# The Early Universe with High-Scale Supersymmetry

Sibo Zheng<EMAIL_ADDRESS>Department of Physics, Chongqing
University, Chongqing 401331, P.R. China

###### Abstract

A small tensor-to-scalar ratio $r$ may lead to distinctive phenomenology of
high-scale supersymmetry. Assuming the same origin of SUSY breaking between
the inflation and visible sector, we show model independent features. The
simplest hybrid inflation, together with a new linear term for the inflaton
field which is induced by large gravitino mass, is shown to be consistent with
all experimental data for $r$ of order $10^{-5}$. For superpartner masses far
above the weak scale we find that the reheating temperature after inflation
might be beneath the value required by thermal leptogenesis if the inflaton
decays to its products perturbatively, but above it if non-perturbatively
instead. Remarkably, the gravitino overproduction can be evaded in such high-
scale supersymmetry because of the kinematically blocking effect.

##  I Introduction

After the discovery Higgs  of standard model (SM) Higgs boson at the Large
Hadron collider (LHC), low-scale supersymmetry (SUSY) susy  which is favored
by the naturalness argument naturalness  has been extensively explored. These
studies show the difficulties in the both theoretic explanation of 125 GeV
Higgs mass and experimental fits to the LHC data. The second run of LHC will
shed light on the prospect of such natural SUSY models. Given above
consideration some efforts have been devoted to the study of high-scale SUSY.

Even though high-scale SUSY cannot be detected at the $14$ TeV LHC, they can
be still studied via their effects on the evolution of early universe.
Measurement on the tensor-to-scalar ratio $r$ via experiments such as WAMP,
Plank and BICEP that are devoted to measure Cosmic Microwave Background (CMB)
temperature anisotropy and polarization during inflation, may probe high-scale
SUSY with mass spectrum far above the weak scale. The measured value of $r$
reported by the Plank Collaboration is of order $r<0.11$ at 95 % CL 1502.02114
; 1303.5076 ; 1303.5082 , from which the energy scale of inflation can be
directly inferred. Since the energy scale of inflation is proportional to
$r^{1/4}$, it is only mildly sensitive to $r$. So the study of high-scale SUSY
remains well motivated as long as $r$ is not extremely small.

In this paper, we consider inflationary models with $r$ far below the Plank
bound value $r_{c}=0.11$. The motivation is mainly based on two facts. At
first, the stability of the SM electroweak vacuum requires $H\leq
0.04~{}h_{\text{max}}$ 1505.04825 , where $h_{\text{max}}$ refers to the value
$h$ at which the Higgs potential is maximal. For the central value of top
quark pole mass $h_{\text{max}}\sim 10^{10}$ GeV, which implies that $H$
should be smaller than Plank bound value $H_{\text{c}}\sim 10^{16}$ GeV
corresponding to $r_{c}$ 1504.08093 . In this sense small $r<<r_{c}$ is more
favored to guarantee the electroweak vacuum stable against quantum fluctuation
during inflationary epoch. Secondly, for $r<<r_{c}$, it can still generate
SUSY mass spectrum large enough to escape the LHC constraints.

For simplicity we adopt the assumption that the inflation and visible (namely
the minimal supersymmetric standard model (MSSM )) sector share the same
origin of SUSY breaking. This assumption is rational, as it can be realized in
model building. Moreover, it allows us to discuss reheating in the early
universe after inflation, once the SUSY mass spectrum and the inflaton decay
are identified explicitly.

The paper is organized as follows. In section 2 we re-analyze the model
independent consequences from above assumption within the range $r<<r_{c}$. In
section 3, we consider hybrid inflation as an example in the course of high-
scale SUSY breaking 111For earlier attempts to address this issue, see, .e.g,
1009.5340 ; 1211.0994 .. We will show that a new linear term for inflaton
field with a large coefficient proportional to $m_{3/2}$ affects the inflation
significantly, and the simplest hybrid inflation is consistent with $r$ of
order $10^{-5}$.

In the second part of this paper we discuss the reheating in the early
universe after inflation in section 4. In particular, reheating temperature
$T_{R}$ after inflation is estimated for superpartner mass spectrum $m_{0}$
above $\mathcal{O}(100)$ TeV. We find that $T_{R}$ might be beneath the value
$\sim 10^{9}$ GeV required by thermal leptogenesis if inflaton decays to its
products perturbatively, but above it if non-perturbatively instead. The
gravitino overproduction in conventional high-scale SUSY can be easily evaded
because of kinematically blocking effect. Finally we conclude in section 5.

##  II Implications of the value of $r$ to Inflation

In this section we revise the model independent implications (together with
Plank and 9-year WAMP data) to single-field inflation for $r<<r_{c}$. These
results provide useful information on the model building of inflation, as a
reliable inflation model should at least explain observable quantities as what
follows.

$(1)$, First of all the scale of energy density during inflation is directly
related to $r$ as,

$\displaystyle{}V^{1/4}=\left(24\pi^{2}M^{4}_{P}\epsilon A_{s}\right)^{1/4}.$
(1)

where $A_{s}$ is the amplitude of the power spectrum of the curvature
perturbation and $\epsilon=r/16$ in the context of single-field inflation.
Recall that $A_{s}^{1/2}=H^{2}(\phi_{*})/2\pi\dot{\phi}$, where $\phi_{*}$ is
the value of $\phi$ when wavenumber $k_{*}=0.05$ Mpc-1 crossed outside the
horizon. Substituting the reported value $A^{1/2}_{S}\simeq 3.089\times
10^{-5}$ by Plank Collaboration 1303.5076  into Eq.(1) gives rises to,

$\displaystyle{}V^{1/4}\simeq 2\times
10^{16}\cdot\left(\frac{r}{0.20}\right)^{1/4}\text{GeV}.$ (2)

Eq.(2) is valid independent of the explicit form of inflation potential. If
the MSSM and inflation share the same origin of SUSY breaking, as we have
assumed in this paper, the high-scale SUSY breaking scale $\sqrt{F}$ in
particle physics will be order $\simeq V^{1/4}$. For example, $\sqrt{F}$ is of
order $\sim 10^{16}$ GeV for $r\sim 0.1$, and slightly reduced to be of order
$\sim 10^{15}$ GeV for $r\sim 10^{-5}$.

$(2)$, In the context of slow roll inflation the spectral index $n_{s}$ (for
scalar) and $n_{t}$ (for tensor) are given by,

$\displaystyle{}n_{s}-1\simeq
2\eta-6\epsilon,~{}~{}~{}~{}~{}~{}n_{t}\simeq-2\epsilon,$ (3)

respectively. Here $\epsilon=\frac{M^{2}_{P}}{2}\left(V_{,\phi}/V\right)^{2}$
and $\eta=M^{2}_{P}V_{,\phi\phi}/V$, with subscript denoting derivative of $V$
over $\phi$. The combination of Plank and 9-year WAMP data measures the value
of $n_{s}$ in high precision 1303.5082 ,

$\displaystyle{}n_{s}=0.9603\pm 0.0073,$ (4)

For $r<<r_{c}$ one finds that $\eta\simeq-0.02$. This tight bound is crucial
to constrain inflation model.

$(3)$, The gravitino mass $m_{3/2}$ can be determined. The constant
superpotential $W_{0}=m_{3/2}M^{2}_{P}$, which is required to cancel out
positive $F^{2}$ term in the potential so as to explain the smallness of cc,
gives rise to

$\displaystyle{}m_{3/2}=\frac{F}{\sqrt{3}M_{P}}$ (5)

$(4)$, Finally the number of e-fold that $k_{*}$ undergoes during inflation is
given by,

$\displaystyle{}N\simeq\int^{\phi_{in}}_{\phi_{end}}\frac{d\phi}{M^{2}_{P}}\frac{V(\phi)}{V_{,}(\phi)}\simeq\int^{x_{in}}_{x_{end}}\frac{dx}{\sqrt{2\epsilon(x)}},$
(6)

where $x=\phi/M_{P}$. Subscript “in" and “end" corresponds to initial and end
value of $x$ during inflation, respectively. For realistic inflation models,
$N$ is bounded as $50\leq N\leq 60$. If $\epsilon$ doesn’t change
significantly during inflation, Eq.(6) can be expressed as
$\Delta\phi/M_{P}\simeq\sqrt{2\epsilon}N\simeq\sqrt{\frac{r}{8}}N$. This is
known as Lyth bound Lythbound , which shows the need of small field inflation
for $r<<r_{c}$.

##  III The Simplest Hybrid Inflation

The section is devoted to the study of inflation building in the course of
high-scale SUSY. We take the simplest hybrid inflation as an explicit
illustration. We will show that a new linear term due to the assumption
adopted in this paper significantly affects the choice on initial condition.
Also this assumption introduces new constraints on parameters in the model,
which make the simplest hybrid inflation only possible with $r$ of order
$10^{-5}$.

###  III.1 Scalar Potential

The scalar potential in hybrid inflation is constructed from superpotential
$W$,

$\displaystyle{}W=\kappa\Phi(\bar{\Psi}\Psi-M^{2}),$ (7)

and Kahler potential $K$,

$\displaystyle{}K=\mid\Phi\mid^{2}+\mid\Psi\mid^{2}+\mid\bar{\Psi}\mid^{2}+k_{1}\frac{\mid\Phi\mid^{4}}{M^{2}_{P}}.$
(8)

Here $\Phi$ denotes the inflaton superfield, with its lowest component
inflaton field $\phi$. $\Phi$ is a singlet of standard model gauge groups
$G=SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$. $\Psi$ and $\bar{\Psi}$ denote
waterfall superfields which are in the bi-fundamental representation of $G$
222 Alternatively, $G$ can be extended to include a local $U(1)_{B-L}$
symmetry so as to explain leptogenesis.. The Kahler potential in Eq.(8) takes
into account the non-canonical term, with $k_{1}$ a real coefficient. The non-
canonical $k_{1}$ term provides the inflaton mass term. $M_{P}=2.4\times
10^{18}$ GeV is the reduced Plank mass, while $M$ is assumed to be far below
$M_{P}$.

