text:
\section{\label{intro}Introduction}
Reheating is one of the important phases of the universe evolution. This is the phase during which all the matter fields of the current universe are assumed to have been
produced from the oscillating inflaton field. Therefore, in order to make any connection with the present universe, the mechanism of inflation \cite{guth, linde1, steinhardt} is important. For this, model building is an important part of the current research in inflationary cosmology.
Over the years a lot of models have been put forward  \cite{encyclopedia}, to explain the observations \cite{PLANCK}. The simplest model of inflation is known to be governed by a single scalar field with power-law potential. However, with the increasing precision of observation, those models turned out to be disfavored. As an alternative, a class of single field $\alpha$ attractor \cite{linde} models have been recently proposed with a modified form of the potential which not only fits well with the observation but also unifies different existing models in a single fold. In view to reconcile power law potentials with observations, we proposed\cite{Maity:2016uyn} another class of models with a plateau potential in the large field limit while the potential during reheating is simply governed by the leading power-law term: $V(\phi) \propto \phi^n$ . Inflationary dynamics have already been studied in detail in the aforementioned references. As emphasized in the beginning, in this paper we will focus on reheating period. Depending upon the inflaton coupling with the decaying fields, reheating phase can be both perturbative\cite{Shtanov:1994ce} and/or non-perturbative\cite{Kofman:1994rk}.
Usually, the reheating period is less constrained by the cosmological observation. Interestingly in the recent papers\citep{Dai:2014jja,Creminelli:2014fca,Martin:2014nya,Domcke:2015iaa}, an important connection has been established between the cosmic microwave background (CMB) and reheating phase. Analysis of those papers was based on one of the important assumptions that the reheating phase is governed by an effective single fluid with a time independent equation of state. Recently we proposed a possible generalization of their work by taking into account the explicit decay of inflaton during reheating. However, we assume the decay to be perturbative. This approach has an interesting consequence that we can go beyond the usual reheating phenomena, and shed light on how one can constrain the dark matter parameter space through CMB anisotropy. Therefore, following the formalism developed in \cite{mhiggs,Maity:2018dgy}, we study the CMB constraints on the inflation model, the reheating phase, and the dark matter production during reheating for our recently proposed minimal inflationary models.

We organize our paper as follows: In section-\ref{model}, we briefly review the minimal plateau inflationary cosmological models studied in detail in \cite{Maity:2018dgy}.
After the end of inflation, generically the inflaton starts to have coherent oscillation at the minimum of the potential, during which the universe will undergo reheating phase. We also compute the effective equation of state of the oscillating inflaton for our subsequent studies. In section-\ref{reheatingprediction}, we have studied the reheating constraints from CMB and production of a heavy dark matter particle during reheating. Finally, we concluded and discussed our future work.

\section{\label{model}Brief account on minimal plateau inflationary potentials and their origin}
In this section we will briefly review the inflationary models proposed in \cite{Maity:2016uyn} specifically mentioning the cosmological in terms of our model parameters. The non-polynomial plateau potential looks like the following form,
\begin{eqnarray}
V_{min}(\phi) =\lambda \frac{m^{4-n} \phi^n}{1+\left(\frac{\phi}{\phi_*}\right)^n}
\end{eqnarray}
The parameters $(n, m~or~\lambda)$ are similar to those of the power-law chaotic potentials. As usual, the parameter $\lambda$ is reserved for $n=4$,\cite{mhiggs}. For other value of $n$, we set $\lambda =1$. The new parameter in this model is a mass scale $\phi_{\ast}$ which controls the shape of the potentials. The potentials are so constructed that for large value of $\phi_{\ast}$, it becomes approximately shift symmetric which is usually dubbed as the plateau potential. The scale $\Lambda = \lambda m^{4-n} \phi_*^n$ can be associated with the inflationary scale which is essentially the height of the potential during inflation.
Considering $\phi_* \leq {\cal{O}}(1)$ in unit of $M_p$, a general expression for the scalar spectral index and the tensor to scalar ratio are calculated as,
\begin{eqnarray} \label{nsrvsN}
1 - n_s &=&
	\label{ns-r-ns}
	\frac{2(n+1)}{(n+2)} \frac{1}{N}
~~;~~
r =
	8n^2 \left(\frac{\phi_*}{M_p}\right)^{\frac{2n}{(n+2)}} \frac{1}{[n(n+2)]^{\frac{2(n+1)}{(n+2)}} N^{\frac{2(n+1)}{(n+2)}}}
\label{ns-r-R}
\end{eqnarray}
For $\phi_*<\cal{O}(1)$, it turns out that the tensor to scalar ratio explicitly depends upon the inflationary scale itself. In all the above expressions, we have taken large-$N$ limit. However, it is important to remember that the aforementioned series expansion is not valid for large $\phi_* > 1~ M_p$. In fact, for large $\phi_{\ast}$ the models reduces to the power-law chaotic models. What we have seen from the detailed analysis is that the this models fit extremely well with the experimental results for a wide range of parameters while during the reheating phase, i.e., around $\phi=0$, the models behaves as the power-law chaotic models. However, the most difficult point would be to understand the selection criteria based on which one can identify a specific model. For all the models we have similar predictions with the same number of external parameters. We all know that usual power-law potential with a single parameter, always predicts large tensor to scalar ratio. Therefore, in terms of inflation model building increasing the number of control parameters may be necessary towards understanding the observation. With this motivation in mind, we had proposed the aforementioned models. In the following sub-sections let us briefly provide the origin of the aforementioned models based on supergravity.

\subsection{$V_{min}(\phi)$ from Supergravity}
Here we will present a brief account of supergravity realization of our model potential which we recently understood \cite{Maity:2016uyn}. We consider the following  form of the K\"ahler and  superpotential with three chiral superfields $S$ and $\{\Phi_1, \Phi_2\}$
\begin{eqnarray}
&&W = \frac{\phi_{\ast} S^2}{2}\left(\Phi_1^p - \Phi_2^p \right)\\\nonumber
&&K = |S|^2 + |\Phi_1|^2 + |\Phi_2|^2 ,
\end{eqnarray}
Assuming a suitable $R$-symmetry, one arrives at the following form of
F-term potential at its minimum with $|\Phi_1|=|\Phi_2| \equiv \phi$, where $\phi$ is a real scalar field as,
\begin{align}
V_F = e^{\frac{\phi}{M_p^2}} |S|^4  p^2 \left(\frac{\phi}{\phi_*}\right)^{2p-2}
\end{align}
\\
In order obtain our minimal form of the potential, we also consider D-term potential coming from the appropriate form of the gauge kinetic function, following the references  \cite{delaMacorra:1995qh,Dimopoulos:2016zhy}
\begin{equation}
V_D = \frac{1}{2} \left( |S|^2 - \sqrt{2}M^2 \right)^2 .
\label{eq:dterm}
\end{equation}
where $M$ is the scale of a grand unified theory(GUT)
\\
Here, we must note that a D-term of form given in (\ref{eq:dterm}) can be generated with an anomalous $U(1)$ symmetry\cite{Binetruy:1996xj,Halyo:1996pp}. These type of symmetries usually appears in the realm of string theories\cite{Dine:1987xk,Atick:1987gy,Dine:1987gj}. Where anomaly cancellation is due to the Green-Schwarz (GS) mechanism\cite{Green:1984sg} which now set the value of the Fayet-Illiopoulos term as
\begin{equation}
\xi_{GS} = \frac{Tr[Q_A]}{192\pi}g^2 M^2
\end{equation}
where $Q_A$ is the charge of the fields under the anomalous $U(1)$ group and trace is taken over fields.
Now let us assign the charges of the fields following the discussion in\cite{Halyo:1996pp}. The charge of the fields $\Phi_1$ and $\Phi_2$ (whose modulus will be identified as the inflaton) must vanish. The charges of $S$, taken to be $-1$, can be non-zero as long as it is opposite of $Tr[Q_A]$.(This will make it necessary to consider the existence of another field field charged under the anomalous $U(1)$ with zero VEV.)
\\
Hence total potential tuns out to be
\begin{align}
V_T =  e^{\frac{\phi}{M_p^2}} |S|^4  p^2 \left(\frac{\phi}{\phi_*}\right)^{2p-2}  + \frac{1}{2} \left( |S|^2 - \sqrt{2}M^2 \right)^2
\end{align}
Minimizing the potential along the $S$ direction reduces the scalar potential into the the following simple form
\begin{equation}
V_{sugra}(\phi) = \frac{m^{4-n}\phi^n}{exp(-\frac{\phi^2}{2M_p^2}) + \left( \frac{\phi}{\phi_{\ast}}\right)^n }
\end{equation}
Where $n=2(p-1)$ and $m=(M^4\phi_{\ast}^{-n})^{1/(4-n)}$. In the limit $\phi_* < M_p$, this plateau potential reduces one of our minimal plateau potential as follows,
\begin{equation}
V_{min}(\phi) = \frac{m^{4-n}\phi^n}{1 + \left( \frac{\phi}{\phi_{\ast}}\right)^n } .
\end{equation}
Hence from our supergravity model, we obtained the model potential in a limit where $\phi_*$ is sub-Planckain.
\iffalse
\subsection{$V_{min}(\phi)$ from non-minimal scalar tensor theory}
In the phenomenological minimal plateau potential, we considered non-polynomial functions, which may be difficult to justify in the perturbative field theory point of view. In this section, we will briefly describe how we can arrive at the same from of the non-polynomial potential from polynomial function with additional conformal transformation \cite{Eshaghi:2015rta, Broy:2016rfg}.
Let us begin with the following non-minimally coupled scalar-tensor theory,
\begin{equation}
S_J = \int d^4x \sqrt{-g} \left[ \frac{\Omega(\varphi)}{2} M_p^2 R - \frac{\Omega(\varphi)}{2} g^{\mu \nu } \partial_{\mu}\varphi \partial_{\nu}\varphi - V(\varphi)     \right],
\end{equation}
where, $\Omega(\varphi)$ is arbitrary function of a scalar field $\varphi$. Applying the  conformal transformation, $
\tilde{g}_{\mu \nu} = \Omega(\varphi) g_{\mu \nu}$ ,
one gets the Einstein frame action \cite{Fujii:2003pa} as follows,
\begin{equation}
S_E = \int d^4x \sqrt{-\tilde{g}} \left[\frac{M_p^2}{2} \tilde{R} -\frac{1}{2} F^2(\varphi) \tilde{g}^{\mu \nu} \partial_{\mu}\varphi \partial_{\nu}\varphi - \tilde{V}(\varphi)      \right],
\end{equation}
Where
\begin{equation}
F^2(\varphi) = \frac{3 M_p^2}{2} \frac{\Omega'^2(\varphi)}{\Omega^2(\varphi)} + 1 ~~;~~\tilde{V}(\varphi) = \frac{V(\varphi)}{\Omega^2(\varphi)}
\label{Funct}
\end{equation}
Now, we choose the following two possible polynomial functional form of coupling functions \cite{Kallosh:2013tua, Galante:2014ifa},
\begin{eqnarray}
\Omega^2(\varphi) =
\begin{cases}
	1+ \xi (\frac{\varphi}{M_p})^n\\
	\left[1+ \xi (\frac{\varphi}{M_p})^2 \right]^{\frac{n}{2}} .
\end{cases}
\label{omega}
\end{eqnarray}
Therefore, applying (\ref{omega}), we find F and $\tilde{V}$ as,
\begin{eqnarray}
F^2(\varphi) =
\begin{cases}
	\frac{3 n^2 \xi ^2 \left(\frac{\varphi }{M_p}\right)^{2 (n-1)}}{8 \left[1 + \xi  \left(\frac{\varphi }{M_p}\right)^n\right]^2}+1\\
	\frac{3  n^2 \xi ^2 \left(\frac{\varphi}{M_p} \right) ^2}{8 \left[1+\xi  \left(\frac{\varphi}{M_p}\right) ^2\right]^2}+1
\end{cases} ~~;~~ \tilde{V}(\varphi) = \begin{cases}
\frac{V(\varphi)}{1 + \xi \left(\frac{\varphi}{M_p}\right)^n}\\
\frac{V(\varphi)}{\left[1+ \xi (\frac{\varphi}{M_p})^2 \right]^{\frac{n}{2}}}
\end{cases}
\end{eqnarray}
\\
We use the following field redefinition
\begin{equation}
\frac{d\phi}{d\varphi} = F(\varphi)
\label{redef}
\end{equation}
At this point we need to integrate eq.(\ref{redef}), to figure the canonical scalar filed potential in terms of  new field. However, for entire range of parameter $\xi$, we will not be able to reproduce our minimal potential. Therefore, we restrict in the weak coupling $\xi << 1$, $F \sim 1$, eq(\ref{redef}) regime and arrive at
\begin{eqnarray}
V_{min}(\phi) =
\begin{cases}
	\frac{\lambda ~m^{4-n} \phi^n}{1 +  \left(\frac{\phi}{\phi_*}\right)^n}\\
	\frac{\lambda ~m^{4-n} \phi^n}{\left[1+  (\frac{\phi}{\phi_*})^2 \right]^{\frac{n}{2}}} ,
\end{cases}
\end{eqnarray}
where, we identify $\phi_*$ as $M_p/\xi^{\frac{1}{n}}$ for Type-I potential and $M_p/\xi^{\frac{1}{2}}$	for type-II potential.
Therefore, in the weak coupling regime, $\xi \ll 1$ or $\phi_* > 1$, the non-minimal scalar tensor theory can give rise
to a large class of minimal cosmologies such as ours which do not belong the $\alpha$-attractor model.}
\fi

