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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "#Get correct dark en
ergy tracking Gamma equation" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "restart;\n\nassume(wq>-1);as
sume(Omq>0);assume(V(t)>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
109 "xsub:=1/2*diff(Q(t),t)^2/V(t);\nx_sub:=simplify((1+wq)/(1-wq));\n
\n\nQdot_sub:=solve(x(t)=xsub,diff(Q(t),t))[1];\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%xsubG,$*&*$)-%%diffG6$-%\"QG6#%#t|irGF/\"\"#\"\"\"F1
-%\"VGF.!\"\"#F1F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&x_subG,$*&,&%
$wq|irG\"\"\"F)F)F),&!\"\"F)F(F)F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%)Qdot_subG*&-%%sqrtG6#\"\"#\"\"\"-F'6#*&-%\"xG6#%#t|irGF*-%\"VGF0F
*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Vdot_sub=(-diff(Qdot
_sub,t)-3*H(t)*Qdot_sub)*Qdot_sub;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/%)Vdot_subG*(,&*&*&-%%sqrtG6#\"\"#\"\"\",&*&-%%diffG6$-%\"xG6#%#t|irG
F6F--%\"VGF5F-F-*&F3F--F16$F7F6F-F-F-F-*$-F*6#*&F3F-F7F-F-!\"\"#F@F,**
\"\"$F--%\"HGF5F-F)F--F*6#F?F-F@F-F)F-FFF-" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "subs(diff(V(t),t)=Vdot_sub,%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%)Vdot_subG*(,&*&*&-%%sqrtG6#\"\"#\"\"\",&*&-%%diffG6$
-%\"xG6#%#t|irGF6F--%\"VGF5F-F-*&F3F-F$F-F-F-F-*$-F*6#*&F3F-F7F-F-!\"
\"#F>F,**\"\"$F--%\"HGF5F-F)F--F*6#F=F-F>F-F)F-FDF-" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 38 "Vdot_sub:=simplify(solve(%,Vdot_sub));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Vdot_subG,$*&*&-%\"VG6#%#t|irG\"\"
\",&-%%diffG6$-%\"xGF*F+F,*(\"\"'F,-%\"HGF*F,F1F,F,F,F,,&F,F,F1F,!\"\"
F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "xdot_sub:=solve(diff(
x(t),t)/x(t)/H(t)=dlnxdlna(t),diff(x(t),t));\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)xdot_subG*(-%)dlnxdlnaG6#%#t|irG\"\"\"-%\"xGF(F*-%\"
HGF(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 352 "varsubs:=proc(xx
)\n subs(diff(Q(t),t)=Qdot_sub,xx):\n\n simplify(subs(diff(V(t),t)=Vdo
t_sub,%)):\n subs(diff(x(t),t)=xdot_sub,%):\n simplify(subs(diff(dlnxd
lna(t),t)=ddlnxdlna(t)*H(t),%)):\n subs(diff(Q(t),t)=Qdot_sub,%):\n\n \+
subs(diff(V(t),t)=Vdot_sub,%):\n subs(diff(x(t),t)=xdot_sub,%):\n subs
(diff(dlnxdlna(t),t)=ddlnxdlna(t)*H(t)*x(t),%):\n\nsimplify(%);\nend;
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(varsubsGR6#%#xxG6\"F(F(C+-%%sub
sG6$/-%%diffG6$-%\"QG6#%\"tGF4%)Qdot_subG9$-%)simplifyG6#-F+6$/-F/6$-%
\"VGF3F4%)Vdot_subG%\"%G-F+6$/-F/6$-%\"xGF3F4%)xdot_subGFB-F86#-F+6$/-
F/6$-%)dlnxdlnaGF3F4*&-%*ddlnxdlnaGF3\"\"\"-%\"HGF3FWFB-F+6$F-FBF:FC-F
+6$/FP*(FUFWFXFWFHFWFB-F86#FBF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 145 "rho(t):=diff(Q(t),t)^2/2+V(t);\nH2_sub:=simplify(kap
pa/3*((1-Omq)/Omq +1)*rho(t));\nH_sub:=simplify(sqrt(kappa/3)*sqrt(((1
-Omq)/Omq + 1)*rho(t)));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%$rhoG
6#%#t|irG,&*$)-%%diffG6$-%\"QGF&F'\"\"#\"\"\"#F1F0-%\"VGF&F1" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'H2_subG,$*&*&%&kappaG\"\"\",&*$)-%%diffG6
$-%\"QG6#%#t|irGF3\"\"#F)F)*&F4F)-%\"VGF2F)F)F)F)%%Omq|irG!\"\"#F)\"\"
'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&H_subG,$*&**-%%sqrtG6#\"\"$\"
\"\"-F)6#%&kappaGF,-F)6#\"\"#F,-F)6#,&*$)-%%diffG6$-%\"QG6#%#t|irGF>F2
F,F,*&F2F,-%\"VGF=F,F,F,F,*$-F)6#%%Omq|irGF,!\"\"#F,\"\"'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "Hdot_sub:=-1/2*kappa*( (1+wb)*(1-O
mq)/Omq*rho(t) + (1+wq)*rho(t));\n%Hdot_sub:=-1/2*kappa*( (1+wb)*((1-O
mq)/Omq+1)*rho(t)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)Hdot_subG,$
*&%&kappaG\"\"\",&*&*(,&F(F(%#wbGF(F(,&F(F(%%Omq|irG!\"\"F(,&*$)-%%dif
fG6$-%\"QG6#%#t|irGF:\"\"#F(#F(F;-%\"VGF9F(F(F(F/F0F(*&,&%$wq|irGF(F(F
(F(F1F(F(F(#F0F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*%Hdot_subG,$**%
&kappaG\"\"\",&F(F(%#wbGF(F(,&*&,&F(F(%%Omq|irG!\"\"F(F.F/F(F(F(F(,&*$
)-%%diffG6$-%\"QG6#%#t|irGF9\"\"#F(#F(F:-%\"VGF8F(F(#F/F:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "Gamma:=diff(diff(V(t),t)/diff(Q(t),
t),t)/diff(Q(t),t) * V(t) / (diff(V(t),t)/diff(Q(t),t))^2;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&GammaG*&*(,&*&-%%diffG6$-%\"VG6#%#t|irG-%
\"$G6$F/\"\"#\"\"\"-F*6$-%\"QGF.F/!\"\"F4*&*&-F*6$F,F/F4-F*6$F7F0F4F4*
$)F5F3F4F9F9F4F5F4F,F4F4*$)F<F3F4F9" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "Gamma:=simplify(Gamma);\nGamma:=varsubs(varsubs(Gamma
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&GammaG*&*&,&*&-%%diffG6$-%\"
VG6#%#t|irG-%\"$G6$F/\"\"#\"\"\"-F*6$-%\"QGF.F/F4F4*&-F*6$F,F/F4-F*6$F
7F0F4!\"\"F4F,F4F4*&F5F4)F:F3F4F>" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>
%&GammaG,$*&,8*(-%*ddlnxdlnaG6#%#t|irG\"\"\")-%\"HGF+\"\"#F--%\"xGF+F-
!\"#**F1F--%)dlnxdlnaGF+F-F2F--%%diffG6$F/F,F-!\"\"**F1F-)F6F1F-F2F-F.
