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\runninghead{Haret C. Rosu, Jose Torres} 
{Grassmann Hubble superfield}

\normalsize\textlineskip
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\copyrightheading{}    %{Vol. 0, No.0 (1992) 000--000}

\vspace*{0.88truein}

%\fpage{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\centerline{Mod. Phys. Lett. A 16 (2001) xxx 
%[gr-qc/0003108 v4]}
\bigskip
\centerline{\bf A GRASSMANN REPRESENTATION OF THE HUBBLE PARAMETER}  
%\centerline{\bf AND THE COSMOLOGICAL ACCELERATION }  
%TIME DEPENDENT ADIABATIC INDICES\footnote{
%This essay received a ``honorable mention'' in the 
%        A E Competition 
%        F for the year 000 --- Ed.}} 
\vspace*{0.035truein}
%\centerline{\bf MANUSCRIPTS USING COMPUTER SOFTWARE\footnote{For
%the title, try not to use more than 3 lines. Typeset the title
%in 10 pt Times Roman, uppercase and boldface.}}
\vspace*{0.37truein}
%\centerline{\footnotesize NAME}
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%\vspace*{0.015truein}
%\centerline{}
%\baselineskip=10pt
%\centerline{\footnotesize\it City, State ZIP/Zone,
%Country\footnote{State completely without abbreviations, the
%affiliation and mailing address, including country. Typeset in 8
%pt Times Italic.}}
\vspace*{10pt}
\centerline{\footnotesize HARET C. ROSU and JOSE TORRES}
\vspace*{0.015truein}
%\centerline{\footnotesize [Received 17 August 1999] }
%\baselineskip=10pt
\centerline{\footnotesize  Instituto de F\'{\i}sica,
Universidad de Guanajuato, Apdo Postal E-143, 37150 Le\'on, Gto, Mexico}
\vspace*{0.225truein}
%\publisher{(May 15, 2000)}{}%{(May 14, 2000)}
%\centerline{\footnotesize Received (16 May 2000) }

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{0.21truein}
\abstracts{The Riccati equation for the Hubble parameter $H$ of
barotropic FRW cosmologies in conformal
time for $\kappa \neq 0$ spatial geometries and in comoving time for the 
$\kappa =0$ geometry, respectively, is generalized to odd Grassmannian time 
parameters. We obtain 
a system of simple differential equations for the four supercomponents 
(two of even type and two of odd type) of the Hubble superfield 
function ${\cal H}$ that is explicitly solved. 
The second even Hubble component does not have an evolution governed by general
relativity although there are effects of the latter upon it. }{}{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\vspace*{10pt}
%\keywords{The contents of the keywords}

\textlineskip                  %) USE THIS MEASUREMENT WHEN THERE IS
\vspace*{12pt}                 %) NO SECTION HEADING

\vspace*{1pt}\textlineskip	%) USE THIS MEASUREMENT WHEN THERE IS
%\section{General Appearance}    %) A SECTION HEADING
\vspace*{-0.5pt}
\noindent

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%PACS number(s):  98.80.Hw, 11.30.Pb

\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newpage

%\pagebreak

%\textheight=7.8truein
%\setcounter{footnote}{0}
%\renewcommand{\thefootnote}{\alph{footnote}}

%\section{The Main Text}

{\bf 1} - Recently, Faraoni,\cite{Far} showed that the equations 
describing barotropic FRW cosmologies can be combined in a simple 
Riccati equation leading in a much easier way 
to the textbook solutions for the FRW scale factors.
Faraoni obtained the following cosmological Riccati equation
%(${\rm c > 0}$ henceforth)
$$
 \frac{dH}{d\eta}=-cH^2-\kappa c~,
\eqno(1a)
$$
for the log derivative of the FRW scale factor, the famous Hubble
parameter $H(\eta)=\frac{da/d\eta}{a}$.
The independent variable is the conformal time $\eta$, 
$c=\frac{3}{2}\gamma -1$ was assumed constant by Faraoni, 
and $\kappa=0,\pm1$ is the curvature index of the flat, closed, open
FRW universe, respectively. The adiabatic index $\gamma$ is defined through 
the barotropic equation of state $P=(\gamma -1)\rho$, where $P$ and $\rho$
are, respectively, the pressure and energy density of the cosmological fluid 
under consideration. It is a constant quantity in the approach of Faraoni.
It is a simple matter to write
the first Friedmann dynamical equation $\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}
(\rho +P)$ in the form $\frac{\ddot{a}}{a}=-\frac{4\pi G\rho}{3}c$ for the 
barotropic case. Thus, the common lore is that
a negative $c$ ($\gamma < \frac{2}{3}$)
implies an accelerating universe as
claimed in the astrophysics of supernovae.  