\begin{table}[t]
\begin{tabular}{|p{0.7cm}|p{0.6cm}|p{1.0cm}|p{1.5cm}|}
\hline
$\frac{\phi_*}{M_p}$& $n$  &\multicolumn{2}{c|}{Model I}  \\
\cline{3-4}
 & & $n_s$ &  $r$  \\
\hline
\hline
 0.01  &  2  & 0.969 & $4\times10^{-5}$ \\
\cline{2-4}
     &  4  & 0.966 &  $2\times10^{-6}$  \\
\cline{2-4}
     &  6  & 0.965 & $3\times10^{-7}$  \\
\cline{2-4}
     &  8  & 0.964 & $1\times10^{-7}$    \\
\hline
\hline
 1    &  2  & 0.969 & $4\times10^{-3}$   \\
\cline{2-4}
     &  4  & 0.966 & $9.6\times10^{-4}$  \\
\cline{2-4}
     &  6  & 0.964 & $3.5\times10^{-4}$   \\
\cline{2-4}
     &  8  & 0.964 & $1.7\times10^{-4}$ \\
\hline
\hline
\end{tabular}
\caption{\scriptsize Sample values of $(n_s,r)$ are given for two values of $\phi_* =(0.01,1) M_p$ corresponding to small field and large field inflation respectively.}
\label{nsrtable}
\end{table}
In any case, the main goal of our previous and the present work is to understand the phenomenological models based on observations. In this regard, one of the important observable quantities which helps us to fix one of the parameters is the amplitude of the scalar field fluctuation which is defined as scalar power spectrum,
$\mathcal{P_R} =
2.4 \times 10^{-9}$, based on PLANCK normalization. The value is fixed at the pivot scale $k/a_0 = 0.05 Mpc^{-1}$. Corresponding observed scalar spectral index is $n_s = 0.9682 \pm 0.0062$, and the upper limit on the tensor to scalar ratio is set to be $r<0.11$.
 In table-(\ref{nsrtable}), we have given some sample values of all the cosmologically relevant quantities for different values of parameters. Our main goal of this paper is to study the reheating dynamics and consequently the dark matter phenomenological constraints based on the formalism developed in
\cite{Dai:2014jja} and its generalization proposed in \cite{ Maity:2018dgy}.
We will show how the inflationary scale $\phi_*$ plays important role in determining the reheating temperature and consequently its influence on the dark matter parameter space. The Only assumption we will make in our analysis is the perturbative decay of the inflaton field into radiation and radiation into the dark matter field.

\section{CMB to dark matter phenomenology via reheating}
\label{reheatingprediction}
It has been a general consensus that the inflation phase must be followed by the reheating phase. During this phase, the inflaton field is assumed to decay into various daughter fields such as radiation, dark matter particles, etc. The quantity which characterizes this phase is known as the reheating temperature $(T_{re})$ and its duration $(N_{re})$. Reheating temperature is defined at the instant when the inflaton decay rate equals the universe expansion rate and the radiation domination sets in.
 Being not an observable quantity, the reheating temperature is usually considered as a free parameter where its upper value could be as high as the GUT scale and the lower value is fixed by the requirement of
  successful Big Bang Nucleosynthesis. However, it has been established in the recent reference \cite{Dai:2014jja} that not just inflationary phase but also  the reheating phase can be constrained by CMB anisotropy. Since then, there has been a surge of research work in this direction\cite{Creminelli:2014fca,Martin:2014nya,Cook:2015vqa,Ellis:2015pla,Ueno:2016dim,Eshaghi:2016kne,DiMarco:2017zek,Bhattacharya:2017ysa,Drewes:2017fmn}. The inherent assumption in all those works is that the inflaton field converts into radiation instantaneously at the end of reheating and the effective fluid describing the reheating dynamics is parameterized by an average equation of state parameter. In our recent work\cite{Maity:2018dgy,mhiggs}, we have shown that by relaxing the aforementioned assumptions we not only can constrain the inflationary model but also shed light on the dark matter
   phenomenology. Through our analysis, we were able to connect the current dark matter relic abundance with the temperature anisotropy in CMB. In this paper, we will follow the aforementioned formalism developed in \cite{Maity:2018dgy} and generalize it for the arbitrary equation of state of the inflaton during inflation. In our model, during reheating phase the inflaton potential can be approximated as $V(\phi) \propto \phi^n$ such that the equation of state of the
    \begin{figure}[t]
    	\centering
    	\includegraphics[scale=0.8]{scales2.pdf}
    	\caption{\scriptsize Evolution of the comoving Hubble horizon starting from inflation (left side black lines), reheating (intermediate lines), radiation domination up to the present time(the shorter period of matter domination and the current dark energy domination is not shown as they don't affect the qualitative conclusion). During the reheating phase when the effective equation of state $w_{eff} = 1/3$, the evolution is indistinguishable from radiation dominated phase as shown in solid brown line. This fact makes reheating parameters $(N_{re}, T_{re})$ indeterministic as observed in \cite{Dai:2014jja}. The evolution with $w_{eff}<1/3$ is shown in blue intermediate lines while that of $w_{eff}> 1/3$ is shown in red intermediate lines. Increase in $n_s$ implies going from solid black line to dotted black line adding extra e-folding number $\Delta N_k$ for horizon crossing of a particular CMB mode $k$. It is clear from the figure for $w_{eff}<1/3$, increasing $n_s$ will result in decreasing reheating e-folding number during reheating $N_{re}$ thereby increasing reheating temperature $T_{re}$. While for $w_{\phi}>1/3$ it will just be the opposite. Hence, one can clearly expect an upper bound on $n_s$ for a particular inflation model with $w_{\phi}<1/3$ where the intermediate line closes indicating an instantaneous reheating with maximum possible reheating temperature. On the other hand for models with $w_{\phi}>1/3$, one expects a lower bound on $n_s$. The bound on the other end of $n_s$ for both the cases will be from the condition of minimum reheating temperature for a successful BBN.}
    	\label{scales}
    \end{figure}
    oscillating inflaton can be approximated as $w_{\phi} = (n-2)/(n+2)$. An illustration of how CMB constraints the reheating temperature as well as how the equation of state parameter play a significant role controlling the reheating period is presented in fig.(\ref{scales}).
    It is also clear from the figure that for a particular scale and the inflaton equation of state during reheating we have maximum possible value of $n^{max}_s$ for $w_{eff} <1/3$ and minimum possible value of $n^{min}_s$ for $w_{eff} > 1/3$. Where $w_{eff}$ is the effective equation of state during reheating. However, for our case, we will consider explicit decay of inflation to radiation and dark matter during reheating. Therefore, the existence of $(n_s^{max},n_s^{min})$ will depend on the equation of state of inflaton $w_{\phi}$ itself. It also implies the existence of maximum reheating temperature $T_{re}^{max}$. Interestingly we will see that the maximum reheating temperature $T_{re}^{max}$ does not depend on a particular model under consideration.

\subsubsection{The Boltzmann Equations during reheating with general inflaton equation of state}
 We will consider three component universe consisting of inflaton($\rho_{\phi}$), radiation($\rho_R$) and a massive dark matter particle with the number density $(n_X)$. For simplicity, we consider dark matter particle production only from radiation annihilation. For the general equation of state of inflaton, we will work with the following rescaled variables,
 \begin{eqnarray}
  \Phi = \frac{\rho_{\phi}a^{3(1+w_{\phi})}}{m_{\phi}^{(1-3w_{\phi})}};~~~~~~~~~~R = \rho_R a^4;~~~~~~~~~~X = n_X a^3.
 \end{eqnarray}
The Boltzmann equations\cite{Giudice:2000ex,Maity:2018dgy} in the rescaled variables with the time variable $A=a/a_I$ reduces to,
\begin{eqnarray}
\label{Inflatoneq}
\frac{d\Phi}{dA} &=& -c_1(1+ w_{\phi}) \frac{A^{1/2} \Phi}{\mathbb{H}};\nonumber\\
\label{Radeq}
\frac{dR}{dA} &=& c_1(1+ w_{\phi}) \frac{A^{3(1-2 w_{\phi})/2}}{\mathbb{H}} \Phi + c_2 \frac{A^{-3/2} 2\left<E_X\right> \left<\sigma v\right> M_{pl}}{\mathbb{H}}\left(X^2 - X_{eq}^2\right);\\
\frac{dX}{dA} &=& - c_2 \frac{A^{-5/2} 2\left<E_X\right> \left<\sigma v\right> M_{pl}}{\mathbb{H}}\left(X^2 - X_{eq}^2\right)\nonumber;
\label{Boltzmann2}
\end{eqnarray}
Where, $\mathbb{H} = \left(\Phi/A^{3 w_{\phi}} + R/A + X \left<E_X\right>/m_{\phi}\right)^{1/2}$ is the Hubble expansion rate and the constants $c_1$ and $c_2$ are given by
\begin{eqnarray}
c_1 = \sqrt{3}\frac{M_P \Gamma}{m_{\phi}^2},~~~~~c_2 = \sqrt{\frac{3}{8\pi}}
\end{eqnarray}
Here, $M_{pl}(=\sqrt{8\pi}M_p)$ is the Planck mass. $\left<E_X\right> = \rho_X/n_X$ is the average energy density of the X-particle which we identify as the candidate dark matter. $X_{eq}$ is the equilibrium (comoving-)number density of the dark matter of mass $M_X$ if it is in equilibrium with background thermal bath at temperature $T$. $\Gamma$ is the inflaton decay rate. We assume the dark matter has an internal degree of freedom $g$. Since we want to study all range of mass of the dark matter, we use the use the relativistic form of the equilibrium distribution function with the modified Bessel function of the second kind as\cite{,book22,Giudice:2000ex}
\begin{eqnarray}
X_{eq} & = & \frac{g T^3}{2\pi^2} \left(\frac{M_X}{T}\right)^2K_2\left(\frac{M_X}{T}\right)   \left(\frac{A}{m_{\phi}} \right)^3
\end{eqnarray}
In all the above equation we introduce an arbitrary mass scale $m_{\phi}$ for convenience in numerical computation. We will take it as the Hubble constant at the end of inflation. At this point, let us emphasize that our analysis is purely perturbative. However the non-perturbative effects may have an important role, we left it for our future studies. In order to solve, the above set of Boltzmann equation, we set the initial condition at the end of inflation as,
\begin{eqnarray}
\Phi(1) = \frac{3}{8\pi} \frac{M_{pl}^2 H_I^2}{m_{\phi}^4};~~~~~R(1)=X(1)=0.
\label{PhiI}
\end{eqnarray}
Where the initial Hubble expansion $H_I^2 = (8\pi/3M_{pl}^2)\rho_{\phi, end}$.
The set of Boltzmann equations can now be easily solved provided we know the inflaton decay with and the annihilation cross section.
The reheating temperature of the universe is identified at the instant of $H(t)=\Gamma_{\phi}$, when all the inflaton field energy is instantaneously converted into the radiation field. However, in general, this is not true.
At any instant of time during the evolution, the temperature of our universe is identified with the radiation temperature with $T \equiv T_{rad}=\left[ {30}/({\pi^2 g_*)}\right]^{1/4} \rho_R^{1/4} $.
 The thermally averaged annihilation cross section of the dark matter from the radiation is $\left< \sigma v \right> $ \cite{Giudice:2000ex}. We will be using this as a free parameter in our subsequent numerical analysis.

\subsubsection{The early time solution and the maximum temperature for general $\omega_{\phi} = (n-2)/(n+2)$}
Before we resort to the numerical solution to find the connection among the CMB anisotropy and the dark matter abundance, let us compute an analytic expression for the maximum possible temperature during reheating phase for general equation of state of the inflaton field. In the early phase of the reheating stage ($H\gg\Gamma_{\phi} )$, considering the initial condition $R(A_I) \simeq X(A_I) \simeq 0$ eq.(\ref{Radeq}) can be solved as
 \begin{eqnarray}
 \frac{dR}{dA} \simeq c_1 (1+w_{\phi}) \Phi_I^{\frac{1}{2}} A^{\frac{3}{2}(1-w_{\phi})}
 \implies R    \simeq 2 c_1\left(\frac{1+w_{\phi}}{5-3w_{\phi}} \right) \left[A^{\frac{5-3w_{\phi}}{2}}  - A_I^{\frac{5-3w_{\phi}}{2}}\right].
\end{eqnarray}
Therefore, by using the above solution and using the definition of temperature in terms of the radiation density, one arrive at the following expression
\begin{eqnarray}
 T = \left[ \frac{60 c_1}{g_* \pi^2}~ \frac{1+w_{\phi}}{5-3w_{\phi}}\right]^{\frac{1}{4}} ~m_{\phi} \left( \frac{\Phi_I}{A_I^{3(1+w_{\phi})}}\right)^{\frac{1}{8}} \left[ \left(\frac{A}{A_I}\right)^{-\frac{3}{2}(1+w_{\phi})}  - \left(\frac{A}{A_I}\right)^{-4}\right]^{\frac{1}{4}}
\end{eqnarray}
By using the the eq.(\ref{PhiI}), the maximum temperature can be found as
\begin{eqnarray}
T_{max} = \left[ \frac{60 c_1}{g_* \pi^2}~ \frac{1+w_{\phi}}{5-3w_{\phi}}\right]^{\frac{1}{4}} ~m_{\phi}^{\frac{1}{2}} \left(\frac{3}{8\pi} \frac{M_{pl}^2 H_I^2}{A_I^{3(1+w_{\phi})}}\right)^{\frac{1}{8}} \left[ \left( \frac{8}{3(1+w_{\phi})}\right)^{-\frac{3(1+w_{\phi})}{5-3w_{\phi}}} - \left( \frac{8}{3(1+w_{\phi})}\right)^{-\frac{8}{5-3w_{\phi}}} \right]
\end{eqnarray}
In the following numerical analysis for the dark matter abundance, we will find maximum temperature plays an important role in constraining the dark matter parameter space. The essential idea behind this temperature is that as the reheating temperature is measured at a later stage of reheating, the radiation production commences at the very early stage. As one can clearly see the non-trivial dependence of the temperature on the inflaton equation of state parameter $(\omega_{\phi})$ and the inflaton decay constant $\Gamma_{\phi}$.
 Depending upon the initial value of Hubble rate which has a direct connection with the CMB anisotropy, the maximum temperature can be many order higher than the reheating temperature\cite{Kolb:1990vq,Chung:1998rq,Giudice:2000ex}. Therefore, this maximum temperature will play as an intermediate scale between the inflationary energy scale and the reheating temperature. Subsequently we show how this will effect the dark matter production mechanism during depending upon the dark matter mass.

\subsubsection{Dark matter relic abundance and its $(T_{re}, T_{max})$ dependence}
As we have emphasized already, we will describe how the present dark matter abundance is controlled by the CMB anisotropy through the inflationary model and its subsequent reheating phase. The present dark matter abundance parametrized by the normalized density parameter $\Omega_X$ can be expressed in terms of present day radiation abundance $\Omega_R$ $(\Omega_R h^2=4.3\times10^{-5})$, as follows
\begin{eqnarray}
\Omega_X h^2 = \left<E_X\right> \frac{X(T_F)}{R(T_F)} \frac{T_F}{T_{now}} \frac{A_F}{m_{\phi}} \Omega_R h^2
\end{eqnarray}
Where, $T_F$ is the temperature at very late time when dark matter and radiation comoving energy densities became constant. The current CMB temperature is given by $T_{now} = 2.35\times10^{-13}GeV$.
For our subsequent discussions, it is important to know the behavior of the dark matter abundance in terms of reheating parameters $(T_{re},M_X, \langle \sigma v\rangle)$. The analytic expressions for the dark matter abundance for different dark matter mass $M_X$ can be calculated following the references \cite{Giudice:2000ex}. As we have considered a generic equation of state parameter for the inflaton, we are interested in the generalized expression for dark matter abundance for arbitrary $w_{\phi}$. The dependence of relic abundance on reheating temperature, the dark matter mass, and annihilation cross section can be expressed as
\begin{eqnarray}
\Omega_X h^2 &\propto&  \left<\sigma|v|\right> M_X^4 Exp\left[-\frac{E(w_{\phi})M_X}{T_{max}}\right]~~ \text{for $M_X \gtrsim T_{max}$} \label{eq28}\\
	\Omega_X h^2 &\propto& \left< \sigma v\right> \frac{T_{re}^{\frac{7+3w_{\phi}}{1+w_{\phi}}}}{M_X^{\frac{9-7w_{\phi}}{2(1+w_{\phi})}}} \qquad \text{for~} T_{max} > M_X > T_{re}.
	\label{eq29}
\end{eqnarray}
One can particularly notice the non-trivial dependence of $w_{\phi}$ for two different dark matter mass range with respect to the $T_{max}$ as derived before. The proportionality factors for the above expressions and $E(w_{\phi})$ has a complicated dependence on $w_{\phi}$, which are important for our subsequent discussions. Furthermore, if we consider the dark matter mass to be less than the reheating temperature, the \textit{freeze-in} happens after the reheating is over. For that case the relic abundance can be expressed as \cite{Dev:2013yza}
\begin{eqnarray}
\Omega_X h^2 \propto \left< \sigma v\right> M_X~T_{re} \qquad \text{When~}M_X < T_{re},
\label{eq30}
\end{eqnarray}
For all the above three expressions for the dark matter abundance, we have considered the freeze-in mechanism. This essentially means that dark matter will never reach the equilibrium with the thermal bath. In the literature, these dark matter is known as feebly interacting dark matter(FIMP)\cite{fimp}. Following our recent work\cite{Maity:2018dgy}, in the next section, we will briefly outline the steps to connect CMB, reheating and dark matter abundance.

\subsection{Connection between CMB and dark matter: Methodology}
In this section we will briefly review the methodology developed in \cite{Maity:2018dgy} which describes the connection between the CMB anisotropy and the dark matter phenomenology.
For the canonical inflation, the important physical quantities are  $(N_k, H_k, V_{end}(\phi_k))$ which we have already defined. Since we would like to compute for a specific CMB  scale $k$ (CMB pivot scale), all those quantities are expressed as
\begin{equation}
H_k = \frac{\pi M_p \sqrt{r A_s}}{\sqrt{2}}~~
;~~	N_k = ln\left(\frac{a_{end}}{a_k} \right) =  \int_{\phi_k}^{\phi_{end}} \frac{1}{\sqrt{2\epsilon_V}} \frac{|d\phi|}{M_p} .
\label{Hk}
\end{equation}
where $(\phi_{end}$ is computed form the condition of end of inflation
\begin{equation}
\epsilon(\phi_{end}) = \frac{1}{2 M_p^2} \left(\frac{V'(\phi_{end})}{V(\phi_{end})}\right)^2 =1,
\end{equation}
While, the value of the scalar spectral index $n_s^k$ at the horizon crossing of a particular scale $k$ is
\begin{eqnarray}
n^k_s = 1 - 6\epsilon(\phi_k) + 2\eta(\phi_k)
\end{eqnarray}
Now, the above expression can be inverted to find $\phi_k$ in terms of the scalar spectral index. This relation provides us the connection between CMB anisotropy and the inflationary dynamics. Thus using this constraint from CMB on inflation, we will set the initial condition for the subsequent reheating dynamics. Next, we will solve the Boltzmann eq.(\ref{Boltzmann2}) for all the three components of energy densities and also the scale factor $a$ during the reheating phase. For numerical analysis one of the important quantities called reheating e-folding number $N_{re}$ measuring the duration of reheating is computed from the standard definition $N_{re} = \ln(a_{re}/a_{end})$. Where $a_{re}$ is the scale factor at the end of reheating which is defined as
\begin{eqnarray}
H(a_{re})^2= \frac{\dot{a}_{re}}{a_{re}} =\frac{8\pi}{3M_{Pl}^2} (\rho_\phi(\Gamma,n_s^k,M_X) +\rho_R(\Gamma,n_s^k,M_X) +\rho_X(\Gamma,n_s^k,M_X)) = \Gamma^2 .
\label{recond}
\end{eqnarray}
Where each of the energy components is written as an implicit function of reheating parameters and inflationary power spectrum $n_s^k$ as they are the solution of the Boltzmann equations with the initial condition at the end of inflation expressed in eq.(\ref{PhiI}).
This is also the scale factor at which we define the radiation temperature as another important parameter called reheating temperature with the following relation
\begin{eqnarray}
T_{re} \equiv T^{end}_{rad}=\left[ {30}/{\pi^2 g_*(T)}\right]^{1/4} \rho_R(\Gamma,n_s,M_X)^{1/4}.
\end{eqnarray}
At this point, it is important to note that in the present case the reheating temperature cannot be straightforwardly expressed in terms of inflaton decay width $\Gamma$, as can be clearly seen from the above equation. Never the less, the cosmological scale $k$ observed in CMB is originated during inflation at the horizon scale $H_k$, and then evolved through the subsequent reheating and radiation phase. Therefore, an important relation has been can established among  $(H_k, T_{re}, N_{re}, T_0)$ in  \cite{Dai:2014jja}, as
\begin{equation}
T_{re} = \left(\frac{43}{11 g_{re}}\right)^{\frac{1}{3}}  \left(\frac{a_0 T_0}{k} \right) H_k e^{-N_k} e^{-N_{re}} .
\label{TreEq}
\end{equation}
where $T_0 = 2.725K$ is the present CMB temperature, $g_{re}$ is the effective number of light species, and $H_0$ is the present value of the Hubble parameter. Hence one gets an important connection among the CMB anisotropy at a particular scale $k$, inflation and the reheating period through the following equation,
\begin{eqnarray}
\left(\frac{43}{11 g_{re}}\right)^{\frac{1}{3}}  \left(\frac{a_0 T_0}{k} \right) H_k e^{-N_k} e^{-N_{re}} =\left[ {30}/{\pi^2 g_*(T)}\right]^{1/4} \rho_R(\Gamma,n_s,M_X)^{1/4}
\end{eqnarray}
However, following the reference \cite{Maity:2018dgy}, in addition to radiation, we also considered the production of dark matter. Therefore,
the observation of present dark matter abundance $\Omega_X h^2 = 0.12$ will lead to the following another important connection between CMB anisotropy, reheating and the dark matter phenomenology. Following the relations given in Eqs.(\ref{eq28},\ref{eq29},\ref{eq30}), the aforementioned connection are expressed as follows,
\begin{eqnarray} \label{abundance}
&&\Omega_X h^2 \propto  \left<\sigma|v|\right> M_X^4 Exp\left[-\frac{E(w_{\phi})M_X}{T_{max}}\right]~~ \text{for $M_X \gtrsim T_{max}$} \nonumber\\
&&\Omega_X h^2 \propto \left< \sigma v\right> \frac{T_{re}^{\frac{7+3w_{\phi}}{1+w_{\phi}}}}{M_X^{\frac{9-7w_{\phi}}{2(1+w_{\phi})}}} \propto  \frac{\left< \sigma v\right>}{M_X^{\frac{9-7w_{\phi}}{2(1+w_{\phi})}}} \left[\left(\frac{a_0 T_0}{k} \right) H_k e^{-N_k} e^{-N_{re}}\right]^{\frac{7+3w_{\phi}}{1+w_{\phi}}}\qquad \text{for~} T_{max} > M_X > T_{re} \nonumber\\
&&\Omega_X h^2 \propto \left< \sigma v\right> M_X~T_{re} \propto  \left< \sigma v\right> M_X \left(\frac{a_0 T_0}{k} \right) H_k e^{-N_k} e^{-N_{re}} \qquad \text{When~}M_X < T_{re}.
\end{eqnarray}
Depending upon the mass of the dark matter we can clearly see the behavior of the dark matter parameter space intimately connected to the CMB scale and the associated temperature.
To this end let us also count the number of independent parameter of our study. At the current epoch of our universe, we have two observables corresponding to the CMB anisotropic power spectrum and the dark matter abundance. Based on our assumption that the radiation and the dark matter are produced through inflaton decay, we have four parameters $(\phi_*, \langle \sigma v\rangle,\Gamma, M_X)$ during reheating. Therefore, given the initial condition set by the CMB on the inflaton and present dark matter abundance, we have two independent parameters $(\phi_*, M_X)$ left.
With this strategy, we will study the Boltzmann equations during reheating and show how for a given set $(\phi_*,M_X)$, we can constrain the dark matter annihilation cross-section through the CMB anisotropy.

\subsubsection{Numerical results, constraints on the model}

With the help of formalism described above, in this section, we would like to describe the constraints on various parameters of our model. As we have already discussed, for a particular model with $n$ value, we have two independent parameters $(\phi_*,M_X)$. The first and foremost point we would like to make is that the production of dark matter does not have much effect on the determination of reheating temperature for all the models. However it is important to mention that specifically for the dark matter mass $M_X > T_{re}$, it is the reheating period which plays important role in determining the present dark matter abundance.
In the following we have considered $n=2,4,6,8$ and three sample values of $\phi_* = 10,1,0.1$ in unit of Planck mass. Before we go to
the quantitative discussion, let us point out the general results for all the models and there generic behavior for different parameters values.
For the class of minimal models we discussed, the spectral index is expressed as
\begin{equation}
n^k_s \simeq 1 - \frac{2n(n+1)M_p^2}{\phi_*^2} \frac{1}{\tilde{\phi}_k^{n+2}}
\end{equation}
Where, $\phi_k$ is the field value of a mode $k$ at horizon crossing. This equation can be inverted to find $\phi_k$ in terms of $n_s$. Once, we know this relation all the quantities such as tensor-to-scalar ratio $r$, the efolding number between the horizon crossing of a particular mode $k$ and the end of inflation $N_k$ and the parameter $m$(for $n \neq 4$) or $\lambda$(for $n = 4$) can be found in terms of $n_s$ and $\phi_*$.

For $n<4$, in $(n_s~vs~N_{re})$ plot, our generic observation is that with the increasing $n_s$, the reheating efolding number $N_{re}$ is decreasing as seen in fig.(\ref{n2_nsnret}a), and as a consequence reheating temperature $T_{re}$ is increasing shown in fig.(\ref{n2_nsnret}b). This behavior suggest  the existence of maximum reheating temperature $T_{re}^{max}$ thereby providing the maximum allowed scalar spectral index $n_s^{max}$. This also indicates the instantaneous reheating for $n_s = n_s^{max}$. On the other hand for $n>4$, what we observed is the opposite that is the reheating efolding number $N_{re}$ is increasing as seen in fig.(\ref{n6_nsnret}a), and as a consequence reheating temperature $T_{re}$ is decreasing with $n_s$, as shown in fig.(\ref{n6_nsnret}b). The maximum reheating temperature corresponding to a minimum possible $n_s^{min}$. In this case the instantaneous reheating occurs for $n_s = n_s^{min}$. This feature also have been observed in work \cite{Dai:2014jja}, with the effective single fluid equation of state formalism.
However, another important fact that we observed is the existence of possible maximum reheating temperature $T_{re}^{max}\simeq 10^{15}$ GeV irrespective the of different parameters in our models. For all values of $n$ and $\phi_*$, we found this temperature has an important connection with the CMB anisotropy. As we have already described in detail, by tuning the value of $\phi_*$, we can go from large field to small field inflation model. However, our prediction of $T_{re}^{max}$ appeared to be independent of the inflation model. We will elaborate on this issue in our future publication.
    Never the less, from the numerical fitting we express the reheating temperature $T_{re}$ in terms of spectral index $n_s$ with the following approximate relation,
\begin{eqnarray}
log_{10}\left( T_{re}\right) \simeq Q_p\left[A + B(n_s - 0.962) + C(n_s - 0.962)^2 \right].
\label{nsTrefor}
\end{eqnarray}
Where, the dimensionless constants $A=4$, $B =1.5\times10^3$ and $C=6\times 10^4$ for $n=2$, $A=20$, $B =-5\times10^3$ and $C=-1.5\times 10^4$ for $n=6$ and $A=20$, $B =-3\times10^3$ and $C=-1.5\times 10^4$ $n=8$. The proportionality constant $Q_p$ is $\phi_*$ dependent constant.
 From the above eqs.(\ref{nsTrefor}) and (\ref{abundance}), we can clearly build an important connection between the dark matter abundance and the CMB anisotropy which is directly connected to the scalar spectral index $n_s$.
In the next section, we will present our detail numerical results.
\begin{figure}[t]
	\centering
	\subfigure[]{\includegraphics[scale=0.4]{nsVsNre_n2}}
	\subfigure[]{\includegraphics[scale=0.4]{nsVsT_n2}}
	\caption{Evolution of (a) Reheating e-folding number $N_{re}$ during reheating and (b) the reheating temperature $T_{re}$(solid lines) and the maximum temperature $T_{max}$(dotted lines) with respect to $n_s$ for $n=2$ for three different values of $\phi_*$. If one extrapolates the solid and dotted lines, one will have maximum reheating temperature $T_{re}^{max}\simeq 10^{15}$ GeV. All the plots are independent of dark matter masses for a set of given initial condition.}
	\label{n2_nsnret}
\end{figure}
\subsubsection{Results and constraints: {\bf Model n=2}}
This is similar to the usual quadratic chaotic model. However, as we have seen before the new parameter $\phi_*$ controls the behavior of cosmological parameters in a significant way. From the Fig.(\ref{n2_nsnret}.a-b) we see the behavior  of $N_{re}$ and $(T_{re}, T_{max}$) in terms of $n_s$. It is evident that the allowed region of $n_s$ shifts towards lower values as we increase $\phi_*$, which in fact resembles the usual chaotic inflation. For $\phi_* < M_p$ we have small field inflation and the values of $(n_s,r)$ are in agreement with the observation.
\begin{figure}[t]
	\centering
	\subfigure[]{\includegraphics[scale=0.3]{nsVsCS_n2a01}}
	\subfigure[]{\includegraphics[scale=0.3]{nsVsCS_n2a1}}
	\subfigure[]{\includegraphics[scale=0.3]{nsVsCS_n2a10}}
	\caption{$(\langle\sigma v\rangle~vs~n_s)$ were plotted for three values of $\phi_*$ and $n=2$. The blue line with green shaded region corresponds to dark matter abundance $\Omega_X h^2 \leq 0.12$. We consider three possible values of $\phi_*$. For all the cases we have chosen a dark matter mass $M_X=200 TeV$. Red horizontal line corresponds unitarity bound described in the main text.}
	\label{n2nssig}
\end{figure}
\begin{figure}[t]
	\centering
	\subfigure[]{\includegraphics[scale=0.3]{MXVsCS_n2a01}}
	\subfigure[]{\includegraphics[scale=0.3]{MXVsCS_n2a1}}
	\subfigure[]{\includegraphics[scale=0.3]{MXVsCS_n2a10}}
	\caption{$(\langle\sigma v\rangle~vs~M_X)$ were plotted for three values of $\phi_*$ and $n=2$. Depending upon the available ranges of temperature within the $1\sigma$ range of $n_S$, we have considered three possible values of $n_s$. Minimum of the blue curve corresponds to the equality $M_X = T_{re}$. It is obvious that at this point the required value of the cross-section is minimum for a given abundance $\Omega_X h^2 = 0.12$}
	\label{n2mxsig}
\end{figure}
 \begin{table}[h!]
	\label{modeln2}
	Model parameters and associated constraints on dark matter parameters for n=2

	\begin{minipage}{.33\linewidth}
		\caption{$\phi_*=0.1M_p$}
		\begin{tabular}{|c|c|c|c}\hline
			{$n_s$} & {$N_{re}$} & {$T_{re}(GeV)$} & $\begin{array}{c}{\fontsize{7}{7}\selectfont \textit{$\langle \sigma v\rangle GeV^{-2}$}}\\ {\fontsize{7}{7}\selectfont \textit{$M_X(200 TeV)$}}\end{array}$\\\hline
			{0.966} & {44} & {$5.8$} & {\textit{unitarity}}\\
			{0.968} & {33} & {$2.4\times 10^{4}$}   & {$3\times 10^{-30}$}\\
			{0.970} & {20} & {$3\times 10^8$}       & {$5\times 10^{-38}$}\\
			{0.972} & {6}  & {$1.5\times 10^{13}$} &  {$1\times 10^{-42}$}\\\hline
		\end{tabular}
	\end{minipage
	\begin{minipage}{.33\linewidth}
		\caption{$\phi_*=1M_p$}
		\begin{tabular}{|c|c|c|c}\hline
			{$n_s$} & {$N_{re}$} & {$T_{re}(GeV)$} & $\begin{array}{c}{\fontsize{7}{7}\selectfont \textit{$\langle \sigma v\rangle GeV^{-2}$}}\\ {\fontsize{7}{7}\selectfont \textit{$M_X(200 TeV)$}}\end{array}$\\\hline
			{0.966}  & {42.5} & {$30.1$}             & {\textit{unitarity}}\\
			{0.968}  & {31}   & {$1.4\times 10^5$}   & {$6\times 10^{-34}$}\\
			{0.970}  & {18}   & {$2\times10^9$}      & {$2\times10^{-40}$}\\
			{0.9722} & {2}    & {$3.8\times10^{14}$} & {$3\times10^{-44}$}\\\hline
		\end{tabular}
	\end{minipage}%
	\begin{minipage}{.32\linewidth}
		\caption{$\phi_*=10M_p$}
		\begin{tabular}{|c|c|c|c|}\hline
			\textit{$n_s$} & \textit{$N_{re}$} & \textit{$T_{re}GeV$} & $\begin{array}{c}{\fontsize{7}{7}\selectfont \textit{$\langle \sigma v\rangle GeV^{-2}$}}\\ {\fontsize{7}{7}\selectfont \textit{$M_X(200 TeV)$}}\end{array}$\\\hline
			{0.962} & {36} & ${3.3\times 10^{3}}$ & {$3\times10^{-24}$}\\
			{0.964} & {26} & ${7.4\times 10^6}$   & {$2\times 10^{-36}$}\\
			{0.966} & {14} & ${4\times 10^{10}}$  & {$4\times10^{-40}$}\\
			{0.968} & {1.5} & ${6\times10^{14}}$  & {$3\times10^{-44}$}\\\hline
		\end{tabular}
	\end{minipage}%

\end{table}
For these small field models, therefore, within the $1\sigma$ range of $n_s$,  very small reheating temperature can be achieved. For example at the central value of $n_s=0.968 $, reheating temperature turned out to be $10^4\sim 10^5$ GeV for $\phi_*=(1,0.1)M_p$, whereas for $\phi_*=10 M_p$, reheating temperature is very high $\sim 10^{14}$ GeV. These fact can be seen from sample values given in the Tab.\ref{modeln2} and also from the fig.(\ref{n2_nsnret}.a-b). The lower limit of the value of $n_s$ is set from BBN constraints\cite{Kawasaki:1999na,Kawasaki:2000en,Steigman:2007xt,Fields:2014uja}, which is $T_{re} \sim 0.1GeV$. On the other hand, we can see from the all the plots as well as from the data tables, there exists a maximum temperature(corresponding to the instantaneous reheating) irrespective of the value of the scale $\phi_*$.

In addition to the one to one correspondence between $n_s$ and $T_{re}$, we simultaneously get constraints on the dark matter parameter space as shown in fig.(\ref{n2nssig}), and fig.(\ref{n2mxsig}). We have chosen a  sample dark matter mass $M_X=200 TeV$. It is clear that once $n_s$ is fixed, for a given $(\phi_*, M_X$), the scattering cross-section is also fixed. Therefore, within the $1\sigma$ range of $n_s$, the dark matter cross-section $\langle \sigma v\rangle$ is bounded given the value of dark matter abundance $\Omega_X h^2 = 0.12$. More importantly depending upon the values of $(M_X, n_s)$ the freeze-in will occur in three different regimes as we have already described in detail and also can be observed in the change of slopes of blue curves as a function of $n_s$ as shown in Figs.(\ref{n2nssig}-\ref{n2mxsig}). From our analysis, we see that for a fixed dark matter mass, annihilation cross-section increases with decreasing $n_s$.
However, as the cross section can not be arbitrarily large, the unitarity limit on $\left<\sigma v\right>_{MAX} = 8\pi/M_X^2$ restricts the allowed region of $n_s$ at its lowest value. From fig.(\ref{n2nssig}.a-b) when, $\phi_* =0.1M_p,~1M_p$ the production of dark matter particle of mass $200TeV$ for lower $n_s$ regime is forbidden due to the unitarity bound(shown as red line). The lowest possible value of $n_s$ turns out to be around $0.9664$ and the highest value $n_s^{max}\simeq 0.968$ for $\phi_*=10M_p$. For $\phi_*=(0.1, 1) M_p$, it turned out that $n_s^{max}\simeq 0.972$ as has already been noted before, depend upon the maximum attainable temperature for the model. This important constraint on the value of $n_s$ coming from dark matter sector could be very important to understand and needs further study.
Figs.(\ref{n2mxsig}.d-f) illustrates the conventional behavior of annihilation cross-section in terms of dark matter mass. However, an important point we would like to make again is that depending upon the value of inflationary power spectrum $n_s$ or CMB anisotropy, we can shed light on the possible production mechanism given the value of dark matter mass.

\subsubsection{Results and constraints: {\bf Model n=4}}
This is a very special case out of all the models. One of the main reasons is that for $n=4$, the inflaton behaves like radiation during reheating with the equation of state parameter $\omega_{\phi}=1/3$. It is known that during reheating if we consider the effective equation of state to be radiation like, then distinguishing the reheating period and the radiation dominated period becomes very difficult as can be seen from the schematic diagram fig.(\ref{scales}). This very fact makes the reheating parameters $(N_{re},T_{re})$ indeterministic which has been observed in \cite{Dai:2014jja}. However, in our analysis, we managed to resolve this fact by considering the explicit decay of inflaton into the analysis. Therefore, even though the inflaton equation of state is radiation like, the effective single fluid equation of state during reheating will not be exactly radiation like rather it has complicated time dependence, expressed as
\begin{figure}[t]
	\centering
	\subfigure[]{\includegraphics[scale=0.6]{nsVsNre_n4}}
	\subfigure[]{\includegraphics[scale=0.6]{nsVsT_n4}}
	\caption{Evolution of (a) Reheating e-folding number $N_{re}$ during reheating and (b) the reheating temperature $T_{re}$(solid lines) and the maximum temperature $T_{max}$(dotted lines) with respect to $n_s$ for $n=4$ for three different values of $\phi_*$. We have will same  $T_{re}^{max}\simeq 10^{15}$ GeV.}
	\label{n4_nsnret}
\end{figure}
 \begin{eqnarray}
 w_{eff} = \left\langle \frac {3 p_{\phi}+\rho_{rad}}{3(\rho_{\phi} + \rho_{rad} + \rho_X)}\right\rangle .
 \end{eqnarray}
 However, in deriving the above equation of state we assumed the perturbative decay of inflation. Keeping this fact in mind, our result for $n=4$ is given in the fig.(\ref{n4_nsnret}). It clearly shows the stiffness of the curve which can be attributed to the fact that effective equation of state varies very close to $1/3$. There exist some flat regions as we lower the value of $n_s$, which we think should be our numerical artefact. Therefore, if we discard that flat region, $n=4$ model predicts a precise value of scalar spectral index $n_s$, however, $N_{re},T_{re}$ become very sensitive to slight variation of it. As we can see from the fig.(\ref{n4_nsnret}), the scalar spectral index assumes $n_s\simeq (0.9575,0.966, 0.967)$ values for $\phi_*=(10,1,0.1) M_p$. Therefore, an important conclusion we can arrive from our analysis that super-Planckian values of $\phi_*>1 M_p$ are disfavored as long as $n_s$ measurement is concerned.
\subsubsection{Results and constraints: {\bf Model n=6}}
As we know all the usual power law inflationary models are large field and consequently predicts large $r$ value. However, because of new scale $\phi_*$, we were able to make our model perfectly compatible with the observation for even stiffer model such as $n=6$ and $n=8$.
\begin{figure}[h!]
	\centering
	\subfigure[]{\includegraphics[scale=0.4]{nsVsNre_n6}}
	\subfigure[]{\includegraphics[scale=0.4]{nsVsT_n6}}
	\caption{Evolution of (a) Reheating e-folding number $N_{re}$ during reheating and (b) the reheating temperature $T_{re}$(solid lines) and the maximum temperature $T_{max}$(dotted lines) with respect to $n_s$ for $n=6$ for three different values of $\phi_*$. One clearly see the change of slop in compared to $n<4$ models. However, we observed maximum reheating temperature to the same $T_{re}^{max}\simeq 10^{15}$ GeV.}
	\label{n6_nsnret}
\end{figure}
\begin{figure}[t]
	\centering
	\subfigure[]{\includegraphics[scale=0.3]{nsVsCS_n6a01}}
	\subfigure[]{\includegraphics[scale=0.3]{nsVsCS_n6a1}}
	\subfigure[]{\includegraphics[scale=0.3]{nsVsCS_n6a10}}
	\caption{$(\langle\sigma v\rangle~vs~n_s)$ were plotted for three values of $\phi_*$ and $n=6$. For all the cases we have chosen a dark matter mass $M_X=2\times 10^{11}  GeV$.}
	\label{n6nssigma}
\end{figure}
\begin{figure}[t]
	\centering
	\subfigure[]{\includegraphics[scale=0.3]{MXVsCS_n6a01}}
	\subfigure[]{\includegraphics[scale=0.3]{MXVsCS_n6a1}}
	\subfigure[]{\includegraphics[scale=0.3]{MXVsCS_n6a10}}
	\caption{$(\langle\sigma v\rangle~vs~M_X)$ were plotted for three values of $\phi_*$ and $n=6$. General descriptions of the plots are same as before.}
	\label{n6mxsigma}
\end{figure}
\begin{table}[h!]
	\label{modeln6}
	Model parameters and associated constraints on dark matter parameters for n=6

	\begin{minipage}{.33\linewidth}
		\caption{$\phi_*=0.1M_p$}
		\begin{tabular}{|c|c|c|c}\hline
			{$n_s$} & {$N_{re}$} & {$T_{re}(GeV)$} & $\begin{array}{c}{\fontsize{7}{7}\selectfont \textit{$\langle \sigma v\rangle GeV^{-2}$}}\\ {\fontsize{7}{7}\selectfont \textit{$M_X(10^{11} GeV)$}}\end{array}$\\\hline
			{0.9678} & {3} & ${4\times 10^{13}}$ & ${3\times 10^{-49}}$\\
			{0.968} & {5} & ${2 \times 10^{12}}$ & $7\times 10^{-48}$\\
			{0.969} & {20} & $2\times 10^5$ & $9\times 10^{-34}$\\
			{0.9698} & {32} & {0.2} & $6\times 10^{-22}$\\\hline
		\end{tabular}
	\end{minipage
	\begin{minipage}{.33\linewidth}
		\caption{$\phi_*=1M_p$}
		\begin{tabular}{|c|c|c|c}\hline
			\textit{$n_s$} & \textit{$N_{re}$} & \textit{$T_{re}(GeV)$} & $\begin{array}{c}{\fontsize{7}{7}\selectfont \textit{$\langle \sigma v\rangle GeV^{-2}$}}\\ {\fontsize{7}{7}\selectfont \textit{$M_X(10^{11} GeV)$}}\end{array}$\\\hline
			{0.9682} & {2} & ${1\times 10^{14}}$ & ${9\times 10^{-50}}$\\
			{0.969} & {14} & ${3\times 10^8}$ & ${1\times 10^{-36}}$\\
			{0.970} & {29} & {10.58} & $unitarity$\\
			{0.972} & {32} & {0.3} & $unitairty$\\\hline
		\end{tabular}
	\end{minipage}%
	\begin{minipage}{.32\linewidth}
		\caption{$\phi_*=10M_p$}
		\begin{tabular}{|c|c|c|c|}\hline
			\textit{$n_s$} & \textit{$N_{re}$} & \textit{$T_{re}(GeV)$} & $\begin{array}{c}{\fontsize{7}{7}\selectfont \textit{$\langle \sigma v\rangle GeV^{-2}$}}\\ {\fontsize{7}{7}\selectfont \textit{$M_X(10^{11} GeV)$}}\end{array}$\\\hline
			{0.9632} & {2} & ${1.9\times 10^{14}}$ & $6\times10^{-50}$\\
			{0.964} & {11} & ${6\times 10^9}$ & ${4\times 10^{-41}}$\\
			{0.965} & {23} & ${8.8\times 10^3}$ & ${unitarity}$\\
			{0.9656} & {30} & {2} & ${unitarity}$\\\hline
		\end{tabular}
	\end{minipage}%

\end{table}
In this sub-section we summarized all our analysis in the figs.(\ref{n6_nsnret},\ref{n6nssigma},\ref{n6mxsigma}) and the following tables (\ref{modeln6}) for $n=6$.

 Important difference with $n<4$ models turned out to be the slope of $(n_s~ vs~T_{re})$ and $(n_s~vs~N_{re})$ curve. This has also been explained in the schematic diagram fig.(\ref{scales}). This is coming from the stiff equation of state of the oscillating inflaton $w_{\phi} = (n-2)/(n+2) = 1/2 > 1/3$. The duration of reheating period ($N_{re}$) increases with increasing $n_s$. Therefore, unlike $n<4$ models, corresponding to the maximum reheating temperature we have minimum possible values of $n_s$.
\subsubsection{Results and constraints: {\bf Model n=8}}

For $n=8$, the qualitative behavior of all the plots will be same as $n=6$. Therefore we summarize all our results in the
table-\ref{modeln8} and the respective plots.
Important to see that very small reheating temperature can be reached in this model. At the central value of $n_s=0.968 \pm 0.006$, we see reheating temperature could be $10^4\sim 10^5$ GeV given in the tab.(\ref{modeln8}). The lowest value of $n_s$ can be set from the minimum possible reheating temperature coming from BBN constraints\cite{Kawasaki:1999na,Kawasaki:2000en,Steigman:2007xt,Fields:2014uja}, which is $T_{re} \sim 0.1GeV$.
\begin{figure}[h!]
	\centering
	\subfigure[]{\includegraphics[scale=0.45]{nsVsNre_n8}}
	\subfigure[]{\includegraphics[scale=0.45]{nsVsT_n8}}
	\caption{Evolution of (a) Reheating e-folding number $N_{re}$ during reheating and (b) the reheating temperature $T_{re}$(solid lines) and the maximum temperature $T_{max}$(dotted lines) with respect to $n_s$ for $n=8$ with the same behavior as $n=6$.}
	\label{n8_nsnret}
\end{figure}
\begin{table}[h!]
	\label{modeln8}
	Model parameters and associated constraints on dark matter parameters for n=8

	\begin{minipage}{.33\linewidth}
		\caption{$\phi_*=0.1M_p$}
		\begin{tabular}{|c|c|c|c}\hline
			{$n_s$} & {$N_{re}$} & {$T_{re}(GeV)$} & $\begin{array}{c}{\fontsize{7}{7}\selectfont \textit{$\langle \sigma v\rangle GeV^{-2}$}}\\ {\fontsize{7}{7}\selectfont \textit{$M_X(10^{11} GeV)$}}\end{array}$\\\hline
			{0.9668} & {2} & ${3 \times 10^{13}}$ & ${4 \times 10^{-49}}$\\
			{0.968} & {13} & ${1 \times 10^8}$ & ${7 \times 10^{-35}}$\\
			{0.969} & {22} & ${2 \times 10^{3}}$ & \textit{unitairty}\\
			{0.9698} & {30} & {0.2} & \textit{unitarity}\\\hline
		\end{tabular}
	\end{minipage
	\begin{minipage}{.33\linewidth}
		\caption{$\phi_*=1M_p$}
		\begin{tabular}{|c|c|c|c}\hline
			\textit{$n_s$} & \textit{$N_{re}$} & \textit{$T_{re}(GeV)$} & $\begin{array}{c}{\fontsize{7}{7}\selectfont \textit{$\langle \sigma v\rangle GeV^{-2}$}}\\ {\fontsize{7}{7}\selectfont \textit{$M_X(10^{11} GeV)$}}\end{array}$\\\hline
			{0.9672} & {1} & ${3 \times 10^{14}}$ & ${4 \times 10^{-50}}$\\
			{0.968} & {8} & ${6 \times 10^{10}}$ & ${2 \times 10^{-45}}$\\
			{0.9692} &{19} & ${1 \times 10^5}$ & \textit{unitarity}\\
			{0.9702} & {29} &{0.72} & \textit{unitarity}\\\hline
		\end{tabular}
	\end{minipage}%
	\begin{minipage}{.32\linewidth}
		\caption{$\phi_*=10M_p$}
		\begin{tabular}{|c|c|c|c|}\hline
			\textit{$n_s$} & \textit{$N_{re}$} & \textit{$T_{re}(GeV)$} & $\begin{array}{c}{\fontsize{7}{7}\selectfont \textit{$\langle \sigma v\rangle GeV^{-2}$}}\\ {\fontsize{7}{7}\selectfont \textit{$M_X(10^{11} GeV)$}}\end{array}$\\\hline
			{0.9638} & {1} & ${4\times 10^{14}}$ & ${3 \times10^{-50}}$\\
			{0.964} & {2} & ${1\times 10^{14}}$ &${1 \times 10^{-49}}$\\
			{0.965} & {10} & ${1 \times 10^{10}}$ & ${3 \times 10^{-43}}$\\
			{0.9674} & {29} & {1} & \textit{unitarity}\\\hline
		\end{tabular}
	\end{minipage}%

\end{table}

\begin{figure}[h!]
	\centering
	\subfigure[]{\includegraphics[scale=0.3]{nsVsCS_n8a01}}
	\subfigure[]{\includegraphics[scale=0.3]{nsVsCS_n8a1}}
	\subfigure[]{\includegraphics[scale=0.3]{nsVsCS_n8a10}}
	\caption{$(\langle\sigma v\rangle~vs~n_s)$ were plotted for $n=8$. Descriptions are same as  previous plots. For all the plots we have chosen dark matter mass $M_X=2\times 10^{11} GeV$.}
	\label{n8_nsnret}
\end{figure}

\begin{figure}[t]
	\centering
	\subfigure[]{\includegraphics[scale=0.3]{MXVsCS_n8a01}}
	\subfigure[]{\includegraphics[scale=0.3]{MXVsCS_n8a1}}
	\subfigure[]{\includegraphics[scale=0.3]{MXVsCS_n8a10}}
	\caption{$(\langle\sigma v\rangle~vs~M_X)$ were plotted for $n=8$.}
	\label{n8_Mxsigma}
\end{figure}

\newpage
\section{\label{conclusion} Conclusions and future directions}
In this paper, we have done the reheating constraint analysis in detail for a class of inflationary models with non-standard potential \cite{Maity:2016uyn}. Our focus was on understanding the inflation model as well as dark matter phenomenology via reheating phase. For both the analysis, the CMB plays a very important role. We, therefore, follow the formalism developed in \cite{Maity:2018dgy} where the dark matter parameter space is constrained via CMB anisotropy and the reheating dynamics. Constraints on dark matter parameter space are dependent upon the specific model of inflation. As mentioned throughout the paper, in our recent work on minimal inflationary cosmologies we had discussed in detail the role of inflationary scale $\phi_*$. By tuning it, we also obtained both high scale as well as low scale inflations which are observationally viable. In this paper we considered those inflationary models and further extended our work on understanding the role of that scale on the reheating phase as well as on dark matter phenomenology.

As we all know, the reheating phase, which is relatively less understood due to lack of direct observables, has potential to explain many unanswered questions related to dark matter, baryogenesis etc. In this paper, we have studied in detail the reheating period considering the perturbative decay of inflaton into radiation and radiation into dark matter component.
One of the important findings from our analysis is the existence of maximum reheating temperature $T_{re}^{max}\simeq 10^{15}$ GeV, which is independent of large or small scale inflation model. Therefore, it is the CMB scale which dictates the value of maximum possible reheating temperature. Corresponding to that maximum $T_{re}$, we also have either maximum possible $n_s^{max}$ for all the models with $n<4$ or minimum possible $n_s^{min}$ for models with $n>4$. This fact can be easily understood from the illustration in fig.\ref{scales}.
As we have already emphasized the goal was to connect the inflation, reheating phase with the CMB anisotropy and the dark matter abundance. Therefore, we further study how the inflationary scalar spectral index fixes the dark matter annihilation cross-section for a given mass of the dark matter once we specify the model. Here is our main findings which tells us how uniquely  we can constrain the dark matter parameter space $(M_X, \langle \sigma v\rangle)$, through the following important relations,
	\begin{eqnarray} \label{abundance}
	&&\Omega_X h^2 \propto  \left<\sigma|v|\right> M_X^4 Exp\left[-\frac{E(w_{\phi})M_X}{T_{max}}\right]~~ \text{for $M_X \gtrsim T_{max}$} \nonumber\\
	&&\Omega_X h^2  \propto  \frac{\left< \sigma v\right>}{M_X^{\frac{9-7w_{\phi}}{2(1+w_{\phi})}}} \left[\left(\frac{a_0 T_0}{k} \right) H_k e^{-N_k} e^{-N_{re}}\right]^{\frac{7+3w_{\phi}}{1+w_{\phi}}}\qquad \text{for~} T_{max} > M_X > T_{re} \nonumber\\
	&&\Omega_X h^2\propto  \left< \sigma v\right> M_X \left(\frac{a_0 T_0}{k} \right) H_k e^{-N_k} e^{-N_{re}} \qquad \text{When~}M_X < T_{re}.
	\end{eqnarray}
The non-trivial relation among the dark matter abundance $(\Omega_X h^2)$ , the general inflaton equation state $(\omega_{\phi})$ , CMB scale $k$, and mass of the dark matter $M_X$ is our main result which has not been reported before.
Here an important point we would like to emphasize again that the dark matter is produced by Freeze-in mechanism. Therefore, detailed analysis for freeze out dark matter based on our formalism will be reported elsewhere.

Nevertheless
considering the central value of $n_s \sim 0.968$ as given by PLANCK our model predictions are summarized in table-\ref{summarycon}.
\begin{table}[t]
	\label{summarycon}
Summary table for central value of $n_s$ with dark matter mass taken to be $M_X =200$ TeV

	\begin{minipage}{.32\linewidth}
		\caption{$\phi_*=0.1M_p$}
		\begin{tabular}{|c|c|c|c|}\hline
			{$n$} & {$N_{re}$} & {$T_{re}(GeV)$} & $\begin{array}{c}{\fontsize{7}{7}\selectfont \textit{$\langle \sigma v\rangle GeV^{-2}$}}\\ {\fontsize{7}{7}\selectfont \textit{$M_X(200 TeV)$}}\end{array}$\\\hline
			{2} & {33} & {$2.4\times 10^{4}$}   & {$3\times 10^{-30}$}\\
			{4} & {-}  & {-} &  {-}\\
			{6} & {5} & ${1.6 \times 10^{12}}$ & $7\times 10^{-42}$\\
			{8} & {-} & {-} & {-}\\
			\hline
		\end{tabular}
	\end{minipage
	\begin{minipage}{.32\linewidth}
		\caption{$\phi_*=1M_p$}
		\begin{tabular}{|c|c|c|c|}\hline
			{$n_s$} & {$N_{re}$} & {$T_{re}(GeV)$} & $\begin{array}{c}{\fontsize{7}{7}\selectfont \textit{$\langle \sigma v\rangle GeV^{-2}$}}\\ {\fontsize{7}{7}\selectfont \textit{$M_X(200 TeV)$}}\end{array}$\\\hline
			{2}  & {31}   & {$1.4\times 10^5$}   & {$6\times 10^{-34}$}\\
			{4} & {-}  & {-} &  {-}\\
			{6} & {13} & ${1.3\times 10^{8}}$ & ${7\times 10^{-38}}$\\
			{8} & {8} & ${6 \times 10^{10}}$ & ${2 \times 10^{-40}}$\\
			\hline
		\end{tabular}
	\end{minipage}%
	\begin{minipage}{.32\linewidth}
		\caption{$\phi_*=10M_p$}
		\begin{tabular}{|c|c|c|c|}\hline
			\textit{$n$} & \textit{$N_{re}$} & \textit{$T_{re}(GeV)$} & $\begin{array}{c}{\fontsize{7}{7}\selectfont \textit{$\langle \sigma v\rangle GeV^{-2}$}}\\ {\fontsize{7}{7}\selectfont \textit{$M_X(200 TeV)$}}\end{array}$\\\hline
			{2} & {1.5} & ${6\times10^{14}}$  & {$3\times10^{-44}$}\\
			{4} & {-}  & {-} &  {-}\\
			{6} & {-}  & {-} &  {-}\\
			{8} & {-}  & {-} &  {-}\\
			\hline
		\end{tabular}
	\end{minipage}%

\end{table}
For $n=4$, the e-folding number during reheating($N_{re}$) and consequently the reheating temperature is highly sensitive to $n_s$ which can also be understood from fig.\ref{scales}. Therefore, we left those parameters blank in the summary table. From the above summary table we can clearly see that given the central value of $n_s = 0.968$ and dark matter mass $M_X =200$ TeV, few models can be ruled out based on the values of the model parameters for which prediction will be out of $1\sigma$ range of $n_s$ given by PLANCK. On the other hand, if one considers $n_s$ within the given range of PLANCK, we can give a bound on the possible range of values of annihilation cross-section for a given mass. An Important conclusion that we can make is that once the CMB anisotropy measurement fixes the value of $n_s$, we can completely fix the dark matter annihilation cross-section for a given mass considering the perturbative reheating process.
Therefore, it is very important to pinpoint the actual value of $n_s$ in the future experiment.
Understanding the dark matter phenomenology is one of the important challenges in the particle physics community. Therefore, our formalism and present analysis may shed some light on this subject. The important assumption of our analysis is the perturbative reheating. It is well known that the non-perturbative effects such as parametric resonance can be very efficient that can initiate the radiation dominance within few efolding numbers after the end of inflation. In that case the CMB will have very little information regarding the reheating phase. However, as shown in\cite{GarciaBellido:2008ab}, incorporating interactions amongst the produced particles, the parametric resonance can be delayed and as a result we will get an extended period of reheating. The validity of the perturbative regime in the context of reheating has been considered in\cite{Drewes:2017fmn} considering the effective equation of state or in \cite{Maity:2017thw} considering the explicit inflaton decay. However, the issue of self-resonance still remains. It has been shown in\cite{Lozanov:2016hid,Lozanov:2017hjm} that when considering the self-resonance, the homogeneous condensate will cease oscillating after a certain period, and ends up fragmenting into higher momentum modes of the inflaton field. For, certain form of the potentials, $V(\phi)\propto \phi^n$, the equation of state eventually ends up becoming $1/3$ instead of $(n-2)/(n+2)$. However, there is a certain period when this equation of state will be maintained and moreover, there are specific parameter range when the self-resonance will be inefficient. We are currently working on these issues of taking into account the non-perturbative effects.
Finally at the end let us point out an another important direction which we will report elsewhere. In our study we have solved for the homogeneous Boltzmann equation for all the energy components. However, it is well known that evolution of perturbation for every individual component would be extremely important to understand the small scale properties of CMB. Interestingly in our unified approach inflation and reheating dynamics control all the evolution. Therefore, any small scale CMB observables will certainly contain valuable information related to inflation. In this regard well known small scale $\mu$-type and $y$-type spectral distortions of CMB are extremely important. At present those distortion parameters are tightly constrained by COBE and FIRAS experiments, $|\mu| < 9 \times 10^{−5}$ and $ y < 1.5 \times 10^{−5}$ \cite{cobe}. However, future projected sensitivity for PIXIE \cite{pixie} and PRISM \cite{prism} experiments are within $ 10^{-8}\sim 10^{-9}$.The standard model of particle physics interactions already predicts those spectral distortion parameters to be $\sim 10^{-8}$ \cite{spectral1,spectral2}. Keeping in mind the future experimental sensitivity, it would be extremely important to understand various other physics processes which can give rise to those distortions. Therefore, by using our unified approach we can further constrain inflation models considering those distortion parameters.

\section{Acknowledgement}
We thank our HEP and Gravity group members for their valuable comments and discussions.

