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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "#Check out ekpyroti
c perturbations: hep-th/0109050\n#AL Feb 02\nrestart;\nH:=1/2/t+h0+h1*
t+h2*t^2+h3*t^3;\nw:=1+w1*t+w2*t^2+w3*t^3+w4*t^4;\ncs2:=1+c1*t+c2*t^2+
c3*t^3+c4*t^4;\nf:=H*(1+3*cs2-6*w);\ng:=H^2*(9*cs2+9/2*w^2-12*w-3/2)+k
^2*cs2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG,,*&\"\"\"F'%\"tG!
\"\"#F'\"\"#%#h0GF'*&%#h1GF'F(F'F'*&%#h2GF')F(F+F'F'*&%#h3GF')F(\"\"$F
'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG,,\"\"\"F&*&%#w1GF&%\"tGF
&F&*&%#w2GF&)F)\"\"#F&F&*&%#w3GF&)F)\"\"$F&F&*&%#w4GF&)F)\"\"%F&F&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$cs2G,,\"\"\"F&*&%#c1GF&%\"tGF&F&*&%
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11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,,*&\"\"\"F(%\"tG!\"\"#F(\"\"#%#h0GF
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\"\"\"F*%\"tG!\"\"#F*\"\"#%#h0GF**&%#h1GF*F+F*F**&%#h2GF*)F+F.F*F**&%#
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "eq1:=series(diff(H,t)+(1+3*w
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}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "h0:=-3/16*w1;\neq2:=series(
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%#h0G,$%#w1G#!\"$\"#;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$eq2G++%\"t
G,&%#w1G!\"#*&\"\"$\"\"\"%#c1GF,F,\"\"!,*%#w2G!\"\"*(F+F,,&F-#F,\"\"#*
&#F,F5F,F(F,F1F,F(F,F,*&F+F,%#c2GF,F,*&#\"\"*\"\")F,*&F(F,,&F-F,F(F1F,
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F5-%\"OG6#F,F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c1:=2/3*w
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"" {XPPMATH 20 "6#>%#c1G,$%#w1G#\"\"#\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#+)%\"tG,(%#w2G!\"\"*&#\"\"\"\"\")F**$)%#w1G\"\"#F*F*F'*
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G,&%#w2G#\"\"\"\"\"$*&#F(\"#CF()%
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#h1G,&*$)%#w1G\"\"#\"\"\"#\"\"*\"$G
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67 "eq2:=simplify(eq2):\nseries(eq2,t,3);\ncoeff(%,t^2);\nc3:=solve(%,
c3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"tG,(*&%#w1G\"\"\"%#w2GF(#
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{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%#w1G\"\"\"%#w2GF&#!\"\"\"\"#*&#F&
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%#c3G,&*&%#w1G\"\"\"%#w2GF(#F(\"\"'*&#F(\"#[F(*$)F'\"\"$F(F(!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "series(simplify(eq1),t,3);\n
coeff(%,t);\nh2:=solve(%,h2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"
tG,**&%#w1G\"\"\"%#w2GF(#!\"$\"\")*&\"\"%F(%#h2GF(F(*&#\"#F\"$c#F()F'
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{XPPMATH 20 "6#,**&%#w1G\"\"\"%#w2GF&#!\"$\"\")*&\"\"%F&%#h2GF&F&*&#\"
#F\"$c#F&)F%\"\"$F&F&*&#F3F*F&%#w3GF&F&" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#h2G,(*&%#w1G\"\"\"%#w2GF(#\"\"$\"#K*&#\"#F\"%C5F(*$)F'F+F(F(!
\"\"*&#F+F,F(%#w3GF(F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "
#Find series solution near bounce. \n#\\epsion_m = eps2 Esol(t) + eps0
 Dsol(t)\nEsol:=t^2+e3*t^3+e4*t^4+e5*t^5;\nsimplify(diff(diff(Esol,t),
t)+f*diff(Esol,t)+g*Esol):\neq4:=series(%,t,4);\ne3:=5/6*w1;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%%EsolG,**$)%\"tG\"\"#\"\"\"F**&%#e3GF*)F(
\"\"$F*F**&%#e4GF*)F(\"\"%F*F**&%#e5GF*)F(\"\"&F*F*" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%$eq4G++%\"tG,&%#w1G#!\"&\"\"#*&\"\"$\"\"\"%#e3GF.F.
