Comments on "Bigger Rip with no dark energy"
A. Kwang-Hua CHU
We make corrections on the paper by Frampton and Takahashi [{Astropart. Phys.} {22} (2004) 307]. Our focus is especially upon Eqs. (12-15) therein. \newline %Keywords. %\noindent %PACS numbers: 98.80.Cq, 98.80.Bp, 98.80.Jk
astro-ph0502012.tex
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\title{Comments on "Bigger Rip with no dark energy"} % Declares the title.
\author{A. Kwang-Hua CHU} %\thanks{Present Address :.  }}  %address%\cite{PKU:1999}
\date{Department of Physics, Xinjiang University, 
Urumqi 830046, PR China}
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\begin{abstract}
We make corrections on the paper by Frampton and Takahashi [{\it Astropart. Phys.}
{\bf 22} (2004) 307]. Our focus is  especially upon Eqs. (12-15) therein. \newline

%Keywords.
%\noindent
%PACS numbers: 98.80.Cq, 98.80.Bp, 98.80.Jk
\end{abstract}
%\pacs{}
%%\end{titlepage}      %%\twocolumn     %%\nopagebreak   %\oddsidemargin=1mm
\doublerulesep=6mm    %
\baselineskip=6mm
\bibliographystyle{plain}               %\section{Introduction}

\noindent
Frampton and Takahashi just presented an interesting paper which shows that a
singularity can occur at a finite future time in an expanding
Friedmann universe [1].
By studying a modified Friedmann equation which arises in an extension of
general relativity (GR) or the modification of GR (due to Dvali,
Gabadadze and Porrati [2] (DGP); considering the
four-dimensional gravity which arises from five-dimensional
general relativity confined to a brane
with three space dimensions; the action is $S=[M(t)]^3 \int d^5 X \sqrt{G} {\cal R}^{(5)}$$+
M^2_{Planck} \int d^4 x \sqrt{g} R$, where ${\cal R}^{(5)}$ and $R$ are the scalar curvature in fiveand four-dimensional spacetime respectively, and
$G$ and $g$ are the determinant of the five- and
four-dimensional spacetime metric) which accommodates a
time-dependent fundamental length $L(t) (\equiv M^2_{Planck}/[M(t)]^3 = [H(t)]^{-1}$, they considered cosmological models where the scale factor diverges with an
essential singularity at a finite future time. Their models have no dark energy in the conventional sense of energy possessing
a simple pressure�Cenergy relationship.  \newline
Frampton and Takahashi  generalized the Schwarzschild
solution to this modification of GR [7] together with the modification of
the Newton potential at
short distances being given by
\begin{displaymath}
 V(r)=-\frac{G m}{r}-\frac{4\sqrt{G m}\sqrt{r}}{L(t)}=-\frac{r_g}{2r}-\frac{2\sqrt{2}\sqrt{r_g r}}{L(t)},
\end{displaymath}
where $r_g = 2Gm$ is the Schwarzschild radius. \newline
The present author would like to point out some mistakes in [1] here.
Firstly, Frampton and Takahashi defined
$L(t)=[L(t_0)]^{-1} [T(t)]^p$ (cf. Eq. (6) in [1]) and then wrote down the modified Friedmann equation for DGP gravity being, in an empty universe, $[H(t)]^2-H(t)/L(t)=0$ (cf. Eq. (11) in [1]). He thus got
\begin{equation}
 \frac{\dot{a}(t)}{a(t)}=H(t)=\frac{H(t_0)}{T^p}
\end{equation}
which is Eq. (12) in [1]. But, as the present author checked, with $L(t)=1/[H(t)]$,
Eq. (12) should be read as
\begin{equation}
  \frac{\dot{a}(t)}{a(t)}=H(t)=\frac{1}{H(t_0) T^p}.
\end{equation}
Even if we follow their assumption : neglecting, for the future
evolution, the term $(\rho_M + \rho_{\gamma})/(3M^2_{Planck})$ on the
right-hand-side of the modified Friedmann equation. Meanwhile, with $\gamma= -dT/dt$$=
1/(t_{rip}-t_0)$, we then have
\begin{equation}
 \ln a(t)=-\int_1^{T(t)} \frac{L(t_0)}{\gamma T^p} dT,
\end{equation}
instead of the expression shown in the right-hand-side of Eq. (13) in [1]. Thus,
for $p = 1$, which is similar to dark energy
with a constant $w < - 1$ equation of state : Eq. (14) in [1] should be changed to
\begin{equation}
 a(t)=T^{-L(t_0)/\gamma},
\end{equation}
while for the Bigger Rip case $p > 1$ one finds
\begin{equation}
 a(t)=a(t_0) \exp[(\frac{1}{T^{p-1}}-1)\frac{L(t_0)}{\gamma(p-1)}],
\end{equation}
instead of the Eq. (15) shown in [1]. \newline
To conclude in brief, as the derivations of Eqs. (12-15) in [1] had something wrong
with the position of $L(t_0)$ (if Eq. (6) is correct in [1]) which the present author just corrected above, the remaining
statements and mathematical derivations associated with Eqs. (12-15) should be corrected
simultaneously for the physical interpretation of Frampton and Takahashi's
subsequent presentation in [1]. {\it Acknowledgements.} The author is partially supported
by the Starting Funds of 2005-XJU-Scholars.