  \hspace{0.5cm}

redpajama_set_name: RedPajamaArXiv


text:
Top Ten Worst Health Stories of 2019 (Part 2)
December 13, 2019 | By Dr. Ronald Hoffman
It's that time of year: I'm reprising my annual rendition of the year's worst health stories. These stories distinguish themselves by perpetuating stereotypes about the dangers or inefficacy of supplements or natural therapies, or alternatively, parroting outmoded concepts about diet.
In last week's article, I rolled out my first 5 picks for Worst Health Stories of 2019; this week, I'm rounding out my Top Ten with the following selections:
Dietary supplement regulation: In 2019, the medical establishment, in league with mainstream media, continued its drumbeat of calls for enhanced regulation of what they term "the Wild West" of the supplement industry. Amidst the relaxed regulatory climate of the Trump administration, editorial writers at The Lancet worry that "the FDA has shifted further in its pro-industry stance, continuing to prioritise unfettered consumerism at the expense of increasing safety and accountability." For a more balanced assessment—and pushback against the claim that the supplement industry is unregulated—listen to my recent podcast with Brian Wommack, Vice President of the Council for Responsible Nutrition (CRN).
Vitamin D backlash: A predictable news cycle is emerging on vitamin D. Once shunned as a potentially toxic vitamin, to be given only in low doses and good merely for prevention of rickets, its potential as a hedge against infections, inflammation, painful conditions, autoimmunity, neurodegenerative disorders, childhood illness and metabolic syndrome dominated headlines during the recent "D Decade".
Now, the doubters are back in force. In an article—typical of its genre—entitled "Is vitamin D hype wishful thinking", Web MD reviews studies which appear to throw shade on D's benefits. They go so far as to recommend that, for those with normal levels of vitamin D, supplements are unnecessary: "The best way to be sure to get enough of the nutrient is to go outside and get some sun."
Good luck if you live anywhere in the U.S. above the latitude of Charleston, South Carolina, especially from October to April. This November, I sat around a pool in southerly San Diego for two hours without sunscreen while conversing with a colleague—and my skin didn't even turn pink! And if you're over 65, your skin may have lost the capacity to synthesize vitamin D with UV exposure. Add to that genetic propensities for people to require more vitamin D, or medical conditions that deplete it. Even many of the nutritionally savvy people who become my patients were found to have suboptimal levels of D.
Let's stipulate that vitamin D is no universal panacea—to claim so would be simplistic. But most assuredly, the recent popularization of vitamin D supplementation has generated far more benefits than harms.
The War Against Keto: The popularization of the ketogenic diet has presented a dilemma for Consumer Reports, who used to dismiss it out of hand. This year they offered a more balanced assessment, admitting the Keto diet helps weight loss and may address certain conditions like Type 2 diabetes and seizure disorders. But they resurrect the worn shibboleth that Keto diets can result in ketoacidosis—a dangerous condition insulin-dependent diabetics get—conflating it with ketosis, a healthy and normal physiological adaptation that occurs during fasting or when carbs are zeroed out.
The Consumer Reports nutritionists admonish: "But carbohydrates as a group shouldn't be vilified . . . A healthy diet includes minimally processed whole grains, legumes, fruit, and vegetables." They just don't get it!
Outmoded diet advice: U.S. News & World Report (they're still in existence?) doubles down on its ratings of popular diets. U.S. News "evaluated and ranked the 41 diets below with input from a panel of health experts", ranking the Mediterranean Diet and DASH Diets #1 and #2 respectively.
Their version of the Mediterranean Diet is "low in red meat, sugar and saturated fat—not withstanding the fact that healthy Spaniards, Provencales, Italians, Greeks, North Africans, Balkan peoples and Turks consume plenty of lamb, goat, and full-fat cheese and yogurt, not to mention pork in non-Muslim countries.
The DASH Diet calls for even more draconian restriction of saturated fat. U.S. News says, "It emphasizes the foods you've always been told to eat (fruits, veggies, whole grains, lean protein and low-fat dairy)."
By contrast, the Paleo Diet and Keto Diet are ranked near the bottom, at #33 and #38 respectively. Their comments about the Paleo Diet reveal a distinct bias: "The high-protein diet is ranked poorly among U.S. News experts, who consider it too restrictive to be healthy or sustainable."
The Whole 30 Diet, which I find enormously helpful for many of my patients (see my review here) is relegated to a tie with the Keto Diet at #38.
These outmoded ratings are at variance with recent studies that exonerate meat and full-fat dairy, and underscore the utility of carbohydrate restriction for combatting the scourges of obesity and metabolic syndrome.
Good fats/bad fats: Non-profit website HelpGuide.org continued to proffer—like many others of its ilk—anachronistic diet recommendations in 2019, repeating the worn shibboleth: "'Bad' fats, such as artificial trans fats and saturated fats, are guilty of the unhealthy things all fats have been blamed for—weight gain, clogged arteries, an increased risk of certain diseases and so forth."
By contrast, they claim, "good fats" include soybean and safflower oils. The authors of these recommendations ignore evidence that refined polyunsaturated oils are potent sources of pro-inflammatory Omega-6 fats, whose presence disrupts metabolism. Yet many "authoritative" sources—including the American Heart Association—continue to erroneously recommend that Americans replace butter, lard, and tropical oils like coconut with corn, soy, safflower and canola oils.
And there you have it! My roundup of the worst health stories of the year. Join me on Facebook or Twitter, and let me know which story you thought really scraped the bottom of the barrel.
5 supplements to use on keto
Ask Leyla: Do paleo/keto diets cause gluconeogenesis ("new glucose")
The plot against Keto

redpajama_set_name: RedPajamaCommonCrawl


The death Tuesday night of BERTHA JOHNSON, 9 years old, in a fall from a window of the children's ward at the City Hospital, resulted in orders Wednesday for iron bars for the windows. They were ordered by Mrs. Ada Belle Harris, acting superintendent, after she had investigated the circumstances of the girl's death.
At the present time four of the nine windows of the porch of the children's ward are without bars, and it was through one of these the JOHNSON girl fell. There was a screen on the window, but whether it gave way when the girl leaned against it or whether she unhooked it has not been determined. There are small hooks at the bottom, easily within the reach of a child.
It was against the rules for the windows not barred to have been open, Mrs. Harris said. However, she added, this rule probably had been violated often during the hot weather.
The bars, which are already on five of the windows, are strong enough to prevent such an accident.
The nurses had been called to the opposite end of the ward when the JOHNSON girl fell. The ward is on the second floor and death resulted almost instantly. Dr. C. J. Armstrong, who issued the statement Tuesday night of the circumstances of the child's death, said she was leaning out of the window in order to wave to her father, who was a patient in the next wing of the hospital. Coroner Roy L. Carter, after an investigation Wednesday, disagreed with the theory of Dr. Armstrong. He said it was impossible for the girl to have seen her father from the window from which she fell.
Dr. Carter said the girl was looking at persons on the street, and that the screen gave way when she leaned against it. This theory is upheld by Mrs. Harris.
Hospital authorities at first refused to give out information. A patrolman sent to the hospital to investigate, was refused particulars. Dr. Armstrong later issued a statement.
The Johnson family lived at 1200 West Kentucky Street. The father was to have been dismissed from the hospital Wednesday, but the shock of his daughter's death prevented his leaving.
Funeral services for the girl will be held at 3 o'clock this afternoon at the residence. Burial will be in Evergreen Cemetery.


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text:
Saudi Arabia executes 47 people in one day including Shia cleric
Scores of 'terrorists' put to death amid warning from Iran that executing prominent Shia cleric Sheikh Nimr al-Nimr would 'cost Saudi Arabia dearly'
Sat 2 Jan 2016 02.40 EST Last modified on Sat 2 Jan 2016 06.46 EST
A Saudi anti-government protester carries a poster with the image of jailed Shia cleric Sheikh Nimr al-Nimr. Photograph: Uncredited/AP
Saudi Arabia has executed 47 people for terrorism, including the prominent Shia cleric Sheikh Nimr al-Nimr.
Most of those executed on Saturday were involved in a series of attacks carried out by al-Qaida from 2003-06, the interior ministry said.
Middle East condemns Saudi execution of Shia cleric - live
However, it also detained hundreds of members of its Shi'ite minority after protests in 2011-13, during which several policemen were killed in shooting and petrol bomb attacks. Several of those held had been sentenced to death.
The interior ministry statement began with Koranic verses justifying the use of execution and state television showed footage of the aftermath of al-Qaida attacks in the last decade. Saudi grand mufti Sheikh Abdulaziz Al al-Sheikh appeared on television soon after to describe the executions as just.
Iran's Shia leadership has warned that executing Nimr "would cost Saudi Arabia dearly".
In October 2015 Saudi Arabia's supreme court rejected an appeal against the death sentence passed earlier on Nimr, who had called for pro-democracy demonstrations and whose arrest in 2012 sparked protests in which three people died.
Nimr had long been regarded as the most vocal Shia leader in the eastern Saudi province of Qatif, willing to publicly criticise the ruling al-Saud family and call for elections. He was, however, careful to avoid calling for violence, analysts say.
That did not prevent the Saudi interior ministry from accusing him of being behind attacks on police, alongside a group of other suspects it said were working on behalf of Shia Iran, the kingdom's main regional rival.
The executions are Saudi Arabia's first in 2016. At least 157 people were put to death last year, a big increase from the 90 people killed in 2014.


Masters of Invention Poetry Workshop with Sophie McKeand, for children and young people aged 8-13.
Creating nonsense words and language was something Roald Dahl delighted in with his phlantasmagozgical use of language and ribble ticklistic plays-on-words.
Working together our group will climb inside words, chew and garble and tear them up in order to create a new poem.
£2.00 Places are limited: please book ahead.
Supported by Literature Wales and Welsh Government as part of the Roald Dahl 100 Wales celebrations.


October 31, 2014- Minnesota activists are pushing the state to move faster to reduce mercury pollution in the St. Louis River. More than 100 people attended a citizen's forum last month.
October 30, 2014- A cap-and-trade provision of California's Assembly Bill 32, also known as AB 32 or the Global Warming Solutions Act of 2006, has become a hot button issue for the state.
Energy Companies Develop Plan To Ship Power From Wyoming To L.A.
October 27, 2014- Four energy companies are collaborating on an $8-billion plan to ship electric power produced in Wyoming to Southern California via transmission lines, with help from an energy-storage and generation system to be built in Utah.
October 22, 2014- Goodyear said it will begin using the ash from burned rice husks as a source of silica for its tires. The rice husk waste, burned to produce electricity, is normally sent to landfills.
October 20, 2014- U.S. engineers invented a new battery made of three molten metals that aims to help store energy on the scale of a national power grid.
October 17, 2014- A former principal at an Australian primary school in Thornbury relinquished his position after an investigation into the exposure of children to asbestos found him negligent.
October 16, 2014- A discarded cigarette is easy to overlook, but tobacco litter can be a big problem in areas around San Francisco Bay.
October 13, 2014- A plan for municipalities and some businesses to make potentially expensive reductions in water pollution in the Milwaukee River basin is two years behind schedule.
October 10, 2014- Montana is working on ways to reduce climate pollution by improving energy efficiency and increasing other forms of cleaner renewable resources -- without closing coal plants. Gov. Steve Bullock recently outlined a list of options that could diversify and clean the state's energy landscape.
October 8, 2014- As the health hazards of radon become better understood by the general public, home testing for the invisible, odorless gas has become more common.
October 6, 2014- The preponderance of mercury is making news as scientists conclude the most comprehensive estimate of mercury in the environment.
October 2, 2014- The El Paso, Texas Department of Health has announced that more than 700 babies and 40 health workers have been exposed to tuberculosis in a local hospital.


Our good friends at the Skirball Cultural Center decided to feature one of our images from Kacey and Vince's wedding on a poster they're going to display on their garden walkway. It's one of our favorite venues so we're excited to be featured by them! If you're having a Skirball ceremony and/or reception give us a shout – we'd love to chat with you about photography for your big day!
How clever this picture. is. I am so proud of you Josh and Jill. Your work is outstanding. Who ever uses you as their Photographers will be very pleased.


Habitat for Humanity Roaring Fork Valley built these three energy efficient homes in Rifle in 2010. The homes at center and on right achieved ENERGY STAR ratings of 39 and 25, meaning they use 61 and 75 percent less energy than a standard home. The home at left achieved a net-zero energy rating because its larger solar electric system is expected to offset the minimal natural gas usage for hot water and speace heating.
Living in a home with no net energy consumption might sound like a high-tech dream in western Colorado, but it's already happening for one family in Rifle.
The latest home built by the nonprofit Habitat for Humanity of the Roaring Fork Valley is predicted to be "net zero" in energy use, meaning the home will produce as much energy as it uses. Organizers say the net zero home may be a first in Garfield County.
The net zero home building process is easier to achieve than one might think, said Silt resident Dan Becker, a certified Home Energy Rating System (HERS) rater and Building Performance Institute analyst, who consulted on the Rifle project. It's as straightforward as closely following the guidelines for ENERGY STAR homes, starting from the design phase, and insuring a continuous thermal barrier, Becker said. Those guidelines can be found at www.EnergyStar.gov.
Above, framing of the net-zero home was thoroughly insulated with foam, including headers, beams and joists.
Below, walls were heavily insulated to keep the home warm in winter and cool in summer.
Below, the Sunsense Solar crew volunteered their labor to install the solar electric system on the Habitat for Humanity home.
A tightly sealed home built with ENERGY STAR windows and doors, upgraded insulation and added exterior foam board in the walls can lead to at least a 20 percent reduction in the size of the heating system, which is less expensive to operate, Becker said. The net zero home in Rifle is so well insulated that air conditioning should not be needed.
"There are no breaks in the thermal envelope, from below grade continuous to the attic," Becker said, noting that the solar electric system is the "whistles and bells" capping off a well-built ENERGY STAR home.
Habitat for Humanity organizers, volunteers and donors completed three, two-story, approximately 1,200-square-foot homes in Rifle's Rim Rock neighborhood, following ENERGY STAR standards. The net zero home has the advantage of a larger 6.3-kilowatt solar electric system. The system was made possible by an additional $6,000 gift from EnCana and donated installation by Sunsense Solar in Carbondale.
With the efficient home, the excess electricity produced by the flush-mounted solar panels on the roof should offset the natural gas energy used for heating and hot water, making the home net-zero for energy use. The homeowners, a single mom and her two children, will be coached by Sunsense employees so that the home operates efficiently.
With less energy required from industrial power plants, the home also will have a lower impact on air pollution. The ENERGY STAR report on the home predicts the residence will save 3.3 tons of carbon dioxide, 4.2 pounds of sulfur dioxide and 9.1 pounds of nitrogen oxides from being emitted each year.
Gilbert said the project has inspired the local Habitat for Humanity to strive for net zero on all of their upcoming housing projects, including a duplex underway in Silt, as another way to keep the cost of living affordable for families. The family now living in the net zero home is relieved to get out from under very high monthly energy bills at their previous residence in a trailer in Dotsero.
Editor's Note: If any other Garfield County homeowners also have a net zero home, please contact CLEER. We'd love to share your story too.