F.,,%#w2G#!\"*F+*&#\"#L\"\")F.*&F(F.F/F.F.!\"\"*$)%\"kGF+F.F.*&F7F.%#e
4GF.F.*&F+F.)F(F+F.F.F+,4*&F@F.F/F.#\"$r\"\"#k*&\"#:F.%#e5GF.F.*&#\"$b
\"\"$G\"F.*$)F(F-F.F.F9*&#\"#^F7F.%#w3GF.F9*(#\"#@\"\"%F.F(F.F1F.F.*(#
F+F-F.F;F.F(F.F.*&#\"#FFVF.*&F1F.F/F.F.F9*&F;F.F/F.F.*&#\"#BFVF.*&F(F.
F>F.F.F9F--%\"OG6#F.FV" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G,$%#w1
G#\"\"&\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Dsol:=1+d1*
t+d3*t^3 +d4*t^4 + c*Esol*ln(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
%DsolG,,\"\"\"F&*&%#d1GF&%\"tGF&F&*&%#d3GF&)F)\"\"$F&F&*&%#d4GF&)F)\"
\"%F&F&*(%\"cGF&,**$)F)\"\"#F&F&*(#\"\"&\"\"'F&%#w1GF&F,F&F&*&%#e4GF&F
0F&F&*&%#e5GF&)F)F:F&F&F&-%#lnG6#F)F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "simplify(diff(diff(Dsol,t),t)+f*diff(Dsol,t)+g*Dsol):
\neq3:=series(%,t,4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$eq3G+-%\"t
G,&%#d1G!\"\"*&#\"\"$\"\"%\"\"\"%#w1GF.F.F),**$)F/\"\"#F.#\"#@\"#K*$)%
\"kGF3F.F.*&#\"\"(\"\")F.*&F/F.F(F.F.F)*&F3F.%\"cGF.F.\"\"!,4*&%#w2GF.
F(F.#!\"*F-*(#F3F,F.F8F.F/F.F.*(#\"#T\"#CF.F@F.F/F.F.*&#F,F-F.%#w3GF.F
)*&F,F.%#d3GF.F.*&#\"$l\"\"$c#F.*$)F/F,F.F.F)*(#\"\"*F-F.F/F.FDF.F.*(#
\"#&)\"#kF.F2F.F(F.F.*&F8F.F(F.F.F.,L*&FWF.F(F.#!$v%\"$7&*&#F,F-F.%#w4
GF.F)*(#\"#\")F6F.F/F.FOF.F.*&#\"#B\"#;F.*(F2F.F@F.-%#lnG6#F&F.F.F)*&#
FZF=F.)FDF3F.F.*&#FZF3F.*(FDF.F@F.FjoF.F.F)*&#\"$,)\"%[?F.)F/F-F.F.*(F
8F.F@F.FjoF.F.**FHF.F8F.F/F.F(F.F.**#\"#:F-F.F/F.FDF.F(F.F.*&FYF.%#c4G
F.F.*(#F.FLF.F8F.F2F.F.*&#\"#dFhoF.*&FOF.F(F.F.F)*&#\"#LF=F.*&F/F.FQF.
F.F)*&#\"#pF6F.*&F2F.FDF.F.F)*(\"\"'F.F@F.%#e4GF.F.*(#F.F,F.F8F.FDF.F.
