2}{T^2}. \qquad (7.43) $$

The reheating consistent slow-roll predictions are displayed in Figs. 339, 341, 343, 345 and 347 for $p = 0.5$, $p = 1$, $p = 1.5$, $p = 2$ and $p = 3$, respectively. For $p > 1$ and $x_{\epsilon_{max}} > 1$, $x_{end}$ is varied between 0 and an upper bound such that $x_{in} < x_{\epsilon_1}^{-}$. One the other hand, if $x_{\epsilon_{max}} < 1$, then one simply takes $x_{end} < 10$. For $p \le 1$, $x_{end}$ is varied on a wider range, with no particular constraints. For $p = 1$, the predictions lie on the line $\epsilon_2 = 4\epsilon_1$ as expected whereas for $p > 1$ one recovers a blue spectral index when $x_{\epsilon_{max}} > 1$, while a red spectral index can be obtained when $x_{\epsilon_1}^{\max} < 1$ and $x_{*} > x_{\epsilon_1}^{\max}$, with $x_{*} < 1$ (that is to say, the large field regime).

## 7.3. Dynamical Supersymmetric Inflation (DSI)

### 7.3.1. Theoretical justifications

This model has been studied in Refs. [695,696]. As for the IMI scenario, see Section 6.18, the model is based on Ref. [633] which has shown that inverse power law potentials naturally arise in supersymmetric theories. The fact that we have an inverse power law behavior, rather than the usual positive power law behavior, can be traced back to the presence of non-perturbative effects, such as for instance gaugino condensation, see Section 6.18. Based on the previous considerations, one can write that

$$ V = V_0 + \frac{\Lambda^{p+4}}{3\phi^p} + \frac{\phi^{q+4}}{M_{Pl}^q}, \qquad (7.44) $$

where the last term encodes a correction to $V(\phi)$ due to a non-renormalizable operator. It is Planck suppressed since $M_{Pl}$ is the only explicit scale present in the theory. This term implies that there is a minimum located at

$$ \phi_{V \text{min}} = \left( \frac{p}{q+4} \Lambda_3^{p+4} M_{Pl}^q \right)^{\frac{1}{p+q+4}}. \qquad (7.45) $$

This means that the extra term can be neglected in the region $\phi \ll \phi_{V \text{min}}$ and, in the following, we assume that this is the case. The difference with the IMI scenario is the presence of the constant term $V_0$ which will be assumed to be dominant.

### 7.3.2. Slow-roll analysis

In this sub-section, we now turn to the slow-roll analysis of the DSI scenario. For this purpose, we rewrite the potential as

$$ V(\phi) = M^4 \left[ 1 + \left( \frac{\phi}{\mu} \right)^{-p} \right], \qquad (7.46) $$

where $p$ is a free index parameter and where we defined

$$ V_0 = M^4, \qquad \mu^p = \frac{\Lambda_3^{p+4}}{M^4}. \qquad (7.47) $$

As already mentioned, in order for inflation to take place in the vacuum dominated regime, we must assume that $\phi \gg \mu$. In Refs. [695,696], it was argued that natural values for $\Lambda_3$ and $M$ are $10^6$GeV and $10^{10}$GeV, respectively. This means that a scale of order $\mu \approx 10^{6+14/p}$ GeV is a reasonable prior for $\mu$.

The potential (7.46), as well as its logarithm, is displayed in Fig. 102. It is a decreasing function of the field, hence inflation proceeds from the left to the right. Defining the quantity

$$ x \equiv \frac{\phi}{\mu}, \qquad (7.48) $$

the first three Hubble flow functions in the slow-roll approximation read

$$ \epsilon_1 = \frac{p^2}{2} \left( \frac{M_{Pl}}{\mu} \right)^2 \frac{x^{-2p-2}}{(1+x^{-p})^2}, \qquad \epsilon_2 = -2p \left( \frac{M_{Pl}}{\mu} \right)^2 \frac{x^{-p} + p + 1}{(1+x^{-p})^2}, \qquad (7.49) $$

and

$$ \epsilon_3 = -p \left( \frac{M_{Pl}}{\mu} \right)^2 x^{-p-2} \frac{[2x^{-2p} + (p+1)(p-4)x^{-p} + (p+1)(p+2)]}{(1+x^{-p})^2 (x^{-p} + p + 1)}. \qquad (7.50) $$
<figure><img src="image_105.png" /><figcaption>Fig. 102. Dynamical Supersymmetric Inflation (DSI) for $p=2$. Upper panels: the potential and its logarithm as a function of $\phi/\mu$. Bottom left panel: slow-roll parameter $\epsilon_1$ rescaled by $M_{Pl}^2/\mu^2$. The shaded area indicates the region in which inflation cannot occur for $\mu = M_{Pl}$. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line), rescaled by $M_{Pl}^2/\mu^2$.</figcaption></figure>

Let us already notice that, from these expressions, one has

$$-2\epsilon_1 - \epsilon_2 = \left(\frac{M_{Pl}}{\mu}\right)^2 \frac{px^{-p-2}}{(1+x^{-p})^2} [px^{-p} + 2p(p+1)x^{-p-2}] > 0, \qquad (7.51)$$

which implies a blue spectral index for the scalar power spectrum since, at first order, $n_s - 1 = -2\epsilon_{1*} - \epsilon_{2*}$. The three slow-roll parameters become very small at large fields $x \gg 1$. There is a value $x_{\epsilon_1=1}$ such that $\epsilon_1 = 1$. For $x$ such that $x < x_{\epsilon_1=1}$, $\epsilon_1 > 1$ and inflation cannot take place. This value has to be determined numerically, but since the natural values for $\mu$ are such that $\mu/M_{Pl} \ll 1$, an approximate expression can be derived

$$x_{\epsilon_1=1} \approx \left( \frac{p}{\sqrt{2}} \frac{M_{Pl}}{\mu} \right)^{1/(p+1)} . \qquad (7.52)$$

Because the potential is decreasing with $x$, inflation can only take place in the domain $x > x_{\epsilon_1=1} \gg 1$ if $\mu \ll M_{Pl}$. It cannot stop by slow-roll violation and another mechanism such as, e.g. a tachyonic instability, has to be introduced. We will denote by $x_{end}$ the field value at which this occurs. It represents an extra parameter of the model. Obviously, it must be such that $x_{\epsilon_1=1} < x_{end} \ll x_{\nu \text{min}}$.

Let us now turn to the slow-roll trajectory. It can be integrated explicitly from Eq. (3.11) and one obtains

$$N_{\text{end}} - N = \frac{\mu^2}{2p M_{\text{Pl}}^2} \left( x_{\text{end}}^2 + \frac{2}{p+2} x_{\text{end}}^{p+2} - x^2 - \frac{2}{p+2} x^{p+2} \right). \qquad (7.53)$$

In the $\mu/M_{Pl} \ll 1$ limit, one has $x > x_{\epsilon_1=1} \gg 1$, and the previous trajectory can be approximated by

$$N_{\text{end}} - N \approx \frac{\mu^2}{p(p+2)M_{\text{Pl}}^2} \left( x_{\text{end}}^{p+2} - x^{p+2} \right). \qquad (7.54)$$

This expression can be analytically inverted to get the observable field value $x_*$ in terms of $\Delta N_* = N_{\text{end}} - N_*$ as

$$x_* \approx \left[ x_{\text{end}}^{p+2} - \frac{M_{\text{Pl}}^2}{\mu^2} p(p+2) \Delta N_* \right]^{\frac{1}{p+2}}. \qquad (7.55)$$

One can notice that the total amount of e-folds is bounded because $x_{end} \ll x_{\nu \text{min}}$ and cannot take infinitely large values. In order to get a number of e-folds, $\Delta N > \Delta N_{\text{min}}$, $x_{end}$ should be sufficiently large with $x_{end} > x_{\text{end}}^{\text{min}}$. More precisely, setting $x_{\text{ini}} = x_{\epsilon_1=1}$, one has

$$x_{\text{end}}^{\text{min}} \approx \left[ p(p+2) \frac{M_{\text{Pl}}^2}{\mu^2} \Delta N_{\text{min}} + \left( \frac{p}{\sqrt{2}} \frac{M_{\text{Pl}}}{\mu} \right)^{\frac{p+2}{p+1}} \right]^{\frac{1}{p+2}} \approx \left[ p(p+2) \frac{M_{\text{Pl}}^2}{\mu^2} \Delta N_{\text{min}} \right]^{\frac{1}{p+2}}. \qquad (7.56)$$
<figure><img src="image_106.png" /><figcaption>Fig. 103. Dynamical Supersymmetric Inflation. Maximal value of $\mu/M_P$ with respect to $p$, and for different values of $q$, such that the condition $x_{\text{end}}^{\text{min}} < x_{\text{end}}^{\text{max}}$ is satisfied. We have fixed $\Delta N_{\text{max}} = 50$. The black dotted line show a typical value for $\mu/M_P \approx 10^{10}$ GeV [695].</figcaption></figure>

In practice one wants $\Delta N_{\text{min}} > 50$ to solve the problems of the standard Big-Bang scenario. Whether this value is compatible, or not, with the condition $x_{\text{end}} \ll x_{\text{v-min}}$ depends on the value of $M^4$ appearing in Eq. (7.45), which is itself determined by the amplitude of the CMB anisotropies. This one reads

$$ \left(\frac{M}{M_{\text{Pl}}}\right)^4 = 720\pi^2 p^2 \left(\frac{M_{\text{Pl}}}{\mu}\right)^2 x_*^{-2p-2} (1+x_*^{-p})^{-3} \frac{Q_{\text{rms-PS}}^2}{T^2}. \qquad (7.57) $$

In the limit $\mu/M_{\text{Pl}} \ll 1$, one has $x_* \gg 1$ and this expression can be approximated by

$$ \frac{M^4}{M_{\text{Pl}}^4} \approx 720\pi^2 p^2 \frac{M_{\text{Pl}}^2}{\mu^2} x_*^{-2p-2} \frac{Q_{\text{rms-PS}}^2}{T^2}. \qquad (7.58) $$

Therefore, from Eq. (7.45), one has

$$ x_{\text{v-min}} \approx \left[ 720\pi^2 \frac{p^3}{q+4} \left(\frac{M_{\text{Pl}}}{\mu}\right)^{6+q} x_*^{-2p-2} \frac{Q_{\text{rms-PS}}^2}{T^2} \right]^{\frac{1}{p+q+4}}, \qquad (7.59) $$

with $x_*$ depending on $x_{\text{end}}$ through Eq. (7.55). One can see that the previous expression decreases with $x_*$ and the condition $x_{\text{end}} \ll x_{\text{v-min}}$ imposes an upper bound on $x_{\text{end}} < x_{\text{end}}^{\text{max}}$ with

$$ x_{\text{end}}^{\text{max}} \approx \left[ 720\pi^2 \frac{p^3}{q+4} \frac{Q_{\text{rms-PS}}^2}{T^2} \left(\frac{M_{\text{Pl}}}{\mu}\right)^{q+6} \right]^{1/(3p+q+6)}. \qquad (7.60) $$

The prior condition on $x_{\text{end}}$ is therefore of the type $x_{\text{end}}^{\text{min}} < x_{\text{end}} \ll x_{\text{end}}^{\text{max}}$, with $x_{\text{end}}^{\text{min}}$ defined by Eq. (7.56) and $x_{\text{end}}^{\text{max}}$ defined by Eq. (7.60). For any $q > 0$, these two equations show that there exists an upper bound $\mu < \mu_{\text{max}}$ under which the condition $x_{\text{end}}^{\text{min}} \ll x_{\text{end}}^{\text{max}}$ is satisfied. It reads

$$ \frac{\mu_{\text{max}}}{M_{\text{Pl}}} \approx \frac{\left( 720\pi^2 \frac{p^3}{q+4} \frac{Q_{\text{rms-PS}}^2}{T^2} \right)^{(p+2)/(pq)}}{\left[ p(p+2)\Delta N_{\text{min}} \right]^{(3p+q+6)/(pq)}}, \qquad (7.61) $$

and has been represented in Fig. 103. One can see that a typical value $\mu/M_{\text{Pl}} \approx 10^{10}$ GeV (see Ref. [695]) is not allowed for realistic values of $p$ and $q$. As such, the prior space for $p, \mu$, and $x_{\text{end}}$ is constrained and should be handled carefully.

The reheating consistent slow-roll predictions of the dynamical supersymmetric models are displayed in Figs. 349, 351 and 353 for $p = 2$, $p = 3$ and $p = 4$, respectively, and with $10^{-10} M_{\text{Pl}} < \mu < \mu_{\text{max}}$ (where $\mu_{\text{max}}$ has been calculated taking $q = 8$ and $\Delta N_{\text{min}} = 60$ to cover a large prior space). The reheating equation of state parameter $\bar{w}_{\text{reh}}$ has been taken to 0 but since there is no potential minimum around which the inflaton field can oscillate at the end of inflation, this parameter is a priori unspecified and can take different values. In any case the reheating temperature is strongly degenerated with the parameter $x_{\text{end}}^{\text{min}} < x_{\text{end}} < x_{\text{end}}^{\text{max}}$ preventing their inference. One can check that the spectral index is blue, as announced earlier, making these models disfavored by the observations. The typical amount of gravitational waves is very small, in agreement with the results of Ref. [695].

## 7.4. Generalized Mixed Inflation (GMLFI)

This model is a generalization of MLFI (see Section 5.3) and is, by definition, the sum of two monomial functions with arbitrary power indices. The corresponding potential can be written as

$$ V = M^4 \left(\frac{\phi}{M_{\text{Pl}}}\right)^p \left[1 + \alpha \left(\frac{\phi}{M_{\text{Pl}}}\right)^q\right], \qquad (7.62) $$

where $\alpha, p$ and $q$ are three dimensionless positive parameters. It can be seen as a generalization of the large field inflation potential (LFI, see Section 5.2), which is recovered when $\alpha \to 0$ or $\alpha \to \infty$. The parameter $\alpha$ therefore controls the relative weight of the two terms. Since the potential
<figure><img src="image_107.png" /><figcaption>Fig. 104. Generalized Mixed Inflation (GMLFI) for $p = 3, q = 2$ and $\alpha = 0.1$. Upper panels: the potential and its logarithm with respect the field value. Bottom left panel: slow-roll parameter $\epsilon_1$, the shaded region is where inflation stops. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line).</figcaption></figure>

is an increasing function of the inflaton vev, inflation proceeds from the right to the left and occurs in the large field regime $\phi/M_{Pl} \gg 1$. Defining the quantity $x$ by

$$x \equiv \frac{\phi}{M_{Pl}}, \qquad (7.63)$$

the first three Hubble flow functions in the slow-roll approximation can be expressed as

$$\epsilon_1 = \frac{1}{2x^2} \left[ \frac{p + \alpha(p+q)x^q}{1 + \alpha x^q} \right]^2, \qquad (7.64)$$

$$\epsilon_2 = \frac{p + \alpha^2 (p+q) x^{2q} + \alpha (2p + q - q^2) x^q}{x^2 (1 + \alpha x^q)^2}, \qquad (7.65)$$

and

$$\epsilon_3 = \frac{1}{x^2 (1 + \alpha x^q)^2} \left[ pq^2 + \alpha^2 q^2 (p+q) x^{2q} + \alpha q^2 (2p+q-q^2) x^q \right]^{-1} \\ \times \left\{ \begin{aligned} & 2q^2 [p^2 + \alpha^4 (p+q)^2] x^{4q} + \alpha^2 q^2 [12p^2 + 6pq(2-q) + (q-2)(q-1)q^2] x^{2q} \\ & + \alpha^3 q^3 (p+q) \left[ 8 \frac{p}{q} + (1-q)(4+q) \right] x^{3q} + \alpha pq^2 [8p+q(4+q^2-3q)] x^q \end{aligned} \right\}. \qquad (7.66)$$

They are decreasing functions of the field, vanishing when $x \to \infty$ and diverging when $x \to 0$. Together with the potential and its logarithm, the Hubble flow functions are represented in Fig. 104.

