{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {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 " 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*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 " 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 " 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 " 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 -%%Omq|irGF-F-F0F1F1*&*,F5F-F)F-F.F-F7F-F/F-F-F0F1F-*&**F5F-F)F-F.F-F6 F-F-F0F1F-*&**\"#=F-F.F-F7F-F/F-F-F0F1F-*&*&\"#aF-F.F-F-F0F1F-*&*(F>F- F.F-F6F-F-F0F1F-*&*(\"\"#F-F.F-)F)FFF-F-F0F1F-*&*(FFF--%*ddlnxdlnaGF+F -F.F-F-F0F1F1*&**F>F-F.F-F6F-F7F-F-F0F1F1F>F1*&F>F-F6F-F1*&F5F-F)F-F-* &FFF-FJF-F-*(F>F-F6F-F7F-F-*(F>F-F7F-F/F-F1**F5F-F)F-F7F-F/F-F1*(F5F-F )F-F6F-F1*\$FGF-F-**F5F-F)F-F6F-F7F-F-F-F0F-F-*&),&F)F-\"\"'F-FFF-F.F-F 1#F-FF" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Gamma:=simplify(% );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&GammaG,\$*&,F)F3#F3F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "coeff(Gamma,ddlnxdlna(t)) ;\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,\$*&\"\"\"F%*&,&%\$wq|irGF%F%F %F%),&-%)dlnxdlnaG6#%#t|irGF%\"\"'F%\"\"#F%!\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "ddterm:=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ddter mG,\$*&\"\"\"F'*&,&%\$wq|irGF'F'F'F'),&-%)dlnxdlnaG6#%#t|irGF'\"\"'F'\" \"#F'!\"\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(ddln xdlna(t)=0,Gamma);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,\$*&,:!#s\"\"\"* &\"#OF'%#wbGF'!\"\"*(\"#CF'-%)dlnxdlnaG6#%#t|irGF'%\$wq|irGF'F+*(\"\"'F 'F.F'F*F'F+*(\"\"\$F')F.\"\"#F'F2F'F+*&\"#=F'F.F'F+*\$F7F'F+**F4F'F.F'F* F'%%Omq|irGF'F'**F4F'F.F'F=F'F2F'F+*&F)F'F2F'F+*(F)F'F=F'F2F'F+*(F)F'F *F'F=F'F'F'*&,&F2F'F'F'F'),&F.F'F4F'F8F'F+#F+F8" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 12 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "tmp:=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$tmpG,\$*&,0*&-%)dlnxdlnaG6#%#t|irG\"\"\"%\$wq|irGF- !\"\$F)!\"\"*(\"\"'F-%%Omq|irGF-F.F-F0*&F2F-F.F-F0*&F2F-%#wbGF-F0\"#7F0 *(F2F-F6F-F3F-F-F-*&,&F)F-F2F-F-,&F.F-F-F-F-F0#F0\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "coeff(tmp,wb);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,\$*&,&!\"'\"\"\"*&\"\"'F'%%Omq|irGF'F'F'*&,&-%)dlnxdlna G6#%#t|irGF'F)F'F',&%\$wq|irGF'F'F'F'!\"\"#F3\"\"#" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 48 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }