Sequestered Inflation

Consider a model of $n$ chiral superfields $T_{i}$ with a superpotential $W^{(I)}(T_{i})$ (the superpotential at Step I) and with a Kähler potential given by some real holomorphic function $K^{(I)}(T_{i},\overline{T}_{i})$. If at some point $T_{i}=t_{i}+{\rm i}a_{i}$ the superpotential $W^{(I)}(T_{i})$ and all of its first derivatives vanish,

this state corresponds to a stable supersymmetric Minkowski vacuum with vanishing scalar potential $V^{(I)}(T_{i})=0$. This property does not depend on the choice of the Kähler potential.

We will be interested in superpotentials such that the conditions (1) are satisfied not only at a single point, but along some flat directions. The potential may have several different flat directions with $V(T_{i})=0$ Gunaydin:2020ric , but the simplest possibility discussed in all examples given in this paper is that the flat direction appears when all fields $T_{i}$ are equal to each other, and real, $T_{i}=t$, $a_{i}=0$.

In this section we will illustrate our general approach using a theory of a single field $T$ with superpotential $W^{(I)}(T)$, for several different choices of the Kähler potential $K^{(I)}(T,\overline{T})$. One should keep in mind that in the theory of a single field supersymmetric flat directions are possible only if $W^{(I)}(T)$ vanishes for all $T$, i.e., if $W^{(I)}(T)=0$. Nevertheless we will keep $W^{(I)}(T)$ in our equations because most of the results to be obtained in this section can be easily generalized for models with many fields $T_{i}$, where flat directions satisfying eqs. (1) may appear for non-trivial superpotentials $W^{(I)}(T_{i})$.

Now we will study a more advanced model, with two fields $T_{1}$ and $T_{2}$, which have a nontrivial superpotential already at Step I:

There is a supersymmetric Minkowski vacuum at $T_{1}=T_{2}$. The canonically normalized variables $T_{i}=\frac{1}{\sqrt{2}}(\phi_{i}+{\rm i}a_{i})$ can be rewritten in terms of canonically normalized mass eigenstates as $\phi=\frac{1}{\sqrt{2}}(\phi_{1}+\phi_{2})$, $a=\frac{1}{\sqrt{2}}(a_{1}+a_{2})$, $\phi_{M}=\frac{1}{\sqrt{2}}(\phi_{1}-\phi_{2})$ and $a_{M}=\frac{1}{\sqrt{2}}(a_{1}-a_{2})$. The mass eigenvalues of $\phi,a$ at $T_{1}=T_{2}=0$ are $0$ and the mass eigenvalues of the massive modes $\phi_{M},a_{M}$ at $T_{1}=T_{2}=0$ are $m^{2}=16M^{2}$. This mass is below the Planck mass $M_{Pl}=1$ for $M<{1\over 4}$.

The absence of tensor modes at the level $r\lesssim 0.05$ implies that the inflaton potential $V$ at the last 60 e-foldings of inflation should be smaller than $2\times 10^{-9}$ Linde:2016hbb . This means that if the field $T_{1}+T_{2}=\phi+{\rm i}a$ deviates from its equilibrium at $T_{1}+T_{2}=0$ by $O(1)$, its potential can become many orders of magnitude higher than the inflationary potential $V$. Thus it is necessary to check whether the scenario outlined in the previous section may remain intact despite the introduction of the large superpotential $W^{(I)}=M(T_{1}-T_{2})^{2}$.

Just as in the previous section, we introduce a nilpotent field $X$ and modify the Kähler potential and the superpotential

Fortunately, the Kähler potential $K^{(I)}$ vanishes along the flat direction $T_{1}=\overline{T}_{1}=t$, $T_{2}=\overline{T}_{2}=t$ along which the superpotential $W^{(I)}$ also vanishes. As a result, one can show that eq. (6) derived for the model with $W^{(I)}=0$ remains valid as well. At this time, instead of a simple quadratic scalar potential we will take a more complicated one

At $T_{M}=0$ and $T_{i}=\overline{T}_{i}=t$ the potential depends only on the canonical inflaton field $T=\frac{1}{\sqrt{2}}\phi\ (=t)$ and (for $\Lambda=0$) is given by

This is the plateau potential of the D-brane inflation model described recently in Kallosh:2018zsi . It was called KKLTI in the Encyclopedia Inflationaris Martin:2013tda because of its possible relation to the KKLT mechanism of vacuum stabilization and inflation in string theory Kachru:2003aw ; Kachru:2003sx . This model is in good agreement with observational data. Its prediction

is very close to the prediction of the $\alpha$-attractors $n_{s}=1-{2\over N_{e}}$. This model and $\alpha$-attractors almost completely cover the area in the $(n_{s},r)$ plane favored by Planck2018 Kallosh:2019hzo , see Fig. 2.