In Fig. 104, one sees that inflation ends by slow-roll violation at $x = x_{end}$, the solution of the equation $\epsilon_1(x_{end}) = 1$. From Eq. (7.64), one obtains

$$\sqrt{2\alpha}x_{\text{end}}^{q+1} + \sqrt{2}x_{\text{end}} = \pm [p + \alpha(p+q)x_{\text{end}}^q]. \qquad (7.67)$$

One can check that, for $\alpha = 0$, one recovers the LFI-$p$ result $x_{end} = p/\sqrt{2}$ (see Section 5.2) and that, for $\alpha \to \infty$, one gets $x_{end} = (p+q)/\sqrt{2}$, which correspond again to the LFI-$p+q$ solution. The above equation cannot be solved analytically for arbitrary values of $p, q$. This is possible only in some particular cases, namely $q = 0, q = 1$ or $q = 2$. For $q = 0$, this is LFI whereas $q = 2$ corresponds to MLFI, both solutions being given in Section 5.2 and Section 5.3, respectively. For $q = 1$, one obtains

$$x_{\text{end}} = \frac{\sqrt{2}}{4}(p+1) - \frac{1}{2\alpha} + \frac{\sqrt{4+4\sqrt{2\alpha}(p-1)+2\alpha^2(p+1)^2}}{4\alpha}, \qquad (7.68)$$
but, in general, $x_{\text{end}}$ has to be determined numerically.

The slow-roll trajectory can be integrated explicitly using Eq. (3.11) and this leads to

$$N_{\text{end}} - N = \frac{1}{2(p+q)} x^2 \left\{ 1 + \frac{q}{p} {}_2F_1 \left[ 1, \frac{2}{q}, 1 + \frac{2}{q}, -\alpha q \left( \frac{1}{p} + \frac{1}{q} \right) x^q \right] \right\} - \frac{1}{2(p+q)} x_{\text{end}}^2 \left\{ 1 + \frac{q}{p} {}_2F_1 \left[ 1, \frac{2}{q}, 1 + \frac{2}{q}, -\alpha q \left( \frac{1}{p} + \frac{1}{q} \right) x_{\text{end}}^q \right] \right\}. \qquad (7.69)$$

Here, ${}_2F_1$ stands for the Gauss hypergeometric function [281,282]. Since it is equal to unity when its last argument vanishes, one can check that, in the limit $\alpha \to 0$, one recovers the slow-roll trajectory for the LFI-$p$ models while the limit $\alpha \to \infty$ leads to the trajectory of the LFI-$(p+q)$ models. Finally, since ${}_2F_1(1, 1, 2, x) = -\ln(1-x)/x$, one can also check that the MLFI case corresponds to $p = q = 2$. The previous expression can only be inverted for $q = 0$ (LFI) and $q = 2$ (MLFI), and they have been already discussed in Section 5.2 and Section 5.3, respectively. The case $q = 1$ can also be simplified using ${}_2F_1(1, 2, 3, x) = -2/x - 2\ln(1-x)/x^2$. In general, one has to inverse this slow-roll trajectory numerically.

The parameter $M$ can be determined from the amplitude of the CMB anisotropies and the Hubble crossing $\text{vev } x_*$. One obtains

$$\frac{M^4}{M_{\text{Pl}}^4} = 720\pi^2 \frac{[p + \alpha(p+q)x_*^q]^2 Q_{\text{rms-PS}}^2}{x_*^{p+2} (1 + \alpha x_*^q)^3 T^2}. \qquad (7.70)$$

The reheating consistent slow-roll predictions for the generalized mixed large field models are displayed in Figs. 355, 356, and 357 for $(p=2$ and $q=1$), $(p=2$ and $q=3$) and $(p=3$ and $q=2$), respectively. As for MLFI, the predictions lie between the LFI-$p$ and LFI-$(p+q)$ models, but can actually exit this region for large enough values of $\alpha$. This means that, if one starts from a pure $V \propto \phi^{p+q}$ potential and adds a small $\propto \phi^p$ term, then this extra term has the effect of increasing the “effective value” of the power index of the potential. Moreover, since for large field inflation models, the $p$-model fits the data better than the $(p+q)$-one, it follows that small values for the parameter $\alpha$ are favored, together with high reheating temperatures.

## 7.5. Logarithmic Potential Inflation (LPI)

### 7.5.1. Theoretical justifications

This class of model assumes that inflation is driven by a composite state in a strongly interacting theory, see Refs. [598,697,698]. Let us consider the following model, see Section 6.14 for more details

$$\mathcal{L}_{\text{GI}} = -\varphi^{-3/2} \partial_{\mu} \varphi \partial^{\mu} \varphi - \frac{\varphi}{2} \ln \left( \frac{\varphi}{A^4} \right), \qquad (7.71)$$

where $\Lambda$ is a mass scale. Moreover, let us consider the situation where the model has a general non-minimal coupling to gravity of the form

$$S = \int d^4 x \sqrt{-g} \left[ -\frac{1}{2} (M^2 + \xi \varphi^{1/2}) R + \mathcal{L}_{\text{GI}} \right]. \qquad (7.72)$$

The coupling to gravity is characterized by the parameter $\xi$. Then, the action in the Einstein frame reads [598,697,698]

$$S = \int d^4 x \sqrt{-g} \left[ -\frac{1}{2} M_{\text{Pl}}^2 R - \Omega^{-2} \left( 1 + \frac{3\xi^2 \varphi^{1/2}}{4 M_{\text{Pl}}^2} \Omega^{-2} \right) \varphi^{-3/2} \partial_{\mu} \varphi \partial^{\mu} \varphi - \Omega^{-4} V_{\text{GI}} \right], \qquad (7.73)$$

where $V_{\text{GI}}$ refers to the potential in Eq. (7.71) and $\Omega^2 = (M^2 + \xi\varphi^{1/2})/M_{\text{Pl}}^2$. If $\xi \neq 0$ and if we are in the large field limit, then $\Omega^2 \simeq \xi\varphi^{1/2}/M_{\text{Pl}}^2$ and the canonically normalized field $\phi$ is such that $\phi \propto \ln \varphi$. In that case the potential reduces to $\Omega^{-4}V_{\text{GI}} \propto \ln \varphi \propto \phi$. Therefore, we have obtained a LFI model with $p=1$, see Section 5.2. On the other hand, if one assumes that $\xi=0$, then $\varphi = \phi^4/(4\sqrt{2})^4$ and

$$V = 2\Lambda^4 \left(\frac{\phi}{\phi_0}\right)^4 \ln\left(\frac{\phi}{\phi_0}\right), \qquad (7.74)$$

with $\phi_0 \equiv 4\sqrt{2}\Lambda$. This resembles the potential found in Section 6.14 which, for $\beta=0$ (see the precise definition in that section), was such that $V \propto \phi^4 \ln^2(\phi/\phi_0)$. These considerations motivate the next section devoted to the slow-roll analysis of this class of scenarios.

### 7.5.2. Slow-roll analysis

Based on the previous discussion, we now turn to the slow-roll analysis of the models described by the following potential

$$V(\phi) = M^4 \left(\frac{\phi}{\phi_0}\right)^p \left(\ln \frac{\phi}{\phi_0}\right)^q. \qquad (7.75)$$

We have just seen that, for $p=4$ and $q=2$, the model discussed in Ref. [598] is recovered, see Section 6.14, while for $p=4$ and $q=1$, this model matches with the so-called Glueball Inflation of Ref. [697]. This class of models has also been studied on general grounds in Ref. [699]. In the following, we keep $p$ and $q$ unspecified. Defining the quantity $x$ by the following relation

$$x \equiv \frac{\phi}{\phi_0}, \qquad (7.76)$$

the potential has a local maximum at $x = x_{V,\text{max}}$ and a local minimum (at which the potential vanishes) at $x = x_{V,0}$ with

$$x_{V,\text{max}} = e^{-q/p}, \qquad x_{V,0} = 1. \qquad (7.77)$$

For $x > x_{V,0}$, $V(x)$ increases and finally diverge when $x$ goes to infinity. The potential is always definite positive in the $x > 1$ branch, whereas it is definite positive in the $x < 1$ branch only if $q$ is an even integer. The first three Hubble flow functions in the slow-roll approximation are given by

$$\epsilon_1 = \frac{M_{\text{Pl}}^2 (q + p \ln x)^2}{\phi_0^2 2x^2 \ln^2 x}, \qquad \epsilon_2 = 2 \frac{M_{\text{Pl}}^2 q + q \ln x + p \ln^2 x}{\phi_0^2 x^2 \ln^2 x}, \qquad (7.78)$$
<figure><img src="image_108.png" /><figcaption>Fig. 105. Logarithmic Potential Inflation (LPI) for $p = 4, q = 2$. Upper panels: the potential and its logarithm. Bottom left panel: slow-roll parameter $\epsilon_1$. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line).</figcaption></figure>

and

$$ \epsilon_3 = \frac{M_{\text{Pl}}^2}{\phi_0^2} (q + p \ln x) \frac{2q + 3q \ln x + 2q \ln^2 x + 2p \ln^3 x}{x^2 \ln^2 x (q + q \ln x + p \ln^2 x)}. \qquad (7.79) $$

Together with the potential, they are displayed in Fig. 105.

As can be checked on this figure, and assuming $q$ is even, the behavior of $\epsilon_1(x)$ exhibits three domains in which inflation can occur and can naturally end. Either $x > 1$ and inflation proceeds from the right to the left (LPI1), or $x_{Vmax} < x < 1$ and inflation proceeds from the left to the right (LPI2), or $0 < x < x_{Vmax}$ and inflation proceeds from the right to the left (LPI3), see the three arrows in Fig. 105. For these three cases, the slow-roll trajectory can be integrated analytically and one has

$$ N - N_{\text{end}} = \left( \frac{\phi_0}{M_{\text{Pl}}} \right)^2 \left\{ -\frac{x^2 - x_{\text{end}}^2}{2p} + \frac{q}{p^2} e^{-2q/p} \left[ \text{Ei}\left(\frac{2q}{p} + 2 \ln x\right) - \text{Ei}\left(\frac{2q}{p} + 2 \ln x_{\text{end}}\right) \right] \right\}. \qquad (7.80) $$

Let us remark that for $x \to +\infty$ (LPI1), one recovers the large field inflation (LFI) trajectory of Section 5.2 with $p$ becoming the same parameter of LFI.

In the three above described regimes, inflation ends at the field value $x_{\text{end}}$ solution of $\epsilon_1(x_{\text{end}}) = 1$, i.e. verifying

$$ p \ln(x_{\text{end}}) + q \mp \sqrt{2} \frac{\phi_0}{M_{\text{Pl}}} x_{\text{end}} \ln x_{\text{end}} = 0. \qquad (7.81) $$

This is a transcendental equation that cannot be solved analytically for any values of $p$ and $q$. It can nevertheless be solved numerically in each of the three above-mentioned situations. Together with Eq. (3.48), Eq. (7.80) uniquely determines the observable field value $x_*$ at which the pivot scale crossed out the Hubble radius during inflation. Therefore, according to our classification, LPI is a three parameters model with $p, q$ and $\phi_0$.

Finally, the parameter $M$ is fixed by the amplitude of the CMB anisotropies to

$$ \frac{M^4}{M_{\text{Pl}}^4} = 720\pi^2 \left(\frac{M_{\text{Pl}}}{\phi_0}\right)^2 \frac{(q + p \ln x_*)^2 Q^2_{\text{rms}-\text{PS}}}{x_*^{2+p} \ln^{2+q} x_*} \qquad (7.82) $$

The reheating consistent slow-roll predictions for the LPI1 models with $p = 4$ are represented in Figs. 359, 358, and 360 for $q = 2, q = 1$ and $q = 3$, respectively. The predictions for LPI2 are displayed in Figs. 361, 362, and 363 for ($p = 1, q = 2$), ($p = 2, q = 2$) and ($p = 3, q = 4$), respectively. For the LPI3 scenario, the predictions have been plotted in Figs. 364, 365, and 366 for ($p = 1, q = 2$), ($p = 2, q = 2$) and ($p = 3, q = 4$), respectively. One can see that the current CMB data generically require LPI inflation to take place with super-Planckian values for $\phi_0$ while some combinations of $p$ and $q$ are already disfavored at more than two-sigma.
<figure><img src="image_109.png" /><figcaption>Fig. 106. Top left panel: constant $n_s$ D inflaton potential for $\alpha = 1$ and two values of $\beta$, namely $\beta = 0.7$ (solid blue line) and $\beta = 1.3$ (solid pink line). Top right panel: logarithm of the potential for the same values of $\alpha$ and $\beta$ and with the same color code. Bottom left panel: first slow-roll parameter $\epsilon_1$ for a potential with $\alpha = 1$ and $\beta = 0.7$ (solid blue line), $\beta = 1.8$ (solid pink line). The shaded area indicates the breakdown of slow-roll inflation (strictly speaking where acceleration cannot occur). Bottom right panel: second and third slow-roll parameters $\epsilon_2$ and $\epsilon_3$ for $\alpha = 0.25$ and the same values of $\beta$ as in the other plots.</figcaption></figure>

### 7.6. Constant $n_s$ D Inflation (CNDI)

This model has been studied in Ref. [596]. Its potential is designed to produce a power law power spectrum $\propto k^n$ (where $n$ is a constant). In this sense, the approach followed here is similar to the one investigated in Sections 5.20, 5.21 and 6.15. The potential studied in this section is given by

$$V(\phi) = \frac{M^4}{\left\{1 + \beta \cos\left[\alpha\left(\frac{\phi - \phi_0}{M_{Pl}}\right)\right]\right\}^2}, \qquad (7.83)$$

where $\alpha$ and $\beta$ are two dimensionless parameters. Since the potential is an even function of $x \equiv (\phi - \phi_0) / M_{Pl}$ and is $2\pi$-periodic, it can be studied without loss of generality in the range $x \in [0, \pi/\alpha]$ only (with $\alpha > 0$, $\beta > 0$). The potential and its logarithm are displayed in Fig. 106 (top panels) for two different representative values of $\beta$. If $\beta < 1$ (blue curve), it is an increasing function of the field, hence inflation proceeds from the right to the left. On the contrary, if $\beta \ge 1$ (pink curve), it diverges at $x_{V \to \infty} = \arccos(-1/\beta)/\alpha$. Then, for $x < x_{V \to \infty}$ it is an increasing function of $x$ and inflation proceeds from the right to the left, whereas for $x > x_{V \to \infty}$ it is a decreasing function of $x$ and inflation proceeds from the left to the right.

