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\title{Supplementary Information for ``Black hole information turbulence and the Hubble tension"}

%\title{Scientific Reports Title to see here}

\author[1,2,*]{Juan Luis Cabrera Fernández}
%\author[2]{Bob Author}
%\author[1,2,+]{Christine Author}
%\author[2,+]{Derek Author}
\affil[1]{Departamento de F\'isica Aplicada, ETSI Aeron\'autica y del Espacio, Universidad Polit\'ecnica de Madrid, Pza. Cardenal    Cisneros 3, 28040, Madrid, Spain.}
\affil[2]{Laboratorio de Din\'amica Estoc\'astica, Centro de F\'isica, Instituto Venezolano de Investigaciones Cient\'ificas,      Caracas 1020-A,  Venezuela.}

%\affil[1]{Affiliation, department, city, postcode, country}
%\affil[2]{Affiliation, department, city, postcode, country}

\affil[*]{juanluis.cabrera[at]upm.es}


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\begin{abstract}
This is the Supplementary Information file for the article ``Black hole information turbulence and the Hubble tension".
 \end{abstract}


\begin{document}
\flushbottom
\maketitle
% * <john.hammersley@gmail.com> 2015-02-09T12:07:31.197Z:
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\section*{An illustrative example of the leading term at generation $\tau = 3$ of the eigenvalues governing the dynamics of the population of qubits}

The leading term at generation $\tau = 3$ is called here $\Lambda_3$. 
We can grasp how the eigenvalues behave by observing the way the
term $\Lambda_3$ is assembled (see \cite{jlc} for the meaning of $\alpha$ and $\beta$),
\begin{eqnarray} \label{A3}
\Lambda_3 &\equiv&\left\{ 
\begin{array}{c}
\left[ 
\begin{array}{c}
\left( 1+\beta ^2\alpha ^2\right) { r_1}^2\\
+\left(\beta ^2+\beta ^4\alpha ^2\right) 
\end{array}
\right] { r_2}^2\\
+\left[ 
\begin{array}{c}
\left(
-2\,\alpha \,\beta ^2-2\,\beta ^4\alpha ^3\right) { r_1}^2
\end{array}
\right] { r_2}\\
+\left[ 
\begin{array}{c}
\left( \beta ^4\alpha ^2+\beta ^6\alpha ^4\right) { r_1}^2
\end{array}
\right] 
\end{array}
  \right\} { r_3}^2 \\ \nonumber
&& +\left\{ 
\begin{array}{c}
\left[ 
\begin{array}{c}
\left( -2\alpha \,\beta ^2\right) \,{ r_1}^2\\
+\left(-2\,\alpha \,\beta ^4\right) 
\end{array}
\right] { r_2}^2\\
+\left[ 
\begin{array}{c}
\left( 2\,\alpha^2\,\beta ^4\right) { r_1}^2
\end{array}
\right] { r_2}
\end{array}
  \right\} { r_3} \\ \nonumber
&& +\left\{ 
\begin{array}{c}
\left[ 
\begin{array}{c}
\left( \beta ^4\alpha ^2\right) { r_1}^2\\
+\left( \alpha^2\beta ^6\right) 
\end{array}
\right] { r_2}^2 
\end{array} 
\right\}
\end{eqnarray}
 
The nested coefficients structures a cascade as illustrated in the main text Fig. 1.
Given the initial coefficient values for 
the second order term: $C_2= 1+\beta^2 \alpha^2$, the linear term : $C_1=0$,  
and the independent term: $C_0=\beta^2$; the generation rule
allow us to calculate the coefficients that will form the leading term of the eigenvalues at the next iteration $\tau=2$, 
shown in the second row of the main text Fig. 1.
Now, this new set of coefficients allows for the calculation of the nested coefficients
of the leading term of the eigenvalues at the iteration $\tau=3$ (third row at  the same figure); 
a full cascade is generated 
by iteratively applying the generation rule. 
On each iteration $\tau$, $3^{\tau}$ new terms are generated. 

\section*{The generation rule}

The coefficients of higher order values of the leading term can be precisely obtained from 
the preceding terms by the following generation rule. 
In general, if we know $\Lambda_1$ and $\Lambda_2$,
we can obtain $\Lambda_{\tau}$ given that, in the generation ${\tau}-1$, the 
term $\left\{ C_{2} r_{i}^2 + C_{1} r_{i} + C_{0}   \right\}
r_{i+1}^{k} $, 
with $k=0,1,2$;  generates the polynomial
\begin{eqnarray}
\left\{  
\begin{array}{c}
\left[ C_{2} r_{i}^2 + C_{1} \chi_1 r_{i} + \beta^2 C_{2} \right] r_{i+1}^{2} \\
+ \left[  -k \alpha \beta^2 C_2 r_i^2 + C_1 \chi_2  r_i + C_1 \chi_3   \right] r_{i+1} \\
+ \left[ \beta^2 \alpha^2 C_0 r_i^2 + C_1 \chi_4 r_i + C_1 \chi_5 \right]  
\end{array}
\right\} r_{i+2}^{k} \label{recurrence},
\end{eqnarray}
\noindent in the next generation ${\tau}$. There is no need to determine the
unknowns, $\chi_1, \chi_2, ... , \chi_5$,
because in the current situation  $C_1 = 0$
 (see details in \cite{jlc}). 

\section*{Fractal construction}

The structure of the $\Lambda_{\tau}$'s  is better represented by the induced fractal it generates: 
to the initial coefficient values $C_2$, $C_1$ and $C_0$, 
same size segments on the unit interval are assigned. 
After repeated iterations  
each subset is divided by a factor of $3$, and
the newly generated coefficients obtained with the generation rule updates
the new $N_{\tau}=3^{\tau}$ subintervals.
A detailed account of the process is available at \cite{jlc}.

\section*{Fitting the cascade's components}\label{section5}

Figure 3 shows the number of coloured $n_C$ and lacunar $n_L$ components in the 
inverted cascade. Both quantities grow exponentially following fits:
\begin{align}
n_C(\tau)  &\sim 0.812 \,e^{0.807\,\tau}\\
n_L(\tau) &\sim 0.561 \,e^{1.142\,\tau}
\end{align}
One may try to fit the fraction of the cascade coloured, $N_C$, and lacunar, $N_L$, components with the equations describing the fractional energy density of nonrelativistic matter and dark energy. 
Such a fitting is exemplified by the arbitrary selection of the following models:
\begin{itemize}
\item The consensus cold dark matter model ($\Lambda$CDM ):  
\begin{align}
     N_L(\tau) &= \frac{ p_m  (1 +  C_z \tau)^3 }{
     p_r (1 + C_z \tau )^4 +
               p_m ( 1 + C_z \tau )^3 +
               p_k ( + C_z \tau )  ^2 + (1 - p_k - p_m - p_r) }
\end{align}
\item The constant $w$ model ($w$CDM) : 
\begin{align}
     N_L(\tau) &= \frac{ p_m  (1 +  C_z \tau)^3 }{
           p_r (1 + C_z \tau )^4 +
               p_m ( 1 + C_z \tau )^3 +
               p_k ( + C_z \tau )  ^2 + (1 - p_k - p_m - p_r) (1 + C_z \tau  )^\omega}
\end{align}
\item The Chevallier–Polarski–Linder  model (CPL): 
\begin{align}
     N_L(\tau) &= 
      \dfrac{ p_m  (1 +  C_z \tau)^3 }{  \splitdfrac{p_r (1 + C_z \tau )^4 +
               p_m ( 1 + C_z \tau )^3 +
               p_k ( + C_z \tau )  ^2 + }{ 
               (1 - p_k - p_m - p_r) (1 + C_z \tau  )^{3(1 + \omega_0 + \omega_a )}
            e{ \frac{-3  \omega_a  C_z \tau }{ 1 + C_z \tau } } }  }
\end{align}
\item The generalised Chaplygin gas model (GCG):
\begin{align}
     N_L(\tau) &= 
      \dfrac{ p_m  (1 +  C_z \tau)^3 }{  \splitdfrac{p_r (1 + C_z \tau )^4 +
               p_b ( 1 + C_z \tau )^3 +
               p_k ( 1 + C_z \tau )  ^2 + }{ 
               (1 - p_k - p_b - p_r) 
               [ ( A_s+ (1 - A_s) (1 + C_z \tau)^{ 3 (1 + \alpha) } ]^{ \frac{1}{1+\alpha} } 
            } }
\end{align}
\item The interacting dark energy model  (IDE):
\begin{align}
	N_L(\tau) &= 
	\dfrac{ p_m  (1 +  C_z \tau)^3 }{  \splitdfrac{p_r (1 + C_z \tau )^4 +
			p_k ( 1 + C_z \tau )  ^2 + }{
			(1 - p_k - p_m - p_r)   ( 1 + C_z \tau )^{3(1+ \omega_x)} +
			\frac{b_m}{\delta + 3 \omega_x}
			[ \delta(1+C_{\tau})^{3(1+\omega_x)} +3 \omega_x(1 + C_z \tau)^{3-\delta }] 
	}}
\end{align}     
\item Early dark energy model (EDE)
\begin{align}
     N_L(\tau) &= 
         \dfrac{ p_m  (1 +  C_z \tau)^3 ( 1 - \Omega_{DE}(z) )}{ 
 p_r (1 + C_z \tau)^4 +
p_m(1 + C_z\tau)^3 +
p_k(1 + C_z\tau)^2 }\\
 \Omega_{DE}(z)  & \equiv  
 ( 
 	(1.0- p_k - p_m  - p_r)- w_e( 1 - (1 + C_z \tau)^{3w_0} ) \nonumber\\
 	&\times [
  (1.0 - p_k - p_m- p_r) +
p_k (1 + C_z \tau)^{-3 w_0 - 1} \nonumber\\
 &+ p_m(1 + C_z \tau)^{-3 w_0}  
+ p_r (1 + C_z \tau)^ {- 3 w_0 + 1} 
]^{-1} \nonumber\\ 
&+
w_e (1 - (1 + C_z \tau)^{3 w_0}).\label{Ode}
\end{align}     
\item Poulin {\it et al.} Early Dark Model (EDEP) 
\begin{align}
N_L(\tau) &=   
 \dfrac{p_m  (1 + C_z \tau )^3}{
 	 \splitdfrac{
  p_r (1 + C_z \tau)^4 +
p_m(1 + C_z \tau)^3 +
p_k(1 + C_z \tau)^2 + }{
(1.0 - p_k - p_m - p_r) +            
\dfrac{ 2 p_a }{\left(\dfrac{ 1 + Z^c}{1 + C_z \tau}\right)^{3(w_n + 1)}  + 1 }  }}
\end{align}     
\end{itemize}
and $N_C(\tau) = 1 - N_L(\tau)$ for all the cases.