%
\begin{thebibliography}{99} %\bibitem{PKU:1999}
\bibitem{Rip:Singul} P.H. Frampton and T. Takahashi, Astropart. Phys. 22 (2004) 307.  %-312
\bibitem{DGP:mGR} G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 485
(2000) 208. Available from: [hep-th/0005016].
%\bibitem{Sudden:S} Barrow JD 2004 {\it Class. Quantum Grav.} {\bf
%21} L79.
%\bibitem{More:S} Barrow JD 2004 {\it Class. Quantum
%Grav.} {\bf 21} 5619.
%\bibitem{DUU:2001} A. K.-H. Chu, {J. Phys. B At. Mol. Opt.
%Phys.} {\bf 34} (2001) L711. V.V. Vedenyapin, I.V. Mingalev, and
%O.V. Mingalev, {Russian Academy of Sciences Sbornik Mathematics}
%{\bf 80} (1995) 271. E. Giglio, E. Suraud  and P.-G. Reinhard, {
%Ann. Physik} {\bf 11} (2002) 291.
%\bibitem{UU:C}  E.A. Uehling and G.E. Uhlenbeck, Phys. Rev. 43 (1933) 552.
%R.L. Liboff, Kinetic Theory, Prentice-Hall, London, 1990.
%\bibitem{DUU:QCP} A. K.-H. Chu, { Phys. Rev. B} {\bf 68} (2003)  205308 .
%\bibitem{UU:2000} A. Rossani and G. Kaniadakis, Physica A {\bf 277} (2000)
%349-358.
%Kinast J, Hemmer SL, Gehm ME, Turlapov A, Thomas JE
%Phys. Rev. Lett. 92 (15): 150402 APR 16 2004.
\end{thebibliography}
\end{document}.
%------------------------------------------
%
\newpage

\vspace{5mm}
%\vspace{4mm}  %chu-fg1.ps
\psfig{file=chu-fg1.ps,bbllx=0cm,bblly=14cm,bburx=22cm,bbury=24cm,rheight=9cm,rwidth=9cm,clip=}
%
\begin{figure}[h]
\hspace*{6mm} {\small Fig. 1 \hspace*{2mm} $ p(t) [\propto
\ddot{a}(t)/a(t)] $ and $\rho(t)$ vs. $t$. Here, $n=1.5$, $q=0.8$,
$t_s =10.0$.
\newline \hspace*{6mm} There might be a singularity for $\rho(t)$
but not for $p(t)$ as $t\sim 0$ \newline \hspace*{6mm} (case of
$a_s \sim 1$; but not for $t\rightarrow t_s$).}
%Comparison of the Navier-slip effects due to  $N_s =0.001, 0.01, 0.05$, and $0.06$
%\newline \hspace*{12mm} on the neutral stability boundary of the plane
%Poiseuille flow. $Re_{cr}$  (the critical \newline \hspace*{12mm}
%Reynolds number of the flow) $\sim \,5772, 2480, 213.6$ for
%different ($N_s$=0, 0.01, 0.06) cases.
%\newline \hspace*{12mm}  $N_s$ is the slip parameter introduced by Navier [23].}
\end{figure}

\vspace{15mm}

%\newpage
\psfig{file=chu-fg2.ps,bbllx=0cm,bblly=14.4cm,bburx=20cm,bbury=23.8cm,rheight=8.8cm,rwidth=8.8cm,clip=}
%
\begin{figure}[h]
\hspace*{6mm} {\small Fig. 2 \hspace*{2mm} $ p(t) [\propto
\ddot{a}(t)/a(t)] $ and $\rho(t)$ vs. $t$. Here, $n=1.5$, $q=0.8$,
$t_s =100$.