Substituting Eq.(7) and Eq.(8) into the SUGRA potential

$\displaystyle{}V=e^{K/M^{2}_{P}}\left[K^{i\bar{j}}D_{i}WD_{\bar{j}}W-3\frac{\mid
W\mid^{2}}{M^{2}_{P}}\right],$ (9)

one obtains the scalar potential of hybrid inflation. Here
$K^{i\bar{j}}=K^{-1}_{i\bar{j}}$ is the Kahler metric and
$D_{i}W=\partial_{i}W+K_{i}W/M^{2}_{P}$. It is well known that the history of
inflation can be naturally divided into two periods. In the first one
inflation usually starts from an initial value of order $M_{P}$ towards to
$\phi_{c}$ which is a critical value separating the two periods. The vacuum in
the first period corresponds to $\Psi=\bar{\Psi}=0$, from which the energy
density reads from Eq.(9) 1007.5152 ,

$\displaystyle{}V=\frac{1}{2}m_{\phi}^{2}\phi^{2}+\kappa^{2}M^{4}\left[1+\gamma\frac{\phi^{4}}{8M^{4}_{P}}+\frac{\kappa^{2}}{16\pi^{2}}\ln\frac{\kappa^{2}\phi^{2}}{2\Lambda^{2}}\right]+2\sqrt{2}\kappa
M^{2}m_{3/2}\phi\cos\theta,$ (10)

where we have defined $\Phi=\phi e^{i\theta}/\sqrt{2}$. Here
$\gamma=1-7k_{1}/2+2k^{2}_{1}$ and inflaton mass $m_{\phi}$ is given by for
negative $k_{1}$,

$\displaystyle{}m_{\phi}=\sqrt{-k_{1}}\kappa M^{2}/M_{P}.$ (11)

The $\log$-term in Eq.(10) represents the contribution due to mass splitting
in waterfall fields 9406319 , with $\Lambda$ the cut-off scale. The linear
term with coefficient proportional to $m_{3/2}$ arises from a constant
superpotential $W_{0}=m_{3/2}M^{2}_{P}$ added to $W$, which is needed to
cancel out positive contribution to energy density due to SUSY breaking, and
explain the smallness of cc.

When $\phi$ approaches to $\phi_{c}=\sqrt{2}M$, $\phi$ becomes massless as
shown from its mass squared
$m^{2}_{\Psi}=-4\kappa^{2}M^{2}+2\kappa^{2}\phi^{2}$. After the time when
$\phi$ is below $\phi_{c}$, $\Psi$ starts to roll towards to its global
minimum value $\Psi=M$ from $\Psi=0$, which is known as the second period of
inflation. The evaluation of field $\phi$ (including angular component
$\theta$) and $\Psi$ during each period is determined by their equations of
motion,

$\displaystyle{}3H\dot{\phi}$ $\displaystyle\simeq$ $\displaystyle
m^{2}_{\phi}\phi+2\sqrt{2}\kappa M^{2}m_{3/2}\cos\theta,$ $\displaystyle
3H\dot{\theta}$ $\displaystyle\simeq$ $\displaystyle 2\sqrt{2}\kappa
M^{2}m_{3/2}\frac{\sin\theta}{\phi},$ (12) $\displaystyle 3H\dot{\Psi}$
$\displaystyle\simeq$ $\displaystyle-2\kappa^{2}(2M^{2}-\phi^{2})\Psi,$

where $H$ the Hubble constant is subject to the Friedmann constraint

$\displaystyle{}H^{2}\simeq 8\pi V/3M^{2}_{P}.$ (13)

The time for each period is controlled by the magnitude of $m_{\phi}$ or $\mid
m_{\Psi}\mid$ relative to Hubble constant $H$. As pointed out in hybrid , the
second period is very short in compared with the first one for wide ranges of
parameter choices. Substituting Eq.(13) into the last equation in Eq.(III.1),
we obtain the constraint for such property,

$\displaystyle{}10^{-4}<\kappa<\mathcal{O}(1).$ (14)

In the next subsection, we will discuss in more details the initial conditions
on inflaton field and the field value of $\phi$ when inflation ends.

###  III.2 Initial Conditions

The inflation usually begins at some field value $\phi_{in}$ near Plank scale.
The choice on $\phi_{in}$ is subtle when the inflaton potential has either a
few local minimums at $\phi_{min}$s, or local maximum at $\phi_{max}$s. If one
adopts $\phi_{in}$ bigger than $\phi_{min}$, inflaton is probably trapped at
these local minimums of inflaton potential along the trajectory, which leads
to inflation with insufficient e-fold number $N\sim 50-60$. In order to avoid
this, one should choose $\phi_{in}<\min\\{\phi_{min}\\}$. On the other hand,
one wants that inflation proceeds with exactly decreasing $\phi$. This is only
allowed if $\phi_{in}$ is less than $\min\\{\phi_{max}\\}$. In other words, we
should impose the initial condition

$\displaystyle{}\phi_{in}<\min\\{\phi_{max},\phi_{min}\\}.$ (15)