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1. Who is just a sweet transvestite from transexual transelvania? besides me.
2. Why is a raven like a writing desk?
3. Who is the best Alice in Wonderland character?
5. Now, what have you learned from #4?
7. What does MER mean?
8. What does tapossessent mean?
9. What would you do if little red dots followed you?
10. What is my name when i'm a man?
5. Call girl or Man slave?
1. What is a mome rath?
2. How many fingers am I holding up?
d) none, i'm typing.. idiot!
3. What is Jean ValJean's #?
4. Why is the mock turtle so sad?
5. Who said "It's all his fancy, that, he hasn't got no sorrows ya know? Come on!" ?
6. What does Psyche mean?
7. Why is the mad hatter insane?
Did you see the pattern in THis or That?


text:
Books Books 1 - 10 of 13 on C. Bundy, The Rise and Fall of the South African Peasantry (London, 1979)..
C. Bundy, The Rise and Fall of the South African Peasantry (London, 1979).
Africans: The History of a Continent - Page 336
by John Iliffe - 2007
Limited preview - About this book
Trilateralism: The Trilateral Commission and Elite Planning for World Management
Holly Sklar - Political Science - 1980 - 604 pages
...Consequence," Southern African Perspectives No. 3 (1978), p. 4. 6. Davis, op. cit., p. 4. 7. Colin Bundy, The Rise and Fall of the South African Peasantry, (London 1979). 8. Francis Wilson, Labour and the South African Gold Mines. (London, 1972), pp. 16-17. 9. African Research...
The African Poor: A History
John Iliffe - History - 1987 - 387 pages
...relations, 6 (1939), 120. 74 See Le Roux, 'Poor Whites'. 75 Wentzel, 'Hard times', p. 29. 76 Colin Bundy, The rise and fall of the South African peasantry (London, 1979), p. 127; South Africa 1984: official yearbook (Johannesburg, 1984), p. 26; above, p. 161. 77 RWM Mettam,...
The Angry Divide: Social and Economic History of the Western Cape
Wilmot Godfrey James, Mary Simons - Cape Town Metropolitan Area (South Africa) - 1989 - 258 pages
...Railway-Building in the Cape Colony, 1870-1885' (Oxford University: unpublished D.Phil, dissertation, 1978); C. Bundy, The Rise and Fall of the South African Peasantry (London: Heinemann, 1979). 4. PJ Le Feuvre, 'Cultural and Theological Factors Affecting Relationships between...
History from South Africa: Alternative Visions and Practices
Joshua Brown, Karin A.. Shapiro - History - 1991 - 469 pages
...in response to the growing mining markets of the late nineteenth and early twentieth centuries; see C. Bundy, The Rise and Fall of the South African Peasantry (London, 1979). Charles van Onselon, like Johnstone, focused on the mining industry (this time in Rhodesia), examining...
Abraham Esau's War: A Black South African War in the Cape, 1899-1902
Bill Nasson - History - 2003 - 272 pages
...(eds.), The Politics of Race, Class and Nationalism in Twentieth Century South Africa, London, 1987; C. Bundy , The Rise and Fall of the South African Peasantry, London, 1 979; W. Beinart, The Political Economy ofPondoland, 1860-1 930, Johannesburg. 1 982; W. Beinart and...
Class, Caste, and Color: A Social and Economic History of the South African ...
Wilmot Godfrey James, Mary Simons - History - 258 pages
Power and Resistance in an African Society: The Ciskei Xhosa and the Making ...
Les Switzer - History - 1993 - 452 pages
...Chapter 3. Peasants and Proletarians in a Transitional Economy 1 . This section is based mainly on C. Bundy, The rise and fall of the South African peasantry (London, 1979), chap. 2; S. Newton-King, "The labour market of the Cape Colony, 1807-28," and T. Kirk, "The Cape economy...
Roland Oliver, Roland Anthony Oliver, Anthony Atmore - History - 1994 - 308 pages
...Frontiers: The Epic of South Africa's Creation and the Tragedy of the Xhosa People, London, 1992. Colin Bundy, The Rise and Fall of the South African Peasantry, London, 1979. Charles van Onselen, Studies in the Social and Economic History of the Witwatersrand 1886-1914, 2 vols.,...
Living with Africa
Jan Vansina - Biography & Autobiography - 1994 - 312 pages
...Production: The Challenge of Africa (Quebec: Safi, 1985). 28. Palmer and Parsons, Roofs of Rural Poverty; C. Bundy, The Rise and Fall of the South African Peasantry (London: Heinemann, 1979). 29. Q. Hoare and GN Smith, eds. and trans., Selections from the Prison Notebooks...
Segregation and Apartheid in Twentieth-century South Africa
William Beinart, Saul Dubow - History - 1995 - 288 pages
...Solly Sachs, Rebel's Daughters (London, 1957). 39 Carnegie Commission, vol. I, 215. 40 As note 39. 41 C. Bundy, The Rise and Fall of the South African Peasantry (London, 1979). 42 Carnegie Commission Report, vol. II, 60. 43 See P. Delius and S. Trapido, 'Inboekselings and Oorlams:...


Change colors, or add an accent wall. The sky is the limit with design options. Add a tinted ceiling, for a new, subtle look that will add class and style to any room. Re-Finish kitchen cabinets, and bathroom vanities, or stain that worn stair case to bring back all that beautiful lustre. If you can dream it, we can do it, with un-matched attention to detail. The possibilities are unlimited.


Literally….he'll be speaking in Columbus.
According to Columbus Retrometro, local author and urban planner, Kyle Ezell, will be presenting a plan, "How to Immediately Begin to Transform the University District into a National Model for Urban Redevelopment" tomorrow night (Monday 29 October). Mix and mingle with the crowd following the presentation.
Kyle Ezell, founder of "Get Urban" is a champion of urban living. His seminars, offered to residents as well as developers around the country, focus on making American towns and cities the best in the world, and believes it can be done one class at a time. And he's been busy doing it! See what he's been up to with a visit to his web site.
Michael Reed of ColumbusArt.org will also be unveiling his plan to transform Columbus into the "Indie Art Capital of the World".
The presentation is open to the public at no cost at United Methodist Church, 299 King Avenue at 7:00pm. Don't miss this opportunity.
The competition entries for the Columbus Rewired event which took place last month will be on display at the Knowlton School of Architecture at OSU from 22 October – 30 November.
While you're there, check out the schools Classic Furniture Collection. Both are open to the public at no cost.
Part III: Just How Far?
Large expanses of dark glass finally put me to sleep. I slept from Cleveland until the bus arrived in Buffalo. From Columbus to Cleveland, the spirit of adventure kept me awake, though there was little to see at that hour of the night.
Greyhound's Columbus terminal was well lit and far more spacious than I would have anticipated. By 10:45p, enough folks were standing in line that a second bus was added to the route to Cleveland. I was on the second of the two, and there was enough room that the seat next to me remained vacant.
At 2:00a, Cleveland's bus terminal was bustling. So much so that additional busses were added to both the Rochester and the Buffalo route. Once again, I got the second bus to Buffalo which offered more personal space.
I found myself with more legroom than any airlines tourist class seating. The seats were comfortable and reclined more than most airline seats. Other than a brief welcome by the driver, there were no interruptions and the cabin was dark and quiet.
The bus to Toronto was more spacious, with even more leg room, better than most domestic first-class airline cabins. The passengers heading across the boarder were of a slightly different decorum. Conversations overheard were spoken with proper English and in full sentences. The children said "please" and "thank you".
Arriving exactly on time in Peterborough, I realized that after twelve hours of travel, I wasn't dehydrated nor were my feet swollen. Sure, I could have been to Stockholm in the same amount of time, but for a trip to the other side of the lake, it beat driving. There were no security fusses, save for x-rayed bags at the boarder.
A quick trip to Tim Horton's for coffee and donuts and we were off to the farm, another 45 minutes, and there I was greeted by the lush acreage of Mayfarm just outside of the town of Warkworth.
Coming back to Columbus, each bus was at full capacity, and still I found the ride reasonably comfortable. Overall, well worth the $90 for round-trip travel. I'll give Greyhound a definite "thumbs up" for easy and comfortable travel.


Called out: How God sets us free from everything that stops us responding to God's call.
Called up: Understanding how God calls us to join in with his loving care of the world.
Called on: how we can grow in our sense of calling beyond the end of the course.


text:
import warnings
import sys
import errno
import functools
import logging
import socket

from nose.plugins.skip import SkipTest

from urllib3.exceptions import MaxRetryError, HTTPWarning
from urllib3.packages import six

# We need a host that will not immediately close the connection with a TCP
# Reset. SO suggests this hostname
TARPIT_HOST = '10.255.255.1'

# (Arguments for socket, is it IPv6 address?)
VALID_SOURCE_ADDRESSES = [(('::1', 0), True), (('127.0.0.1', 0), False)]
# RFC 5737: 192.0.2.0/24 is for testing only.
# RFC 3849: 2001:db8::/32 is for documentation only.
INVALID_SOURCE_ADDRESSES = [('192.0.2.255', 0), ('2001:db8::1', 0)]

def clear_warnings(cls=HTTPWarning):
    new_filters = []
    for f in warnings.filters:
        if issubclass(f[2], cls):
            continue
        new_filters.append(f)
    warnings.filters[:] = new_filters

def setUp():
    clear_warnings()
    warnings.simplefilter('ignore', HTTPWarning)

def onlyPy26OrOlder(test):
    """Skips this test unless you are on Python2.6.x or earlier."""

    @functools.wraps(test)
    def wrapper(*args, **kwargs):
        msg = "{name} only runs on Python2.6.x or older".format(name=test.__name__)
        if sys.version_info >= (2, 7):
            raise SkipTest(msg)
        return test(*args, **kwargs)
    return wrapper

def onlyPy27OrNewer(test):
    """Skips this test unless you are on Python 2.7.x or later."""

    @functools.wraps(test)
    def wrapper(*args, **kwargs):
        msg = "{name} requires Python 2.7.x+ to run".format(name=test.__name__)
        if sys.version_info < (2, 7):
            raise SkipTest(msg)
        return test(*args, **kwargs)
    return wrapper

def onlyPy3(test):
    """Skips this test unless you are on Python3.x"""

    @functools.wraps(test)
    def wrapper(*args, **kwargs):
        msg = "{name} requires Python3.x to run".format(name=test.__name__)
        if not six.PY3:
            raise SkipTest(msg)
        return test(*args, **kwargs)
    return wrapper

def requires_network(test):
    """Helps you skip tests that require the network"""

    def _is_unreachable_err(err):
        return getattr(err, 'errno', None) in (errno.ENETUNREACH,
                                               errno.EHOSTUNREACH) # For OSX

    @functools.wraps(test)
    def wrapper(*args, **kwargs):
        msg = "Can't run {name} because the network is unreachable".format(
            name=test.__name__)
        try:
            return test(*args, **kwargs)
        except socket.error as e:
            # This test needs an initial network connection to attempt the
            # connection to the TARPIT_HOST. This fails if you are in a place
            # without an Internet connection, so we skip the test in that case.
            if _is_unreachable_err(e):
                raise SkipTest(msg)
            raise
        except MaxRetryError as e:
            if _is_unreachable_err(e.reason):
                raise SkipTest(msg)
            raise
    return wrapper

class _ListHandler(logging.Handler):
    def __init__(self):
        super(_ListHandler, self).__init__()
        self.records = []

    def emit(self, record):
        self.records.append(record)

class LogRecorder(object):
    def __init__(self, target=logging.root):
        super(LogRecorder, self).__init__()
        self._target = target
        self._handler = _ListHandler()

    @property
    def records(self):
        return self._handler.records

    def install(self):
        self._target.addHandler(self._handler)

    def uninstall(self):
        self._target.removeHandler(self._handler)

    def __enter__(self):
        self.install()
        return self.records

    def __exit__(self, exc_type, exc_value, traceback):
        self.uninstall()
        return False

redpajama_set_name: RedPajamaGithub