The three first slow-roll parameters are given by the following expressions

$$\epsilon_1 = \frac{2\alpha^2\beta^2\sin^2(\alpha x)}{[1 + \beta \cos(\alpha x)]^2}, \qquad \epsilon_2 = \frac{-4\alpha^2\beta[\beta + \cos(\alpha x)]}{[1 + \beta \cos(\alpha x)]^2}, \qquad (7.84)$$

and

$$\epsilon_3 = \frac{-2\alpha^2\beta[2\beta^2 - 1 + \beta \cos(\alpha x)]\sin^2(\alpha x)}{[\beta + \cos(\alpha x)][1 + \beta \cos(\alpha x)]^2}. \qquad (7.85)$$

They are displayed in Fig. 106 (bottom panels). Let us now study in more detail the behavior of $\epsilon_1$ and $\epsilon_2$. It depends on whether $\beta$ is larger or smaller than 1. If $\beta < 1$, the first slow-roll parameter $\epsilon_1$ vanishes at $x = 0$ and $x = \pi/\alpha$, and reaches a maximum in between at $x_{\epsilon_2=0}$. This maximum is larger than one provided $\alpha > \alpha_{min}(\beta)$, where

$$\alpha_{\min}(\beta) = \sqrt{\frac{1 - \beta^2}{2\beta^2}}. \qquad (7.86)$$
In that case, inflation can stop by slow-roll violation, at the position $x_{\text{end}}$ given by

$$x_{\text{end}} = x_{\epsilon_1=1}^{+} = \frac{1}{\alpha} \arccos \left[ \frac{\alpha \sqrt{2\beta^2 (1 + 2\alpha^2) - 2 - 1}}{\beta + 2\alpha^2\beta} \right], \qquad (7.87)$$

and proceeds in the range $[x_{\text{end}}, \pi/\alpha]$ (from the right to the left). On the other hand, the second slow-roll parameter $\epsilon_2$ is a monotonic increasing function of $x$, which vanishes at $x_{\epsilon_2=0} = \arccos(-\beta)/\alpha$. If $\beta \ge 1$, as can be seen in Fig. 106, the first slow-roll parameter $\epsilon_1$ diverges at $x_{V \to \infty} = \arccos(-1/\beta)/\alpha$, so that inflation cannot stop by slow-roll violation in that case. This means that inflation must end by another mechanism and, therefore, that the model depends on an additional parameter. The second slow-roll parameter $\epsilon_2$ is always negative and also diverges at $x_{V \to \infty}$. Let us notice that, for $\beta < 1$ and $\alpha > \alpha_{\min}(\beta)$, and for $\beta > 1$ (for any $\alpha$), we will need below the other solution of $\epsilon_1 = 1$, namely

$$x_{\epsilon_1=1}^{-} = \frac{1}{\alpha} \arccos \left[ - \frac{\alpha \sqrt{2\beta^2 (1 + 2\alpha^2) - 2 + 1}}{\beta + 2\alpha^2\beta} \right]. \qquad (7.88)$$

We are now in a position where the slow-roll trajectory can be determined. It turns out that this one can be integrated analytically and reads

$$N - N_{\text{end}} = \frac{1}{2\alpha^2} \left\{ -\ln[\sin(\alpha x)] - \frac{1}{\beta} \ln\left[\tan\left(\alpha \frac{x}{2}\right)\right] + \ln[\sin(\alpha x_{\text{end}})] + \frac{1}{\beta} \ln\left[\tan\left(\alpha \frac{x_{\text{end}}}{2}\right)\right] \right\}. \qquad (7.89)$$

Because of the logarithmic functions, a sufficient number of e-folds can be realized only if the initial conditions are fine-tuned and $x_{\text{ini}}$ is chosen to be extremely close to $\pi/\alpha$.

Indeed, inserting Eq. (7.87) into Eq. (7.89), the total number of e-folds during inflation becomes a function of $x_{\text{ini}}$ and of the two parameters $\alpha$ and $\beta$. For given values of those parameters, one can check that $(N_{\text{end}} - N_{\text{ini}})(x_{\text{ini}})$ remains always small compared to unity, unless $x_{\text{ini}} \to \pi/\alpha$ where it blows up. Let us write $x_{\text{ini}}$ as $\pi/\alpha + \delta x_{\text{ini}}$ with $\delta x_{\text{ini}} \ll 1$ and defining $A \equiv \ln[\sin(\alpha x_{\text{end}})] + \ln[\tan(\alpha x_{\text{end}}/2)]/\beta$, one arrives at

$$\delta x_{\text{ini}} \approx \left[ \alpha \left( \frac{\alpha}{2} \right)^{-1/\beta} e^{-A} \right]^{\beta/(1-\beta)} e^{-2\alpha^2 \beta (N_{\text{end}} - N_{\text{ini}})/(1-\beta)}. \qquad (7.90)$$

The coefficient between the squared brackets only depends on $\alpha$ and $\beta$ which are, a priori, coefficients of order one. On the other hand, the argument of the exponential is $2(N_{\text{end}} - N_{\text{ini}}) > 120$, times a negative term of order one. This means that $\delta x_{\text{ini}}$ must be exponentially small to obtain a significant number of e-folds and one can question the physical relevance of such a fine-tuning. The typical predictions of the model (taking $x_* \simeq \pi/\alpha$) actually are $\epsilon_1 \simeq 0$, $\epsilon_2 \simeq 4\alpha^2\beta/(1-\beta)$, and $\epsilon_3 \simeq 0$. It follows that the condition $\alpha > \alpha_{\min}(\beta)$ implies $\epsilon_2 > 2(1+\beta)/\beta > 4$, which is completely ruled out by the observations. Therefore, we conclude that the case $\beta < 1$ is not of cosmological interest.

The only remaining possibility is $\beta > 1$. Inflation cannot end by slow-roll violation and $x_{\text{end}}$ is an additional parameter, making the model a three parameters one. In the range $\alpha x_{\text{end}} \ll 1$, one has $\epsilon_1 \ll 1$ and $\epsilon_2 \simeq -4\alpha^2\beta/(1+\beta)$ such that the spectral index is given by $n_s \simeq 1+4\alpha^2\beta/(\beta+1)$. Therefore, it is indeed a constant.

The CMB normalization gives the mass scale $M$ as

$$\left(\frac{M}{M_{\text{Pl}}}\right)^4 = 2880\alpha^2\beta^2\pi^2\sin^2(\alpha x_*) \frac{Q^2}{T^2}, \qquad (7.91)$$

which has to be numerically evaluated when if $\alpha x_*$ is not small. The predictions of CNDI inflation are displayed in Figs. 369 and 367. We see that, in the regime $\alpha x_{\text{end}} \ll 1$, the spectral index is constant, as expected. However, this occurs in a regime where the predictions are not consistent with the observations (the spectrum is too blue). On the other hand, when $\alpha x_{\text{end}}$ is no longer small, we observe strong deviations from $n_s \simeq 1+4\alpha^2\beta/(\beta+1)$ but, for intermediate values of $\alpha \simeq 0.3$, this renders the predictions compatible with the data. Obviously, these considerations bear some resemblance with the findings of Sections 5.20, 5.21 and 6.15.

## 7.7. String Axion Inflation II (SAIII)

This model shares the same theoretical construction as String Axion Inflation I (SAII) presented in Section 6.20 and has been proposed in Ref. [655]. Compared to SAII, a mass term coming from higher-order corrections associated with instanton effects, proportional to $\phi^2$, appears in the potential. This mass term could also be viewed as yet another correction to the potential of Natural Inflation (NI), see Section 5.6, and this has been further discussed in Ref. [700]. For reasons that will be clearer later on, it is convenient to write the potential of SAIII under the form

$$V(\phi) = M^4 \left\{ 1 - \cos\left(\frac{\phi}{\mu}\right) + \alpha \left[ \frac{\phi}{\mu} \sin\left(\frac{\phi}{\mu}\right) + \frac{1}{2}\beta \left(\frac{\phi}{\mu}\right)^2 \right] \right\}, \qquad (7.92)$$

where $\mu$ is a vev, and $\alpha$ and $\beta$ are two dimensionless constants that are not required to be small. However, the new mass term is required to be positive, which implies the constraint $\alpha\beta \ge 0$, i.e. $\alpha$ and $\beta$ must be of the same sign.

The potential is symmetric with respect to $\phi = 0$ and the analysis can thus be restricted to the domain $\phi \ge 0$. As for SAII, depending on the value of $\alpha$ and $\beta$, the potential can become negative in some domains that are separated by a local maximum. The inflationary domains existing on both side of the first maximum of $V(\phi)$ will be referred to as SAIII1 and SAIII2, by analogy with the treatment of SAII carried out in Section 6.20. However, and as opposed to SAII, the additional mass term implies that, for large enough values of $\beta$, the potential can become a strictly monotonic increasing function of $\phi$. In this regime, the model becomes similar to Large-Field Inflation with $p = 2$ (LFI$_2$, see Section 5.2), plus some small modulations. This regime will be denoted SAIII3 and its existence is mutually exclusive with SAIII1 and SAIII2. The potential and its logarithm have been represented in Figs. 107 and 108 for those three inflationary regimes.
<figure><img src="image_110.png" /><figcaption>Fig. 107. String Axion Inflation II (SAIII) for $\alpha = 0.8$, $\beta = 0.1$ (black curve) and $\alpha = -1.2$, $\beta = -0.5$ (blue curve). Top panels: the potential and its logarithm. Only the SAIII1 and SAIII2 regimes exist for $\alpha = 0.8$, $\beta = 0.1$ (black) whereas inflation gracefully ends only for SAIII1 in the case $\alpha = -1.2$, $\beta = -0.5$. Bottom left panel: slow-roll parameter $\epsilon_1$ for $\alpha = 0.8$, $\beta = 0.1$ and $\mu = 20 M_\odot$, with the two inflationary regimes annotated with an arrow indicating the direction to which the field evolves. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line) for the same parameters value. When $\epsilon_3$ becomes negative, the plot shows $|\epsilon_3|$ as a red dotted line, the black dotted line corresponds to positive values. The regime SAIII3 is represented in Fig. 108.</figcaption></figure>

### 7.7.1. Parameter space analysis

Let us first derive the conditions on $\alpha$ and $\beta$ under which SAIII3 exists, and SAIII1/SAIII2 do not exist. From the above considerations, the potential must not have any local extremum. Defining $x \equiv \phi/\mu$, and deriving Eq. (7.92) with respect to $x$, one gets

$$ \frac{V'(x)}{M^4} = (1 + \alpha) \sin(x) + \alpha x [\beta + \cos(x)]. \qquad (7.93) $$

The potential possesses local extrema if solutions to $V'(x) = 0$ exist. This is a transcendental equation that can be recast into the form $f(x) = x$ by defining the function

$$ f(x) = \frac{(1 + \alpha) \sin(x)}{\alpha [\cos(x) + \beta]} \qquad (7.94) $$

The location of the separatrices in parameter space that delineate the regions where solutions to $f(x) = x$ exist cannot be obtained analytically, but they can be worked out numerically as follows. Along such a separatrix, at the location $x = x_f$ where $f(x_f) = x_f$, the two functions $x$ and $f(x)$ must tangent each other. In other words, one should have $f'(x_f) = 1$ along the separatrices. Given that

$$ f'(x) = -\frac{1 + \alpha}{\alpha} \frac{1 + \beta \cos(x)}{[\beta + \cos(x)]^2}, \qquad (7.95) $$

this implies that

$$ \cos(x_f) = -\left(3 + \frac{1}{\alpha}\right) \frac{\beta}{2} \pm \sqrt{\left(3 + \frac{1}{\alpha}\right) \frac{\beta}{2} - \beta^2 - 1 - \frac{1}{\alpha}} \qquad (7.96) $$

This formula can be plugged into the relation $f(x_f) = x_f$, using the fact that $x_f = \arccos(\cos x_f)$ or $2\pi - \arccos(\cos x_f)$ and that $\sin x_f = \pm \sin[\arccos(\cos x_f)]$. One obtains an equation that involves $\alpha$ and $\beta$ and that must be solved numerically in order to obtain the functions $\alpha(\beta)$ that delineate the regions where SAIII3 exists.

In order to gain further analytical insight, the following constraints can be put on the location of the separatrices. First, from Eq. (7.94), one notices that $f(x)$ has poles for $\cos(x) = -\beta$ provided $|\beta| \le 1$. Since the sign of the numerator in $f$ necessarily flips between two such poles (and given that the sign of the denominator remains the same), $f$ interpolates between $\pm\infty$ between two consecutive poles and necessarily crosses $x$. This ensures that $V(x)$ is extremal somewhere, and that SAIII3 does not exist. The separatrices need therefore to be looked for at values of $\beta$ such that $|\beta| > 1$ and we will thus focus on this region hereafter.
<figure><img src="image_111.png" /><figcaption>Fig. 108. String Axion Inflation II in the SAIII3 regime where the potential does not exhibit positive extrema, for $\alpha = 1$, $\beta = 2$ (black curve) and $\alpha = -2$, $\beta = -1.2$ (blue curve). Top panels: the potential and its logarithm. Bottom left panel: slow-roll parameter $\epsilon_1$ for $\alpha = 0.8$, $\beta = 0.1$ and $\mu = 40 M_\text{pl}$, with the field evolution annotated with an arrow indicating the direction to which it evolves. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line) for the same parameters value. When $\epsilon_3$ becomes negative, the plot shows $|\epsilon_3|$ as a red dotted line, the blue dotted line corresponds to positive values. Similarly, negative values of $\epsilon_2$ are represented as green solid curves. Notice that $\epsilon_3$ becomes singular at the points where $\epsilon_2 = 0$, but the product $\epsilon_2\epsilon_3$ remains finite. The other regimes SAIII1 and SAIII2 are represented in Fig. 107.</figcaption></figure>

<figure><img src="image_112.png" /><figcaption>Fig. 109. The parameter space $(\beta, \alpha)$ of String Axion Inflation II and its various inflationary regimes. SAIII3 exists only in the upper right and lower left corner separated by the red curves, which are $\alpha_1(\beta)$, $\alpha_2(\beta)$ and $\alpha_3(\beta)$ (see text). SAIII3 is mutually exclusive with SAIII1, therefore SAIII1 always exists on the left of the positive red curve and on the right of the negative one. However, SAIII2 can take place only if the potential becomes negative at field values larger than the one corresponding to the first maximum of the potential. This occurs only within the central domain bounded by the blue curves. For convenience, we have also represented in dashed green the separatrix for which the potential is negative, or positive, around the origin. Parameters can be accommodated such that all the three inflationary regimes may encounter a positive or negative potential around $\phi = 0$.</figcaption></figure>

Second, from Eq. (7.95), the function $f$ is extremal where $f'(x) = 0$, i.e. where $\cos(x) = -1/\beta$, and the two extrema of $f$ in $[0, 2\pi]$ occur at

$$x_{f^{-}} = \arccos\left(-\frac{1}{\beta}\right), \quad x_{f^{+}} = \pi + \arccos\left(\frac{1}{\beta}\right). \qquad (7.97)$$

Because $f(x)$ is $2\pi$-periodic, and since $f(0) = 0$, if there is no solution to $f(x) = x$ within the domain $[0, 2\pi]$, none can exist for $x > 2\pi$.
In order to identify which of $x_{f-}$ and $x_{f+}$ is a minimum or a maximum, one can consider the sign of

$$f'(0) = -\frac{1+\alpha}{\alpha(1+\beta)}. \qquad (7.98)$$

If $\beta > 1$ (hence $\alpha > 0$, since $\alpha\beta > 0$), $f'(0) < 0$, the function $f$ initially decreases away from the origin, and its first extremum is a minimum: in this case, $x_{f-}$ is a negative minimum and $x_{f+}$ is a positive maximum, given that $f(\pi) = 0$. If $\beta < -1$ (hence $\alpha < 0$), $f'(0) < 0$ if $\alpha > -1$, in which case $x_{f-}$ is a minimum and $x_{f+}$ is a maximum, while $f'(0) > 0$ if $\alpha < -1$, in which case $x_{f-}$ is a maximum and $x_{f+}$ is a minimum. In passing, let us note that $f'(0) > 1$ if and only if $-2 < \beta < -1$ and $\alpha < -1/(2+\beta)$. When this happens, $f(x) > x$ when $x$ approaches 0 while $f(x) < x$ at $x = \pi$ since $f(\pi) = 0$, hence solutions to the equation $f(x) = x$ can be found and SAIII3 does not exist. Otherwise, let us derive the condition for the maximum of the function $f(x)$ to lie above the $x$ function. This will provide a sufficient condition for solutions to exist, which is however not a necessary condition but that will still allow us to better bound the location of the separatrices. From the above considerations, three subcases need to be distinguished.

If $\beta > 1$, the condition $f(x_{f+}) \ge x_{f+}$ reads

$$\frac{1+\alpha}{\alpha\sqrt{\beta^2-1}} \ge \pi + \arccos\left(\frac{1}{\beta}\right). \qquad (7.99)$$

Solving for $\alpha$, the condition for $V$ to have an extremum translates into $\alpha \le \alpha_2(\beta)$, where

$$\alpha_2(\beta) \equiv \frac{1}{\sqrt{\beta^2 - 1} \left[ \pi + \arccos\left(\frac{1}{\beta}\right) \right] - 1}. \qquad (7.100)$$

Conversely, the necessary (but in principle not sufficient) condition for SAIII3 to exist if $\beta > 1$ is then $\alpha > \alpha_2(\beta)$.