In Fig.1 we show how extremes in $V$ depend on $\cos\theta$ and $\kappa$ by
evaluating $\sqrt{2\epsilon}=V_{,\phi}/V$. The sign of $V_{,\phi}/V$ changes
when $x\sim 0.10$ for $\cos\theta=-0.002$ and $x\sim 0.15$ for
$\cos\theta=-0.003$. This implies that $\cos\theta\simeq 0$ for realistic
inflation. Otherwise, $\phi_{in}<<M_{P}$, which is too small to provide enough
e-fold number $N$. This observation has been noted in 1007.5152  for $m_{3/2}$
of order electroweak scale, and further verified for larger value of
$m_{3/2}\sim 10^{13}$ GeV. With initial value $\theta\simeq\pi/2$, the initial
value $\phi_{in}$ can be chosen in wide range, as shown in Fig.1. The
evaluation of $\phi$ from $\phi_{in}$ is the same as original hybrid model
because of absence of linear term in the first equation in Eq.(III.1). In this
sense, inflation mainly ends at field value $\phi_{end}=\sqrt{2}M$.

Figure 1: Initial condition on $x\equiv\phi/M_{P}$ as function of $\cos\theta$
for $\kappa=0.01,0.001$.

The e-fold number $N$ produced during inflation and $n_{s}$ can be both
estimated in terms of slow roll parameters $\epsilon$ and $\eta$ in the model,
which are given by respectively,

$\displaystyle{}\eta(x)$ $\displaystyle\equiv$ $\displaystyle
M^{2}_{P}\frac{V_{,\phi\phi}}{V}\simeq-
k_{1}+\frac{3}{2}x^{2}-\frac{\kappa^{2}}{8\pi^{2}}\frac{1}{x^{2}},$
$\displaystyle\epsilon(x)$ $\displaystyle\equiv$
$\displaystyle\frac{M^{2}_{P}}{2}\left(\frac{V_{,\phi}}{V}\right)^{2}\simeq\frac{1}{2}\left(\sqrt{\frac{8}{3}}\cos\theta-
k_{1}x+\frac{1}{2}x^{3}+\frac{\kappa^{2}}{8\pi^{2}}\frac{1}{x}\right)^{2}.$
(16)

With $k_{1}\simeq-0.01$ and $\cos\theta\simeq 0$, $\eta(x)$ and $\epsilon(x)$
mainly depend on parameter $\kappa$.

Figure 2: $N$ as function of $\kappa$ for initial value $x_{in}=0.3$ (red),
0.5 (black) respectively. Dotted lines represent the uncertainty on
experimental value. Slow roll condition $\mid\eta\mid<1$ leads to the bound
$\phi_{in}\leq M_{P}$.

Figure 3: $n_{s}$ as function of $\kappa$ for initial value $x_{in}=0.3$
(red), 0.5 (black) respectively. Note that $x_{end}\geq\phi_{c}/M_{P}$. Dotted
lines represent the uncertainty on experimental value.

Fig.2 shows the bound on $\kappa$ for two choices of $x_{in}=0.3$ (red curve)
and 0.5 (black curve) respectively. Note that $x$ is constrained from slow
roll condition $\mid\eta\mid<1$. Given the range shown in Eq.(14) for
$\kappa$, $x$ should be below unity, which implies that large field inflation
is excluded under our assumption. Fig.2 shows that $\kappa\sim 0.1$ for $N\sim
50-60$.

In Fig.3 we show how $n_{s}$ changes for two typical choices of $\kappa$
subtracted from Fig. 2. It clearly indicates that for observed value of
$n_{s}$, $r$ is of order $\sim 10^{-5}$. The simplest hybrid inflation can
provide large e-folds number $N\sim 50-60$ and small $r\sim 10^{-5}$.
Nevertheless, large $N$ and large $r>>10^{-5}$ can not be induced at the same
time. In the next section, we focus on reheating after (or during) inflation.

##  IV Reheating in High-Scale SUSY

When inflation ends the conversion of energy to the MSSM matters from the
inflaton begins immediately. The efficiency of energy transfer depends on how
inflaton is coupled to the MSSM matters, the magnitude of their couplings, and
the SUSY mass spectrum. In general, the ways of energy transfer include the
perturbative and non-perturbative decay of inflaton 333For reviews, see, e.g.,
0507632 ; 1001.0993 .. The later way is known as preheating 9405187 ; 9704452
; 9711360 . The conditions between these two ways of energy transfer are
rather different. In the later case, parameter resonance requires a quartic
interaction term $\sim g^{2}\phi^{2}\chi^{2}$ with large magnitude of $g$.
This only happens if one allows renormalizable superpotential term of mass
dimension 4 0603244 ,

$\displaystyle{}R_{\Phi}=+1:\Phi\mathbf{H}_{u}\mathbf{H}_{d};~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{\Phi}=-1:\Phi\mathbf{H}_{u}\mathbf{L},$
(17)

where $\Phi$ is the inflaton superfield, and $\mathbf{H}_{u,d}$ are Higgs
doublet superfields. $R_{\Phi}$ denotes the $R$-parity of inflaton, which is
useful to keep the dark matter stable. In contrast, in the SM the inflaton
couples to SM chiral fermions and gauge bosons in terms of non-renormalizable
interactions of mass dimension 5. Quartic term above doesn’t exist in the SM,
and therefore the way of energy transfer in the SM is perturbative decay. In
what follows, we consider these two ways separately.

###  IV.1 Perturbative Decay

As briefly mentioned above, perturbative decay happens either when there is no
renormalizable interaction in Eq.(17) or the quartic coupling constant is
tiny. Instead, the inflaton only decays to SM matters via five-dimensional
operators such as

$\displaystyle{}\\{\frac{\phi}{M}F_{\mu\nu}F^{\mu\nu},~{}~{}~{}~{}\frac{\phi}{M}\phi(H\bar{q}_{L})q_{R},~{}~{}~{}~{}\cdots\\}.$
(18)

Here $F_{\mu\nu}$s refer to strengths of SM gauge fields, $q$s refer to SM
fermions and $M$ represents the mass scale appearing in the five-dimensional
operators. The plasma will be MSSM-like if the reheating temperature is larger
than the typical scale of superpartner mass, $m_{0}$. Otherwise, the plasma is
actually SM-like.

Now we calculate the reheating temperature. We organize the decay width of
inflaton to SM particles as,

$\displaystyle{}\varGamma_{d}\equiv\frac{\lambda}{16\pi^{2}}\left(\frac{m_{\phi}}{M}\right)^{2}m_{\phi}.$
(19)

We simply take $M=M_{P}$ but leave $\lambda$ as a free parameter. The thermal
equilibrium of relativistic plasma is dominated by $\varGamma$, the rate for
SM inelastic scatterings of $2\rightarrow 3$ processes perturbative  .
$\varGamma$ is related to $\varGamma_{d}$ as ,

$\displaystyle{}\varGamma\sim\alpha^{3}\left(\frac{M_{P}}{m_{\phi}}\right)\varGamma_{d},$
(20)

where $\alpha\sim 1/30$ is the SM fine structure constant. Therefore the
reheating temperature $T_{R}$ for the perturbative decay isn’t equal to the
conventional one, $T_{rth}$ defined as $T_{rth}\simeq
0.3\times\left(\frac{100}{g_{*}}\right)^{1/4}\sqrt{\varGamma_{d}M_{P}}$,
$g_{*}$ denotes the number of relativistic number. Instead, in terms of
Eq.(20) we have

$\displaystyle{}T_{R}\simeq\alpha^{3/2}\cdot\left(\frac{2\pi
N_{c}}{0.09}\frac{M_{P}}{m_{\phi}}\right)^{1/2}\cdot\left(\frac{g^{susy}_{*}}{g^{sm}_{*}}\right)^{1/4}\cdot
T_{rth}\simeq
0.01\cdot\left(\frac{g^{susy}_{*}}{g^{sm}_{*}}\right)^{1/4}\cdot\left(\frac{\lambda^{1/2}}{4\pi}\right)\cdot
m_{\phi},$ (21)

where $N_{c}$ denote the quantum numbers of SM gauge groups. Note that Eq.(21)
is valid for $m_{0}<T_{R}$, which implies that the reheating temperature can
serve as the upper bound on $m_{0}$.

Figure 4: Reheating temperature (also the upper bound on $m_{0}$) as function
of inflaton mass. The inflaton mass range is inferred as $\sim F/M_{P}$ up to
a coefficicent.

In Fig.4 we show reheating temperature $T_{R}$ as function of inflaton mass.
Note that $\lambda$ captures the magnitude of coupling between inflaton and
“mediate " field, which also couples to the SM matter and gauge fields. Here a
few comments are in order. (1), For $\lambda<10^{-3}$, $T_{R}$ is below the
lower bound $\sim 1\times 10^{9}$ GeV required by thermal leptogenesis in the
whole range of $m_{\phi}$. (2), Since $T_{R}$ is the upper bound on
superpartner mass spectrum $m_{0}$, one finds that $m_{0}$ is upper bounded as
$1$ TeV $<<m_{0}<10^{7}$ TeV for the case of perturbative decay. (3), As
$T_{R}$ is far below the gravitino mass of Eq.(5), there is no overproduction
problem of gravitino in high-scale SUSY. Superheavy gravitino mass of order
$\sim 10^{14}$ GeV kinetically blocks its production in the thermal bath.

###  IV.2 Non-perturbative Decay

If it admits renormalizable superpotential Eq.(17), there exists quartic
interaction between inflaton and its decay products $\chi_{i}$, Non-
perturbative decay can happen in wide range of parameter space for potential
of type444Assuming inflaton and MSSM matters share the same origin of SUSY
breaking, inflaton mass is dynamical induced by SUSY breaking. In this sense,
the mass term $m^{2}_{\phi}\phi^{2}$ is a soft SUSY-breaking term other than
arises from SUSY tree-level mass superpotential $\sim m\Phi\Phi$.
Consequently, there is no cubic interaction $\phi\chi^{2}$ in compared with
earlier discussions in 0603244 ; 0512227 .,

$\displaystyle{}V(\phi,\chi)=\frac{1}{2}m^{2}_{\phi}\phi^{2}+g^{2}\phi^{2}\chi^{2}_{i}+m^{2}_{\chi_{i}}\chi^{2}_{i},$
(22)

where the inflaton mass term is included and $m_{\chi_{i}}$ is the mass for
$\chi_{i}$. We would like to mention that $m_{\chi_{i}}$ include soft SUSY
breaking contribution of order $\sim m_{0}$ and dynamical mass
$\sim\lambda^{1/2}\left<\varphi\right>$ induced by VEV of flat direction
$\varphi$ negative  through the quartic interaction 0512227 ,

$\displaystyle{}V(\chi,\varphi)=\lambda\chi_{i}^{2}\varphi^{2},$ (23)

where $\lambda$ is the quartic coupling constant. The magnitude of
$\left<\varphi\right>$ is determined by the self-interaction potential for
flat direction $V(\varphi)$.