We also notice that from the mathematical point of view, Faraoni`s FRW 
Riccati equation being of constant coefficients is directly integrable. 
The solutions are 
$$
H^+=-\tan c\eta ~, \qquad H^-={\rm coth}(c\eta)~,
\eqno(1b)
$$ 
 for the closed and open FRW universes, respectively.

In the following we perform a very simple generalization of the 
cosmological Riccati equation
($1a$) by considering the Hubble parameter as a function not only of time 
but also  of the so-called odd ``time" parameters, $\theta _1$ and 
$\bar{\theta}_1$
(where $\bar{\theta}_1$ is the complex conjugate of $\theta _1$).\cite{tk}
In this way, the Hubble parameter becomes a superfield in the terminology of 
superanalysis.\cite{ber} This is only a toy generalization since we do not
relate it to any sophisticated formalism, such as supergravity. However, 
we obtain some simple analytic formulas for all the components
of the Hubble superfield. We mention that similar types of 
supersymmetric extensions (affecting in addition the operatorial structure)
are of much interest in the theory of integrable 
models that appear naturally in the study of strings in the matrix 
model approach.\cite{das}

\bigskip

{\bf 2} - 
We shall write the Riccati equation for the superfield ${\cal H}$ in the 
following form
%%%%%%%%%%%%%%%%%%%%%%%%%
$$
\frac{d{\bf {\cal H}}}{d\eta}\equiv \frac{i}{2}
\{ D_{\theta _1},D_{\bar{\theta}_1}\}{\bf {\cal H}}
=\alpha {\bf{\cal H}}^2 + \beta~,
\eqno(2)
$$
%%%%%%%%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%%%%%%%
$$
{\bf {\cal H}}(\eta , \theta _1, \bar{\theta}_1)=H_0(\eta) +i\theta _1 
\bar{H}_1(\eta)+
i\bar{\theta}_1H_1(\eta) +\theta _1 \bar{\theta}_1H_{11}(\eta) ~,
\eqno(3)
$$
%%%%%%%%%%%%%%%%%%%%
$\alpha =-c$, $\beta =-\kappa c$, and 
$$
D_{\theta _1}=\frac{d}{d\theta _1}+i\bar{\theta}_1\frac{d}{d\eta}~, \qquad 
D_{\bar{\theta}_1}=-\frac{d}{d\bar{\theta} _1}-i\theta _1\frac{d}{d\eta}~.  
\eqno(4)
$$
%%%%%%%%%%%%%%%%%%%%%%%
The superfield ${\bf {\cal H}}$ satisfies the reality condition 
${\bf {\cal H}}^{\dagger} ={\bf {\cal H}}$ and is
assumed to be dimensionless implying the following $\eta$ dimensions
$$
[\eta]=1~,\qquad [\theta _{1}]=1/2~, \qquad [H_0]=0~,\qquad 
[H_1]=-1/2~, \qquad [H_{11}]=-1~.
$$
After simple (super)calculations we get the following system of equations
$$
\frac{dH _0}{d\eta}=\alpha H_{0}^{2}+\beta,
\eqno(5a)
$$
$$
\frac{d\bar{h} _1}{d\eta}=2\alpha H_{0}\bar{h} _1,
\eqno(5b)
$$
$$
\frac{dh _1}{d\eta}=2\alpha H_{0}h _1,
\eqno(5c)
$$
$$
\hskip 1.6cm\frac{dH _{11}}{d\eta}=2\alpha ( H_{0}H_{11}+\bar{h} _1h _1
\bar{\epsilon}\epsilon),
\eqno(5d)
$$
where $\bar{h}_1\bar{\epsilon}=\bar{H}_1$,  $h_1\epsilon =H_1$, and 
$\epsilon$ is a Grassmann parameter.

%has the following 2 x 2 matrix representation
%$\epsilon=\left( \begin{array}{cc}
%0 & 0 \\
%1 & 0 \\
%\end{array} \right)$ and 
%$\bar{\epsilon}=\left(
%\begin{array}{cc}
%0 & 1 \\
%0 & 0 \\
%\end{array} \right)$, i.e., we use Clifford spinors. If instead, one uses 
%Grassmann 2 x 2 spinors of the form 
%$\epsilon=\left( \begin{array}{cc}
%0 & 0 \\
%\theta & 0 \\
%\end{array} \right)$ and 
%$\bar{\epsilon}=\left(
%\begin{array}{cc}
%0 & \theta \\
%0 & 0 \\
%\end{array} \right)$ the last term in ($5d$) disappears resulting in a 
%particular case of the results in the following.
 
One can see that equation (5a) 
is identical to Faraoni's Riccati equation.
Therefore, the solutions are
$$
H_{0}^{+}=-{\rm tan}(c\eta), \qquad H_{0}^{-}={\rm coth}(c\eta),
\eqno(6)
$$
For the other components, we get simple first order differential equations.
The equations (5b) and (5c) are identical since we consider only the 
possibility of real superpartners. The solutions are easily obtained 
$$
%\begin{equation} \label{u1}
h_1^+=\frac{1}{\cos ^2(c\eta)}~,\qquad h_1^-=\frac{1}{{\rm sinh} ^2(c\eta)}~, 
\eqno(7)
$$
%%%%%%%%%%%%%%%%%%%%%%%
for the closed and open spatial geometries, respectively.

Finally, the solution of the equation for $H_{11}$ can be obtained by the 
method of the variation of constants leading to
$$
H_{11}^{\pm}=e^{2\alpha\int _{\eta _0}^{\eta} H_0 d\eta}\Bigg[A_{11}^{\pm}+
\bar{\epsilon}\epsilon
\int _{\eta _0}^{\eta}2h_1^2e^{-2\alpha\int ^{\eta} H_0d\eta}d\eta\Bigg]~,
\eqno(8)
$$
where the parameters $A_{11}^{\pm}$ are related to the initial conditions.
We thus obtain
$$
H_{11}^{+}=\cos ^{-2}(c\eta)\left( A_{11}^{+}-
2\bar{\epsilon}\epsilon{\rm tan}(c\eta)
%\int \cos ^{-(\frac{2}{c}+4)}(c\eta)d\eta 
\right)
\eqno(9)
$$
and
$$
H_{11}^{-}={\rm sinh} ^{-2}(c\eta)\left( A_{11}^{-}+
2\bar{\epsilon}\epsilon {\rm coth}(c\eta)
%\int {\rm sinh} ^{-(\frac{2}{c}+4)}(c\eta)d\eta 
\right)~.
\eqno(10)
$$


\bigskip

{\bf 3} - The case $\kappa =0$ is special in the sense that $\alpha =
-(c+1)$, $\beta =0$ 
and the independent variable is the comoving time $t$.\cite{0} Therefore,
the odd time parameters refer to the comoving time in this case.

\bigskip

 (i) For $c+1\neq 0$ ($\alpha \neq 0, \, \beta =0$) we get
$$
H_0^0=\frac{1}{c+1}\frac{1}{t},\qquad 
h_1^0=\bar{h}^{0}_{1}=\frac{1}{t^2}~,\qquad
H_{11}^{0}=A_{11}^{0}t^{-2}
+2\bar{\epsilon}\epsilon (c+1)
% \frac{2}{c+1}}
t^{-3}~.
%+\frac{2}{c+1})}\right)~.
\eqno(11)
$$
%Plots involving the function $H_{11}^{0}$ are presented in Figs.~(1) and (2).

\bigskip 

 (ii) The case $c+1=0$ ($\alpha = 0, \, \beta =0$) leads to
$$
H_0=C_0, \qquad h_1=\bar{h}_{1}=C_1, \qquad H_{11}
=C_{11}~.
\eqno(12)
$$ 
 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bigskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%$$
%{\rm c_{\kappa,susy}(\eta)=-u_{p}^{'}+cu_{p}^2=
%\left\{ \begin{array}{ll}
%c(1+2{\rm tan}^2 c\eta) & \mbox{if $\kappa >1$}\\
%c(-1+2{\rm tanh}^2 c\eta) & \mbox{if $\kappa <1$}
%\end{array} \right.}
%\eqno(7d)
%$$
%\end{equation}