\newline \hspace*{6mm} There is no singularity for $p(t)$ or
$\rho(t)$ once $t\rightarrow t_s (\equiv 100)$.}
\end{figure}

\newpage
%\newpage
\psfig{file=chu-fg3.ps,bbllx=0cm,bblly=14.4cm,bburx=20cm,bbury=23.8cm,rheight=8.8cm,rwidth=8.8cm,clip=}
%
\begin{figure}[h]
\hspace*{6mm} {\small Fig. 3 \hspace*{2mm} $ p(t) [\propto
\ddot{a}(t)/a(t)] $ and $\rho(t)$ vs. $t$. Here, $n=1.5$, $q=0.8$,
$t_s =100$.
\newline \hspace*{6mm} There might be no singularity for $p(t)$ or
$\rho(t)$ once $t\rightarrow t_s (\equiv 100)$.}
\end{figure}
%
%\end{equation}
%$T(t)=(t_{rip}-t)/(t_{rip}-t_0)=\gamma(t_{rip}-t)$, $t_{rip}$ is the time of the Rip.
%In particular we study the values p = 2,3,. . ., as alternative to the 'dark energy' case
%p = 1.
%\begin{displaymath}
% t_{rip}-t_U = \frac{1}{\gamma}(\frac{8 l_0^3}{L^2_0 r_g})^{1/(2p)}
%\en{displaymath}
%
%even when $\rho > 0$ and $\rho + 3\,p > 0$ [1].
%He has identified conditions that are sufficient to avoid the
%appearance of this type of sp-curvature singularity at finite
%future time. These results can either be used as a more
%fine-grained method to search for finite time singularities and
%evaluate their likelihood of occurrence with realistic fluids or,
%if such singularities are regarded as causal pathologies, to
%impose realistic conditions which exclude them by fiat. \newline
%Consider the Friedmann universe with expansion scale factor
%$a(t)$, curvature parameter $k$, and Hubble expansion rate $H =
%\dot{a}/a$; then ($8\pi G = c = 1$) the Einstein equations reduce
%to
%\begin{equation}
% 3 H^2=\rho-\frac{k}{a^2}, \hspace*{24mm} \dot{\rho}+3 H
% (\rho+p)=0,
%\end{equation}
%and
%\begin{equation}
% \frac{\ddot{a}}{a}=-\frac{\rho+3 p}{6}.
%\end{equation}
%He first looked informally at whether it is possible for any type
%of singularity to develop in which a physical scalar becomes
%infinite at a finite future comoving proper time $t_s$ , when,
%say, $a(t) \rightarrow a(t_s) \not = 0$ or $\infty$, and $H(t)
%\rightarrow H_s < \infty$ and $H_s > 0$. \newline Guided by these
%heuristic arguments, he then constructed an explicit example by
%seeking, over the time interval $0 < t < t_s$ , a solution for the
%scale factor $a(t)$ of the form $a(t) = A + B \,t^q + C(t_s -
%t)^n$, where $A > 0$,$B > 0$, $q > 0$, $C$ and $n > 0$ are free
%constants to be determined. \newline He can obtain (Eq. (14) in
%[1])
%\begin{displaymath}
% \frac{\ddot{a}}{a} \rightarrow -\infty,
%\end{displaymath}
%whenever $1 < n < 2$ and $0 < q \le 1$; the solution exists on the
%interval $0 < t < t_s$ . Hence, as $t \rightarrow t_s$ we have $a
%\rightarrow a_s$ ; $H_s$ and $\rho_s > 0$ (for $3 q^2(a_s - 1)^2
%t^{-2}_s > -k$) are finite but $p_s \rightarrow \infty$. When $2 <
%n < 3$, he noted that $\ddot{a}$ remains finite but $d^3 a/dt^3
%\rightarrow \infty$ as $t \rightarrow t_s$ as $p_s$ remains finite
%but $\dot{p}_s \rightarrow \infty$. By contrast, there is an
%initial all-encompassing strong-curvature singularity, with $\rho
%\rightarrow ��\infty$ and $p \rightarrow \infty$, as $t
%\rightarrow 0$. However, from Eqs. (14) and (4) of [1], he can see
%that $\rho$ and $\rho +3p$ remain positive. \newline After
%considering two approximate forms of $a(t)$ as $t\rightarrow 0$
%and  $t_s$, he then claimed that this specific family of solutions
%shows that it is possible for an expanding universe to develop a
%'big-rip' singularity at a finite future time even if the matter
%fields in the universe satisfy the strong-energy conditions $\rho
%> 0$ and $\rho + 3 p > 0$. \newline  % amalgam
%The present author would like to make comments about his
%derivations (especially about the equation (4) and its subsequent
%variations in [1]) and his results, say, equations (11) or (13) in
%[1]. Our second remark could be induced by the first remark!
