On the $\alpha$-attractor T-models

An interesting phenomenon occurs near the end of inflation and we illustrate it below for the case $p=2$. By looking at the third panel in Fig. 2 (in Fig. 2 we plot $-\eta$) we see that the curve $-\eta$ just touches the horizontal line at 1 for $\lambda=\lambda_{l}$. This means that the condition $\eta=-1$ has been marginally satisfied signaling the end of slow-roll. In the Fig. 3 this detail is amplified together with a second curve for $-\eta$ corresponding to a value slightly greater than $\lambda_{l}$ ($\lambda=0.966$). We can see that at the point $a=(1.0374,1)$ the condition $-\eta=1$ is saturated but the curve enters again the region $-\eta<1$ at $b=(1.0245,1)$ when $\epsilon$ is still less than 1 i.e., slow-roll resets to eventually end inflation at $c=(0.8956,1)$ when $\epsilon$ is finally equal to one.

In Fig. 3 we also show the potential for $\lambda\geq\lambda_{l}$ during this phenomenon: the small box under the number 1 in Fig. 3 shows the region during which slow-roll has ended and in the square above 2 the region where slow-roll is restored to eventually end permanently in the l.h.s. corner of the square corresponding to $\epsilon=1$. The total number of e-folds between points $b$ and $c$ is $0.104$ corresponding to an $11\%$ increase of the size of the universe in $b$. For slightly higher values of $\lambda$ slow-roll will be interrupted for a longer period. While the number of e-folds after slow-roll is restored is negligible, the phenomenon itself is interesting and worth reporting.

The equations for $N_{k}$ and $N_{e}$, Eqs. (2.6) and (2.7) respectively, are obtained by substituting $\phi_{k}$ and $\phi_{e\eta}$ from (2.1) and (2.2). Together with the large $\lambda$ expansion they are given by

and

We see that both terms are $\lambda$-independent for infinitely large $\lambda$, the term associated with the end of slow-roll is negligible contributing with less than 1 e-fold. We will see in Subsection 2.3 that when $\lambda$ is small this is not the case and $N_{e}$ is important.

Using Eqs. (2.8) and (2.9), the number of e-folds $N_{ke}$ can be written in terms of the observable $n_{s}$ as follows

where $R_{1}=\sqrt{1+4p^{2}\lambda^{2}(\lambda^{2}-1)}$ and the subindex $\eta$ appears here to remind us that $N_{ke}$ is obtained from the solution to the $\eta=-1$ condition but then it is dropped (as well as the argument) from the following expressions. For large $\lambda$, $N_{ke}$ has the following expansion

Thus, $N_{ke}$ is $p$-independent for infinitely large $\lambda$. We can solve Eq. (2.10) for $n_{s}$ in terms of $N_{ke}$ (see Fig. 4)

Thus, the leading order term in the large-$N_{ke}$ expansion of the scalar spectral index $n_{s}$ is also p-independent

for the mean value $n_{s}=0.9649$ [16] Eq. (2.13) implies $N_{ke}\approx 57$.

One can notice that $N_{ke}$ as given by Eq. (2.10) above depends only on the observable $n_{s}$, this is so because $\phi_{k}$ was obtained by solving the equation (1.4) written in the form $\delta_{n_{s}}+2\eta-6\epsilon=0$. We could also obtain $\phi_{k}$ by solving Eq. (1.5) or $r=16\epsilon$ and write an equivalent expression for $N_{ke}$ involving only the observable $r$ with the result

which together with Eq. (2.2) implies that the number of e-folds can now be written as

Solving for $r$

as shown by Fig. 4 for the $p=2$ case and for various values of $N_{ke}$. In the large-$N_{ke}$ limit $r$ is given by

The end of inflation is given by the solution to the condition $\epsilon=1$

while $\phi_{k}$ in terms of $n_{s}$ is given, as before, by Eq. (2.1).