**F=F.F@F.FjoF.F_rF.F.*&#FZF-F.*&FDF.F@F.F.F)*&F=F.%#d4GF.F.*&#\"$J\"
\"$#>F.*&F2F.F@F.F.F)F3-%\"OG6#F.F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "d1:=3/4*w1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d1G
,$%#w1G#\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(
t=0,eq3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"kG\"\"#\"\"\"F(*&
F'F(%\"cGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "c:=-k^2/2;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG,$*$)%\"kG\"\"#\"\"\"#!\"\"F
)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "series(eq3,t,2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#+'%\"tG,,*&%#w1G\"\"\"%#w2GF(#\"\"*\"#
;*(F*F()%\"kG\"\"#F(F'F(F(*&#\"\"$\"\"%F(%#w3GF(!\"\"*&F3F(%#d3GF(F(*&
#\"#X\"$G\"F()F'F3F(F(F(-%\"OG6#F(F0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "coeff(%,t);\nd3:=solve(%,d3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,,*&%#w1G\"\"\"%#w2GF&#\"\"*\"#;*(F(F&)%\"kG\"\"#F&F%F&
F&*&#\"\"$\"\"%F&%#w3GF&!\"\"*&F1F&%#d3GF&F&*&#\"#X\"$G\"F&)F%F1F&F&" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d3G,**&%#w1G\"\"\"%#w2GF(#!\"$\"#
;*&#\"\"$F,F(*&)%\"kG\"\"#F(F'F(F(!\"\"*&#F(\"\"%F(%#w3GF(F(*&#\"#:\"$
G\"F(*$)F'F/F(F(F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "serie
s(simplify(eq4),t,3);\ne4:=solve(coeff(%,t^2),e4);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#+'%\"tG,*%#w2G#!\"*\"\"#*&#\"#B\"#;\"\"\"*$)%#w1GF)F.
F.!\"\"*$)%\"kGF)F.F.*&\"\")F.%#e4GF.F.F)-%\"OG6#F.\"\"$" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#e4G,(%#w2G#\"\"*\"#;*&#\"#B\"$G\"\"\"\")%#w1G
\"\"#F.F.*&#F.\"\")F.*$)%\"kGF1F.F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "Anz:=eps0*Dsol+eps2*Esol;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%$AnzG,&*&%%eps0G\"\"\",,F(F(*(#\"\"$\"\"%F(%#w1GF(%\"
tGF(F(*&,**&F.F(%#w2GF(#!\"$\"#;*&#F,F6F(*&)%\"kG\"\"#F(F.F(F(!\"\"*&#
F(F-F(%#w3GF(F(*&#\"#:\"$G\"F(*$)F.F,F(F(F=F()F/F,F(F(*&%#d4GF()F/F-F(
F(*&#F(F<F(*(F:F(,**$)F/F<F(F(*(#\"\"&\"\"'F(F.F(FGF(F(*&,(F3#\"\"*F6*
&#\"#BFDF()F.F<F(F(*&#F(\"\")F(*$F:F(F(F=F(FJF(F(*&%#e5GF()F/FSF(F(F(-
%#lnG6#F/F(F(F=F(F(*&%%eps2GF(FNF(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "chi:=simplify(H/(1+w)/k^2*(diff(Anz,t)+3/2*H*(1-w)*An
z)):\nexpand(simplify(series(chi,t,1)));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#+'%\"tG,,*&%%eps2G\"\"\"*$)%\"kG\"\"#F(!\"\"#F(F,*&#\"\"$\"#;F(*
&*&%#w2GF(%%eps0GF(F(*$F*F(F-F(F-*&#\"\"*\"$G\"F(*&*&F6F()%#w1GF,F(F(*
$F*F(F-F(F-*&#F(\"\"%F(*&F6F(-%#lnG6#F$F(F(F-*&#F(\"\")F(F6F(F-\"\"!-%
\"OG6#F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "#That is the \+
answer for the curvature perturbation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "#Other series are" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "H;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&\"\"\"F%%\"tG!\"\"#F%\"\"#*
&#\"\"$\"#;F%%#w1GF%F'*&,&*$)F.F)F%#\"\"*\"$G\"*&#F%\"\")F%%#w2GF%F'F%
F&F%F%*&,(*&F.F%F9F%#F,\"#K*&#\"#F\"%C5F%*$)F.F,F%F%F'*&#F,F>F%%#w3GF%
F'F%)F&F)F%F%*&%#h3GF%)F&F,F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "#E(tau)\nEsol;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"tG\"
\"#\"\"\"F(*(#\"\"&\"\"'F(%#w1GF()F&\"\"$F(F(*&,(%#w2G#\"\"*\"#;*&#\"#
B\"$G\"F()F-F'F(F(*&#F(\"\")F(*$)%\"kGF'F(F(!\"\"F()F&\"\"%F(F(*&%#e5G
F()F&F+F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#D(tau)\nDso
l-c*Esol*ln(t) +c*E(t)*ln(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"
\"\"F$*(#\"\"$\"\"%F$%#w1GF$%\"tGF$F$*&,**&F)F$%#w2GF$#!\"$\"#;*&#F'F1
F$*&)%\"kG\"\"#F$F)F$F$!\"\"*&#F$F(F$%#w3GF$F$*&#\"#:\"$G\"F$*$)F)F'F$
F$F8F$)F*F'F$F$*&%#d4GF$)F*F(F$F$*&#F$F7F$*(F5F$-%\"EG6#F*F$-%#lnGFKF$
F$F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "exp(int(H,t))^2;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$expG6#,2-%#lnG6#%\"tG#\"\"\"\"
\"#*&#\"\"$\"#;F.*&%#w1GF.F,F.F.!\"\"*(#\"\"*\"$c#F.)F,F/F.)F5F/F.F.*&
#F.F3F.*&%#w2GF.F;F.F.F6**#F.\"#KF.)F,F2F.F5F.F@F.F.*&#F9\"%C5F.*&FDF.