F-F-**\"#=F-F.F-F6F-F2F-F-*(\"#7F-F8F-F2F-F;*(\"#OF-F.F-F2F-F-*(F1F-F)
F-F.F-F;*&F=F-F.F-F;*(F1F-F6F-F8F-F;*(\"\"'F-F.F-F6F-F;*&FAF-F8F-F;F-*
(F2F-F.F-),&F6F-FHF-F1F-F;#F-F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 66 "subs(diff(H(t),t)=Hdot_sub,Gamma):\nsimplify(%):\nGamma:=varsu
bs(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&GammaG,$*&,jn*&%&kappaG\"
\"\"-%\"VG6#%#t|irGF*!\"'**\"\"'F*F)F*)-%\"xGF-\"\"#F*F+F*!\"\"*.\"#7F
*F)F*F3F*F+F*%%Omq|irGF*%$wq|irGF*F6*,F5F*-%*ddlnxdlnaGF-F*)-%\"HGF-F5
F*F3F*F9F*F**,F5F*F>F*F3F*)-%)dlnxdlnaGF-F5F*F9F*F6*,\"#=F*F>F*F3F*FCF
*F9F*F6*,F5F*FCF*F3F*F)F*F+F*F6*,F8F*F)F*F3F*F+F*%#wbGF*F6**FCF*F)F*F+
F*FIF*F6*.F5F*FCF*F3F*F)F*F+F*FIF*F6*0F5F*FCF*F3F*F)F*F+F*FIF*F9F*F**0
F5F*FCF*F3F*F)F*F+F*F9F*F:F*F6**F1F*F)F*F+F*FIF*F6*,F1F*F)F*F+F*FIF*F9
F*F**,F1F*F)F*F+F*F9F*F:F*F6*.F8F*F)F*F3F*F+F*FIF*F9F*F**,FCF*F)F*F+F*
FIF*F9F*F**,FCF*F)F*F+F*F9F*F:F*F6**\"#OF*F>F*F3F*F9F*F6*(FBF*F>F*F9F*
F***F1F*F>F*FCF*F9F*F***F8F*F)F*F3F*F+F*F6*(FCF*F)F*F+F*F6**F5F*F<F*F>
F*F9F*F***FCF*F2F*F)F*F+F*F6*,FCF*F2F*F)F*F+F*FIF*F6*.F1F*F)F*F2F*F+F*
F9F*F:F*F6*,F1F*F)F*F2F*F+F*FIF*F6*.F1F*F)F*F2F*F+F*FIF*F9F*F**.FCF*F2
F*F)F*F+F*F9F*F:F*F6*.FCF*F2F*F)F*F+F*FIF*F9F*F*F***F9F*F3F*F>F*),&FCF
*F1F*F5F*F6#F6F5" }}}{PARA 12 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#,$*&,jn*&%&kappaG\"\"\"-%\"VG6#%#t|irGF(!\"'**\"\"'F(F'F()-%\"xG
F+\"\"#F(F)F(!\"\"*.\"#7F(F'F(F1F(F)F(%%Omq|irGF(%$wq|irGF(F4*,F3F(-%*
ddlnxdlnaGF+F()-%\"HGF+F3F(F1F(F7F(F(*,F3F(F<F(F1F()-%)dlnxdlnaGF+F3F(
F7F(F4*,\"#=F(F<F(F1F(FAF(F7F(F4*,F3F(FAF(F1F(F'F(F)F(F4*,F6F(F'F(F1F(
F)F(%#wbGF(F4**FAF(F'F(F)F(FGF(F4*.F3F(FAF(F1F(F'F(F)F(FGF(F4*0F3F(FAF
(F1F(F'F(F)F(FGF(F7F(F(*0F3F(FAF(F1F(F'F(F)F(F7F(F8F(F4**F/F(F'F(F)F(F
GF(F4*,F/F(F'F(F)F(FGF(F7F(F(*,F/F(F'F(F)F(F7F(F8F(F4*.F6F(F'F(F1F(F)F
(FGF(F7F(F(*,FAF(F'F(F)F(FGF(F7F(F(*,FAF(F'F(F)F(F7F(F8F(F4**\"#OF(F<F
(F1F(F7F(F4*(F@F(F<F(F7F(F(**F/F(F<F(FAF(F7F(F(**F6F(F'F(F1F(F)F(F4*(F
AF(F'F(F)F(F4**F3F(F:F(F<F(F7F(F(**FAF(F0F(F'F(F)F(F4*,FAF(F0F(F'F(F)F
(FGF(F4*.F/F(F'F(F0F(F)F(F7F(F8F(F4*,F/F(F'F(F0F(F)F(FGF(F4*.F/F(F'F(F