With the exception of $\Omega_{DE}(\tau)$ - being a function - 
we avoided the conventional use of the sign for the energy densities ($\Omega_x$)
as -- at this point in our discourse --,
the fitting parameters have no physical meaning.
Even so, we have kept the same subscripts to 
maintain a certain parallelism with the original meaning, i.e., 
we deal with parameters 
$p_k$, $p_m$, $p_r$ in the $\Lambda$CDM model. The $w$CDM adds the parameter $w$, 
the CPL, GCG, IDE and EDE models include the additional 
parameter sets $(\omega_0, \omega_a)$, $(A_s, \alpha)$, 
$(\omega_x, \delta)$ and 	$(\omega_0, \Omega_e)$, respectively, 
while the EDEP includes the parameters $(p_a,w_n,Z^c)$. 
Used models where inspired by \cite{shi2012} 
where all of them included a contribution from curvature. 
In particular, the EDE model is based on 
\cite{doran2001,doran2006}.  
However the EDEP model was assembled using equations (5) and (15)
in \cite{poulin} and, for fitting purposes, we decided to took the previous triple 
as parameters. 
We incorporated the additional parameter $C_z$ 
which intents to fit the relationship between the iteration step $\tau$ and the cosmological redshift $z$.
Best fit parameters  
using the Turing.jl package for Bayesian inference with the No-U-Turn sampler \cite{HoffmanGelman} 
are summarised in Table 1 and the corner plots 
for the parameters and their covariance are shown in  Figs. \ref{SIfig3}-\ref{SIfig8}. 
No corner plot for the GCG model are included as our fitting results were quite unsatisfactory 
With the exception of GCG all the models yielded a $\sigma^2$ measure of the order of $\sim 10^{-4}$ 
with $w$CDM reaching a best value of  $\sim 10^{-5}$. 
For all the models $p_{\Lambda} \sim 0.8$ and $p_{m}$ is 
in the range $\sim 0.14 - 0.18$. 
While GCG differs substantially, most models produced $p_k$ $\,\,\sim 0.00027$, 
excepting EDE with $p_k \sim 0.005726 $ and EDEP with the only slightly negative value $p_k \sim  -0.001418$ (positive in the error margin, reported in the Fig. \ref{SIfig8}.).
Meanwhile, wCDM, CPL, GCG and IDE reported $p_r$ negative values in the range  $\sim [-5.8  ,  - 0.2] \times 10^{-3}$, while LCDM, EDE and EDEP models have 
positive $p_r \sim 0.000137$, $ 0.000209 $ and $0.000148$ 
(whose $\sigma^2$ were $0.000228$, $0.000181$ and $0.000226 $, respectively).
The obtained values for the parameter pairs 
$(\omega_0, \omega_a)$, $(A_s, \alpha)$ and $(\omega_x, \delta)$ in the 
CPL, GCG and IDE models are very close to those obtained in \cite{shi2012} 
 - i.e., $(-0.966, 0.202)$, $(0.733,-0.011)$ and $(-1.001, -0.0043)$, respectively -  
 \footnote{
 \cite{shi2012} makes use of data from the  
 Union2.1 SNe compilation and the WiggleZ BAO measurements, 
together with the WMAP 7-yr distance priors and the observational Hubble data.
}. 
 Given that the only models yielding  
 positive values for $p_r$ are the $\Lambda$CDM, EDE and EDEP 
 models we conclude these are the ones best fitting the $N_L$ data. 
 Remarkably, the best values obtained for the parameter $C_z$
 for these models are quite close to each other:  
 $0.365496$, $0.389605$ and $0.361704$. 
 Thus,  the relationship between 
 iteration steps and the redshift may be 
 written as $z \sim C_z \tau$.
 Plots with the fitting results for the 
 $\Lambda$CDM, $w$CDM, CPL, IDE, EDE and EDEP models 
are shown in the Fig. 13. 
 