%\newline The first remark has been partially addressed, as Barrow
%noticed that, there are two independent Friedmann equations for
%the three quantities $a$, $p$ and $\rho$. In the absence of a
%relation between any two of them there is an unconstrained degree
%of freedom. If we introduce an equation of state $p(\rho)$ which
%bounds the pressure by some well-behaved function of the density��
%for example, $p < C \rho$ with $C > 0$-then the pressure
%singularity at a finite future time is eliminated if we require
%$\rho > 0$ and $\rho +3p> 0$. An upper bound on the ratio $p/\rho$
%corresponds to a bound on signal propagation of waves associated
%with small changes in pressure.
%\newline The second remark will be demonstrated by my illustration
%of figures once we plot $\ddot{a(t)}/a(t) [\propto p(t)$] and
%$\rho(t)$ with respect to $t$ (from the equation (13) of [1]). As
%Barrow mentioned, $1 < n < 2$ and $0 < q \le 1$, thus we choose
%$n=1.5$, $q=0.8$, and $t_s =10$. But, the trouble is that $a_s$ is
%unknown. We then try three cases : $a_s =1.00,1.05,50.00$. We can
%easily observe from Fig. 1 that there might be a singularity for
%$\rho(t)$ but not for $p(t)$ as $t \sim 0$ once $a_s \sim 1$.  We
%cannot observe a singularity for $\rho(t)$ or $p(t)$ once
%$t\rightarrow t_s$ which Barrow claimed in [1] that there will be
%a singularity for $p(t)$! These can be demonstrated in Figs. 1, 2
%and 3 for either $t_s =10$ or $t_s=100$ once $t\rightarrow t_s$.
%\newline
%Let us consider the expression of $\ddot{a}(t)/a(t)$, the important term is
%$t_s -t = K\, t_s$. If $K \ll 1$, then there will be no singularity
%for $p(t)$ as $t \rightarrow t_s$ once $t_s^2 \times K^{2-n} \sim O(1)$!
%\newline
%To conclude in brief, our remarks as demonstrated in Figs. 1, 2
%and 3 could not only be valid to the false statement which Barrow
%made in [1] but also be possible be valid to his subsequent paper
%(quite similar works) in [2].
%
%Hannestad gave a presentation [1] of the possible effect
%of the residual annihilations of heavy particles after freeze-out
%from equilibrium in the early universe. An error in previous work
%on this subject was pointed out and the correct method of solving
%the Boltzmann equation for this case was developed. For Majorana
%particles there are significant differences relative to previous
%work.\newline %%
%As this paper was published in a high-impact-factor Journal ({\it
%Phys. Lett. B}), its influence to the subsequent readers in
%relevant fields will be significant. Thus, the present author,
%based on the similar experiences in the quantum kinetic thoery
%(especially the Uehling-Uhlenbeck approach) [2], would like to
%point out some fundamental mistakes made in this paper which are
%related to the essential definition. These wrong definitions
%appeared at the page 2 (the right-hand-side column, just after the
%description of Eq. (4) for the collisional terms) : "$\cdots
%\Lambda (f_1, f_2, f_3, f_4)=f_3 f_4(1\pm f_1)(1\pm f_2)-f_1 f_2
%(1\pm f_3)(1\pm f_4)$  is the phase-space factor with $+$
%corresponding to Fermi-Dirac statistics and $-$ to Bose-Einstein
%statistics". In fact, Eq. (4) reads
%\begin{displaymath}
% C_{coll} [f]=\frac{1}{2 E_1} \int d^3 \tilde{p}_2 d^3 \tilde{p}_3
%d^3 \tilde{p}_4 \Lambda (f_1, f_2, f_3, f_4) \,\, \Sigma
%|M|^2_{12\rightarrow 34} \delta^4  \,\,(p_1 +p_2 -p_3 -p_4)
%(2\pi)^4,
%\end{displaymath}
%as long as they were only concerned with 2-body collisions of the
%type $1 + \,2 \rightarrow 3 + \,4$. Here we have $d^3\tilde{p}=
%d^3 p/( (2\pi)^3 2 E)$.$S$ is a symmetrisation factor of 1/2! for
%each pair of identical particles in initial or  final states [1],
%and $\Sigma |M|^2$ is the interaction matrix element squared and
%spin-summed. $p_i$ is the fourth momentum of particle $i$.
%\newline Similar mistakes also appeared in the following sentences :
%"$\cdots$ Further, a significant simplification is possible if one
%uses Boltzmann statistics for all particles instead of the
%appropriate quantum statistics, so that $1 \pm f \rightarrow 1$,
%(Eq. (5) in [1]), where, again, $+$ corresponds to Fermi-Dirac
%statistics and $-$ to Bose-Einstein statistics".