If $\beta < -1$ and $-1 < \alpha < 0$, the condition $f(x_{f+}) \ge x_{f+}$ now reads

$$-\frac{1+\alpha}{\alpha\sqrt{\beta^2-1}} \ge \pi + \arccos\left(\frac{1}{\beta}\right). \qquad (7.101)$$

Solving again for $\alpha$, this translates into the condition $\alpha \ge \alpha_1(\beta)$ where

$$\alpha_1(\beta) \equiv \frac{-1}{\sqrt{\beta^2 - 1} \left[ \pi + \arccos\left(\frac{1}{\beta}\right) \right] + 1}. \qquad (7.102)$$

The corresponding necessary condition for SAIII3 to exist here is $-1 < \alpha < \alpha_1(\beta)$.

Finally, if $\beta < -1$ and $\alpha < -1$, the condition $f(x_{f-}) \ge x_{f-}$ reads

$$\frac{1+\alpha}{\alpha\sqrt{\beta^2-1}} \ge \arccos\left(-\frac{1}{\beta}\right), \qquad (7.103)$$

or, $\alpha \le \alpha_3(\beta)$ where we have defined

$$\alpha_3(\beta) \equiv \frac{1}{\sqrt{\beta^2 - 1} \arccos\left(-\frac{1}{\beta}\right) - 1}. \qquad (7.104)$$

In this parameter space corner, for SAIII3 to exist one must ensure that $\alpha_3(\beta) < \alpha < -1$.

In practice, one can check that the three curves $\alpha_1(\beta)$, $\alpha_2(\beta)$ and $\alpha_3(\beta)$ provide very good approximations to the exact solutions of $f(x_f) = x_f$ with $x_f$ given by Eq. (7.97), hence they can be used as proxies for the separatrices in parameter space. They have been represented in Fig. 109 as red curves, where the domains in which SAIII3 is defined have been explicitly labeled.

Conversely, in all domains in which SAIII3 does not exist, the potential has, at least, one positive maximum and this ensures that the SAIII1 inflationary regime can occur for $x < x_{Vmax}$, where $x_{Vmax}$ has to be numerically determined to solve $V'(x) = 0$, see Eq. (7.93). The regime SAIII2, occurring for $x > x_{Vmax}$ at increasing field values, gracefully ends only if the potential does not admit a de-Sitter minimum at a larger field value. In other words, denoting by $x_{Vmin}$ the value at which the potential is minimum, with $x_{Vmin} > x_{Vmax}$, one should have $V'(x_{Vmin}) = 0$ and $V(x_{Vmin}) \le 0$, i.e.,

$$1 - \cos(x_{Vmin}) + \alpha \left[ x_{Vmin} \sin(x_{Vmin}) + \frac{1}{2} \beta x_{Vmin}^2 \right] \le 0. \qquad (7.105)$$

One cannot determine an analytical condition on $\alpha$ and $\beta$ such that this condition is fulfilled. However, in the limit $|\alpha| \gg 1$, the condition $V(x_{V=0}^+) = 0$ simplifies to

$$\frac{\sin(x_{V=0}^+)}{x_{V=0}^+} = -\frac{\beta}{2}, \qquad (7.106)$$

and assessing if a solution exists boils down to comparing the amplitude of the second and third extremum of the function $\sin(x)/x$ to $|\beta/2|$. In the domain of interest, we find that a solution exists, for $\alpha \to \infty$, only if $\beta \in [-0.257, 0.434]$. In general, Eq. (7.105) has to be solved numerically and the solutions have been represented as blue curves in Fig. 109.

Let us finally notice that, expanding the potential around $x = 0$, one has

$$\frac{V(x)}{M^4} = \left( \alpha + \frac{1+\alpha\beta}{2} \right) x + \mathcal{O}(x^3). \qquad (7.107)$$

The potential is therefore a negative decreasing function of $x$ around the origin for $\alpha < -1/(\beta + 2)$. This is the same behavior as discussed for SAII, which is recovered by taking $\beta = 0$. When this occurs, the inflationary domains are defined only for $x > x_{V=0}^-$, where $x_{V=0}^-$ is the first positive solution of $V(x) = 0$. The separatrix $\alpha = -1/(\beta + 2)$ has been represented as a green dashed curve in Fig. 109.
<figure><img src="image_113.png" /><figcaption>Fig. 110. Field domains over which SAIII1 and SAIII2 are defined, as a function of $\alpha$. Each panel corresponds to, clockwise, $\beta = -0.6$, $\beta = -0.1$, $\beta = 0.6$ and $\beta = 0.1$. The regime SAIII2 appears only in a limited domain of $\alpha$ for $\beta = \pm 0.1$ and does not exist for $\beta = \pm 0.6$, see also Fig. 109. When they appear, the blue and red curves labeled $x_{\vec{V}=0}^+$ and $x_{\vec{V}=0}^-$ determine the field range over which the potential is positive. The lower and upper dashed lines represent the field value (in unit of $\mu$, and for $\mu = 10 M_{\text{pl}}$) at which inflation gracefully ends, namely the relevant solution of $\epsilon_1(x) = 1$.</figcaption></figure>

<figure><img src="image_114.png" /><figcaption>Fig. 111. Field domains over which SAIII3 is defined, as a function of $\alpha$. The left panel is for $\beta = -1.5$ while the right one is for $\beta = 1.5$. When it is non-vanishing, the blue curve labeled $x_{\vec{V}=0}^+$ gives the field value below which the potential is negative around the origin. The lower dashed line represents the field value (in unit of $\mu$, and for $\mu = 10 M_{\text{pl}}$) at which inflation gracefully ends, namely the smallest solution of $\epsilon_1(x) = 1$. For $\beta = 1.5$, $x_{\text{end}}(\alpha)$ is almost horizontal and weakly depends on $\alpha$ because the potential shape is quite close to the one of LFI2. In that case, to a very good approximation, one has $x_{\text{end}} \approx \sqrt{2/\mu}$.</figcaption></figure>

### 7.7.2. Slow-roll analysis

The first slow-roll parameter reads

$$ \epsilon_1 = \frac{1}{2\mu^2} \left[ \frac{(1+\alpha)\sin(x) + \alpha x \cos(x) + \alpha \beta x}{1 - \cos(x) + \alpha x \sin(x) + \frac{1}{2}\alpha \beta x^2} \right]^2, \quad (7.108) $$
while the second one is given by

$$ \epsilon_2 = \frac{1}{\mu^2} \frac{1}{[1 - \cos(x) + \alpha x \sin(x) + \frac{1}{2} \alpha \beta x^2]^2} \times (2 + \alpha \{-2\beta + \alpha [(\beta^2 + 2) x^2 + 1] + 4\} + \alpha x \sin(x) \{2 + \beta [\alpha (x^2 + 2) + 4]\} + \cos(x) \{-2 - 4\alpha + \alpha \beta [(2\alpha - 1)x^2 + 2]\} - \alpha^2 - \cos(2x)). \quad (7.109) $$

and, finally, the third one is

$$ \begin{aligned} \epsilon_3 = & -\frac{1}{2\mu^2} \frac{(1+\alpha)\sin(x) + \alpha x [\beta + \cos(x)]}{[1 - \cos(x) + \alpha x \sin(x) + \frac{1}{2}\alpha\beta x^2]^2} \\ & \times \left[ (\alpha x \cos(x) \{\beta [\alpha (x^2 + 6) + 2] + 2\} + \sin(x) [6\alpha + (\alpha + 1)\alpha\beta (x^2 + 2) + 2] \right. \\ & \qquad \left. + 2\alpha^2 [(\beta^2 + 2) x + \sin(2x)] \right) [\alpha\beta x^2 + 2\alpha x \sin(x) - 2\cos(x) + 2] \\ & - 4 \{\alpha x [\beta + \cos(x)] + (\alpha + 1) \sin(x)\} \\ & \times (\alpha \{-2\beta + \alpha [(\beta^2 + 2) x^2 + 1] + 4\} + \alpha x \sin(x) \{\beta [\alpha (x^2 + 2) + 4] + 2\} \\ & + \cos(x) \{-4\alpha + \alpha\beta [(2\alpha - 1)x^2 + 2] - 2\} - \alpha^2 \cos(2x) + 2) \\ & \times \left( \alpha \{-2\beta + \alpha [(\beta^2 + 2) x^2 + 1] + 4\} + \alpha x \sin(x) \{\beta [\alpha (x^2 + 2) + 4] + 2\} \right. \\ & \qquad \left. + \cos(x) \{-4\alpha + \alpha\beta [(2\alpha - 1)x^2 + 2] - 2\} - \alpha^2 \cos(2x) + 2 \right)^{-1}. \end{aligned} \quad (7.110) $$

The three slow-roll parameters have been represented as a function of x for SAIII1 and SAIII2 in Fig. 107, and for SAIII3 in Fig. 108. In the latter regime, the modulation of the potential implies that $\epsilon_2$ may change sign during inflation (see lower right panel in Fig. 108). As a result, $\epsilon_3(x)$ becomes singular when $\epsilon_2(x) = 0$ and we are, a priori, in the presence of slow-roll violations at second order. In fact, they are not really problematic as the product $\epsilon_2\epsilon_3$, which is the quantity entering into the second-order slow-roll power spectra, remains always finite, but they do signal potentially large running of the spectral index.

For the three regimes inflation gracefully ends at the field value $x_{end}$ solution of $\epsilon_1(x_{end}) = 1$, in the domain of interest. Because $\epsilon_1$ diverges for $V(x) \to 0$, this always occurs in a domain in which the potential is positive definite, and we have $x_{end} \in ]x_{V=0}^-, x_{Vmax}[$ for SAIII1, $x_{end} \in ]x_{Vmax}, x_{V=0}^[$ for SAIII2 and $x_{end} \in ]x_{V=0}^-, +\infty[$ for SAIII3 (if $x_{V=0}^-$ does not exist, it has to be replaced by $x = 0$). The actual value for $x_{end}$ has to be determined numerically, in the previous domains, by solving

$$ (1 + \alpha) \sin(x) + \alpha x \cos(x) + \alpha \beta x = \pm \mu \sqrt{2} [1 - \cos(x) + \alpha x \sin(x) + \frac{1}{2} \alpha \beta x^2]. \quad (7.112) $$

In Fig. 110, we have represented, for various values of $\beta$, the values of $x_{V=0}^-$, $x_{V=0}^+$ and $x_{end}$ as functions of $\alpha$ for the regimes SAIII1 and SAIII2. For $\beta = 0$ one recovers the results of SAI1 and this is displayed in Fig. 87. For SAIII3, the potential does not have any maximum, but for $\beta < 0$, there are some values of $\alpha$ for which $x_{V=0}^-$ exists and this is represented in Fig. 111. As soon as $\beta$ becomes large, the potential of SAIII3 is essentially dominated by the mass term and a very good approximation is $x_{end}(\alpha) \approx \sqrt{2}/\mu$, the field value at which Large Field Inflation with $p = 2$ ends. This can be seen on the right panel of Fig. 111 where the dashed curve representing the function $x_{end}(\alpha)$ is essentially a horizontal line. A word of caution is however in order. There are some values of $\alpha$ and $\beta$ within SAIII3 for which $\epsilon_1(x)$ may transiently exceed unity with $x > x_{end}$, precisely due to the modulation around the LFI2 potential. In these situations, inflation ends with “hicups”, it stops and restarts within a few e-folds before definitely stopping at the value of $x_{end}$ we have calculated. Therefore, slow-roll violations can occur, but being at the end of inflation, they are unlikely to be directly observable. They could nonetheless have other interesting effects, e.g. for primordial black holes formation.

The slow-roll trajectory cannot be analytically solved and requires a numerical integration. It reads

$$ N_{\text{end}} - N = \mu^2 \int_{x_{\text{end}}}^{x} \frac{1 - \cos(x) + \alpha x \sin(x) + \frac{1}{2} \alpha \beta x^2}{(1 + \alpha) \sin(x) + \alpha x [\cos(x) + \beta]} dx, \quad (7.113) $$

where $x_{end}$ is obtained from the solution of Eq. (7.112) in the relevant field domain for the inflationary regime under scrutiny (SAIII1, SAIII2 or SAIII3).

The normalization of the potential $M^4$ is obtained from the amplitude of the CMB anisotropies once the field value $x_*$ at which the pivot mode crossed the Hubble radius is determined. One gets

$$ \left(\frac{M}{M_{\text{Pl}}}\right)^4 = \frac{720\pi^2}{\mu^2} \frac{[(1+\alpha)\sin(x_*) + \alpha x_* \cos(x_*) + \alpha \beta x_*]^2 Q^2_{\text{rms}-\text{PS}}}{[1 - \cos(x_*) + \alpha x_* \sin(x_*) + \frac{1}{2} \alpha \beta x_*^2]^3 T^2}. \quad (7.114) $$

The reheating consistent slow-roll predictions for SAIII1, SAIII2 and SAIII3 are represented in Figs. 373 to 408. Because the parameter space in $(\alpha, \beta)$ is disjoint between positive and negative values, the models have been split into two sub-regimes according to the sign of $\alpha\beta$. Notice the strong running of the predictions associated with SAIII3, for some values of $\alpha$ and $\beta$, they essentially explore the whole plane $(n_s, r)$ while varying $\mu$.

## 7.8. Radiatively Corrected Large Field Inflation (RCLFI)

This model is based on Ref. [701] and considers the radiative corrections of bosonic and fermionic fields onto a monomial potential, i.e., a large-field model (see Section 5.2). Compared to the RCMI and RCQI models already discussed in Sections 5.4 and 5.5 respectively, both proposed in Ref. [310], RCLFI is not restricted to the quadratic and quartic monomial terms and the renormalization scale $\mu$ is not fixed at the Planck mass
<figure><img src="image_115.png" /><figcaption>Fig. 112. Radiatively Corrected Large Field Inflation in the RCLFI1, RCLFI2 and RCLFI3 regimes for $\alpha = 8$, $p = 2.5$. Top panels: the potential and its logarithm. Bottom left panel: slow-roll parameter $\epsilon_1$ for $\mu = 100 M_\mu$, with the three inflationary regimes annotated with an arrow indicating the direction to which the field evolves. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line) for the same parameters value. When $\epsilon_3$ becomes negative, the plot shows $|\epsilon_3|$ as a red dotted line, the black dotted line corresponds to positive values. The perturbative regime RCLFI4 is represented in Fig. 113.</figcaption></figure>

but becomes a free parameter. Bosonic and fermionic loop corrections yield a potential of the Coleman–Weinberg type, see Section 5.11, which reads [701]

$$V(\phi) = \frac{1}{2} \lambda m_{\text{Pl}}^4 \left(\frac{\phi}{m_{\text{Pl}}}\right)^p + \frac{g^4 - 4h^4}{64\pi^2} \phi^4 \ln\left(\frac{\phi^2}{\mu^2}\right). \qquad (7.115)$$

Here $g$ is the renormalized coupling constant coming from a bosonic interaction term of the form $\phi^2\chi^2$, and $h$ is the one coming from a Yukawa coupling between $\phi$ and a fermionic field. Depending on the relative strength of these couplings, the coefficient in front of the logarithmic term can be positive or negative.

For our purpose, it is more convenient to recast the potential in a simpler form

$$V(\phi) = M^4 \left[ \left(\frac{\phi}{\mu}\right)^p + \alpha \left(\frac{\phi}{\mu}\right)^4 \ln\left(\frac{\phi}{\mu}\right) \right], \qquad (7.116)$$

where $\lambda = (M/m_{\text{Pl}})^4(m_{\text{Pl}}/\mu)^p$ and $g^4 - 4h^4 = 32\pi^2(M/\mu)^4\alpha$.

Let us notice that the potential is renormalizable only for $p \le 4$, and, if one wants to keep the loop corrections under control, the coupling constants should be small, namely the combination $\alpha M^4/\mu^4$ should not exceed unity. In the following, these considerations are relaxed and we will also consider the predictions coming from Eq. (7.116) in full generality. Another point worth mentioning concerns the peculiar value $p = 4$. In that case, as one can check in Eq. (7.116), the renormalization scale $\mu$ can be re-absorbed in the normalization of the potential $M^4$ and, up to a redefinition of the parameter $\alpha$, the model ends up being equivalent to RCQI, see Section 5.5.