Now we consider the potential for flat direction. $V(\varphi)$ includes soft
breaking mass, Hubble parameter induced term and high dimensional operators,

$\displaystyle{}V(\varphi)\simeq(m^{2}_{0}+c_{H}H^{2})\varphi^{2}+c_{6}\frac{\varphi^{6}}{M^{4}}+\cdots,$
(24)

where $c_{H}$ is real coefficient. Since $m_{0}$ isn’t far beneath the Hubble
constant $H\sim m_{\phi}$ at the beginning of inflation, VEV
$\left<\varphi\right>$ depends on the sign of $c_{H}$, which can be either
positive or negative negative ; positive . In particular,
$\left<\varphi\right>=0$ for the case of either positive $c_{H}$ or negative
$c_{H}$ but with $\mid c_{H}\mid<<1$. It implies that SM gauge symmetry is
unbroken during the whole history of early universe. On the other hand,
$\left<\varphi\right>\neq 0$ for negative $c_{H}$ but with $\mid c_{H}\mid>1$.
It implies that SM gauge symmetry is broken in the eary universe, with gauge
boson mass of order $\left<\varphi\right>$, then restored after the epoch of
reheating.

The potentials we define in Eq.(22) to Eq.(23) are rather general, which can
be applied to both cases in Eq.(17). To discuss the condition for parameter
resonance, one starts with the modified Klein-Gordon equation for Fourie modes
$\chi$. Whether WKB approximation is viable for the study can be analyzed in
term of a quantity $R$ defined as 0507632 ,

$\displaystyle{}R\equiv\frac{\dot{\omega}_{k}}{\omega_{k}^{2}},~{}~{}~{}~{}~{}~{}~{}~{}\omega^{2}=k^{2}/a^{2}+g^{2}\left<\phi(t)\right>^{2},$
(25)

where dot refers to derivative over time and $\omega$ is the frequency, with
$a$ the expansion factor and $k$ the momentum. If $\mid R\mid<<1$, the WKB
approximation is valid, the produced particle number of $\chi$ doesn’t grow in
this case. If $\mid R\mid>1$ instead, the WKB approximation isn’t valid, which
leads to significant production of $\chi$. In long wavelengths limit, this
constraint is given by 555There is a coefficient of order one in front of
$m_{0}$ for either $R_{\Phi}=1$ or $R_{\Phi}=-1$. Here we simply take it equal
to unity.,

$\displaystyle{}m^{2}_{\chi_{i}}\simeq
m^{2}_{0}+\lambda\left<\varphi\right>^{2}<g^{2}\left<\phi(t)\right>^{2},$ (26)

where we have used Eq.(22) and Eq.(23). Moreover, in order to keep that the
parameter resonance isn’t spoiled by expansion, an additional constraint must
be imposed,

$\displaystyle{}q\equiv g^{2}\bar{\phi}^{2}(t)/4m_{\phi}^{2}>>1,$ (27)

where $\bar{\phi}(t)\sim M_{P}$ refers to the amplitude of inflaton
oscillations. For more details, we refer to reader to 0507632  and references
therein.

Figure 5: Reheating temperature as function of $m_{0}$ for $m_{\phi}=10^{13}$
GeV and $\left<\varphi\right>\neq 0$. ($\mathbf{Left}$: $m_{0}<\varGamma_{d}$
with $\left<\varphi\right>=M_{P}$, $\mathbf{Right}$: $m_{0}>\varGamma_{d}$
with $\lambda=10^{-3}$ and $\left<\varphi\right>=0.01M_{P}$). In the
$\mathbf{Right}$ panel, $\left<\varphi\right>\geq 0.1M_{P}$ is excluded by
condition Eq.(26) for $\lambda\geq 10^{-3}$. The bounds on $g$ and $m_{0}$ are
explained in the text.

Fig.5 shows reheating temperature for the case in which
$\left<\varphi\right>\neq 0$. In this case, SM gauge bosons are massive and
the rate for thermal equilibrium $\varGamma$ 0512227  depends on its magnitude
relative to $m_{0}$. For $m_{0}<\varGamma_{d}$, $T_{R}$ depends on both $g$
and $\left<\varphi\right>$, whereas it mainly depends on
$\left<\varphi\right>$ for $m_{0}>\varGamma_{d}$. In this figure, we take
$m_{\phi}=10^{13}$ GeV and $\bar{\phi}=M_{P}$. The bounds on $g$ are due to a
few considerations. The first one is that condition Eq.(27) from parameter
resonance requires $g>>10^{-5}$. The second one is that overproduction problem
gravitino  of gravitino in high-scale SUSY can be kinematically blocked if
$g<10^{-2}$ such that non-perturbatively induced mass during parameter
resonance is beneath $m_{3/2}$ 666 Ref. 1404.1914  provides an example how
gravitino problem in high-scale SUSY is evaded in the context of mini-Split
SUSY. In comparison with 1404.1914 , gravitino mass is far heavier in this
paper, and kinematically blocking is the solution to the overproduction of
gravitino.. The bound on $\lambda\left<\varphi\right>$ arises from condition
Eq.(26) which shows $\lambda\left<\varphi\right>^{2}<g^{2}M_{P}^{2}$.

The $\mathbf{Left}$ panel in Fig.5 shows that in the range $10^{5}$ GeV
$<m_{0}<1.5\times 10^{7}$ GeV reheating temperature $T_{R}\geq 10^{9}$ GeV in
the allowed range of $g$ for $\left<\varphi\right>=M_{P}$. With modifying
$\left<\varphi\right><M_{P}$,

$\displaystyle{}T_{R}\rightarrow\left(\frac{\left<\varphi\right>}{M_{P}}\right)^{-1}T_{R},$
(28)

which is always above the value required by thermal leptogenesis. The
$\mathbf{Right}$ panel in Fig.5 shows that in the range $10^{11}$ GeV $\leq
m_{0}<4\times 10^{13}$ GeV reheating temperature $10^{11}$ GeV $\leq T_{R}\leq
10^{13}$ GeV if $\left<\varphi\right>=10^{-2}M_{P}$, and changes similarly to
Eq.(28) for modifying $\left<\varphi\right>$. This implies that $T_{R}$ is
also always above the the value required by thermal leptogenesis. Note that
$\left<\varphi\right>\geq 10^{-1}M_{P}$ is excluded by condition Eq.(26) from
parameter resonance for $\lambda\simeq 10^{-3}$.

Figure 6: Reheating temperature as function of $g$ for
$\left<\varphi\right>=0$, with the definition $\varGamma_{d}\equiv
g^{2}m_{\phi}/8\pi$.

Fig.6 shows reheating temperature for the case in which
$\left<\varphi\right>=0$. In this case, SM gauge symmetries are unbroken in
the epoch of reheating. Thermalization cannot occur before the inflaton decay
has completed. Due to $\left<\varphi\right>=0$ the $2\rightarrow 3$
scatterings are already efficient when $H\simeq\varGamma_{d}$. The reheat
temperature in this case is given by the standard expression: $T_{R}\simeq
0.3\sqrt{\varGamma_{d}M_{P}}$. However, $\varGamma_{d}$ calculated via
renormalizable couplings is rather different from Eq.(19) calculated via non-
renormazible couplings. This difference between Fig.4 and Fig.6 is obvious.
Typically, we have $T_{R}\geq 10^{10}$ GeV in non-perturbative decay into MSSM
and $T_{R}\leq 10^{10}$ GeV in perturbative decay into SM.

Both Fig.6 and Fig.5 show that $T_{R}$ is above $\sim 10^{9}$ GeV but beneath
$m_{3/2}$ in a wide range of parameter space. Due to kinematically blocking
effect this evades the overproduction of gravitino in conventional high-scale
SUSY.

##  V Conclusions

In the light of both LHC data and Plank bound on $r$, High-scale SUSY is more
favored in compared with low-scale SUSY. In this paper, we discussed the
implications of high-scale SUSY to the early universe. In particular, we
assumed that the inflation and visible sector share the same origin of SUSY
breaking, and derived model independent consequences based on this assumption.
We find that the reheating temperature for superpartner mass spectrum above
$\mathcal{O}(100)$ TeV might be beneath the value required by thermal
leptogenesis if inflaton decays to its products perturbatively but above it if
non-perturbatively instead. We also observed that problem of gravitino
overproduction can be evaded through kinematically blocking in a wide range of
parameter space in the later way.

As an illustration for the model building of inflation in the course of high-
scale SUSY, in section 3 we revise the simplest hybrid inflation that includes
a new linear term for inflaton with coefficient proportional to $m_{3/2}$. It
is shown that this term significantly affects the choices on initial condition
of inflaton fields. We found that with the assumption the simplest hybrid
inflation is consistent with present experimental data for $r$ of order
$10^{-5}$.

Under our assumption only the dark matter is a light SUSY state with mass near
the weak scale Zheng , which is the target of LUX and Xenon experiments, etc.
Hopefully, it can be addressed in the near further.

###### Acknowledgements.

We would like to thank J.-h, Huang for discussion. This work is supported in
part by the Fundamental Research Funds for the Central Universities under
Grant No.CQDXWL-2013-015.

## References

  * (1) G. Aad et al, [$\mathbf{ATLAS~{}Collaboration}$], “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B 716, 1 (2012), arXiv:1207.7214 [hep-ex];
S. Chatrchyan et al, [$\mathbf{CMS~{}Collaboration}$], “Observation of a new
boson at a mass of 125 GeV with the CMS experiment at the LHC,” Phys. Lett. B
716, 30 (2012), arXiv:1207.7235 [hep-ex].