The equations for $N_{k}$ and $N_{e}$, Eqs. (2.6) and (2.7) respectively, and their small $\lambda$ expansion, are given by

and

For small $\lambda$ the first term in the expansion of $N_{k}$ grows large thus, (contrary to the large $\lambda$ case where $N_{e}$ is less than 1) here the end of inflation is important and necessary to cancel the leading term in $N_{k}$ so that the number of e-folds of inflation goes like $N_{ke}\approx(p+2)/2\delta_{n_{s}}$, to first approximation.

The number of e-folds Eq. (2.5) is

where $R_{2}=\sqrt{1+2p^{2}\lambda^{2}}$ and, in analogy with Eq. (2.10), the subindex $\epsilon$ appears here to remind us that $N_{ke}$ is obtained from the solution to the $\epsilon=1$ condition but then it is dropped from the following expressions. From Eq. (2.21) we can solve for $n_{s}$ (see Fig. 5)

for large $N_{ke}$

To obtain $N_{ke}$ in terms of $r$ we proceed in a similar way as in Eq. (2.16). From Eqs. (2.5), (2.14) and (2.3) we get

solving for $r$ (see Fig. 5)

with large $N_{ke}$ expansion

The previous results for $n_{s}$ in the large $N_{ke}$ expansion Eqs. (2.13) and (2.23) give a leading term $n_{s}=1-\frac{2}{N_{ke}}+\cdot\cdot\cdot$. Also Eqs. (2.17) and (2.26) give a leading term $r=\frac{2}{\lambda^{2}N_{ke}^{2}}+\cdot\cdot\cdot$ for the large $N_{ke}$ expansion of $r$.

We can eliminate the parameter $\lambda$ in all the previous equations in such a way that only the observables $n_{s}$ and $r$ appear. First we separately discuss the $p=2$ case by substituting $\lambda$ from Eq. (3.1) into the expressions for $N_{ke}$ given by Eqs. (2.10), (2.15), (2.21), (2.24) and solving for $r$ obtaining the following $two$ independent solutions: one for the case $\lambda\geq\lambda_{l}$ where the end of slow-roll is given by the condition $\eta=-1$ and labeled by the symbol $\eta$ and one for the case $\lambda<\lambda_{l}$ where the end of inflation is given by the condition $\epsilon=1$, labeled by the symbol $\epsilon$

The approximations for large $N_{ke}$ are (see Figs. 7 and 8 for the $p=2$ case).

For the $p=2$ case, the bound $\lambda\geq\lambda_{l}=(2+\sqrt{3})^{1/2}/2\approx 0.9659$ is equivalent to a bound on $r$ as $r<0.00082$ for $n_{s}=0.9607$. Thus, the expression for $r_{\eta}$ is valid for very small $r$ only. In this regime the solution $r_{\epsilon}$ (which should be used for $\lambda<\lambda_{l}$ or $r>r_{l}\approx 0.00051$ at $n_{s}=0.9691$) is very close to the $r_{\eta}$ solution being possible to use it also for the very small $r$ regime (large $\lambda$ regime) with negligible error. The case $p\neq 2$ is not very different thus, in what follows, we study only the $r_{\epsilon}$ solution for all possible values of $r$. In Fig. 7 we show the solutions (3.6) and (3.7) for the $p=2$ case reproducing in Fig. 8 the numerical solution which appear in Fig. 8 of the Planck 2018 Collaboration [16] (shown here as Fig. 9). This solution contains the monomial solution for $r(n_{s})$ of the $\phi^{2}$ model. In Fig. 10 we plot several other cases of Eq. (3.11) by giving $p$ the values $p=2,4/3,1$ and $2/3$ containing all the monomial solutions shown in Fig. 9. These solutions are contained exactly in the $\alpha$-attractor models as shown analytically in the following subsection.