The potential will be studied for $\phi > 0$ only, and it is displayed in the top panels of Figs. 112 and 113 for $\alpha = 8$ and $p = 3/2$, and $\alpha = 4$ and $p = 2$, respectively. Depending on the value of $\alpha$, one can see that the potential can be negative in some intermediate field domains separated by a maximum (see Fig. 112). The appearance of a local maximum implies that, in addition to the large-field inflationary regime, inflation can also occurs close to the top of the new local maximum, on both sides, and we will be referring to these regimes as RCLFI1 and RCLFI2. The large-field regime will be referred to as RCLFI3. Finally, there are parameter values for which the local maximum disappears and all these three regimes become unified in what will be referred to as RCLFI4 (see Fig. 113). This one implicitly requires the logarithmic term to be small everywhere and is prototypical of what one should expect from perturbative loop corrections.

### 7.8.1. Parameter space analysis

Let us first discuss the existence of the four afore-mentioned inflationary regimes with respect to the model parameters. As already discussed, the case $p = 4$ corresponds to RCQI and will thus not be considered.
<figure><img src="image_116.png" /><figcaption>Fig. 113. Radiatively Corrected Large Field Inflation in the perturbative regime RCLFI4, where the potential does not exhibit extrema, for $\alpha = 4$, $p = 2$. Top panels: the potential and its logarithm. Bottom left panel: slow-roll parameter $\epsilon_1$ for $\mu = 100 M_{\text{pl}}$, with the field evolution annotated with an arrow indicating the direction to which it evolves. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line) for the same parameters value. When $\epsilon_3$ becomes negative, the plot shows $|\epsilon_3|$ as a red dotted line, the black dotted line corresponds to positive values. Similarly, negative values of $\epsilon_2$ are represented as blue solid curves. Notice that $\epsilon_3$ becomes singular at the points where $\epsilon_2 = 0$, but the product $\epsilon_2\epsilon_3$ remains finite. The other regimes RCLFI1 to RCLFI3 are represented in Fig. 112.</figcaption></figure>

<figure><img src="image_117.png" /><figcaption>Fig. 114. The parameter space $(p, \alpha)$ of Radiatively Corrected Large Field Inflation and its various inflationary regimes. The perturbative regime, RCLFI4, exists only in the domain between the horizontal axis and $\alpha_1(p)$ represented by the red curve. RCLFI3 can occur if a large-field domain exists. Finally, RCLFI1 and RCLFI2 are hilltop-like models and can occur only if the potential has a local maximum. Moreover, for RCLFI2, inflation may never ends if the local minimum towards which the field evolves is positive and this case is excluded. The corresponding regions are represented with the green and blue labels, see also the main text.</figcaption></figure>

Close to the origin, the potential behaves as $V(x) \propto x^p$ for $p < 4$ or $V(x) \propto \alpha x^4 \ln x$ for $p > 4$, where we have introduced

$$x = \frac{\phi}{\mu} \qquad (7.117)$$

Therefore, it is positive and increasing with $x$ for $p < 4$, and also for $p > 4$ and $\alpha < 0$. For $p > 4$ and $\alpha > 0$, it is negative and decreasing with $x$ in a neighborhood of $x = 0$. In the regime $x \gg 1$, we have $V(x) \propto x^p$ for $p > 4$ and $V(x) \propto \alpha x^4 \ln x$ for $p < 4$. The potential is thus positive and growing in all situations but one: $p < 4$ and $\alpha < 0$ where it becomes unbounded from below (RCLFI3 does not exist in this case).
The (non-vanishing) field values at which the potential vanishes are solution of

$$1 + \alpha x^{4-p} \ln x = 0. \qquad (7.118)$$

The above equation has solutions for $(p-4)/\alpha > -1/e$, given in terms of the Lambert function $W$ by

$$x_{V=0}^{\pm} = \left[ \frac{\alpha}{p-4} W \left( \frac{p-4}{\alpha} \right) \right]^{1/(p-4)}. \qquad (7.119)$$

At a fixed value of $p$, we can thus define the limiting values of $\alpha$ for which these solutions may exist as

$$\alpha_0 \equiv -e(p-4). \qquad (7.120)$$

There is only one solution given by the principal branch $W_0$ for $(p-4)/\alpha > 0$, and it will be referred to as $x_{V=0}^+$. For $-1/e < (p-4)/\alpha < 0$, there are two solutions, one still given by the principal branch $W_0$, and a new one given by the branch $W_{-1}$ that will be referred to as $x_{V=0}^-$. In view of the asymptotic behavior of the potential, the cases where there is only one zero corresponds to the potential transitioning from negative values close to the origin to asymptotically positive growth, and conversely. The cases where two zeros appear correspond to a potential with a local maximum and a negative local minimum.

There is also the possibility that the local minimum is positive and in order to discuss this situation one should determine the field values for which $V'(x) = 0$, i.e.,

$$px^{p-4} + \alpha(1 + 4 \ln x) = 0. \qquad (7.121)$$

The solutions are again given in terms of the Lambert function, they exist only for

$$\frac{p(p-4)}{4\alpha} e^{1-p/4} \geq -\frac{1}{e}. \qquad (7.122)$$

This leads us to define another boundary function $\alpha_1(p)$ by

$$\alpha_1 \equiv -\frac{p(p-4)}{4} e^{2-p/4}. \qquad (7.123)$$

When the condition Eq. (7.122) is satisfied, the solutions are given by

$$x_{V'=0}^{\pm} = \left\{ \frac{4\alpha}{p(p-4)} W \left[ \frac{p(p-4)}{4\alpha} e^{1-p/4} \right] \right\}^{1/(p-4)}. \qquad (7.124)$$

There is only one solution, $x_{V'=0}^+$, given by the principal branch $W_0$, for $[p(p-4)/(4\alpha)]e^{1-p/4} > 0$. Another solution appears for $-1/e < [p(p-4)/(4\alpha)]e^{1-p/4} < 0$ that will be referred to as $x_{V'=0}^-$ given by the branch $W_{-1}$. As for the zeros of the potential, when the solution is unique, it represents a local maximum, or minimum, in a transitioning regime between a negative potential close to the origin to a positive one asymptotically, or the converse. When there are two solutions, we have both a local maximum and a local minimum.

From these considerations we can determine the parameter space in which RCLFI1, RCLFI2, RCLFI3 and RCLFI4 exist. RCLFI1 and RCLFI2 are hilltop-like models and inflation takes place close to the local maximum of the potential, at decreasing field values for RCLFI1 and at increasing field values for RCLFI2. Moreover, in the case of RCLFI2, the field evolves towards the local minimum of the potential and this one can actually be positive: this would prevent inflation to end. Therefore, we add the requirement that this local minimum is negative, or null, to ensure a graceful exit of RCLFI2. Combining the above considerations, we find that the RCLFI1 regime can take place for $p > 4$ and $\alpha < \alpha_1(p)$, or, $p < 4$ and $\alpha < 0$, or, $p < 4$ and $\alpha > \alpha_1(p)$. For RCLFI2, the conditions are identical but one should replace $\alpha_1$ by $\alpha_0$ to ensure graceful exit. RCLFI3 describes inflation in the large-field domain and requires the potential to grow positive at large-field values. It exists for $p > 4$ and $\alpha > 0$, or, $p > 4$ and $\alpha < \alpha_0(p)$, or, $p < 4$ and $\alpha > \alpha_0(p)$. Finally, the perturbative regime demands that the potential has no extrema and this corresponds to $p > 4$ and $\alpha_1(p) < \alpha < 0$, or, $p < 4$ and $0 < \alpha < \alpha_1(p)$. These domains have been represented in Fig. 114.

### 7.8.2. Slow-roll analysis

The slow-roll parameters associated with RCLFI read

$$\epsilon_1 = \frac{M_{\mathrm{Pl}}^2}{2x^2\mu^2} \left[ \frac{px^p + \alpha x^4 (1 + 4 \ln x)}{x^p + \alpha x^4 \ln x} \right]^2, \qquad (7.125)$$

together with

$$\epsilon_2 = \frac{2 M_{\mathrm{Pl}}^2 - \alpha x^{p+4} \{[p(p-9) + 12] \ln x - 2p + 7\} + px^{2p} + \alpha^2 x^8 (4 \ln^2 x + \ln x + 1)}{(x^p + \alpha x^4 \ln x)^2}, \qquad (7.126)$$

and

$$\epsilon_3 = \frac{M_{\mathrm{Pl}}^2}{x^2 \mu^2 (x^p + \alpha x^4 \ln x)^2} \times \frac{px^p + \alpha x^4 + 4\alpha x^4 \ln x}{\alpha x^4 \ln x [\alpha x^4 - (p^2 - 9p + 12) x^p] - 7\alpha x^{p+4} + px^p (x^p + 2\alpha x^4) + \alpha^2 x^8 + 4\alpha^2 x^8 \ln^2 x} \times \left\{ 3\alpha p^2 x^{2p+4} + \alpha x^4 \ln x [-\alpha (3p^2 - 30p + 68) x^{p+4} - (p^3 - 9p^2 + 20p - 24) x^{2p} + 3\alpha^2 x^8] + \alpha^2 x^8 \ln^2 x [(p^3 - 15p^2 + 74p - 96) x^p + 2\alpha x^4] + 2px^p (-9\alpha x^{p+4} + x^{2p} + 3\alpha^2 x^8) + \alpha x^4 (-21\alpha x^{p+4} + 26x^{2p} + 2\alpha^2 x^8) + 8\alpha^3 x^{12} \ln^3(x) \right\}. \qquad (7.127)$$

The three slow-roll parameters have been plotted as a function of $x$ for RCLFI1, RCLFI2 and RCLFI3 in Fig. 112, and for RCLFI4 in Fig. 113. For the perturbative regime, even though the potential does not exhibit extrema, the logarithmic correction modulates the shape of the potential
<figure><img src="image_118.png" /><figcaption>Fig. 115. Non-Renormalizable Corrected Loop Inflation (NCLI). Top left panel: Non-Renormalizable Corrected Loop Inflation potential Eq. (7.132) as a function of $\phi/M_n$, for $\alpha = 10^{-3}$, $\phi_0 = 10M_n$ and $n = 2$. Top right panel: logarithm of the potential. Bottom left panel: slow-roll parameter $\epsilon_1$. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid line) and $\epsilon_3$ (dotted line) for the same parameters value. Notice that $\epsilon_3$ becomes singular at the points where $\epsilon_2 = 0$, but the product $\epsilon_2\epsilon_3$ remains finite.</figcaption></figure>

and this implies that $\epsilon_2$ may change sign during inflation (see lower right panel in Fig. 113). As a result, $\epsilon_3(x)$ becomes singular when $\epsilon_2(x) = 0$ and we are, a priori, in the presence of slow-roll violations at second order. In fact, they are not really problematic as the product $\epsilon_2\epsilon_3$, which is the quantity entering into the second-order slow-roll power spectra, remains always finite, but they do signal potentially large running of the spectral index.

For the four regimes, RCLFI inflation gracefully ends at the field value $x_{end}$ solution of $\epsilon_1(x_{end}) = 1$, in the domain of interest. Because $\epsilon_1$ diverges for $V(x) \to 0$, this always occur in a domain in which the potential is positive definite and ensures that inflation remains confined in these domains. There is no analytical solution of the equation $\epsilon_1(x) = 1$ and $x_{end}$ has to be determined numerically by solving

$$px^p + \alpha x^4 (1 + 4 \ln x) = \pm \sqrt{2x\mu} (x^p + \alpha x^4 \ln x). \qquad (7.128)$$

The slow-roll trajectory cannot be analytically solved and also requires a numerical integration. It reads

$$N_{\text{end}} - N = \frac{\mu^2}{M_{\text{Pl}}^2} \int_{x_{\text{end}}}^{x} \frac{y^{p+1} + \alpha y^5 \ln y}{py^p + \alpha y^4 (1 + 4 \ln y)} dy, \qquad (7.129)$$

where $x_{end}$ is obtained from the solution of Eq. (7.128) in the relevant field domain for the inflationary regime under scrutiny (RCLFI1, RCLFI2, RCLFI3 or RCLFI4).

The normalization of the potential $M^4$ is obtained from the amplitude of the CMB anisotropies once the field value $x_*$ at which the pivot mode crossed the Hubble radius is determined. This one obtained from the trajectory, the value of $x_{end}$, and the reheating equation (3.48). One gets

$$\left(\frac{M}{M_{\text{Pl}}}\right)^4 = \frac{720\pi^2 M_{\text{Pl}}^4}{\mu^4} \frac{[px_*^p + \alpha x_*^4 (1 + 4 \ln x_*)]^2}{x_*^2 (x_*^p + \alpha x_*^4 \ln x_*)^3} \frac{Q_{\text{rms}-\text{PS}}^2}{T^2}. \qquad (7.130)$$

The reheating consistent slow-roll predictions for the four RCLFI regimes are represented in Figs. 409 to 452. Because the parameter space in $(p, \alpha)$ is disjoint, the models have been split in two sub-regimes according to the sign of $p-4$ and/or the sign of $\alpha$.

## 7.9. Non-Renormalizable Corrected Loop Inflation (NCLI)

### 7.9.1. Theoretical justifications

This model is based on Ref. [702], where the potential is assumed to be exactly flat at tree level, with two types of corrections considered: (i) corrections that do not spoil the flatness of the potential and which correspond to radiative modulations of the potential, and (ii) corrections that do spoil the flatness, and which correspond to non-renormalizable operators in the tree-level potential. These are suppressed by $(\phi/\Lambda)^{2n}$, where $n$
is a positive integer and $\Lambda$ is the scale where new physics becomes relevant and is assumed to be larger than the energy at which inflation takes place. The potential reads [702]

$$V(\phi) = \rho + \beta \ln \left[ \frac{m(\phi)}{Q} \right] + \phi^4 \frac{\phi^{2n}}{\Lambda^{2n}}. \qquad (7.131)$$

In this expression, $\rho$ corresponds to the tree-level flat potential, $\beta$ is a positive coupling constant, $m^2(\phi) = \lambda^2 \phi^2 / 2$ is a mass term, and $Q$ corresponds to a renormalization energy scale.

### 7.9.2. Slow-roll analysis

For our purpose, it is more convenient to recast the potential into the form

$$V(\phi) = M^4 \left[ 1 + \alpha \ln \left( \frac{\phi}{M_{Pl}} \right) + \left( \frac{\phi}{\phi_0} \right)^{4+2n} \right], \qquad (7.132)$$

with $M^4 = \rho + \beta \ln[\lambda^2 M_{Pl}^2/(2Q)]$, $\alpha = 2\beta/M^4$ and $\phi_0^{4+2n} = \Lambda^{2n} M^4$. This model can be seen as a correction to (or an extended version of) Loop Inflation, see Section 5.12, which is recovered in the limit $\phi_0 \to \infty$. The potential is displayed in Fig. 115. It is an increasing function of the field value, and is positive for $\phi > \phi_{V=0}$, with

$$\phi_{V=0} = \phi_0 \left\{ \frac{\alpha}{4+2n} W_0 \left[ \frac{4+2n}{\alpha} e^{-(4+2n)\left(\frac{1}{\alpha}+\ln\frac{\phi_0}{M_{Pl}}\right)} \right] \right\}^{\frac{1}{4+2n}}, \qquad (7.133)$$

and where $W_0$ is the 0-branch of the Lambert function. The potential has a concave shape below its inflection point $\phi < \phi_{V''=0}$, where

$$\phi_{V''=0} = \phi_0 \left[ \frac{\alpha}{(4+2n)(3+2n)} \right]^{\frac{1}{4+2n}}, \qquad (7.134)$$

and is convex above.