)F5F2F.F.F6*&#F.FCF.*&%#w3GF.FDF.F.F6*(#F.\"\"%F.%#h3GF.)F,FPF.F.F/F.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "series(%,t);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#+/%\"tG\"\"\"F%,$%#w1G#!\"$\"\")\"\"#,&*$)F'
F+F%#\"\"*\"#k*&#F%F*F%%#w2GF%!\"\"\"\"$,**&,&F-#\"#F\"$7&*&#F%\"#;F%F
4F%F5F%F'F%F(*(#\"#6\"$G\"F%F'F%F4F%F%*&#\"$N\"\"%'4%F%*$)F'F6F%F%F5*&
#F%F?F%%#w3GF%F5\"\"%,0%#h3G#F%F+*&#\"#vFGF%*&F.F%F4F%F%F5*&#\"%R:\"'W
@EF%)F'FMF%F%*(#F6\"$c#F%F'F%FLF%F%*&#F%FfnF%)F4F+F%F%*&#F6F*F%*&,(*&F
'F%F4F%#FBFfn*&#FF\"%#>)F%FHF%F5*&#F%\"#KF%FLF%F5F%F'F%F%F5*$)F9F+F%F%
\"\"&-%\"OG6#F%\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "#a^
2\na2_sub:=a02*%;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'a2_subG*&%$a02
G\"\"\"+/%\"tGF'F',$%#w1G#!\"$\"\")\"\"#,&*$)F+F/F'#\"\"*\"#k*&#F'F.F'
%#w2GF'!\"\"\"\"$,**&,&F1#\"#F\"$7&*&#F'\"#;F'F8F'F9F'F+F'F,*(#\"#6\"$
G\"F'F+F'F8F'F'*&#\"$N\"\"%'4%F'*$)F+F:F'F'F9*&#F'FCF'%#w3GF'F9\"\"%,0
%#h3G#F'F/*&#\"#vFKF'*&F2F'F8F'F'F9*&#\"%R:\"'W@EF')F+FQF'F'*(#F:\"$c#
F'F+F'FPF'F'*&#F'FjnF')F8F/F'F'*&#F:F.F'*&,(*&F+F'F8F'#FFFjn*&#FJ\"%#>
)F'FLF'F9*&#F'\"#KF'FPF'F9F'F+F'F'F9*$)F=F/F'F'\"\"&-%\"OG6#F'\"\"'F'
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "#Now find w^(2) before \+
and after\nphidsq:=2*(H^2-diff(H,t))-4/3*R/a2_sub:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 12 "simplify(%):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "phidot_
sub:=series(sqrt(%),t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+phidot_s
ubG+/%\"tG,$*$-%%sqrtG6#\"\"'\"\"\"#F-\"\"#!\"\",$*&*&F)F-,&*&%$a02GF-
%#w1GF-\"\"**&\"#KF-%\"RGF-F-F-F-F6F0#F0\"$W\"\"\"!,$*&F)F-,&*&*&F7F-,
&F5F8*&\"#kF-F;F-F0F-F-F6F0\"\"%*&#FG\"#FF-*&*$)F4F/F-F-*$)F6F/F-F0F-F
0F-#F-\"%sIF-,$*&F)F-,(*&,(*(F6F-F7F-%#w2GF-F8*(\"#=F-F6F-%#w3GF-F0*(
\"#;F-F;F-FXF-F-F-F6F0#!#;\"\"$*&**#F-FZF-F4F-F7F-FDF-F-*$FOF-F0F-*&#F
-\"$'[F-*&*$)F4FjnF-F-*$)F6FjnF-F0F-F0F-FPF/,$*&F)F-,,*&,0*&%#h3GF-F6F
-!&/$)***\"%#f#F-F6F-)F7F/F-FXF-F0*(\"$H(F-F6F-)F7FGF-F-**\"%G<F-F6F-F
7F-FenF-F-*(\"$o(F-F6F-)FXF/F-F-**\"$7&F-F;F-F7F-FXF-F-*(\"%[?F-F;F-Fe
nF-F0F-F6F0#F-\"#[*&#F/FJF-*&*&F4F-FVF-F-*$FOF-F0F-F0*&#F-\"$#>F-*&*&F
apF-)FDF/F-F-*$FOF-F0F-F0*&**#F-\"$k)F-FMF-F7F-FDF-F-*$FfoF-F0F-*&#\"