  * (2) See, e.g., S. P. Martin, “A Supersymmetry primer,” Adv. Ser. Direct. High Energy Phys. 21, 1 (2010). [hep-ph/9709356].
  * (3) See, e.g., J. L. Feng, “Naturalness and the Status of Supersymmetry,” Ann. Rev. Nucl. Part. Sci. 63, 351 (2013), arXiv:1302.6587 [hep-ph], and references therein.
  * (4) P. A. R. Ade et al. [Planck Collaboration], arXiv:1502.02114 [astro-ph.CO].
  * (5) P. A. R. Ade et al. [$\mathbf{Plank}$ Collaboration], “Planck 2013 results. XVI. Cosmological parameters,” arXiv:1303.5076 [astro-ph.CO].
  * (6) P. A. R. Ade et al. [$\mathbf{Plank}$ Collaboration], “Planck 2013 results. XXII. Constraints on inflation,” arXiv:1303.5082 [astro-ph.CO].
  * (7) J. R. Espinosa, G. F. Giudice, E. Morgante, A. Riotto, L. Senatore, A. Strumia and N. Tetradis, “The cosmological Higgstory of the vacuum instability,” arXiv:1505.04825 [hep-ph].
  * (8) See, e.g., S. Zheng, “Can Higgs Inflation be Saved with High-scale Supersymmetry ?,” arXiv:1504.08093 [hep-ph].
  * (9) Q. Shafi and J. R. Wickman, Phys. Lett. B 696, 438 (2011), arXiv:1009.5340 [hep-ph].
  * (10) T. Higaki, K. S. Jeong and F. Takahashi, JHEP 1212, 111 (2012), arXiv:1211.0994 [hep-ph].
  * (11) D. H. Lyth, “What would we learn by detecting a gravitational wave signal in the cosmic microwave background anisotropy?,” Phys. Rev. Lett. 78, 1861 (1997), [hep-ph/9606387].
  * (12) K. Nakayama, F. Takahashi and T. T. Yanagida, “Constraint on the gravitino mass in hybrid inflation,” JCAP 1012, 010 (2010), arXiv:1007.5152 [hep-ph].
  * (13) G. R. Dvali, Q. Shafi and R. K. Schaefer, “Large scale structure and supersymmetric inflation without fine tuning,” Phys. Rev. Lett. 73, 1886 (1994). [hep-ph/9406319]
  * (14) A. D. Linde, “Hybrid inflation,” Phys. Rev. D 49, 748 (1994), [astro-ph/9307002].
  * (15) B. A. Bassett, S. Tsujikawa and D. Wands, “Inflation dynamics and reheating,” Rev. Mod. Phys. 78, 537 (2006), [astro-ph/0507632].
  * (16) A. Mazumdar and J. Rocher, “Particle physics models of inflation and curvaton scenarios,” Phys. Rept. 497, 85 (2011), arXiv:1001.0993 [hep-ph].
  * (17) L. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. Lett. 73, 3195 (1994), [hep-th/9405187].
  * (18) L. Kofman, A. D. Linde and A. A. Starobinsky, “Towards the theory of reheating after inflation,” Phys. Rev. D 56, 3258 (1997), [hep-ph/9704452].
  * (19) J. Garcia-Bellido and A. D. Linde, “Preheating in hybrid inflation,” Phys. Rev. D 57, 6075 (1998), [hep-ph/9711360].
  * (20) R. Allahverdi and A. Mazumdar, “Reheating in supersymmetric high scale inflation,” Phys. Rev. D 76, 103526 (2007), [hep-ph/0603244].
  * (21) S. Davidson and S. Sarkar, “Thermalization after inflation,” JHEP 0011, 012 (2000), [hep-ph/0009078];
R. Allahverdi and M. Drees, “Thermalization after inflation and production of
massive stable particles,” Phys. Rev. D 66, 063513 (2002), [hep-ph/0205246].

  * (22) M. Dine, L. Randall and S. D. Thomas, “Supersymmetry breaking in the early universe,” Phys. Rev. Lett. 75, 398 (1995), [hep-ph/9503303]; “Baryogenesis from flat directions of the supersymmetric standard model,” Nucl. Phys. B 458, 291 (1996), [hep-ph/9507453].
  * (23) M. K. Gaillard, H. Murayama and K. A. Olive, “Preserving flat directions during inflation,” Phys. Lett. B 355, 71 (1995), [hep-ph/9504307].
  * (24) R. Allahverdi and A. Mazumdar, “Supersymmetric thermalization and quasi-thermal universe: Consequences for gravitinos and leptogenesis,” JCAP 0610, 008 (2006), [hep-ph/0512227].
  * (25) M. Kawasaki, F. Takahashi and T. T. Yanagida, “Gravitino overproduction in inflaton decay,” Phys. Lett. B 638, 8 (2006), [hep-ph/0603265].
  * (26) J. Fan, B. Jain and O. Ozsoy, “Heavy Gravitino and Split SUSY in the Light of BICEP2,” JHEP 07, 073 (2014), arXiv:1404.1914 [hep-ph].
  * (27) S. Zheng, “On Dark Matter Selected High-Scale Supersymmetry,” JHEP 1503, 062 (2015), arXiv:1409.2939 [hep-ph].