 

\section*{$E(z)$ used in the calculations of  the Hubble constant}

The analysis was restricted to the best behavior models from  the above section, i.e., $E(\tau)$, 
\begin{itemize}
\item for the $\Lambda$CDM model  is given by,   
\begin{align}
	E(\tau) &= 		
		p_r (1 + C_z \tau )^4 +
		p_m ( 1 + C_z \tau )^3 +
		p_k (1 + C_z \tau )^2 + (1 - p_k - p_m - p_r) \label{eLCDM},
\end{align}
\item for the EDE model  is given by,   
\begin{align}
	E(\tau) &= 
	\dfrac{		p_r (1 +C_z \tau)^4 +
		p_m(1 + C_z \tau)^3 +
		p_k(1 + C_z \tau)^2 }{1 - \Omega_{DE}(\tau)} \label{eEDE},
\end{align}     
with $\Omega_{DE}(\tau)$	 expressed by eq. (\ref{Ode}), and 
\item for the EDEP model is given by,   
\begin{align}
	E(\tau) &=   
			p_r (1 + C_z \tau)^4 +
			p_m(1 + C_z \tau)^3 +
			p_k(1 + _z \tau)^2 + \nonumber\\
			&\,\,(1.0 - p_k - p_m - p_r) +            
			\dfrac{ 2 p_a }{\left(\dfrac{ 1 + Z^c}{1 + C_z \tau}\right)^{3(w_n + 1)}  + 1} \label{eEDEP}.
\end{align}     
\end{itemize}
The parameter values in these models are those given in Table 1. 

\section*{$H_0$ error determination}
 
Native Interpolations.jl doesn't provide built-in confidence intervals. 
To calculate an error estimation for $H_0$ we took the mean value of the 
propagated error of the equation (28) in the main text, i.e., 
	\begin{align} 
	\epsilon(H_0) &\sim \left< \epsilon (\Upsilon_x) \right > = \dfrac{1}{2} \dfrac{\epsilon (p_{h})}{ p_h } +    
	\dfrac{1}{2}  \left< \dfrac{\epsilon (\mathcal{M}_x)}{\mathcal{M}_x} \right>. \label{eh0}
\end{align}
In this expression $\epsilon(\mathcal{M}_x)$ is the 
root mean square deviation, i.e. 
$\epsilon (\mathcal{M}_x) \equiv \sqrt{ \left<  \dfrac{\Delta D_x}{D_x} -  \mathcal{M}^{1/2}_x  \right>^2 }$. 
Note that $\epsilon \left(\dfrac{\Delta D_x}{D_x}\right) = 0$, as the involved quantities were  measured directly on the fractal. 
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