During inflation the mass squared of the heavy fields $\phi_{M},a_{M}$ remain approximately the same as it was at Step I, so it can take any value up to the Planck mass. This means that the fields $\phi_{M}$ and $a_{M}$ are strongly stabilized at $\phi_{M}=a_{M}=0$. The mass squared of the imaginary component $a$ of the inflaton field field $T={1\over\sqrt{2}}(\phi+{\rm i}a)$ is positive. Thus, the inflaton trajectory $T={1\over\sqrt{2}}\phi$, $\phi_{M}=a_{M}=a=0$ is stable. Importantly, the potential masses of the light fields $\phi$ and $a$ and the inflaton potential (32) do not depend on $M$. This ensures sequestering of the inflaton potential from the high energy scale physics related to the superpotential $W^{(I)}=M(T_{1}-T_{2})^{2}$.

The development of inflation depends on our choice of potential $V(T_{i},\overline{T}_{i})$ in eq. (31). Note that one can add to this potential various terms stabilizing the inflaton trajectory without affecting the inflaton potential $V(\phi)$. For example, any term proportional to $-(T_{i}-\overline{T}_{i})^{2}$ vanishes along the inflaton trajectory, but it allows to increase the mass squared of the axion $a$. We also note that the mass of $T_{M}$ is not much affected by the inflaton dynamics in this model because the Kähler potential is independent of the inflaton $\phi$. This might not be the case in more general setups, and we will show an example in the next section.

The next model describes three fields $T_{1}$, $T_{2}$ and $T_{3}$:

There is a supersymmetric Minkowski flat direction $T_{1}=T_{2}=T_{3}$, corresponding to a massless field

There are also two heavy fields $T_{M_{1}}$, $T_{M_{2}}$, which correspond to some other combinations of the fields $T_{1},T_{2},T_{3}$.

The fields $T_{i}$ can be represented as $T_{i}=e^{-\sqrt{2}\,\phi_{i}}\,(1+{\rm i}\sqrt{2}a_{i})$, where the fields $\phi_{i}$ and the fields $a_{i}$ are canonical in the small $a_{i}$ limit. When $\phi_{i}={1\over\sqrt{3}}\phi$ and $a_{i}=0$ the mass squared eigenvalues of the canonically normalized fields $\phi_{i}$ are:

The masses of the heavy fields change depending on $\phi$. Therefore, we must check at Step II whether the heavy fields remain heavy during inflation and decouple from the inflationary dynamics.

The Kähler  potential and superpotential are

Our choice of the potential in equation (38) with $T$ defined in equation (35) is

In the first approximation, let us follow the supersymmetric trajectory $T_{1}=T_{2}=T_{3}=T$ on which the Kähler and superpotential are reduced to

At $T_{1}=T_{2}=T_{3}$ and $T_{i}=\overline{T}_{i}$ the potential depends on the field $T=t=e^{-\sqrt{2\over 3}\phi}$ and is given by

Here with $\Lambda=F_{X}^{2}-3W_{0}^{2}>0$, inflation ends at $t=1,\phi=0$ in a de Sitter vacuum. This is the $\alpha=1$ attractor model, whose bosonic part coincides with that of the Starobinsky model.

In order to make sure that the sequestering mechanism works consistently, we need to check the behaviour of the two heavy fields during inflation. The masses of them calculated by using eqs. (38) are presented in Fig. 3, for the choice of parameters $W_{0}=F_{X}/\sqrt{3}=10^{-5},\ \mu=10^{-5}$. The axion masses of these heavy superfields are only slightly different from the saxion masses plotted here, this difference is due to the small values of the parameters $W_{0}=F_{X}/\sqrt{3}=10^{-5},\ \mu=10^{-5}$.

The mass squared of the inflaton is negative at the plateau of the potential and flips sign at $\phi\approx 0.8$, at which the absolute value of the inflaton mass squared shows a singular behavior in Fig. 4.

The heavy masses of the directions orthogonal to the inflationary trajectory ensure the validity of the sequestering mechanism, and we can safely use the effective description given by eq. (41). It is also worth emphasizing that the effective Kähler potential acquires the different Kähler  curvature radius, $\alpha_{1,2,3}=\frac{1}{3}\to\alpha_{T}=1$. Thus, the sequestering not only removes the heavy degrees of freedom, but also affects the inflationary dynamics.