%\newline
%Considering the derivation of this collision term especially after
%taking into account the spin characteristic and the Pauli-blocking
%effect in quantum-mechanic sense [2-4], the above mentioned
%arithmetic operators $+$ and $-$ were falsely attributed to wrong
%particle-statistics. They should be appeared in reverse order
%following the presentations in [1]. \newline To make comparison
%with the correct definition, we shall cite the relevant work below
%for introducing the Uehling-Uhlenbeck equation [5], which extends
%its applicability to particles which obey non-conventional
%statistics. The Uehling-Uhlenbeck equation (UUE) is a quantum
%extension of the Boltzmann equation [5]. Consider particles (mass
%m) which obey the Bose-Einstein or Fermi-Dirac statistics. These
%particles interact elastically by means of binary encounters
%(cross section $\sigma$). In the space homogeneous case the UUE
%reads as follows ($\lambda$=1 for bosons, $\lambda$ = $-1$ for
%fermions):
%\begin{displaymath}
%\frac{\partial f}{\partial t} = \frac{1}{h^3} \int d{\bf p}_1  d
%{\bf \Sigma}' [f' f'_1 (1 + \lambda f)(1 + \lambda f_1)-f f_1(1 +
%\lambda f')(1 + \lambda f'_1 )] \times g \sigma(g;\xi)) ;
%\end{displaymath}
%where $f = f({\bf p}; t)$ is the quantum distribution function
%(dimensionless) and $f' = f({\bf p}'; t)$; $f_1=f({\bf p}_1; t)$;
%$f'_1 =f({\bf p}'_1 ; t)$. We denote by ${\bf p}$ and ${\bf p}_1$
%the momenta before collision, while ${\bf p}'$ and ${\bf p}'_1$
%are the momenta after collision:
%\begin{displaymath}
%  {\bf p}' = ({\bf p}+{\bf p}_1+m g{\bf \Omega}')/2; \hspace*{6mm}
%   {\bf p}'_1 = ({\bf p}+{\bf p}_1-m g {\bf \Omega}')/2 ;
%\end{displaymath}
%where $g=|{\bf p}-{\bf p}_1|/m$ and ${\bf \Omega}'$ is a unit
%vector. The cross section $\sigma$, in general, depends on both
%$g$ and $\xi$, where $\xi ={\bf \Omega}\cdot {\bf \Omega}'$; ${\bf
%\Omega} = 1/ ({\bf p}- {\bf p}_1)$. \newline To conclude in brief,
%for those readers of interest, we should read : '$-$' corresponds
%to Fermi-Dirac statistics and '$+$' to Bose-Einstein statistics in
%[1] instead of the above mentioned forms.
\bibitem{Tao:1999} S. Hannestad, Phys. Lett. B {\bf 452} (1999)
23-27.

astro-ph0502012.extracted_bib
\begin{thebibliography}{99} %\bibitem{PKU:1999}
\bibitem{Rip:Singul} P.H. Frampton and T. Takahashi, Astropart. Phys. 22 (2004) 307.  %-312
\bibitem{DGP:mGR} G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 485
(2000) 208. Available from: [hep-th/0005016].
%\bibitem{Sudden:S} Barrow JD 2004 {\it Class. Quantum Grav.} {\bf
%21} L79.
%\bibitem{More:S} Barrow JD 2004 {\it Class. Quantum
%Grav.} {\bf 21} 5619.
%\bibitem{DUU:2001} A. K.-H. Chu, {J. Phys. B At. Mol. Opt.
%Phys.} {\bf 34} (2001) L711. V.V. Vedenyapin, I.V. Mingalev, and
%O.V. Mingalev, {Russian Academy of Sciences Sbornik Mathematics}
%{\bf 80} (1995) 271. E. Giglio, E. Suraud  and P.-G. Reinhard, {
%Ann. Physik} {\bf 11} (2002) 291.
%\bibitem{UU:C}  E.A. Uehling and G.E. Uhlenbeck, Phys. Rev. 43 (1933) 552.
%R.L. Liboff, Kinetic Theory, Prentice-Hall, London, 1990.
%\bibitem{DUU:QCP} A. K.-H. Chu, { Phys. Rev. B} {\bf 68} (2003)  205308 .
%\bibitem{UU:2000} A. Rossani and G. Kaniadakis, Physica A {\bf 277} (2000)
%349-358.
%Kinast J, Hemmer SL, Gehm ME, Turlapov A, Thomas JE
%Phys. Rev. Lett. 92 (15): 150402 APR 16 2004.
\end{thebibliography}