\begin{figure*}[hbt]
	\centering
	\includegraphics[width=\textwidth]{./SIfig1.pdf}
	\caption{{\bf   Determination of the Hubble constant from the relative growth of the fractal space.} 
		Same as in the main text Fig. 7, 
		but using EDE $E(t)$, i.e., given by
		the SI equation \ref{eEDE}. In this case 	{\bf b} $H_0 = \Upsilon(1) \sim (78.91 \pm 0.91)$ Km/s/Mpc and {\bf d}
		$H_0 = \Upsilon(1) \sim (37.57 \pm 0.69)$ Km/s/Mpc.
	}\label{EF_fig13}
\end{figure*} 




\begin{figure*}[hbt]
	\centering
	\includegraphics[width=\textwidth]{./SIfig2.pdf}
	\caption{{\bf   Determination of the Hubble constant from the relative growth of the fractal space.} 
		Same as in the main text Fig. 7, 
		 but using EDEP $E(t)$, i.e., given by
		the SI equation \ref{eEDEP}. In this case 	{\bf b} $H_0 = \Upsilon(1) \sim (73.55 \pm 2.75)$ Km/s/Mpc and {\bf d}
		$H_0 = \Upsilon(1) \sim (36.91 \pm 0.53)$ Km/s/Mpc.
	}\label{EF_fig14}
\end{figure*} 




\begin{figure*}[hbt]
	\centering
	\includegraphics[width=\textwidth]{./SI_ppLCDM.pdf}
		\caption{{\bf  Parameters distribution and their covariance for the $\Lambda$CDM model.}
	}\label{SIfig3}
\end{figure*}


\begin{figure*}[hbt]
	\centering
	\includegraphics[width=\textwidth]{./SI_ppwCDM.pdf}
			\caption{{\bf  Parameters distribution and their covariance for the  $w$CDM model.}
	}\label{SIfig4}
\end{figure*}

\begin{figure*}[hbt]
	\centering
	\includegraphics[width=\textwidth]{./SI_ppCPL.pdf}
			\caption{{\bf  Parameters distribution and their covariance for the  CPL model.}
	}\label{SIfig5}
\end{figure*}

 
 \begin{figure*}[hbt]
 	\centering
 	\includegraphics[width=\textwidth]{./SI_ppIDE}
				\caption{{\bf  Parameters distribution and their covariance for the  IDE model.}
 	}\label{SIfig6}
 \end{figure*}
 
 
 
 \begin{figure*}[hbt]
 	\centering
 	\includegraphics[width=\textwidth]{./SI_ppEDE}
					\caption{{\bf Parameters distribution and their covariance for the   EDE model.}
 	}\label{SIfig7}
 \end{figure*}
 
 
 
\begin{figure*}[hbt]
	\centering
	\includegraphics[width=\textwidth]{./SI_ppEDEP}
					\caption{{\bf  Parameters distribution and their covariance for the   EDEP model.}
	}\label{SIfig8}
\end{figure*}


\begin{figure*}[hbt]
	\centering
	\includegraphics[width=\textwidth]{./SI_fig9.pdf}
					\caption{{\bf All cosmological models fitting.} From left to right and from top to bottom $\Lambda$CDM, $w$CDM, CPL, IDE, EDE and EDEP models fitted using Julia's Turing.jl Bayesian inference. 
	}\label{fig9}
\end{figure*}