\bigskip


{\bf 4} - Through a simple generalization of the barotropic FRW Riccati 
evolution equation 
to odd Grassmann time parameters ($\theta _1$, $\bar{\theta}_1$) 
a Grassmann
(superfield) representation of the Hubble parameter is given in this work. The 
superfield components are explicitly
obtained under the simple assumption of a real Hubble superfield.
%One is free to identify $H_{11}$ as the logarithmic derivative of a 
%`scale factor' $a_{11}$. 
We notice that the evolution equation of the second even component
$H_{11}$ is a 
linear first order differential equation not a nonlinear Riccati equation as 
required by
Friedmann equations. Thus, we are out of the realm of general relativity
although there is an induced effect of the latter as seen from the presence of
$H_0$ in equation (5d).  




\nonumsection{Acknowledgements}

\noindent
The authors thank Prof. V. Tkach and Dr. J.J. Rosales for discussions.
%This work was supported by a Project from CONACyT (No.~x).

\newpage
\bigskip
\noindent
{\bf References}

\begin{thebibliography} {000}
\bibitem{Far} V. Faraoni, 
%``Solving for the dynamics of the universe",
Am. J. Phys. {\bf 67}, 732 (1999) [physics/9901006].
See also, J.A.S. Lima,  Am. J. Phys., to appear, [astro-ph/0109215].
For nonrelativistic supersymmetric approaches with application to 
cosmological acceleration, see H.C. Rosu, Mod. Phys. Lett. A {\bf 15},
979 (2000) [gr-qc/0003108].

\bibitem{tk} V.I. Tkach, J.J. Rosales and J. Socorro,
Mod. Phys. Lett. A {\bf 14}, 1209 (1999) and the references therein.

\bibitem{ber} F.A. Berezin, {\em Introduction to Superanalysis}, Reidel,
Dordrecht, 1987.

\bibitem{das} A. Das, {\em Integrable Models}, World Scientific, 1989;
For recent work, see e.g., A. Das and Z. Popowicz, Phys. Lett. A {\bf 274},
30 (2000) and references therein;
R. Jackiw, hep-th/0110057 and physics/0010079; 
see also, A. Turbiner, hep-th/9209080.


%\bibitem{562} See formulas 15.4.13 and 15.4.26 at page 562 in 
%M. Abramowitz and I.A. Stegun, {\em Handbook of Mathematical Functions},
%ninth edition, 1972.

\bibitem{0} H.C. Rosu,  Mod. Phys. Lett. A {\bf 16}, 1147 (2001)
[gr-qc/0003108].  

\end{thebibliography}

%\newpage


\end{document}



gr-qc0109071.extracted_bib
\begin{thebibliography} {000}
\bibitem{Far} V. Faraoni, 
%``Solving for the dynamics of the universe",
Am. J. Phys. {\bf 67}, 732 (1999) [physics/9901006].
See also, J.A.S. Lima,  Am. J. Phys., to appear, [astro-ph/0109215].
For nonrelativistic supersymmetric approaches with application to 
cosmological acceleration, see H.C. Rosu, Mod. Phys. Lett. A {\bf 15},
979 (2000) [gr-qc/0003108].

\bibitem{tk} V.I. Tkach, J.J. Rosales and J. Socorro,
Mod. Phys. Lett. A {\bf 14}, 1209 (1999) and the references therein.

\bibitem{ber} F.A. Berezin, {\em Introduction to Superanalysis}, Reidel,
Dordrecht, 1987.

\bibitem{das} A. Das, {\em Integrable Models}, World Scientific, 1989;
For recent work, see e.g., A. Das and Z. Popowicz, Phys. Lett. A {\bf 274},
30 (2000) and references therein;
R. Jackiw, hep-th/0110057 and physics/0010079; 
see also, A. Turbiner, hep-th/9209080.


%\bibitem{562} See formulas 15.4.13 and 15.4.26 at page 562 in 
%M. Abramowitz and I.A. Stegun, {\em Handbook of Mathematical Functions},
%ninth edition, 1972.

\bibitem{0} H.C. Rosu,  Mod. Phys. Lett. A {\bf 16}, 1147 (2001)
[gr-qc/0003108].  

\end{thebibliography}