In the slow-roll approximation, the first Hubble-flow function is given by

$$\epsilon_1 = \frac{1}{2} \frac{M_{Pl}^2}{\phi_0^2} \left[ \frac{\alpha \frac{\phi_0}{\phi} + (4+2n) \left( \frac{\phi}{\phi_0} \right)^{3+2n} \right]^2 \left[ 1 + \alpha \ln \left( \frac{\phi}{M_{Pl}} \right) + \left( \frac{\phi}{\phi_0} \right)^{4+2n} \right]^2, \qquad (7.135)$$

while the second one reads

$$\epsilon_2 = \frac{2 \frac{M_{Pl}^2}{\phi_0^2}}{\left[ 1 + \alpha \ln \left( \frac{\phi}{M_{Pl}} \right) + \left( \frac{\phi}{\phi_0} \right)^{4+2n} \right]^2} \left\{ \left[ \alpha \frac{\phi_0}{\phi} + 2(2+n) \left( \frac{\phi}{\phi_0} \right)^{3+2n} \right]^2 + \left[ \alpha \left( \frac{\phi_0}{\phi} \right)^2 - 2(2+n)(3+2n) \left( \frac{\phi}{\phi_0} \right)^{2+2n} \right] \left[ 1 + \alpha \ln \left( \frac{\phi}{M_{Pl}} \right) + \left( \frac{\phi}{\phi_0} \right)^{4+2n} \right] \right\}. \qquad (7.136)$$

and, finally,

$$\epsilon_3 = \frac{M_{Pl}^2}{\phi_0^2} \left[ \alpha \frac{\phi_0}{\phi} + 2(2+n) \left( \frac{\phi}{\phi_0} \right)^{3+2n} \right] \left\{ 2 \left[ \alpha \frac{\phi_0}{\phi} + 2(2+n) \left( \frac{\phi}{\phi_0} \right)^{3+2n} \right]^3 \right. \\
+ 3 \left[ \alpha \frac{\phi_0^2}{\phi^2} - 2(2+n)(3+2n) \left( \frac{\phi}{\phi_0} \right)^{2+2n} \right] \left[ \alpha \frac{\phi_0}{\phi} + 2(2+n) \left( \frac{\phi}{\phi_0} \right)^{3+2n} \right] \\
\times \left[ 1 + \alpha \ln \left( \frac{\phi}{M_{Pl}} \right) + \left( \frac{\phi}{\phi_0} \right)^{4+2n} \right] \\
+ \left[ 2\alpha \frac{\phi_0^3}{\phi^3} + 4(1+n)(2+n)(3+2n) \left( \frac{\phi}{\phi_0} \right)^{1+2n} \right] \left[ 1 + \alpha \ln \left( \frac{\phi}{M_{Pl}} \right) + \left( \frac{\phi}{\phi_0} \right)^{4+2n} \right]^2 \\
\left. \times \left( \left[ 1 + \alpha \ln \left( \frac{\phi}{M_{Pl}} \right) + \left( \frac{\phi}{\phi_0} \right)^{4+2n} \right]^2 \left\{ \left[ \alpha \frac{\phi_0}{\phi} + 2(2+n) \left( \frac{\phi}{\phi_0} \right)^{3+2n} \right]^2 + \left[ \alpha \frac{\phi_0^2}{\phi^2} - 2(2+n)(3+2n) \left( \frac{\phi}{\phi_0} \right)^{2+2n} \right] \left[ 1 + \alpha \ln \left( \frac{\phi}{M_{Pl}} \right) + \left( \frac{\phi}{\phi_0} \right)^{4+2n} \right] \right\} \right]^{-1}. \qquad (7.137)$$

These parameters are displayed in the lower panels of Fig. 115, and, clearly feature two different regimes. At large-field values, the potential is dominated by the monomial correction, proportional to $\phi^{4+2n}$, and the Hubble-flow parameters increase as the field decreases, i.e. as inflation
proceeds. However, in that region of the potential, higher-order corrections may become important, which is why inflation is meant to take place at smaller-field values in that model. At small-field values, the potential is dominated by the logarithmic term and the constant term, and is therefore of the same type as Loop Inflation, see Section 5.12. In that region, the Hubble-flow parameters again increase as inflation proceeds.

When transiting between these two regions, the behavior of the Hubble-flow parameters is more involved. For instance, $\epsilon_1$ first reaches a maximum, then decreases for a transient period and increases again. If $\phi_0$ is large enough, this local maximum is such that $\epsilon_1 < 1$ and inflation does not end before $\epsilon_1$ increases again when $\phi$ approaches 0. Otherwise, inflation could terminate at the end of the first phase, and resume afterwards, but we do not consider this possibility any further since, as already stressed, the model is reliable only in the second phase.

Inflation ends when $\epsilon_1 = 1$, and the corresponding field value $\phi_{end}$ can be obtained by solving the equation

$$ \alpha \frac{\phi_0}{\phi_{end}} + (4 + 2n) \left( \frac{\phi_{end}}{\phi_0} \right)^{3+2n} = \sqrt{2} \frac{\phi_0}{M_{Pl}} \left[ 1 + \alpha \ln \left( \frac{\phi_{end}}{M_{Pl}} \right) + \left( \frac{\phi_{end}}{\phi_0} \right)^{4+2n} \right] \quad (7.138) $$

Unfortunately, this equation does not have analytical solutions and it must be solved numerically. Likewise, the slow-roll trajectory,

$$ N_{\text{end}} - N = \frac{\phi_0}{M_{\text{Pl}}} \int_{\phi_{\text{end}}}^{\phi} \frac{1 + \alpha \ln \left( \frac{\chi}{M_{\text{Pl}}} \right) + \left( \frac{\chi}{\phi_0} \right)^{4+2n}}{\alpha \frac{\phi_0}{\chi} + (4 + 2n) \left( \frac{\chi}{\phi_0} \right)^{3+2n}} \frac{d\chi}{M_{\text{Pl}}}. \quad (7.139) $$

cannot be integrated analytically and must be computed numerically.

The normalization of the potential $M^4$ is obtained from the amplitude of the CMB anisotropies once the field value $\phi_*$ at which the pivot mode crossed the Hubble radius is determined. One gets

$$ \left(\frac{M}{M_{Pl}}\right)^4 = 720\pi^2 \frac{M_{Pl}^2}{\phi_0^2} \frac{\left[\alpha \frac{\phi_0}{\phi_*} + 2(n+2) \left(\frac{\phi_*}{\phi_0}\right)^{2n+3}\right]^2}{\left[\alpha \log \left(\frac{\phi_*}{M_{Pl}}\right) + \left(\frac{\phi_*}{\phi_0}\right)^{2(n+2)} + 1\right]^3} \frac{Q_{\text{rms}-\text{PS}}^2}{T^2}. \quad (7.140) $$

The reheating consistent slow-roll predictions for NCLI are represented in Figs. 453 to 456, for $n=2$ and $n=3$, plus a few values of $\alpha$ and $\phi_0$. One can check that when $\phi_0$ is large, one recovers the same predictions as in Loop Inflation. When $\phi_0$ decreases, the spectral index increases, and quickly leaves the region allowed by the data. As a consequence, the amplitude of the corrective monomial term is bounded from above in this model.

The way this upper bound varies with $\alpha$ can be understood as follows. In Loop Inflation, in the regime $\alpha \ll 1$, the slow-roll trajectory is approximately given by $\phi_*/M_{Pl} \simeq \sqrt{2\alpha(N_{end} - N_*)}$, see the relation given below Eq. (5.183). By performing an expansion of Eqs. (7.135), (7.136) and (7.137) in $1/\phi_0$, one obtains for the first Hubble flow function(still at leading order in $\alpha$)

$$ \epsilon_{1*} \approx \frac{\alpha}{4\Delta N_*} + \alpha(4+2n)(2\alpha\Delta N_*)^{n+1} \left(\frac{M_{Pl}}{\phi_0}\right)^{4+2n}. \quad (7.141) $$

The second reads

$$ \epsilon_{2*} \approx \frac{1}{\Delta N_*} - 4(n+2)(3+2n)(2\alpha\Delta N_*)^{n+1} \left(\frac{M_{Pl}}{\phi_0}\right)^{4+2n}, \quad (7.142) $$

while the third one is

$$ \epsilon_{3*} \approx \frac{1}{\Delta N_*} + 4(n+2)(2n^2+7n+7)(2\alpha\Delta N_*)^{n+1} \left(\frac{M_{Pl}}{\phi_0}\right)^{4+2n}, \quad (7.143) $$

where $\Delta N_* = N_{end} - N_*$. One can check that the leading terms of these expressions match the approximate predictions of Loop Inflation (LI), see Eq. (5.185). The first-order corrections have a relative amplitude controlled by $(\Delta N_*)^{n+2}\alpha^{n+1}(M_{Pl}/\phi_0)^{4+2n}$ for all three parameters, so one concludes that the predictions of NCLI are close to the ones of LI provided

$$ \frac{\phi_0}{M_{Pl}} \gg \alpha^{\frac{n+1}{2n+4}}. \quad (7.144) $$

From this expression, one can see that the smaller $\alpha$, the smaller $\phi_0$ can be. The above equation also provides a correct estimate for the value of $\phi_0$ below which the predictions of NCLI deviate from the ones of LI (and are therefore disfavored by the data), as can be seen in Figs. 453 to 456.

## 7.10. Hybrid Natural Inflation (HNI)

### 7.10.1. Theoretical justifications

This scenario is an extension of Natural Inflation (NI, see Section 5.6) in which the end of inflation can be triggered by a waterfall mechanism as in Hybrid Inflation (see Section 7.2). This idea, as well as explicit supersymmetric and non-supersymmetric two-fields constructions of the model, have been presented in Refs. [333,703–707]. In addition to stopping inflation, the waterfall field distorts the effective potential along the inflationary direction such that the potential for HNI reads

$$ V(\phi) = M^4 \left[ 1 + \alpha \cos \left( \frac{\phi}{f} \right) \right]. \quad (7.145) $$
<figure><img src="image_119.png" /><figcaption>Fig. 116. Hybrid Natural Inflation for $\alpha = 0.5$. Top panels: the potential and its logarithm. Bottom left panel: slow-roll parameter $\epsilon_1 f^2/M_{Pl}^2$. Notice that, depending on $\alpha$ and $f$, the maximal value of $\epsilon_1$ can become larger than unity. Bottom right panel: slow-roll parameters $\epsilon_2 f^2/M_{Pl}^2$ (solid line) and $\epsilon_3 f^2/M_{Pl}^2$ (dotted line) for $\alpha = 0.5$. As for SFI2 (see Section 6.1), the potential at $\phi = 0$ has a tachyonic mass and $\epsilon_2$ goes to a constant at small-field values, which violates slow roll for sub-Planckian $f$.</figcaption></figure>

The typical vacuum expectation value for the inflaton, $f$, can be made super-Planckian as HNI describes a multiple-field model (see Section 5.6). The parameter $0 < \alpha < 1$ encodes the distortions induced by the waterfall field onto the inflationary direction, and as can be seen in Fig. 116, this implies that the minimum of the potential is non-vanishing. Note however that the true minimum of the potential in the two-field space is elsewhere, and, exactly as for Hybrid Inflation, the ground state of the waterfall field is expected to cancel this apparent residual cosmological constant. For our purpose, it means that the potential of Eq. (7.145) cannot be trusted for $\phi/f \simeq \pi$, and we will only consider the inflationary domain connected to the maximum of the potential at $\phi = 0$.

Finally, let us remark that the potential of NI is recovered for $\alpha \to 1$ but, even in this limit, both models may have different observable predictions as HNI may end by waterfall instability instead of slow-roll violations. As such, HNI has three parameters, $\alpha$, $f$ and the field value at which inflation stops, $x_{end}$.

### 7.10.2. Slow-roll analysis

Introducing $x = \phi/f$, the Hubble-flow functions in the slow-roll approximation are

$$ \epsilon_1 = \frac{\alpha^2 M_{Pl}^2}{2f^2} \frac{\sin^2(x)}{[1 + \alpha \cos(x)]^2}, \qquad \epsilon_2 = \frac{2\alpha M_{Pl}^2}{f^2} \frac{\alpha + \cos(x)}{[1 + \alpha \cos(x)]^2}, \qquad (7.146) $$

and

$$ \epsilon_3 = - \frac{\alpha M_{Pl}^2}{f^2} \frac{\sin^2(x)}{[1 + \alpha \cos(x)]^2} \frac{1 - 2\alpha^2 - \alpha \cos(x)}{\alpha + \cos(x)}. \qquad (7.147) $$

They have been represented in the lower panels of Fig. 116 for $\alpha = 0.5$. The first Hubble-flow function exhibits a maximum within the inflationary domain, at a field value that is solution of $\epsilon_2(x) = 0$, and given by

$$ x_{\text{max}} = \arccos(-\alpha). \qquad (7.148) $$

Plugging this value into Eq. (7.146) gives the maximal value reached by $\epsilon_1(x)$ over the domain $x \in ]0, \pi[$,

$$ \epsilon_1^{\text{max}} = \frac{M_{Pl}^2}{2f^2} \frac{\alpha^2}{1 - \alpha^2}. \qquad (7.149) $$

It is larger than unity if $\alpha > \alpha_1$, or, equivalently, if $f < f_1$, where

$$ \alpha_1 \equiv \frac{1}{\sqrt{1 + M_{Pl}^2/(2f^2)}}, \qquad f_1 \equiv \frac{\alpha}{\sqrt{2\alpha^2 - 2}}. \qquad (7.150) $$
When this happens, the inflationary domain becomes disconnected, and, a priori, one could have inflation close to the top of the potential (around $\phi = 0$), or close to the bottom of the potential at $\phi/f \le \pi$. As we have mentioned in the previous section, the latter situation will be discarded as one cannot trust anymore Eq. (7.145) to describe the two-field dynamics (see also the discussion regarding this possibility for VHI in Section 7.2). Nonetheless, this creates the possibility that HNI gracefully ends if the field value at which the waterfall instability develops is in the domain for which $\epsilon_1 > 1$. This scenario will be referred to as HNII and the smallest root of $\epsilon_1(x) = 1$ gives the field value at which inflation ends in that case,

$$x_{\epsilon_1=1} = \arccos \left\{ \frac{M_{\mathrm{Pl}} \sqrt{\frac{a^2 M_{\mathrm{Pl}}^2}{4f^2} + \frac{a^2-1}{2}} - 1}{\alpha f \left( 1 + \frac{M_{\mathrm{Pl}}^2}{2f^2} \right)} \right\}. \qquad (7.151)$$

This solution exists provided $\alpha > \alpha_1$ (or $f < f_1$) and one has $x_{\mathrm{end}} = x_{\epsilon_1=1}$, making HNII a two-parameter model.

In the other, and more generic, scenario, referred to as HNII, inflation will be assumed to always end by instability, for all values of $\alpha$ and $f$. As a result, for $\alpha > \alpha_1$, there is an upper bound for the field value at which the instability occurs, namely $x_{\mathrm{end}} < x_{\epsilon_1=1}$. For $\alpha < \alpha_1$, $x_{\mathrm{end}}$ can be taken up to $x_{\mathrm{end}}^{\mathrm{max}} = \pi$, the minimum of the potential.

The slow-roll trajectory can be explicitly integrated using Eq. (3.11) and reads

$$N_{\mathrm{end}} - N = \frac{f^2}{\alpha M_{\mathrm{Pl}}^2} \left\{ (1 - \alpha) \ln \left[ \frac{\cos\left(\frac{x}{2}\right)}{\cos\left(\frac{x_{\mathrm{end}}}{2}\right)} \right] - (1 + \alpha) \ln \left[ \frac{\sin\left(\frac{x}{2}\right)}{\sin\left(\frac{x_{\mathrm{end}}}{2}\right)} \right] \right\}. \qquad (7.152)$$

The logarithmic divergence at $x \to 0$ shows that the number of $e$-folds that can be realized around the top of the potential is unbounded, so $x_{\mathrm{end}}$ can be arbitrarily small. From Eq. (7.146), one also sees that although $\epsilon_1 \to 0$ at the top of the potential, the second slow-roll parameter reaches a constant value

$$\epsilon_2(x=0) = \frac{2\alpha}{1+\alpha} \frac{M_{\mathrm{Pl}}^2}{f^2}. \qquad (7.153)$$

As such, for $\alpha = \mathcal{O}(1)$, slow roll inflation at the top of the HNI potential can only take place for super-Planckian values of $f$, a situation in all points similar to the small-field model SFI2 (see Section 6.1). It is however possible to accommodate sub-Planckian values of $f$ with slow roll but only at the expanse of having $\alpha \ll 1$. The scenario HNII in the limit $x_{\mathrm{end}} \ll 1$ is therefore a constant-spectral-index model with vanishing running and highly suppressed tensor-to-scalar ratio.

Using the reheating equation (3.48), together with the field value at which inflation ends, $x_{\mathrm{end}}$, Eq. (7.152) uniquely determines $x_*$, the field value at which the pivot mode crossed the Hubble radius during inflation. The mass scale $M$ of the potential is fixed by the CMB normalization and verifies

$$\left(\frac{M}{M_{\mathrm{Pl}}}\right)^4 = 720\pi^2\alpha^2 \frac{M_{\mathrm{Pl}}^2}{f^2} \frac{\sin^2(x_*)}{[1 + \alpha \cos(x_*)]^3} \frac{Q_{\mathrm{rms}-\mathrm{PS}}^2}{T^2}. \qquad (7.154)$$

The reheating consistent slow-roll predictions for the two HNI models (HNI1 and HNII) are represented in Figs. 457 to 468.

## 7.11. N-Formalism Inflation (NFI)

### 7.11.1. Theoretical justifications

This model is phenomenological and motivated by the search of “universality classes” for slow-roll inflation as originally proposed in Ref. [708]. There, it was argued that most of the observationally relevant inflationary models should lead to a first Hubble flow function varying as

$$\epsilon_1 \propto \frac{1}{\Delta N^a}, \qquad (7.155)$$

the higher-order terms of an expansion in $1/\Delta N_*$ being neglected. This idea was later proven to not encompass not all relevant models, and to lead to insufficiently accurate predictions in light of the precision of the CMB data in Ref. [28]. However, it is still possible to search for models verifying Eq. (7.155) at leading order in slow-roll. Plugging Eq. (7.155) into the definition of the first Hubble-flow function given in Eq. (3.8), one gets $\Delta\phi/M_{\mathrm{Pl}} \propto \Delta N^{1-\frac{a}{2}}$ with $\Delta\phi = \phi_{\mathrm{end}} - \phi$, which implies

$$\frac{1}{M_{\mathrm{Pl}}} \frac{d\phi}{dN} \propto \frac{1}{\Delta N^{\frac{a}{2}}} \propto \left( \frac{\Delta\phi}{M_{\mathrm{Pl}}} \right)^{\frac{a}{a-2}}. \qquad (7.156)$$

From the slow-roll trajectory given in Eq. (3.10), one gets

$$\frac{d \ln V}{d \phi} \propto \frac{1}{M_{\mathrm{Pl}}} \left( \frac{\Delta \phi}{M_{\mathrm{Pl}}} \right)^{\frac{a}{a-2}}. \qquad (7.157)$$

Ignoring the singular case $\alpha = 2$, the potential verifies

$$\ln \left( \frac{V}{M_{\mathrm{Pl}}^2} \right) \propto \left( \frac{\Delta \phi}{M_{\mathrm{Pl}}} \right)^{\frac{2(\alpha-1)}{\alpha-2}}, \qquad (7.158)$$

and is an exponential function of some power of the field.
<figure><img src="image_120.png" /><figcaption>Fig. 117. N-Formalism Inflation in the regime NFI1 for $a = 1$, $b = 1.2$ (black curves) and NFI2 for $a = -1$, $b = 1.2$ (blue curves). Top panels: the potential and its logarithm. Bottom left panel: slow-roll parameter $\epsilon_1/a^2$, which is the same in both regime. Bottom right panel: slow-roll parameters $\epsilon_2/|a|$ (solid line) and $\epsilon_3/|a|$ (dotted line). Their signs are different in the NFI1 and NFI2 regime. Within the range $1 < b < 2$, NFI2 exhibit slow-roll violations at small field values but only for $\epsilon_2$ and $\epsilon_3$. This does not stop inflation as $\epsilon_1 \to 0$ for $\phi \to 0$. For $b > 2$, all slow-roll parameters are regular and vanish at the origin.</figcaption></figure>

### 7.11.2. Slow-roll analysis

Let us write the potential of NFI under the more convenient form

$$V(\phi) = M^4 e^{-a \left(\frac{\phi}{M_{Pl}}\right)^b}, \qquad (7.159)$$

where $a$ and $b$ are two dimensionless parameters. The peculiar case $b = 1$ is Power Law Inflation (PLI) discussed in Section 5.8 and will not be further discussed in the following. Using

$$x \equiv \frac{\phi}{M_{Pl}}, \qquad (7.160)$$

the Hubble flow functions in the slow-roll approximation read

$$\epsilon_1 = \frac{1}{2} a^2 b^2 x^{2(b-1)}, \qquad \epsilon_2 = 2ab(b-1)x^{b-2}, \qquad \epsilon_3 = ab(b-2)x^{b-2}. \qquad (7.161)$$

When $b$ is not an integer, the potential is well-defined only for $x > 0$ but could be extended in the negative domain otherwise (see below). The solution to $\epsilon_1 = 1$ is given by

$$x_{\epsilon_1=1} = \left( \frac{2}{a^2 b^2} \right)^{\frac{1}{2(b-1)}}, \qquad (7.162)$$

and one can see that inflation can only take place in the domains $x < x_{\epsilon_1=1}$ for $b > 1$ and in the domains $x > x_{\epsilon_1=1}$ for $b < 1$. Moreover, depending on the sign of both $a$ and $b$, the potential, within the positive domain $x > 0$, can either be an increasing or a decreasing function of the field value. If the potential drives the field away from $x_{\epsilon_1=1}$ then inflation requires an additional mechanism to end. In this situation, $\phi_{end}$, the field value at which inflation ends is an additional model parameter making NFI a three-parameter model. However, if the potential drives the field towards $x_{\epsilon_1=1}$ then inflation can gracefully end and the model has only two parameters in that case.

For these reasons we distinguish four regimes. The first, NFI1, obtained for $a > 0$ and $b > 1$, is associated with a decreasing potential with respect to $x$, inflation proceeds at increasing field values and naturally ends at $x_{\epsilon_1=1}$. The second regime, referred to as NFI2, corresponds to $a < 0$ and $b > 1$. The potential is an increasing function of the field, inflation proceeds at decreasing field values moving away from $x_{\epsilon_1=1}$. The third regime, NFI3, is obtained for either $a < 0$ and $0 < b < 1$, or, $a > 0$ and $b < 0$. It is associated with a increasing potential with respect to $x$, inflation occurring at decreasing field values and moving towards $x_{\epsilon_1=1}$ with a graceful ending. Finally, NFI4 is obtained for either $a > 0$ and $0 < b < 1$, or, $a < 0$ and $b < 0$. Inflation proceeds at increasing field values, moving away from $x_{\epsilon_1=1}$ and requiring an additional mechanism to stop. The potential and
<figure><img src="image_121.png" /><figcaption>Fig. 118. N-Formalism Inflation in the regime NFI3 for $a = -1$, $b = 0.4$ (black curves) and NFI4 for $a = 1$, $b = 0.4$ (blue curves). Top panels: the potential and its logarithm. Bottom left panel: slow-roll parameter $\epsilon_1/a^2$, which is the same in both regime. Bottom right panel: slow-roll parameters $\epsilon_2/|a|$ (solid line) and $\epsilon_3/|a|$ (dotted line). They differ in sign between the NFI3 and NFI4 regimes.</figcaption></figure>

<figure><img src="image_122.png" /><figcaption>Fig. 119. The four inflationary regimes of N-Formalism Inflation in the parameter space (a, b). Inflation naturally ends for NFI1 and NFI3 whereas an additional mechanism has to be invoked for NFI2 and NFI4. The vertical dotted line at $b = 2$ is a separatrix in the behavior of the Hubble flow functions at small-field values. For $b < 2$, both $\epsilon_2$ and $\epsilon_3$ diverge for $x \to 0$ but vanish otherwise.</figcaption></figure>

the Hubble-flow functions for the regimes NFI1 and NFI2 have been represented in Fig. 117 while NFI3 and NFI4 are shown in Fig. 118. Fig. 119 shows the location of these four inflationary regimes in the parameter space (a, b).

The integer values of $b$ allow for the potential to be extended into the domain $x < 0$. For even values of $b$, the potential of Eq. (7.159) is symmetric under the transformation $\phi \to -\phi$ such that, at negative field values, inflation proceeds exactly as in the positive domain. For odd values of $b$, the potential remains unchanged under the transformation $\phi \to -\phi$ combined with $a \to -a$. As a result, inflation in the negative domain of NFI1 is the same as inflation in the positive domain of NFI2, and conversely. This symmetry also unifies the negative domain of NFI3 with the positive domain of NFI4, and conversely. All in all, we can restrict the analysis to $x > 0$ in all possible situations.

The slow-roll trajectory can be integrated analytically and reads

$$N_{\text{end}} - N = \frac{x_{\text{end}}^{2-b} - x^{2-b}}{ab(2-b)}. \qquad (7.163)$$
The case $b=2$ is included as the regular logarithmic limit of this expression. The trajectory can be inverted into

$$x = \left[ x_{\text{end}}^{2-b} - ab(2-b)(N_{\text{end}} - N) \right]^{\frac{1}{2-b}}, \qquad (7.164)$$

which matches the one of Small Field Inflation (SFI) in the deep sub-Planckian limit, see Eq. (6.12). The case $b=2$ can be dealt by starting with the logarithmic limit of Eq. (7.163) and one gets a pure exponential term

$$x = x_{\text{end}}e^{-2a(N_{\text{end}}-N)}. \qquad (7.165)$$

As already mentioned, for NFI1 and NFI3 we have $x_{\text{end}} = x_{\epsilon_1=1}$ given in Eq. (7.162) whereas $x_{\text{end}}$ is an additional model parameter for NFI2 and NFI4. Let us notice that for $b>2$, Eq. (7.163) diverges for $x \to 0$ and the maximal number of e-folds achievable at the origin of the potential is unbounded. In the opposite situation, for $b<2$, this implies that there is a maximal acceptable value for $|a|$ to have enough e-folds of NFI1 inflation between $x=0$ and $x_{\text{end}} = x_{\epsilon_1=1}$. Similarly, $|a|$ should be small enough to have enough e-folds of NFI2 inflation between $x_{\text{ini}}^{\text{max}} = x_{\epsilon_1=1}$ and $x_{\text{end}}^{\text{min}} = 0$. In the regime NFI3 one always has $b<1$ and the number of e-folds is unbounded. The regime NFI4 proceeds at $x > x_{\epsilon_1=1}$ and the number of e-folds can be made as large as needed by choosing the free parameter $x_{\text{end}}$ large enough.

From the trajectory of Eq. (7.163), combined with the reheating equation (3.48) and the field value $x_{\text{end}}$ at which inflation ends, one obtains $x_*$, the field value at which the pivot mode crossed the Hubble radius during inflation. The mass scale $M$ of the potential is then fixed by the CMB normalization and verifies

$$\left(\frac{M}{M_{\text{Pl}}}\right)^4 = 720\pi^2 a^2 b^2 e^{ax_*^b} x_*^{2(b-1)} \frac{Q_{\text{rms-PS}}^2}{T^2}. \qquad (7.166)$$

The reheating consistent slow-roll predictions for the four NFI regimes are represented in Figs. 469 to 487. Let us remark that plugging Eq. (7.164) within the expression of the Hubble-flow functions given in Eq. (7.161), one obtains

$$\epsilon_{1*} = \frac{a^2 b^2}{2 \left[ x_{\text{end}}^{2-b} + ab(b-2)\Delta N_* \right]^{\frac{2b-2}{b-2}}}. \qquad (7.167)$$

For $b>2$, in the regime NFI1, for a small enough one can neglect $x_{\text{end}} = x_{\epsilon_1=1}$ compared to the term containing $\Delta N_*$. In this limit one recovers the power-law behavior of Eq. (7.155), namely

$$\epsilon_{1*} \propto \frac{1}{\Delta N_*^{\frac{2b-2}{b-2}}}. \qquad (7.168)$$

Notice that the case $\epsilon_{1*} \propto 1/\Delta N_*^2$ (which appeared as singular in the introductory discussion of this section) corresponds to the asymptotic limit $b \to +\infty$ while other values of $b>2$ produce higher-than-two power-law exponents. Less-than-unity positive exponents can be generated within NFI2 with $0 < b < 1$ and $a < 0$ while choosing $x_{\text{end}}$ very small. Exponents between one and two can be generated within NFI3 for $b < 0$ and a small enough $a > 0$. When the contribution from $x_{\text{end}}$ is not negligible, the model predictions deviate from this simple power law. It is also interesting to remark that the peculiar case $b=2$, generating the exponential trajectory of Eq. (7.165), yields

$$\epsilon_{1*} = \frac{2a^2 x_{\text{end}}^2}{e^{4a\Delta N_*}}, \qquad (7.169)$$

which is quite different from a power-law dependence in $1/\Delta N_*$. There is nothing surprising in this result as the potential of NFI written in Eq. (7.159) has been derived from Eq. (7.155) as a leading-order approximation only. The proper way to devise an inflationary model generating a given functional shape for $\epsilon_1(N)$ exactly is what is presented for VFMI in Section 6.21.

## 7.12. Radiatively Corrected Inflection Point Inflation (RCIPI)

### 7.12.1. Theoretical justifications

This class of models has been introduced in Ref. [709] and constitutes yet another implementation of the radiative corrections induced by bosonic and fermionic loops onto a monomial potential of the large-field type (see Sections 5.2, 5.4, 5.5 and 7.8). More precisely, Ref. [709] explores the possibility that the bosonic and fermionic degrees of freedom are somehow compensated to generate an inflection-point potential<fnref n="13" /> while never triggering an instability at large-field values. In the case of a quartic tree-level potential, $V_4(\phi) = \lambda\phi^4$, the Coleman–Weinberg corrections (see Section 5.11), at large values of the field $\phi$, are expected to generate an effective quartic coupling

$$\lambda(\phi) = \lambda(\phi_0) + \frac{c_1(\phi_0)}{2} \ln \left( \frac{\phi}{\phi_0} \right)^2 + \frac{c_2(\phi_0)}{8} \left[ \ln \left( \frac{\phi}{\phi_0} \right)^2 \right]^2 + \dots, \qquad (7.170)$$

where $\phi_0$ is the typical field value associated with the renormalization scale. Requiring the corrected effective potential to have an inflection point at $\phi_0$, namely $V'(\phi_0) = V''(\phi_0) = 0$, demands that

$$c_2(\phi_0) = -4c_1(\phi_0) = 16\lambda(\phi_0), \qquad (7.171)$$

and the resulting inflationary potential reads

$$V_4(\phi) = \lambda(\phi_0)\phi^4 \left\{ 1 - 2 \ln \left( \frac{\phi}{\phi_0} \right)^2 + 2 \left[ \ln \left( \frac{\phi}{\phi_0} \right)^2 \right]^2 + \dots \right\}. \qquad (7.172)$$

<footnote n="13">¹³ In Ref. [709], this inflection point is unfortunately referred to as a “plateau”, which it is not. See Ref. [9] for the difference between plateau inflation and other models.</footnote>
Because the relation in Eq. (7.171) requires some tuning between the bosonic and fermionic loop corrections, it is not expected to be exact, simply by the existence of higher order-corrections. One can therefore introduce two distortion parameters, $b_1$ and $b_2$, encoding by how much Eq. (7.171) is violated. One finally obtains an effective quartic potential

$$V_4(\phi) = \lambda \phi^4 \left\{ 1 - 2(1 - b_1) \ln \left( \frac{\phi}{\phi_0} \right)^2 + 2(1 + b_2) \left[ \ln \left( \frac{\phi}{\phi_0} \right)^2 \right]^2 \right\}. \quad (7.173)$$

As done in Ref. [709], the same reasoning can be applied to a monomial large-field potential being a pure mass term $V_2(\phi) = \lambda M_{\text{Pl}}^2 \phi^2$. In that case, the condition for the effective potential to exhibit an inflection point at $\phi_0$ is modified to

$$c_2(\phi_0) = -2c_1(\phi_0) = 4\lambda(\phi_0). \quad (7.174)$$

Including the distortion parameters, the effective quadratic potential becomes

$$V_2(\phi) = \lambda \phi^2 M_{\text{Pl}}^2 \left\{ 1 - (1 - b_1) \ln \left( \frac{\phi}{\phi_0} \right)^2 + \frac{1}{2}(1 + b_2) \left[ \ln \left( \frac{\phi}{\phi_0} \right)^2 \right]^2 \right\}. \quad (7.175)$$

In both Eqs. (7.173) and (7.175), the renormalization scale $\phi_0$ can be completely reabsorbed into a redefinition of the coupling $\lambda$ and of the distortion parameters $b_1$ and $b_2$. We therefore define the potential of RCIPI as

$$V(\phi) = M^4 \left( \frac{\phi}{M_{\text{Pl}}} \right)^p \left[ 1 + \alpha \ln \left( \frac{\phi}{M_{\text{Pl}}} \right) + \beta \ln^2 \left( \frac{\phi}{M_{\text{Pl}}} \right) \right], \quad (7.176)$$

which has three parameters, $p$, $\alpha$ and $\beta$. It is defined for positive field values $\phi \ge 0$ and we assume $p > 0$.