0F(F)F(FGF(F7F(F(*.FAF(F0F(F'F(F)F(F7F(F8F(F4*.FAF(F0F(F'F(F)F(FGF(F7F
(F(F(**F7F(F1F(F<F(),&FAF(F/F(F3F(F4#F4F3" }}}{EXCHG {PARA 12 "" 1 "" 
{TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "subs(H(t)
^2=H2_sub,%):\nsimplify(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&*&%
&kappaG\"\"\",bo*&-%\"xG6#%#t|irGF'-%\"VGF,F'!$W\"*(\"#OF'F*F')-%%diff
G6$-%\"QGF,F-\"\"#F'!\"\"*(\"\"'F'-%)dlnxdlnaGF,F'F.F'F'*(F2F'F.F'%#wb
GF'F:*&)F=F9F'F3F'F'*(\"\"%F'-%*ddlnxdlnaGF,F'F.F'F'*(F9F'FEF'F3F'F'*(
F<F'F=F'F3F'F'*,F2F')F*F9F'F.F'F@F'%%Omq|irGF'F'*&F2F'F.F'F:*.F<F'F=F'
FJF'F.F'FKF'%$wq|irGF'F:*.\"#7F'F=F'F*F'F.F'FKF'FNF'F:**F9F'FEF'F*F'F3
F'F'**FDF'FEF'F*F'F.F'F'**F9F'F*F'FBF'F3F'F:**FDF'F*F'FBF'F.F'F:**\"#=
F'F=F'F*F'F3F'F:**F2F'F.F'FKF'FNF'F:**\"#sF'F*F'F.F'F@F'F:**F<F'F=F'F.
F'F@F'F:**F2F'F.F'F@F'FKF'F'**\"#[F'F=F'F*F'F.F'F:**F2F'FJF'F.F'F@F'F:
*.FPF'F=F'F*F'F.F'F@F'FKF'F'**F<F'F=F'FJF'F.F'F:*(F9F'FBF'F.F'F'*.F<F'
F=F'FJF'F.F'F@F'FKF'F'*,F<F'F=F'F.F'F@F'FKF'F'*,FYF'F*F'F.F'FKF'FNF'F:
*,F<F'F=F'FJF'F.F'F@F'F:*,F2F'FJF'F.F'FKF'FNF'F:*,FPF'F=F'F*F'F.F'F@F'
F:*,FYF'F*F'F.F'F@F'FKF'F'*,F<F'F=F'F.F'FKF'FNF'F:*(F2F'FJF'F.F'F:F'F'
**FKF'F*F')-%\"HGF,F9F'),&F=F'F<F'F9F'F:#F:FP" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 11 "varsubs(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,
$*&*(%&kappaG\"\"\"-%\"VG6#%#t|irGF',hn!#=F'*&\"\"$F'-%)dlnxdlnaGF*F'F
'*$)F0\"\"#F'F'*&\"#=F'%#wbGF'!\"\"*&F4F'-%*ddlnxdlnaGF*F'F'*(F6F')-%
\"xGF*F4F'F7F'F8*(F4F'F=F'F3F'F8*(F4F'F:F'F=F'F'*(\"\"%F'F:F'F>F'F'*(
\"#@F'F0F'F=F'F8*&F>F'F3F'F8*(\"#OF'F>F'F7F'F8*(F/F'F0F'F7F'F8*&\"#aF'
F=F'F8*(F6F'F0F'F>F'F8**\"\"'F'F0F'F>F'F7F'F8*(F6F'%%Omq|irGF'%$wq|irG
F'F8*(F6F'F7F'FPF'F'*,F/F'F0F'F=F'FPF'FQF'F8*,FNF'F0F'F>F'FPF'FQF'F8*,
F/F'F0F'F=F'F7F'FPF'F'**FHF'F>F'FPF'FQF'F8**F/F'F0F'F=F'F7F'F8**F6F'F=
F'FPF'FQF'F8*,FNF'F0F'F>F'F7F'FPF'F'**F/F'F0F'FPF'FQF'F8**FHF'F>F'F7F'
FPF'F'**F/F'F0F'F7F'FPF'F'**F6F'F=F'F7F'FPF'F'*&\"#sF'F>F'F8F'F'**FPF'
F>F')-%\"HGF*F4F'),&F0F'FNF'F4F'F8#F8FN" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "subs(H(t)=H_sub,%);\n" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#,$*&*&-%\"VG6#%#t|irG\"\"\",hn!#=F**&\"\"$F*-%)dlnxdlnaGF(F*F**$
)F/\"\"#F*F**&\"#=F*%#wbGF*!\"\"*&F3F*-%*ddlnxdlnaGF(F*F**(F5F*)-%\"xG
F(F3F*F6F*F7*(F3F*F<F*F2F*F7*(F3F*F9F*F<F*F**(\"\"%F*F9F*F=F*F**(\"#@F
*F/F*F<F*F7*&F=F*F2F*F7*(\"#OF*F=F*F6F*F7*(F.F*F/F*F6F*F7*&\"#aF*F<F*F
7*(F5F*F/F*F=F*F7**\"\"'F*F/F*F=F*F6F*F7*(F5F*%%Omq|irGF*%$wq|irGF*F7*
(F5F*F6F*FOF*F**,F.F*F/F*F<F*FOF*FPF*F7*,FMF*F/F*F=F*FOF*FPF*F7*,F.F*F
/F*F<F*F6F*FOF*F***FGF*F=F*FOF*FPF*F7**F.F*F/F*F<F*F6F*F7**F5F*F<F*FOF
*FPF*F7*,FMF*F/F*F=F*F6F*FOF*F***F.F*F/F*FOF*FPF*F7**FGF*F=F*F6F*FOF*F
***F.F*F/F*F6F*FOF*F***F5F*F<F*F6F*FOF*F**&\"#sF*F=F*F7F*F**(F=F*,&*$)
-%%diffG6$-%\"QGF(F)F3F*F**&F3F*F&F*F*F*),&F/F*FMF*F3F*F7F7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "varsubs(%);" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#,$*&,J*&-%)dlnxdlnaG6#%#t|irG\"\"\"-%\"xGF)F+!#@*,
\"\"$F+F'F+F,F+%#wbGF+%%Omq|irGF+F+*,F0F+F'F+F,F+F2F+%$wq|irGF+!\"\"**
F0F+F'F+F,F+F1F+F5**\"#=F+F,F+F2F+F4F+F5*&\"#aF+F,F+F5*(F8F+F,F+F1F+F5
*(\"\"#F+F,F+)F'F=F+F5*(F=F+-%*ddlnxdlnaGF)F+F,F+F+**F8F+F,F+F1F+F2F+F
+F8F5*&F8F+F1F+F5*&F0F+F'F+F+*&F=F+F@F+F+*(F8F+F1F+F2F+F+*(F8F+F2F+F4F
+F5**F0F+F'F+F2F+F4F+F5*(F0F+F'F+F1F+F5*$F>F+F+**F0F+F'F+F1F+F2F+F+F+*
&),&F'F+\"\"'F+F=F+F,F+F5#F5F=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 9 "Gamma:=%;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&GammaG,$*&,J*&-
%)dlnxdlnaG6#%#t|irG\"\"\"-%\"xGF+F-!#@*,\"\"$F-F)F-F.F-%#wbGF-%%Omq|i
rGF-F-*,F2F-F)F-F.F-F4F-%$wq|irGF-!\"\"**F2F-F)F-F.F-F3F-F7**\"#=F-F.F
-F4F-F6F-F7*&\"#aF-F.F-F7*(F:F-F.F-F3F-F7*(\"\"#F-F.F-)F)F?F-F7*(F?F--
%*ddlnxdlnaGF+F-F.F-F-**F:F-F.F-F3F-F4F-F-F:F7*&F:F-F3F-F7*&F2F-F)F-F-
*&F?F-FBF-F-*(F:F-F3F-F4F-F-*(F:F-F4F-F6F-F7**F2F-F)F-F4F-F6F-F7*(F2F-
F)F-F3F-F7*$F@F-F-**F2F-F)F-F3F-F4F-F-F-*&),&F)F-\"\"'F-F?F-F.F-F7#F7F
?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(x(t)=x_sub,Gamma)
;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&*&,J*&*&-%)dlnxdlnaG6#%#t|irG
\"\"\",&%$wq|irGF-F-F-F-F-,&!\"\"F-F/F-F1\"#@*&*,\"\"$F-F)F-F.F-%#wbGF
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