For monomials of the form $V_{mon}=\frac{1}{2}m^{4-p}\phi^{p}$ we get $\phi_{k}=\frac{2\sqrt{2}\,p}{\sqrt{r}}$ and $\phi_{e}=\frac{p}{\sqrt{2}}$ from where it follows that

also

Eqs. (3.16) and (3.17) follow from the condition $\epsilon=1$, this condition ends inflation whenever $2/3\leq p\leq 2$. Por $p$ outside this range slow-roll is terminated by $\eta=-1$ for $p<2/3$ in which case $r=8p/(2N_{ke}+p-p^{2})$ or $\eta=+1$ for $p>2$ in this case $r=8p/(2N_{ke}+p-1)$. In any case for large $N_{ke}$ and $p<<N_{ke}$ they all have the same limit $\sim 4p/N_{ke}$. Thus for $p=2$ and $N_{ke}=50$, $(n_{s},r)=(\frac{97}{101},\frac{16}{101})\approx(0.9604,0.1584)$ and so on. In this way we can calculate all the points appearing as circles in Fig. 9. The point $(0.9604,0.1584)$, for example, is reached by the solution $r_{\epsilon}(p=2)$ Eq. (3.7) for $N_{ke}=50$ as shown by the left line in Fig. 8 and similarly for the $N_{ke}=60$ line on the r.h.s. For $p\neq 2$ Eq. (3.11) applies. In this way we draw Fig. 10 connecting the $\alpha$-attractor T-models with all the monomial as shown. Analytically, we can see this as follows: for the $p=2$ case we sustitute Eq. (3.17) in Eq. (3.7) and we get $r_{\epsilon}(p=2)=16/(2N_{ke}+1)$ which is Eq. (3.16) (with $p=2$). In general (for $p\neq 2$) substituting Eq. (3.17) in Eq. (3.11) gives

which is Eq. (3.16), exactly. Thus, all the monomial models are contained as particular cases by the $\alpha$-attractor T-models. This is illustrated by Fig. 10 only for the monomials shown in Fig. 8 (reproduced here as Fig. 9) of the Planck 2018 Collaboration [16] . Even more to the point: the monomials are the ending points of the attractors [6]. This can be seen as follows: if we substitute Eq. (3.17) in the equation defining $\lambda$ (Eq.  (3.1)) we find that $\lambda=0$ becoming imaginary after this point. To have a clear understanding of this phenomenon let us substitute equation (3.1) for $\lambda$ in Eq. (1.1); the resulting potential can then be written as

where $V_{0}$ has been calculated from Eq. (1.6). If we study the potential (3.19) in the limit where $\delta_{n_{s}}\rightarrow\frac{p+2}{8p}r$ we find that it reduces to the following expression

One can easily check that the potential (3.20) corresponds exactly to monomials of the form $V_{mon}=\frac{1}{2}m^{4-p}\phi^{p}$. Thus, the fact that we reach the monomials in Fig. 10 is now easily understood because our original potential transitions to the monomials potential when $\lambda=0$, that is, when $r=8p\delta_{n_{s}}/(p+2)$ which is precisely the relation between $r$ and $n_{s}$ for the monomials potential. The potential (3.19) is shown in Fig. 11 as a function of $\phi$ for $p=2$ and the mean value $n_{s}=0.9649$ [16] (or $\delta_{n_{s}}=0.0351$) for various values of $r$ reaching $r=8p\delta_{n_{s}}/(p+2)$ (dashed curve) signaling the transition of the $\tanh^{p}(\lambda\phi)$ potential (3.19) to the monomial $\phi^{p}$ potential of Eq. (3.20). The expresion $r=8p\delta_{n_{s}}/(p+2)$ makes $\lambda=0$ but $\lambda=0$ in what we saw before takes us to the ends of the curves in Fig. 10 reaching the predictions in the $n_{s}$-$r$ plane for the monomials potential. A similar study can be done with $\alpha$-attractor E-models defined by the potential (1.2) which generalize the Starobinsky potential in the Einstein frame. In this case the generalized potential written in terms of $n_{s}$ and $r$