Expanding the logarithmic terms of the effective potential $V_4(\phi)$ in Eq. (7.173), and matching them to Eq. (7.176), one gets for $p = 4$

$$M^4 = \lambda M_{\text{Pl}}^4 \left[ 1 + 4(1 - b_1) \ln \left( \frac{\phi_0}{M_{\text{Pl}}} \right) + 8(1 + b_2) \ln^2 \left( \frac{\phi_0}{M_{\text{Pl}}} \right) \right], \quad (7.177)$$

with

$$\alpha = - \frac{4(1 - b_1) + 16(1 + b_2) \ln \left( \frac{\phi_0}{M_{\text{Pl}}} \right)}{1 + 4(1 - b_1) \ln \left( \frac{\phi_0}{M_{\text{Pl}}} \right) + 8(1 + b_2) \ln^2 \left( \frac{\phi_0}{M_{\text{Pl}}} \right)}, \quad (7.178)$$ $$\beta = \frac{8(1 + b_2)}{1 + 4(1 - b_1) \ln \left( \frac{\phi_0}{M_{\text{Pl}}} \right) + 8(1 + b_2) \ln^2 \left( \frac{\phi_0}{M_{\text{Pl}}} \right)}. \quad (7.179)$$

For the quadratic potential $V_2(\phi)$, expanding Eq. (7.175) and identifying the resulting terms with Eq. (7.176) yields for $p = 2$

$$M^4 = \lambda M_{\text{Pl}}^4 \left[ 1 + 2(1 - b_1) \ln \left( \frac{\phi_0}{M_{\text{Pl}}} \right) + 2(1 + b_2) \ln^2 \left( \frac{\phi_0}{M_{\text{Pl}}} \right) \right], \quad (7.179)$$

with

$$\alpha = - \frac{2(1 - b_1) + 4(1 + b_2) \ln \left( \frac{\phi_0}{M_{\text{Pl}}} \right)}{1 + 2(1 - b_1) \ln \left( \frac{\phi_0}{M_{\text{Pl}}} \right) + 2(1 + b_2) \ln^2 \left( \frac{\phi_0}{M_{\text{Pl}}} \right)}, \quad (7.180)$$ $$\beta = \frac{2(1 + b_2)}{1 + 2(1 - b_1) \ln \left( \frac{\phi_0}{M_{\text{Pl}}} \right) + 2(1 + b_2) \ln^2 \left( \frac{\phi_0}{M_{\text{Pl}}} \right)}. \quad (7.181)$$

Let us notice that a convenient renormalization scale for simplifying these expressions is $\phi_0 = M_{\text{Pl}}$, for which an inflection point appears at $\alpha_4 = -4$, $\beta_4 = 8$ and $\alpha_2 = -2$, $\beta_2 = 2$ (and, more generally, at $\alpha_p = -p$ and $\beta_p = p^2/2$). These expressions also show that, as soon as the inflection point is detuned, a tachyonic mass term appears and slow-roll violations are present in a very same fashion as for SFI2 (see Section 6.1).

In order for the potential of RCIPI to remain positive, as requested from its desired stability, one has to impose some restrictions on the parameters $\alpha$ and $\beta$. At large field values, the last term of Eq. (7.176) dominates and the potential is positive only for $\beta \ge 0$. Moreover, to prevent the potential to become negative in some intermediate domain, one has to impose

$$\alpha^2 \le 4\beta. \quad (7.181)$$

In the following, we study the generic potential of RCIPI under these conditions while its observable predictions can be narrowed down to the quadratic and quartic cases by using Eqs. (7.177) to (7.180). Let us however notice that the regimes for which the potential could become negative, when driven by the first-order loop corrections, have already been studied within the RCLFI scenario in Section 7.8.

### 7.12.2. Parameter space analysis

For $\alpha^2 \le 4\beta$ and $\beta \ge 0$, the potential is well-defined and positive for $\phi \ge 0$. It has been constructed to possibly have an inflection point, and moving away from this situation, one can have either a monotonic increasing potential with respect to $\phi$, or the appearance of both a local maximum and minimum. Defining

$$x \equiv \frac{\phi}{M_{\text{Pl}}}, \quad (7.182)$$
<figure><img src="image_123.png" /><figcaption>Fig. 120. Location of the inflationary regimes RCIPI1 and RCIPI2 in the parameter space $(\beta, \alpha)$ for $p = 2$. The hatched region associated with $\alpha^2 > 4\beta$ is not considered as the potential would become negative in some field domains. The red ellipse, defined by $\alpha = \pm \alpha_0$, separates the domain in which the potential is a monotonic increasing function of the field (inside) from the ones in which it develops a false vacuum (outside). The RCIPI1 regimes are hilltop models starting from a local maximum of the potential and evolving towards the origin. The RCIPI2 regime is a large field-like inflationary model, running from large to small field values. On the ellipse, the potential has a flat inflection point and only RCIPI1 is considered. This is because, since the number of e-folds diverges at the flat inflection point [see the discussion below Eq. (7.199)], inflation never ends in RCIPI2 on the ellipse.</figcaption></figure>

the potential is extremal when $V'(x) = 0$, i.e. for $x$ solution of

$$x^{p-1} [p + \alpha + (p\alpha + 2\beta) \ln(x) + p\beta \ln^2(x)] = 0. \qquad (7.183)$$

For $p > 1$, there is an obvious solution at $x = 0$, which is also the global vanishing minimum of the potential. We are rather interested in the roots of the second term, which is a quadratic equation in $\ln(x)$. The existence of these roots depends on the sign of the determinant

$$\Delta = 4\beta^2 - p^2 (4\beta - \alpha^2). \qquad (7.184)$$

The inflection point is recovered when there is only one root to this equation, namely for $\Delta = 0$ or $\alpha = \pm \alpha_0$ with

$$\alpha_0 \equiv 2\sqrt{\beta} \sqrt{1 - \frac{\beta}{p^2}}. \qquad (7.185)$$

As a result, the inflection point exists only in the domains for which $\beta \le p^2$. If $|\alpha| < \alpha_0$ then Eq. (7.184) becomes negative and the potential is a monotonic increasing function of the field values. For $|\alpha| > \alpha_0$ we have $\Delta > 0$ and Eq. (7.183) admits the two roots

$$x_{V'=0}^{\pm} = \exp \left[ \frac{-(p\alpha + 2\beta) \pm \sqrt{4\beta^2 - p^2 (4\beta - \alpha^2)}}{2p\beta} \right]. \qquad (7.186)$$

For all the other cases, associated with $\beta > p^2$, $\Delta > 0$ for all values of $\alpha$ and the potential always develops two extrema at the locations given by Eq. (7.186). Let us notice that for $\beta \le p^2$ and $\alpha = \pm \alpha_0$ the position of the infection point is also given by Eq. (7.186) in which the square root term vanishes and reads

$$x_0 = \exp \left( - \frac{p\alpha + 2\beta}{2p\beta} \right). \qquad (7.187)$$

As soon as Eq. (7.186) has two distinct solutions, the potential of RCIPI develops a false vacuum at $x = x_{V'=0}^+$ in which inflation would become eternal. Within slow-roll, and without resorting to an additional mechanism ending such a phase of inflation (which would make RCIPI a four-parameter model), this implies that slow-roll inflation may gracefully end only if it moves away from this false vacuum by going towards the global minimum at $x = 0$. We refer to this regime as RCIPI1 and it exists only in the domain $x < x_{V'=0}^-$. As such, it requires either $\beta > p^2$, or $|\alpha| \ge \alpha_0$ when $\beta \le p^2$. RCIPI1 also encompasses the case of a flat inflection point. Let us notice that non-slow roll solutions may exist, as for instance if the field acquires enough kinetic energy in a previous inflationary phase to climb out of the false vacuum, but these situations necessarily violate slow roll. The other slow-roll inflationary regime corresponds to the situation in which the potential is a monotonic increasing function of the field.
<figure><img src="image_124.png" /><figcaption>Fig. 121. Radiatively Corrected Inflection Point Inflation for $p = 2$ in the regime RCIPI1. Inflation proceeds at decreasing field values from an inflection point (black curves), which can be detuned to a hilltop (blue curves). Top panels: the potential and its logarithm. Bottom left panel: slow-roll parameter $\epsilon_1$. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid curves) and $\epsilon_3$ (dotted curves). The opposite of the negative values of $\epsilon_2$ and $\epsilon_3$ have been represented in red for $\alpha = -2$, $\beta = 2$ and in green for the detuned case with $\alpha = -2.5$, $\beta = 2$. Notice that as soon as the inflection point becomes a hilltop, $\epsilon_2$ can become large due to the appearance of a tachyonic mass term (see also SFI2, Section 6.1).</figcaption></figure>

This happens only for $\beta \le p^2$ with $|\alpha| < \alpha_0$ and inflation proceeds from large to small field values. This regime will be referred to as RCIPI2. The location of these two regimes in the parameter space $(\beta, \alpha)$ have been represented in Fig. 120 for $p = 2$.

### 7.12.3. Slow-roll analysis

The first Hubble-flow function in the slow-roll approximation, stemming from the potential of Eq. (7.176), reads

$$ \epsilon_1 = \frac{[p + \alpha + (p\alpha + 2\beta) \ln(x) + p\beta \ln^2(x)]^2}{2x^2 [1 + \alpha \ln(x) + \beta \ln^2(x)]^2}. \qquad (7.188) $$

The second and third Hubble-flow functions are given by

$$ \epsilon_2 = \frac{2}{x^2} \left\{ p + \frac{\alpha^2 - 4\beta}{[1 + \alpha \ln(x) + \beta \ln^2(x)]^2} + \frac{\alpha + 2\beta + 2\beta \ln(x)}{1 + \alpha \ln(x) + \beta \ln^2(x)} \right\}, \qquad (7.189) $$

and

$$ \begin{aligned} \epsilon_3 = & \frac{p + \alpha + \ln(x) [p\alpha + 2\beta + p\beta \ln(x)]}{x^2 \{1 + \ln(x) [\alpha + \beta \ln(x)]\}^4} \\ & \times \left\{ \begin{aligned} & p + \frac{\alpha^2 - 4\beta}{[1 + \alpha \ln(x) + \beta \ln^2(x)]^2} + \frac{\alpha + 2\beta + 2\beta \ln(x)}{1 + \alpha \ln(x) + \beta \ln^2(x)} \\ & - 2\beta [1 + \alpha \ln(x) + \beta \ln^2(x)]^2 + 2(\alpha^2 - 4\beta) [1 + \alpha \ln(x) + \beta \ln^2(x)] \\ & + 2[\alpha + 2\beta + 2\beta \ln(x)] [1 + \alpha \ln(x) + \beta \ln^2(x)]^2 + 2p [1 + \alpha \ln(x) + \beta \ln^2(x)]^3 \end{aligned} \right\}. \end{aligned} \qquad (7.190) $$
<figure><img src="image_125.png" /><figcaption>Fig. 122. Radiatively Corrected Inflection Point Inflation for $p = 2$ in the regime RCIPI2. The potential is a monotonic increasing function of the field values and inflation may proceed at large field values. Top panels: the potential and its logarithm. Bottom left panel: slow-roll parameter $\epsilon_1$. Bottom right panel: slow-roll parameters $\epsilon_2$ (solid curves) and $\epsilon_3$ (dotted curves). The opposite of the negative values of $\epsilon_2$ and $\epsilon_3$ have been represented in red. Notice that inflation ends with “hiccups”, slow-roll violations occur but remain confined at the end of inflation.</figcaption></figure>

In the limit $x \to 0$ one has $\epsilon_1(x) \approx p^2 \ln^2(x)/(2x^2)$, which is divergent, and this ensures that inflation always gracefully ends when approaching the global minimum of the potential at $x = 0$. The field value $x_{end}$ at which this occurs is solution of the equation $\epsilon_1(x) = 1$, i.e.

$$p + \alpha + (p\alpha + 2\beta) \ln(x) + p\beta \ln^2(x) = \pm\sqrt{2x} [1 + \alpha \ln(x) + \beta \ln^2(x)]. \quad (7.191)$$

It does not admit analytical solution and has to be solved numerically. Moreover it has multiple roots in domains that can be determined by studying the sign of $\epsilon_2(x)$. This one vanishes for

$$p [1 + \alpha \ln(x) + \beta \ln^2(x)]^2 + [\alpha + 2\beta + 2\beta \ln(x)] [1 + \alpha \ln(x) + \beta \ln^2(x)] + \alpha^2 - 4\beta = 0. \quad (7.192)$$

This is a quartic equation in $\ln(x)$ and it admits, at maximum, four roots that can be determined analytically and can be found in the ASPIC library. When the potential develops a false vacuum one of these roots lies between the locations of the two potential extrema because, there, $\epsilon_1$ vanishes so it must have a local maximal (where $\epsilon_2$ vanishes) in between. The other roots correspond to other local extrema of $\epsilon_1(x)$, which can potentially exceed unity. However, RCIPI1 proceeds at decreasing field values, always with $x < x_{\bar{V}}=0$, and one can show that it never encounters one of the local extrema of $\epsilon_1$. Therefore, only the smallest of all roots of Eq. (7.191) gives $x_{end}$, the field value at which RCIPI1 ends. Concerning RCIPI2, the situation is more complex. The potential is a monotonic increasing function of $x$ and RCIPI2 inflation proceeds at decreasing field values, from large to small values of $x$. Although there is no inflection point, the potential exhibits a change of convexity and $\epsilon_2$ may still vanish before the end of inflation. Because $\epsilon_1$ then develops a local maximum, possibly exceeding unity, inflation can be potentially halted before restarting for a few e-folds to definitely end at $x_{end}$, this one being still defined as the lowest root of Eq. (7.191). These “hiccups” are necessarily associated with slow-roll violations, but, being located close to the end of inflation, they are not directly observable in the CMB. As it is the case for SAIII3 (see Section 7.7), they might however be of some interest for the formation of primordial black holes.

The potential, and the Hubble-flow functions, in the RCIPI1 regimes have been represented in Fig. 121 in the case of an inflection point, for $p = 2$, $\alpha = -2$, $\beta = 2$, as well as when it is strongly detuned to a hilltop ($p = 2$, $\alpha = -2.5$ and $\beta = 2$). The potential and Hubble flow functions for RCIPI2, when the potential is monotonic, are represented in Fig. 122. As can be seen on the lower left panel of this figure, $\epsilon_1$ may transiently exceed unity before inflation definitely ends.