\begin{table*}[ht]
	\begin{center}
			\resizebox{1\columnwidth}{!}{
			\begin{tabular}{cccccccc}
				\toprule
				Parameter & LCDM & wCDM & CPL & GCG & IDE & EDE & EDEP\\
				\midrule
				$ C_{z} $ &  $ 0.366038 $  &  $ 0.728667 $  &  $ 0.423091 $  &  $ 0.917701 $  &  $ 0.528297 $ & $ 0.389605 $&$0.361704$\\
				$ p_k $ &  $ 0.000268 $  &  $ 0.00027 $  &  $ 0.00027 $  &  $ 0.334792 $  &  $ 0.00027 $ & $ 0.005726$&$-0.001418$\\
				$ p_m $ & $ 0.177603$  &  $0.141675$  &  $0.164878$  &  $ p_b \equiv 0.0451 $  &  $ 0.163759 $ & $ 0.179941$&$0.180455$\\
				$ p_r $ &  $ 0.000137 $  &  $ -0.000225 $  &  $ -0.00035 $  &  $ -0.005757 $  &  $ -0.000459 $ & $ 0.000209$&$0.000148$\\
				$ p_{ \Lambda} $ &  $ 0.821993 $  &  $ 0.85828 $  &  $ 0.835201 $  &  $ 0.77703 $  &  $ 0.821993 $&$0.814124$&$0.820815$\\
				$ p_5 $ & -- &  $ 0.821767 $  &  $ -1.006021 $  &  $ 0.747408 $  &  $ -0.822978 $  & $ -0.953692$&$0.219839$ \\
				$ p_6 $ & -- & -- &  $ 0.205858 $  &  $ -0.011034 $  &  $ -0.00453 $ & $ 0.814125$&$0.565335$\\
				$ p_7 $ & -- & -- &  $  --  $  &  $  --  $  &  $  --  $ & $  -- $& $10^6$\\
				$ \sigma^2$ &  $ 0.000228 $  &  $ 0.000097 $  &  $ 0.000126 $  &  $ 0.432337 $  &  $ 0.000134 $ & $0.000181$&$0.000226$\\
				\bottomrule
			\end{tabular}
		}
		\caption{{\bf  Best fit parameters obtained using Julia's Turing.jl Bayesian inference}. Here $p_{\Lambda} \equiv 1 - p_k - p_m - p_r$; 
			$p_5 = w$ in the $w$CDM,  $(p_5, p_6)$ are $(\omega_0, \omega_a)$, $(A_s, \alpha)$, $(\omega_x, \delta)$ and 
			$(\omega_0, \Omega_e)$ in the 
			CPL, GCG, IDE and EDE models, respectively; the EDEP model extra parameters are 
			$(p_5, p_6,p_7)=(p_a,w_n,Z^c)$; $ \sigma^2$ means residuals. In the GCG model $p_m$ was substituted by 
			the present density parameter of baryonic matter $p_b = \Omega_b = 0.0451$, according to the WMAP 7yr results 
			\cite{Komatsu2011} following the example in  \cite{shi2012}.	
		}\label{table1}
	\end{center}
\end{table*}



\begin{table*}[hbt] 
	\begin{center}
		\begin{tabular}{ccccc}
		\toprule
		Model& $D_x$ & $p_h$ & $p_e$  & $H_0$\\
		\midrule
		$\Lambda$CDM& $D_{cg}$ & $116.11 \pm 20.21$ & $1.54 \pm 0.03$  & $\boldsymbol{62.79 \pm 5.48}$\\
		$\Lambda$CDM& $D_{m}$ & $74.00 \pm 0.00$ & $1.41 \pm 0.00$  & $\boldsymbol{70.07 \pm 0.09}$\\
		EDE& $D_{cg}$ & $127.87 \pm 2.87$ & $1.50 \pm 0.04$  & $78.91 \pm 0.91$\\
		EDE& $D_{m}$ & $35.69 \pm 1.23$ & $1.50 \pm 0.01$  & $37.57 \pm 0.69$\\
		EDEP& $D_{cg}$ & $134.60 \pm 9.99$ & $1.52 \pm 0.01$  & $73.55 \pm 2.75$\\
		EDEP& $D_{m}$ & $39.81 \pm 1.08$ & $1.51 \pm 0.00$  & $36.91 \pm 0.53$\\
				\midrule
	\end{tabular}
	\caption{{\bf  Best fitted parameters for models given by the equation (21)}. Bayesian inference with a Markov Chain Monte Carlo 
		and $300000$ priors. $H_0$ was obtained from the estimate posterior distributions according to equations 
		(28) and (29 ). 
		Cero errors means they are smaller than $10^{-2}$.
		Results for $H_0$ are in units of $H(\tau)$, i.e. in Km/s/Mpc, as expected. 
	}\label{table1}
\end{center}
\end{table*}




 \clearpage
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