\"&\"'o*R\"F-*&*$)F4FGF-F-*$)F6FGF-F0F-F0F-FPFjn-%\"OG6#F-FG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 50 "V_sub:=(3*H^2-1/2*phidot_sub^2)/a2_sub-R/a2_sub^2:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "V_sub:=simplify(series(
%,t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&V_subG+-%\"tG,$*&,&*&%$a0
2G\"\"\"%#w1GF,\"\"**&\"\")F,%\"RGF,F,F,*$)F+\"\"#F,!\"\"#F5\"#C!\"#,$
*&,(*&F-F,F1F,!#;*(F.F,F+F,)F-F4F,F,*(F7F,F+F,%#w2GF,F5F,*$F3F,F5#F,\"
#kF5,$*&,,*&F1F,F?F,\"#s*(\"$G\"F,F1F,FAF,F,**\"$K%F,F+F,F-F,FAF,F5*(
\"$w&F,F+F,%#w3GF,F,*(\"#\")F,F+F,)F-\"\"$F,F,F,*$F3F,F5#F5\"%O:\"\"!,
$*&,0*&F+F,)FAF4F,\"$%Q*(\"'!)G7F,%#h3GF,F+F,F5**\"%OFF,F+F,F?F,FAF,F5
**\"%/BF,F+F,F-F,FPF,F,*(\"$H(F,F+F,)F-\"\"%F,F,*(\"%C5F,F1F,FPF,F,**
\"$7&F,F1F,F-F,FAF,F,F,*$F3F,F5#F5\"&wX#F,-%\"OG6#F,F4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "Q:=-a2_sub*phidot_sub/sqrt(6):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "Q_ser:=simplify(series(%,t));" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%&Q_serG+/%\"tG,$%$a02G#!\"\"\"\"#\"\"!,&*&F(\"\"
\"%#w1GF/#F/\"\"%*&#F+\"\"*F/%\"RGF/F/F/,$*&,**&)F(F+F/)F0F+F/\"%`5**
\"$)GF/F(F/F0F/F6F/F**(\"$['F/F;F/%#w2GF/F**&\"$7&F/)F6F+F/F*F/F(F*#F*
\"&o.\"F+,$*&,**&)F(\"\"$F/)F0FMF/\"&^Q\"**\"&'\\<F/FLF/F0F/FBF/F***\"
&CQ\"F/F(F/F0F/FEF/F/*&\"%#>)F/)F6FMF/F/F/*$F;F/F*#F/\"'[KPFM,$*&,:*$)
F6F2F/!(S9i#**\"(k!)['F/F(F/F0F/FVF/F***\"'Cm=F/FLF/FNF/F6F/F/**\"(c1*
>F/F;F/F<F/FEF/F**(\")?&z&*)F/)F(F2F/%#h3GF/F**(\"(zj`$F/FboF/)F0F2F/F
/*(\"(_&f>F/FboF/)FBF+F/F/**\"(Oz-*F/FboF/F<F/FBF/F***\"(S!*>%F/FboF/F
0F/%#w3GF/F/**\"(%)f)HF/FLF/F6F/F^pF/F/**\"(/rK\"F/F;F/FEF/FBF/F**,\"(
#*H\\\"F/FLF/F6F/F0F/FBF/F*F/*$FLF/F*#F*\"*[3*\\@F2-%\"OG6#F/\"\"&" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "U:=a2_sub^2*V_sub:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "U_ser:=simplify(series(U,t,4
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&U_serG+)%\"tG,&*&%$a02G\"\"
\"%#w1GF*#!\"$\"\")*&#F*\"\"$F*%\"RGF*!\"\"\"\"!,&*&F)F*)F+\"\"#F*#\"#
F\"#k*&#F1F.F**&F)F*%#w2GF*F*F3F*-%\"OG6#F*F8" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "w2_sub:=solve(coeff(%,t)=U1,w2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'w2_subG,$*&,&*&%$a02G\"\"\")%#w1G\"\"#F*!#F*&\"
#kF*%#U1GF*F*F*F)!\"\"#F2\"#C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "W2:=w2_sub+3/8*w1^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#W2G,&
*&,&*&%$a02G\"\"\")%#w1G\"\"#F*!#F*&\"#kF*%#U1GF*F*F*F)!\"\"#F2\"#C*&#
\"\"$\"\")F*F+F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "w1_su
b:=solve(U0=subs(t=0,U_ser),w1);\nsubs(w1=w1_sub,W2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'w1_subG,$*&,&%#U0G\"\"$%\"RG\"\"\"F+%$a02G!\"\"
#!\")\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,**$)%#U0G\"\"#\"\"
\"\"#=*(\"#7F*F(F*%\"RGF*F**&F)F*)F.F)F*F**(\"\"*F*%#U1GF*%#Q0GF*F*F**
$)F4F)F*!\"\"#\"\"%\"#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "
W2:=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#W2G,$*&,**$)%#U
0G\"\"#\"\"\"\"#=*(\"#7F,F*F,%\"RGF,F,*&F+F,)F0F+F,F,*(\"\"*F,%#U1GF,%
#Q0GF,F,F,*$)F6F+F,!\"\"#\"\"%\"#F" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "solve(Q0=subs(t=0,Q_ser),a02);\nsubs(a02=%,W2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#Q0G!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&,**$)%#U0G\"\"#\"\"\"!#O*(\"#CF*F(F*%\"RGF*!\"\"*&
\"\"%F*)F.F)F*F/*(\"#=F*%#U1GF*%#Q0GF*F/F**$)F6F)F*F/#!\"#\"#F" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "W2:=simplify(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#W2G,$*&,**$)%#U0G\"\"#\"\"\"\"#=*(\"#7F,F
*F,%\"RGF,F,*&F+F,)F0F+F,F,*(\"\"*F,%#U1GF,%#Q0GF,F,F,*$)F6F+F,!\"\"#
\"\"%\"#F" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "simplify(subs(
R=0,%));\n###For w^(2<)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&*$)%#
U0G\"\"#\"\"\"F)*&%#U1GF*%#Q0GF*F*F**$)F-F)F*!\"\"#\"\"%\"\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "simplify(subs(\{U0=0,U1=0\},
W2));\n#For w^(2>) if V=0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*$)%
\"RG\"\"#\"\"\"F)*$)%#Q0GF(F)!\"\"#\"\")\"#F" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Ph
i:=diff(a2_sub,t)/2/a2_sub^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 24 "simplify(series(%,t,4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%
\"tG,$*&\"\"\"F'%$a02G!\"\"#F'\"\"#!\"#,$*&%#w2GF'F(F)#F)\"#;\"\"!,$*&
,&*&%#w1GF'F/F'F'*&\"\"%F'%#w3GF'F)F'F(F)#F'\"#kF'-%\"OG6#F'F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "epsilon:=simplify(2*Phi*k^2/
3/H^2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(epsilonG,$*&*(+/%\"tG\"
\"\"\"\"!,$%#w1G#!\"$\"\"%F*,&*$)F-\"\"#F*#\"#F\"#k*&#\"\"$\"\")F*%#w2
GF*!\"\"F4,(*$)F-F:F*#!#F\"$G\"*(#\"\"(\"#;F*F-F*F<F*F**&#F*F0F*%#w3GF
*F=F:,,%#h3G#\"\"&F4*&#\"$0\"\"$7&F**&F3F*F<F*F*F=*&#\"%:7\"&%Q;F*)F-F
0F*F**(#\"#:FCF*F-F*FJF*F**&#FNFCF*)F<F4F*F*F0-%\"OG6#F*FNF*)%\"kGF4F*
)F)F4F*F**(%$a02GF*)+/F)F*F*,$F-#F/F;F4,&F2#\"\"*F7*&#F*F;F*F<F*F=F:,(
F?#FBFR*(#FFF7F*F-F*F<F*F**&#F*FGF*FJF*F=F0,,FL#F*F4*&#\"#@FRF*FSF*F=*
&#\"$V#FWF*FXF*F**(#F:FCF*F-F*FJF*F**&#F*FCF*FhnF*F*FNFin\"\"'F4F*),2F
RF**(\"$#>F*F-F*F)F*F=*(\"#sF*F^oF*F3F*F**(FCF*F<F*F^oF*F=**\"#'*F*)F)
F:F*F-F*F<F*F**(F6F*FfqF*F@F*F=*(FeqF*FJF*FfqF*F=*(\"%C5F*FLF*)F)F0F*F
*F4F*F=#\"(w&[5F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "series
(epsilon,t,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"tG,$*&*$)%\"kG
\"\"#\"\"\"F+%$a02G!\"\"#\"\"%\"\"$\"\"!*&*&F(F+%#w1GF+F+F,F-F+,&*&*&F
(F+,&%#w2G#F+\"')GC&*&#\"\"*\");sx;F+)F4F*F+F+F+F+F,F-#\"(w&[5F0*(F.F+
,(*&,&F9#F+F/*&#F>\"#kF+F@F+F+F+F,F-F+*&#F>\"#;F+*&*$F@F+F+F,F-F+F-*&,
&*$F@F+#\"#FFJ*&#F0\"\")F+F9F+F-F+F,F-F+F+F(F+F+F*-%\"OG6#F+F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "epsilon0:=simplify(coeff(%,t
,0)*k^(-1-v)/sqrt(p)/2);\n#p factor from relation for tau and a'/a^3" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)epsilon0G,$*&)%\"kG,&\"\"\"F*%\"v
G!\"\"F**&%$a02GF*-%%sqrtG6#%\"pGF*F,#\"\"#\"\"$" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 97 "#Can check expression for a02 in terms of a0 \+
and a1...\nphi_sub:=simplify(int(phidot_sub,t))+phi0;" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%(phi_subG,&+/%\"tG,$*&-%%sqrtG6#\"\"'\"\"\"-%#ln
G6#F'F.#F.\"\"#\"\"!,$*&*&F*F.,&*&%$a02GF.%#w1GF.\"\"**&\"#KF.%\"RGF.F
.F.F.F:!\"\"#F@\"$W\"F.,$*&*&F*F.,(*&)F:F3F.)F;F3F.\"#\")**\"%_6F.F:F.
F;F.F?F.F@*&\"$7&F.)F?F3F.F@F.F.*$FHF.F@#F.\"&O2#F3,$*&*&F*F.,0*()F:\"
\"$F.F;F.%#w2GF.!&k;\"*(\"&GL#F.FXF.%#w3GF.F.**FRF.FHF.F?F.FZF.F@*(\"$
H(F.FXF.)F;FYF.F.**\"%wxF.FHF.FIF.F?F.F@**\"&s9%F.F:F.F;F.FOF.F@*&\"&%
Q;F.)F?FYF.F@F.F.*$FXF.F@#F.\"()[RAFY,$*&*&F*F.,8*&)F:\"\"%F.%#h3GF.!)
3=$e$**\"(g(\\5F.F\\pF.FIF.FZF.F@*(\"',!p#F.F\\pF.)F;F]pF.F.**\"'3)R)F
.F\\pF.F;F.FhnF.F.*(\"'O*z#F.F\\pF.)FZF3F.F.*,\"'[KPF.FXF.F?F.F;F.FZF.
F@**\"'_NmF.FHF.FOF.FZF.F@**\"&cm%F.FXF.F\\oF.F?F.F.**\"(gTC\"F.FHF.FI
F.FOF.F@**\"(%Qk?F.F:F.F;F.FcoF.F@*&\"'g`lF.)F?F]pF.F@F.F.*$F\\pF.F@#F
.\"*[3*\\@F]p-%\"OG6#F.\"\"&F.%%phi0GF." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 103 "a0sq:=simplify(4*(a2_sub)*cosh(phi_sub/sqrt(6))^2):
\na1sq:=simplify(4*(a2_sub)*sinh(phi_sub/sqrt(6))^2):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(series(a0sq,t)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a0:=series(sqrt(%),t,3);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#a0G++%\"tG*$-%%sqrtG6#*&%$a02G\"\"
\"-%$expG6#,$*&-F)6#\"\"'F-%%phi0GF-#!\"\"\"\"$F-F-\"\"!,$*&*&F(F-,(*&
F,F--F/6#,$F2#F-F9F-!#s*(\"\"*F-F,F-%#w1GF-F-*&\"#;F-%\"RGF-F8F-F-F,F8
#F8\"#sF-*&F(F-,&*&,,*()F,\"\"#F-F@F-FGF-!$['*(\"#\")F-FRF-)FGFSF-F-*(
\"$3\"F-FRF-%#w2GF-F8*(\"$k)F-FRF--F/6#,$F2#FSF9F-F-*&\"$G\"F-)FJFSF-F
-F-*$FRF-F8#F-\"%G<*&#F-\"&o.\"F-*&*$)F>FSF-F-*$FRF-F8F-F8F-FS-%\"OG6#
F-F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "simplify(series(a1s
q,t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "a1:=simplify(seri
es(sqrt(%),t,3));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#a1G++%\"tG*$-%
%sqrtG6#*&%$a02G\"\"\"-%$expG6#,$*&-F)6#\"\"'F-%%phi0GF-#!\"\"\"\"$F-F
-\"\"!,$*&*&F(F-,(*&F,F--F/6#,$F2#F-F9F-\"#s*(\"\"*F-F,F-%#w1GF-F-*&\"
#;F-%\"RGF-F8F-F-F,F8#F8FDF-,$*&*&F(F-,.*()F,\"\"#F-F@F-FGF-\"%#f#*(\"
$0%F-FQF-)FGFRF-F-*(\"$['F-FQF-%#w2GF-F8*&\"$7&F-)FJFRF-F-**\"%/BF-F,F
-F@F-FJF-F-**\"$)GF-F,F-FGF-FJF-F-F-F-*$FQF-F8#F-\"&o.\"FR-%\"OG6#F-F9
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "simplify(series(a0*(dif
f(a0-a1,t)),t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"tG,$%$a02G\"
\"#\"\"!,(*&F&\"\"\"%#w1GF+#!\"&\"\"%*(F'F+F&F+-%$expG6#,$*&-%%sqrtG6#
\"\"'F+%%phi0GF+#F+\"\"$F+F+*&#F/\"\"*F+%\"RGF+!\"\"F+-%\"OG6#F+F'" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "#so a0*(a0'-a1') = 2*a02, f
actor of 4 then agrees to zeroth order in p. \n" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "#After collision..." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simp
lify(subs(w1=w1_sub,series(diff(a0,t)^2-diff(a1,t)^2,t)));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#+)%\"tG,&%#U0G#\"\"%\"\"$*&F'\"\"\"%\"RGF+F+
\"\"!,$*&,**$)F,\"\"#F+\"\")*(\"\"*F+)%$a02GF3F+%#w2GF+!\"\"*&\"#sF+)F
&F3F+F+*(\"#[F+F&F+F,F+F+F+F8F:#F+\"#=F+-%\"OG6#F+F3" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "diff(a1,t)^4/a02^2:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 39 "simplify(series(subs(\{w1=0,R=0\},%),t));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"tG-%$expG6#,$*&-%%sqrtG6#\"\"'\"
\"\"%%phi0GF.#\"\"#\"\"$\"\"!,$*&-F&6#,$F)#F.F2F.%#w2GF.#F.F1F.-%\"OG6
#F.F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "simplify(subs(\{w1
=0,R=0\},series(diff(a1,t)^2/a1^2,t)));" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"tG-%$expG6#,$*&-%%sqrt
G6#\"\"'\"\"\"%%phi0GF.#\"\"#\"\"$\"\"!,&-F&6#F)F1*(#F.\"\"%F.-F&6#,$F
)#F.F2F.%#w2GF.F.F.-%\"OG6#F.F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "#So w^(2>)=2/3*zeta^2*a1'^2/a1^2" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "#I then get no factor of 2 in Eq. 62 and 1/6 in Eq 63
 instead of 5/192" }}}}{MARK "65 0 0" 30 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }