{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 2511 "#Do series expansi on without using tight coupling approximation\n#The vector perturbatio n equations, in A=0 gauge\n#See astro-ph/0403583\n#AL April 04\n\n# Us e conformal time. \n# Includes cdm, baryons, lambda\n\n#The vector per turbation equations, in A=0 gauge\n#See astro-ph/0403583\n#AL March 04 \n\n# Use conformal time. \n# Includes cdm, baryons, lambda\n\nrestart ;\nno_quint:=\{psi(t)=0,V(t)=0,V1(t)=0,V2(t)=0,rhopsi(t)=0,ppsi(t)=0\} ;\nno_numassive:=\{pinu(t)=0,qnu(t)=0,rhonu(t)=0,pnu(t)=0\};\nassign(n o_quint);\nassign(no_numassive);\n\nH_t:=diff(S(t),t)/S(t);\n\ndS:=S(t )*H(t);\ndH:=-1/6*S(t)^2*kappa*(rho(t)+3*p(t));\n\nK:=0;\nFriedmann:=H (t)^2=1/3*S(t)^2*kappa*rho(t)-K;\n\nphi(t):=-kappa*S(t)^2/2/k^3*(k*rho pi(t) + 4 * H(t) * rhoq(t));\n\ndrag_t:=opac(t)*(4/3*v(t)-qg(t));\npho tbar_t:=rhog(t)/rhob(t);\n\ndqr:=-1/2*k*pir(t);\ndqg:=-1/2*k*pig(t)+dr ag(t);\ndv:=-(1-3*c2(t))*H(t)*v(t)-photbar(t)*drag(t) - k/2*B0*rhog(t) /rhob(t);\ndvc=-H(t)*vc(t);\n\n\n#dsigma:=simplify(-H(t)*sigma(t)+k*ph i(t)-1/2*kappa*S(t)^2/k*(rhopi(t)));\ndsigma:=simplify(-2*H(t)*sigma(t )-kappa*S(t)^2/k*(rhopi(t)));\n\n\nrho_t:=rhob(t)+rhoc(t)+rhor(t)+rhog (t)+rhonu(t)+rhopsi(t)+rhov(t);\np_t:=1/3*(rhor(t)+rhog(t))+pb(t)+pnu( t)+ppsi(t)-rhov(t);\ndpb:=c2(t)*drhob;\n\n\ndpig:=-opac(t)*(pig(t)-pol ter(t)) - 8/15*k*J_3(t) + 2/5*k*qg(t) + 8/15*k*sigma(t);\ndJ_3:=k*(3/7 *pig(t)-15/28*J_4(t))-opac(t)*J_3(t);\n\ndpir:=- 8/15*k*G_3(t) + 2/5*k *qr(t) + 8/15*k*sigma(t);\ndG_3:=k*(3/7*pir(t)-15/28*G_4(t));\ndG_4:=k *(4/9*G_3(t)-24/45*G_5(t));\n\nG_eq:=proc(l)\n local Gl,Eq;\n Gl:=cat( 'G_',l)(t);\n Eq:=diff(Gl,t)+k*l/(2*l+1)*( (l+2)/(l+1)*cat('G_',l+1)(t ) - cat('G_',l-1)(t));\n if (l = 2) then\n Eq:=Eq - 8/15*k*sigma(t); \n fi;\n simplify(subs(\{G_1(t)=qr(t),G_2(t)=pir(t)\},Eq));\nend;\n\nr hopi_t:=rhog(t)*pig(t)+rhor(t)*pir(t) + rhonu(t)*pinu(t) + rhog(t)*B0; \nrhoq_t:=rhoc(t)*vc(t) + rhog(t)*qg(t)+rhor(t)*qr(t)+(rhob(t)+pb(t))* v(t) + rhonu(t)*qnu(t) +k*diff(psi(t),t)*clv(t)/S(t)^2;\n\nsigma(t):=2 *kappa*S(t)^2*rhoq_t/k^2;\n#rhoq(t):=k^2*sigma(t)/2/kappa/S(t)^2;\n\n \nsubtots:=\{rhopi(t)=rhopi_t,rho(t)=rho_t,p(t)=p_t\};\n\ndrhoq:=-4*H( t)*rhoq(t)-1/2*k*rhopi(t);\n\npolter_t:=2/15*(3*pig(t)/4 + 9*E2(t)/2); \n\ndE2:=-opac(t)*(E2(t) - polter(t)) - 8/27*k*E3(t) + 1/3*k*B2(t);\nd B2:=-opac(t)*B2(t) - 8/27*k*B3(t) - 1/3*k*E2(t);\n\nsublist:=\{diff(S (t),t)=dS,diff(qr(t),t)=dqr,diff(qg(t),t)=dqg,diff(v(t),t)=dv,diff(H(t ),t)=dH,diff(exptau(t),t)=g(t),diff(pig(t),t)=dpig\};\n\nsubtotderivs: =\{diff(S(t),t)=dS,diff(H(t),t)=dH\};\n\n#End of main definitions and \+ equations\n#########################################\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)no_quin tG<(/-%\$psiG6#%\"tG\"\"!/-%\"VGF)F+/-%#V1GF)F+/-%#V2GF)F+/-%'rhopsiGF) F+/-%%ppsiGF)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-no_numassiveG<&/ -%%pinuG6#%\"tG\"\"!/-%\$qnuGF)F+/-%&rhonuGF)F+/-%\$pnuGF)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$H_tG*&-%%diffG6\$-%\"SG6#%\"tGF,\"\"\"F)! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dSG*&-%\"SG6#%\"tG\"\"\"-% \"HGF(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dHG,\$*()-%\"SG6#%\"tG\" \"#\"\"\"%&kappaGF-,&-%\$rhoGF*F-*&\"\"\$F--%\"pGF*F-F-F-#!\"\"\"\"'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*FriedmannG/*\$)-%\"HG6#%\"tG\"\"#\"\"\",\$*()-%\"SGF*F ,F-%&kappaGF--%\$rhoGF*F-#F-\"\"\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>- %\$phiG6#%\"tG,\$*&*(%&kappaG\"\"\")-%\"SGF&\"\"#F,,&*&%\"kGF,-%&rhopiGF &F,F,*(\"\"%F,-%\"HGF&F,-%%rhoqGF&F,F,F,F,*\$)F3\"\"\$F,!\"\"#F?F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'drag_tG*&-%%opacG6#%\"tG\"\"\",&-% \"vGF(#\"\"%\"\"\$-%#qgGF(!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% *photbar_tG*&-%%rhogG6#%\"tG\"\"\"-%%rhobGF(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$dqrG,\$*&%\"kG\"\"\"-%\$pirG6#%\"tGF(#!\"\"\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$dqgG,&*&%\"kG\"\"\"-%\$pigG6#%\"tGF( #!\"\"\"\"#-%%dragGF+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dvG,(*(, &\"\"\"F(*&\"\"\$F(-%#c2G6#%\"tGF(!\"\"F(-%\"HGF-F(-%\"vGF-F(F/*&-%(pho tbarGF-F(-%%dragGF-F(F/*&#F(\"\"#F(*&*(%\"kGF(%#B0GF(-%%rhogGF-F(F(-%% rhobGF-F/F(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\$dvcG,\$*&-%\"HG6#% \"tG\"\"\"-%#vcGF)F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dsigma G,\$*&,&*(-%\"HG6#%\"tG\"\"\"-%&sigmaGF+F-%\"kGF-\"\"#*(%&kappaGF-)-%\" SGF+F1F--%&rhopiGF+F-F-F-F0!\"\"F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&rho_tG,,-%%rhobG6#%\"tG\"\"\"-%%rhocGF(F*-%%rhorGF(F*-%%rhogGF(F*-% %rhovGF(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$p_tG,*-%%rhorG6#%\"tG #\"\"\"\"\"\$*&F*F+-%%rhogGF(F+F+-%#pbGF(F+-%%rhovGF(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$dpbG*&-%#c2G6#%\"tG\"\"\"%&drhobGF*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dpigG,**&-%%opacG6#%\"tG\"\"\",&-%\$ pigGF)F+-%'polterGF)!\"\"F+F1*&#\"\")\"#:F+*&%\"kGF+-%\$J_3GF)F+F+F1*(# \"\"#\"\"&F+F7F+-%#qgGF)F+F+*(#F4F5F+F7F+-%&sigmaGF)F+F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%dJ_3G,&*&%\"kG\"\"\",&-%\$pigG6#%\"tG#\"\"\$\" \"(*&#\"#:\"#GF(-%\$J_4GF,F(!\"\"F(F(*&-%%opacGF,F(-%\$J_3GF,F(F7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dpirG,(*&%\"kG\"\"\"-%\$G_3G6#%\"tGF (#!\")\"#:*(#\"\"#\"\"&F(F'F(-%#qrGF+F(F(*(#\"\")F/F(F'F(-%&sigmaGF+F( F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dG_3G*&%\"kG\"\"\",&-%\$pirG6# %\"tG#\"\"\$\"\"(*&#\"#:\"#GF'-%\$G_4GF+F'!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dG_4G*&%\"kG\"\"\",&-%\$G_3G6#%\"tG#\"\"%\"\"**&#\"\" )\"#:F'-%\$G_5GF+F'!\"\"F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%G_eqGR 6#%\"lG6\$%#GlG%#EqG6\"F+C&>8\$--%\$catG6\$.%#G_G9\$6#%\"tG>8%,&-%%diffG6\$F .F7\"\"\"*&*(%\"kGF>F5F>,&*&*&,&F5F>\"\"#F>F>--F16\$F3,&F5F>F>F>F6F>F>F J!\"\"F>--F16\$F3,&F5F>F>FKF6FKF>F>,&F5FFF>F>FKF>@\$/F5FF>F9,&F9F>*&#\" \")\"#:F>*&FAF>-%&sigmaGF6F>F>FK-%)simplifyG6#-%%subsG6\$<\$/-%\$G_1GF6-% #qrGF6/-%\$G_2GF6-%\$pirGF6F9F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %(rhopi_tG,(*&-%%rhogG6#%\"tG\"\"\"-%\$pigGF)F+F+*&-%%rhorGF)F+-%\$pirGF )F+F+*&F'F+%#B0GF+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'rhoq_tG,**& -%%rhocG6#%\"tG\"\"\"-%#vcGF)F+F+*&-%%rhogGF)F+-%#qgGF)F+F+*&-%%rhorGF )F+-%#qrGF)F+F+*&,&-%%rhobGF)F+-%#pbGF)F+F+-%\"vGF)F+F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>-%&sigmaG6#%\"tG,\$*&*(%&kappaG\"\"\")-%\"SGF&\" \"#F,,**&-%%rhocGF&F,-%#vcGF&F,F,*&-%%rhogGF&F,-%#qgGF&F,F,*&-%%rhorGF &F,-%#qrGF&F,F,*&,&-%%rhobGF&F,-%#pbGF&F,F,-%\"vGF&F,F,F,F,*\$)%\"kGF0F ,!\"\"F0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(subtotsG<%/-%&rhopiG6#% \"tG,(*&-%%rhogGF)\"\"\"-%\$pigGF)F/F/*&-%%rhorGF)F/-%\$pirGF)F/F/*&F-F/ %#B0GF/F//-%\$rhoGF),,-%%rhobGF)F/-%%rhocGF)F/F3F/F-F/-%%rhovGF)F//-%\" pGF),*F3#F/\"\"\$*&FGF/F-F/F/-%#pbGF)F/FA!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&drhoqG,&*&-%\"HG6#%\"tG\"\"\"-%%rhoqGF)F+!\"%*&#F+\" \"#F+*&%\"kGF+-%&rhopiGF)F+F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%)polter_tG,&-%\$pigG6#%\"tG#\"\"\"\"#5*&#\"\"\$\"\"&F+-%#E2GF(F+F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$dE2G,(*&-%%opacG6#%\"tG\"\"\",&-%#E 2GF)F+-%'polterGF)!\"\"F+F1*&#\"\")\"#FF+*&%\"kGF+-%#E3GF)F+F+F1*(#F+ \"\"\$F+F7F+-%#B2GF)F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$dB2G,(*& -%%opacG6#%\"tG\"\"\"-%#B2GF)F+!\"\"*&#\"\")\"#FF+*&%\"kGF+-%#B3GF)F+F +F.*&#F+\"\"\$F+*&F4F+-%#E2GF)F+F+F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%(sublistG<)/-%%diffG6\$-%'exptauG6#%\"tGF--%\"gGF,/-F(6\$-%\"SGF,F-*& F3\"\"\"-%\"HGF,F6/-F(6\$-%#qrGF,F-,\$*&%\"kGF6-%\$pirGF,F6#!\"\"\"\"#/-F (6\$-%#qgGF,F-,&*&F@F6-%\$pigGF,F6FC-%%dragGF,F6/-F(6\$-%\"vGF,F-,(*(,&F6 F6*&\"\"\$F6-%#c2GF,F6FDF6F7F6FTF6FD*&-%(photbarGF,F6FOF6FD*&#F6FEF6*&* (F@F6%#B0GF6-%%rhogGF,F6F6-%%rhobGF,FDF6FD/-F(6\$F7F-,\$*()F3FEF6%&kappa GF6,&-%\$rhoGF,F6*&FZF6-%\"pGF,F6F6F6#FD\"\"'/-F(6\$FMF-,**&-%%opacGF,F6 ,&FMF6-%'polterGF,FDF6FD*&#\"\")\"#:F6*&F@F6-%\$J_3GF,F6F6FD*(#FE\"\"&F 6F@F6FIF6F6*&**#\"#;F_qF6FioF6FhoF6,**&-%%rhocGF,F6-%#vcGF,F6F6*&F_oF6 FIF6F6*&-%%rhorGF,F6F%-subtotderivsG<\$/-%%diffG6\$-%\"SG6#% \"tGF-*&F*\"\"\"-%\"HGF,F//-F(6\$F0F-,\$*()F*\"\"#F/%&kappaGF/,&-%\$rhoGF ,F/*&\"\"\$F/-%\"pGF,F/F/F/#!\"\"\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "#this mode is singular so can do\nvc(t):=0;\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#vcG6#%\"tG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "opac(t):= ne0/S(t)^2;\n#define delta_tig ht as small effect of non-total coupling\nne0:=omm^2*H0^4/omega^3/delt a_tight;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%%opacG6#%\"tG*&%\$ne0G \"\"\"*\$)-%\"SGF&\"\"#F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$ne 0G*&*&)%\$ommG\"\"#\"\"\")%#H0G\"\"%F*F**&)%&omegaG\"\"\$F*%,delta_tight GF*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 737 "#Definitions a nd background\npb(t):=0;\ndrag(t):= drag_t;\nphotbar(t):=photbar_t;\np olter(t):=polter_t;\n\n\n#B0:=0;\n\n#include terms up to t^Ord\nOrd:=5 ;\nDoQuint = \"F\";\n\nOrder:=Ord+3;\n\nmakeseries:=proc(v,x,ord)\n lo cal Res,i;\n Res:=0;\n for i from 0 to ord do\n Res:=Res + cat('s',v )[i]*x^i;\n end do;\n v(x)=series(Res,x,ord); \nend;\n \n\nexpandfu lly:=proc(x)\n eval(x);\n %;\n %;\nend;\n\n\nRb:=1-Rc;\n\n#v(t):=3/4*q g(t);\n\nomb:=Rb*omm;\nomc:=Rc*omm;\nomr:=Rv*omm^2*H0^2/omega^2;\nomg: =omm^2*H0^2/omega^2-omr;\n\npb(t):=0;c2(t):=0;\n\n#pig(t):=0;\n\n\nrho g(t):=3*omg*H0^2/kappa/S(t)^4;\nrhor(t):=3*omr*H0^2/kappa/S(t)^4;\nrho b(t):=3*omb*H0^2/kappa/S(t)^3;\nrhoc(t):=3*omc*H0^2/kappa/S(t)^3;\nrho v(t):=3*omv*H0^2/kappa;\nH(t):=H_t;\n\nassign(subtots);\n\nK:=0;\nKfac :=beta[2];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#pbG6#%\"tG\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%%dragG6#%\"tG*&*&%\$ne0G\"\"\",&-% \"vGF&#\"\"%\"\"\$-%#qgGF&!\"\"F+F+*\$)-%\"SGF&\"\"#F+F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%(photbarG6#%\"tG*&-%%rhogGF&\"\"\"-%%rhobGF&!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%'polterG6#%\"tG,&-%\$pigGF&#\" \"\"\"#5*&#\"\"\$\"\"&F,-%#E2GF&F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\$OrdG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(DoQuintGQ\"F6\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OrderG\"\")" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+makeseriesGR6%%\"vG%\"xG%\$ordG6\$%\$ResG%\"iG6\"F-C%>8 \$\"\"!?(8%F1\"\"\"9&%%trueG>F0,&F0F4*&&-%\$catG6\$.%\"sG9\$6#F3F4)9%F3F4F 4/-F@6#FC-%'seriesG6%F0FCF5F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %,expandfullyGR6#%\"xG6\"F(F(C%-%%evalG6#9\$%\"%GF.F(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#RbG,&\"\"\"F&%#RcG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$ombG*&,&\"\"\"F'%#RcG!\"\"F'%\$ommGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$omcG*&%#RcG\"\"\"%\$ommGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$omrG*&*(%#RvG\"\"\")%\$ommG\"\"#F()%#H0GF+F(F(*\$)%&om egaGF+F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$omgG,&*&*&)%\$ommG \"\"#\"\"\")%#H0GF*F+F+*\$)%&omegaGF*F+!\"\"F+*&*(%#RvGF+F(F+F,F+F+*\$F/ F+F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#pbG6#%\"tG\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#c2G6#%\"tG\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>-%%rhogG6#%\"tG,\$*&*&,&*&*&)%\$ommG\"\"#\"\"\")%#H0GF 0F1F1*\$)%&omegaGF0F1!\"\"F1*&*(%#RvGF1F.F1F2F1F1*\$F5F1F7F7F1F2F1F1*&%& kappaGF1)-%\"SGF&\"\"%F1F7\"\"\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-% %rhorG6#%\"tG,\$*&*(%#RvG\"\"\")%\$ommG\"\"#F,)%#H0G\"\"%F,F,*()%&omegaG F/F,%&kappaGF,)-%\"SGF&F2F,!\"\"\"\"\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%%rhobG6#%\"tG,\$*&*(,&\"\"\"F,%#RcG!\"\"F,%\$ommGF,)%#H0G\"\"#F,F, *&%&kappaGF,)-%\"SGF&\"\"\$F,F.F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>- %%rhocG6#%\"tG,\$*&*(%#RcG\"\"\"%\$ommGF,)%#H0G\"\"#F,F,*&%&kappaGF,)-% \"SGF&\"\"\$F,!\"\"F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%%rhovG6#%\" tG,\$*&*&%\$omvG\"\"\")%#H0G\"\"#F,F,%&kappaG!\"\"\"\"\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"HG6#%\"tG*&-%%diffG6\$-%\"SGF&F'\"\"\"F,!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%KfacG&%%betaG6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "\n#Start series solving\ndsolve(\{expandfully(Friedm ann),S(0)=0,D(S)(0)=H0^2*omm/omega\},S(t),type=series);\nassign(%);\n \n# S(t):=omm*H0^2/omega^2*(omega*t+(omega*t)^2/4-K/6*omega*t^3-K/48*o mega^2*t^4);\n\n\n\nRb:=1-Rc;\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-% \"SG6#%\"tG+/F'*&*&)%#H0G\"\"#\"\"\"%\$ommGF.F.%&omegaG!\"\"F.,\$*&F+F.F /F.#F.\"\"%F-,\$*&*()F,\"\")F.%\$omvGF.)F/\"\"\$F.F.*\$)F0F=F.F1#F.\"#5\" \"&,\$*&*(F9F.F;F.F%#RbG,& \"\"\"F&%#RcG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "omb: =Rb*omm;\nomc:=Rc*omm;\nomr:=Rv*omm^2*H0^2/omega^2;\nomg:=omm^2*H0^2/o mega^2-omr;\n\npb(t):=0;c2(t):=0;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\$ombG*&,&\"\"\"F'%#RcG!\"\"F'%\$ommGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$omcG*&%#RcG\"\"\"%\$ommGF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\$omrG*&*(%#RvG\"\"\")%\$ommG\"\"#F()%#H0GF+F(F(*\$)%&omegaGF+F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$omgG,&*&*&)%\$ommG\"\"#\"\"\")%#H0 GF*F+F+*\$)%&omegaGF*F+!\"\"F+*&*(%#RvGF+F(F+F,F+F+*\$F/F+F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#pbG6#%\"tG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#c2G6#%\"tG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 428 "solvevars:= \{B2,J_3,E2,pig,v,qr,qg,pir,G_3,G_4\};\n \neqs:=\{simplify(diff(J_3(t),t)-dJ_3),simplify(diff(B2(t),t)-dB2),sim plify(diff(E2(t),t)-dE2),simplify(diff(pig(t),t)-dpig),simplify(diff(v (t),t)-dv),simplify(dqr-diff(qr(t),t)),simplify(dqg-diff(qg(t),t)),sim plify(diff(pir(t),t)-dpir),simplify(diff(G_3(t),t)-dG_3),simplify(diff (G_4(t),t)-dG_4)\};\n\nknownvars:=\{H0,omm,omv,k,beta[2],Kf[2],Kf[3],K f[4],omega,Rv,Rc,B0,delta_tight\};\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*solvevarsG<,%\"vG%#qgG%\$pirG%\$pigG%\$J_3G%\$G_3G%#qrG%\$G_4G%#E2 G%#B2G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\$eqsG<,,\$*&,***-%%diffG6\$- %\$J_3G6#%\"tGF0\"\"\")%&omegaG\"\"\$F1%,delta_tightGF1)+/F0*&*&)%#H0G\" \"#F1%\$ommGF1F1F3!\"\"F1,\$*&F:F1F=F1#F1\"\"%F<,\$*&*()F;\"\")F1%\$omvGF1 )F=F4F1F1*\$F2F1F>#F1\"#5\"\"&,\$*&*(FFF1FHF1FIF1F1*\$)F3F#F1\"#?\" \"',\$*&*(FFF1FHF1FIF1F1F3F>#F1\"\$7\"\"\"(-%\"OG6#F1FGF*.\"#:F1F\\oF1F2F1F5F1F6F1-%\$J_4GF/F1F1 **FinF1)F=F#F1Fin,\$*&,6*,-F+6\$F]oF0F 1F2F1F5F1F6F1F\\oF1\"#I*,\"#FF1FdoF1FeoF1F\\oF1F]oF1F1*,\"#=F1FdoF1Feo F1F\\oF1-%#E2GF/F1F>*.\"#;F1)F\\oF*.\"#'*F1FeoF1FdoF1F3F1F5F1FipF1F>*0F\\qF1FeoF1Fd oF1F3F1F5F1FipF1%#RvGF1F1*0F\\qF1FeoF1FdoF1F3F1F5F1F^qF1-%#qrGF/F1F>*0 F\\qF1F:F1F=F1F2F1F5F1-%\"vGF/F1F7F1F>*2F\\qF1F:F1F=F1F2F1F5F1FcqF1F7F 1%#RcGF1F1F1**F6F1F2F1F5F1F\\oF1F>#F1F^p,\$*&,2**-F+6\$-%\$pirGF/F0F1F\\o F1FRF1F6F1F`o*,FGF1FgpF1-%\$G_3GF/F1FRF1F6F1F1*,FUF1FgpF1F`qF1FRF1F6F1F >**\"#[F1FdoF1FeoF1FipF1F>*,FfrF1FeoF1FdoF1FipF1F^qF1F1*,FfrF1F^qF1Fdo F1FeoF1F`qF1F>*.FfrF1F:F1F=F1FcqF1FRF1F7F1F>*0FfrF1F:F1F=F1FcqF1FRF1F7 F1FfqF1F1F1*(F6F1F\\oF1FRF1F>#F1F`o,\$*&,***-F+6\$-%#B2GF/F0F1F2F1F5F1F6 F1F`p**F`pF1FdoF1FeoF1FcsF1F1*.FGF1F\\oF1-%#B3GF/F1F2F1F5F1F6F1F1*.\" \"*F1F\\oF1FcpF1F2F1F5F1F6F1F1F1*(F2F1F5F1F6F1F>#F1F`p,\$*&,6**-F+6\$Fcq F0F1)F3FMF1)F7F4F1F5F1!\"'*.FUF1FatF1FctF1FdtF1F5F1FfqF1F1*.FUF1+/F0F8 \"\"!,\$F@#F1F*0FUF1FhtF1FcqF1FctF1F6F1F5F1FfqF1F1**FGF1FIF1)F;FUF1FcqF1F>**FU F1FIF1FcuF1FipF1F1*,FGF1FIF1FcuF1F^qF1FcqF1F1*,FUF1FIF1FcuF1F^qF1FipF1 F>*2F4F1F\\oF1%#B0GF1F=F1F:F1F2F1F6F1F5F1F>*4F4F1F\\oF1FhuF1F=F1F:F1F2 F1F6F1F5F1F^qF1F1F1**FctF1FdtF1,&F>F1FfqF1F1F5F1F>#F1FU,&*&F\\oF1F_rF1 #F>F<-F+6\$F`qF0F>,\$*&,**,F\\oF1F2F1F5F1F6F1F]oF1F4**FGF1FdoF1FeoF1FcqF 1F>**FUF1FdoF1FeoF1FipF1F1*,FUF1-F+6\$FipF0F1F2F1F5F1F6F1F1F1*(F2F1F5F1 F6F1F>#F>FU,\$*&,,**-F+6\$FcpF0F1F2F1F5F1F6F1\"\$q#**\"\$3\"F1FdoF1FeoF1Fc pF1F1**F`pF1FdoF1FeoF1F]oF1F>*.\"#!)F1F\\oF1-%#E3GF/F1F2F1F5F1F6F1F1*. \"#!*F1F\\oF1FcsF1F2F1F5F1F6F1F>F1*(F2F1F5F1F6F1F>#F1Fcw,(-F+6\$-%\$G_4G F/F0F1*&#FBFjsF1*&F\\oF1FbrF1F1F>*(#FGF`oF1F\\oF1-%\$G_5GF/F1F1,(-F+6\$F brF0F1*&#F4FenF1F^vF1F>*(#F`oFinF1F\\oF1FbxF1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*knownvarsG " 0 "" {MPLTEXT 1 0 81 "#Set up series ansatz\nserie s_subs:=map(makeseries,solvevars,t,Ord+3);\nassign(%);\n" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%,series_subsG<,/-%\$J_3G6#%\"tG+5F*&%%sJ_3G6#\" \"!F/&F-6#\"\"\"F2&F-6#\"\"#F5&F-6#\"\"\$F8&F-6#\"\"%F;&F-6#\"\"&F>&F-6 #\"\"'FA&F-6#\"\"(FD-%\"OGF1\"\")/-%\$pigGF)+5F*&%%spigGF.F/&FMF1F2&FMF 4F5&FMF7F8&FMF:F;&FMF=F>&FMF@FA&FMFCFDFEFG/-%#qgGF)+5F*&%\$sqgGF.F/&FZF 1F2&FZF4F5&FZF7F8&FZF:F;&FZF=F>&FZF@FA&FZFCFDFEFG/-%\$pirGF)+5F*&%%spir GF.F/&FaoF1F2&FaoF4F5&FaoF7F8&FaoF:F;&FaoF=F>&FaoF@FA&FaoFCFDFEFG/-%\$G _3GF)+5F*&%%sG_3GF.F/&F^pF1F2&F^pF4F5&F^pF7F8&F^pF:F;&F^pF=F>&F^pF@FA& F^pFCFDFEFG/-%#qrGF)+5F*&%\$sqrGF.F/&F[qF1F2&F[qF4F5&F[qF7F8&F[qF:F;&F[ qF=F>&F[qF@FA&F[qFCFDFEFG/-%\$G_4GF)+5F*&%%sG_4GF.F/&FhqF1F2&FhqF4F5&Fh qF7F8&FhqF:F;&FhqF=F>&FhqF@FA&FhqFCFDFEFG/-%\"vGF)+5F*&%#svGF.F/&FerF1 F2&FerF4F5&FerF7F8&FerF:F;&FerF=F>&FerF@FA&FerFCFDFEFG/-%#E2GF)+5F*&%\$ sE2GF.F/&FbsF1F2&FbsF4F5&FbsF7F8&FbsF:F;&FbsF=F>&FbsF@FA&FbsFCFDFEFG/- %#B2GF)+5F*&%\$sB2GF.F/&F_tF1F2&F_tF4F5&F_tF7F8&F_tF:F;&F_tF=F>&F_tF@FA &F_tFCFDFEFG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "#Here only solve for modes with zero initial octopole or higher\nsG_3[0]:=0;\nsG _4[0]:=0;\nsG_4[1]:=0;\nG_5(t):=O(t^3);\nE3(t):=O(t^2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%sG_3G6#\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%sG_4G6#\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%%sG_4G6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\$G_5G6#%\" tG-%\"OG6#*\$)F'\"\"\$\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#E3G6 #%\"tG-%\"OG6#*\$)F'\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "series_eqs:=map(expandfully,eqs):\n\nmap(series,series_eqs,t): \nmap(simplify,%):\nseries_eqs:=%;\n" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#>%+series_eqsG<,+3%\"tG,&&%%sG_3G6#\"\"\"F,*&#\"\"\$\"\"(F,*&%\"kGF,& %%spirG6#\"\"!F,F,!\"\"F6,&&F*6#\"\"#F;*&#F/F0F,*&F2F,&F4F+F,F,F7F,,(& F*6#F/F/*&#F/F0F,*&F2F,&F4F:F,F,F7*(#\"#:\"#GF,F2F,&%%sG_4GF:F,F,F;,(& F*6#\"\"%FP*&#F/F0F,*&F2F,&F4FBF,F,F7*(FHF,F2F,&FLFBF,F,F/,(*&F2F,&FLF OF,FH*&#F/F0F,*&F2F,&F4FOF,F,F7*&\"\"&F,&F*6#FinF,F,FP,(*&F2F,&FLF[oF, FH*&\"\"'F,&F*6#F`oF,F,*&#F/F0F,*&F2F,&F4F[oF,F,F7Fin,(&F*6#F0F0*&#F/F 0F,*&F2F,&F4FboF,F,F7*(FHF,F2F,&FLFboF,F,F`o-%\"OGF+F0+-F',\$*&,&&%%spi gGF5F,*&FPF,&%\$sE2GF5F,F7F,*&%&omegaGF,%,delta_tightGF,F7#F7\"#5!\"#,\$ 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,*&FitF,F]qF,F7#F7FienFPF`pFin+3F'*&F`sF,*&F\\qF,F]qF,F7F`q,\$*&,&*&F`s F,F\\qF,F,*&F;F,FctF,F7F,*&F\\qF,F]qF,F7F]hmF7,\$*&,.*&F`sF,FbrF,F_el*& FiclF,&FasF:F,F,*,FedlF,F2F,FipF,F\\qF,F]qF,F,**FiclF,FctF,F\\qF,F]qF, F,*,Fi]nF,F2F,-%#B3GF5F,F\\qF,F]qF,F,*(FhdlF,FctF,F\\qF,F7F,*&F\\qF,F] qF,F7#F,FiclF6,\$*&,0*&FctF,FbrF,F_el**\"\$k)F,Fc^pF,F\\qF,F]qF,F,*,Fi]n F,F2F,--F[ho6#Fh^pF5F,F\\qF,F]qF,F,*,FedlF,F2F,FjqF,F\\qF,F]qF,F,*&Fic lF,&FasFBF,F,*(FhdlF,F\\qF,Fc^pF,F7*(F]glF,F`sF,FitF,F7F,*&F\\qF,F]qF, F7F[_pF,,\$*&,4*&FixF,Fc^pF,Fb[m**\"'!o.\"F,FitF,Fh_pF,F]qF,F,*,F_yF,Fi tF,F2F,--F^ioFe_pF5F,F]qF,F,*,FbyF,FitF,F2F,F[sF,F]qF,F,*(FbjlF,FbrF,& FasFOF,F,*(Fe[mF,F`sF,FhyF,F,*,FhhlF,F`sF,F[zF,F]zF,F_zF,F7*(F][mF,Fit F,Fh_pF,F7*(FjjlF,FctF,FizF,F7F,*&FitF,F]qF,F7#F,FbjlF;,\$*&,8*&F`sF,Fj [lF,!%:7*(Fa`pF,FitF,Fg`pF,F7*(\"%]SF,FctF,FhyF,F,*(\"'gt?F,FbrF,&FasF [oF,F,*(\"&!))QF,FixF,Fh_pF,F,*(F_]mF,FizF,Fc^pF,F7*.\"&o.\"F,F`sF,F\\ qF,F[zF,F]zF,F_zF,F,*,Fe\\lF,FitF,F2F,--F^[pFe_pF5F,F]qF,F,*,F_^mF,Fit F,F2F,FatF,F]qF,F,*,\"&s9%F,FctF,F[zF,F]zF,F_zF,F7**\"'S%H)F,FitF,Fg`p F,F]qF,F,F,*&FitF,F]qF,F7#F,FgapF/,\$*&,<*(FhapF,FitF,F]qF,\")+31e*(\"& X)>F,F`sF,Fi_lF,F,*(\")g@h6F,&FasFboF,FbrF,F,*(\"&S!oF,FctF,Fj[lF,F7*( F_dmF,FizF,Fh_pF,F7*,\"(KCK#F,Fc^pF,F[zF,F]zF,F_zF,F7*,FienF,F2F,--Fh \\pFe_pF5F,FitF,F]qF,F,*(\"(!Gx@F,FixF,Fg`pF,F,*.FgapF,F`sF,FbrF,F[zF, F]zF,F_zF,F7*,F_gmF,F2F,FdjlF,FitF,F]qF,F,*(\"'+oAF,Fc^pF,FhyF,F,*.\"' 31eF,FctF,F\\qF,F[zF,F]zF,F_zF,F,*(\"(!31eF,FitF,FhapF,F7F,*&FitF,F]qF ,F7#F,F`cpFPF`pFin" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "#sol ve by looking at lowest powers\ndosolve:=proc(ineqs)\n local tmp,vars; \n map(series,ineqs,t);\n map(simplify,%);\n map(convert,%,polynom);\n map(tcoeff,%,t):\n tmp:=map(simplify,%);\n vars:=indets(tmp) minus kn ownvars;\n print(vars);\n solve(tmp,vars);\n assign(%);\nend;\n\n" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%(dosolveGR6#%&ineqsG6\$%\$tmpG%%varsG6 \"F+C+-%\$mapG6%%'seriesG9\$%\"tG-F.6\$%)simplifyG%\"%G-F.6%%(convertGF6% (polynomG-F.6%%'tcoeffGF6F2>8\$F3>8%-%&minusG6\$-%'indetsG6#F?%*knownvar sG-%&printG6#FA-%&solveG6\$F?FA-%'assignG6#F6F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "#get the solution\nfor i from 1 to Ord+1 do \n dosolve(series_eqs);\nend do;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# <-&%%sJ_3G6#\"\"!&%%spirGF&&%%spigGF&&%\$sqgGF&&%%sG_3G6#\"\"\"&%\$sqrGF &&F3F0&%%sG_4G6#\"\"#&%#svGF&&%\$sB2GF&&%\$sE2GF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<.&%%sJ_3G6#\"\"\"&%%spirG6#\"\"!&F)F&&%%spigGF&&%\$sqgG F&&%%sG_3G6#\"\"#&%\$sqrGF3&%%sG_4G6#\"\"\$&%#svGF*&F " 0 "" {MPLTEXT 1 0 54 "sig_ser:=simplify(series(expandfully(sigma(t)),t,7 ));\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(sig_serG++%\"tG,\$*&*&,&%#R vG\"\"\"F,!\"\"F,&%#svG6#\"\"!F,F,,&F+\"\"%\"\"&F,F-!\"%F1,\$*&,0*(%\"k GF,)F+\"\"#F,%#B0GF,\"#g**\"\$S\"F,%&omegaGF,F+F,F.F,F,**\"#:F,F+F,F=F, F:F,F,*,\"\$[%F,FAF,%,delta_tightGF,F.F,F+F,F,*(F@F,FAF,F.F,F-**FEF,FAF ,FFF,F.F,F-*(\"#vF,F=F,F:F,F-F,*&F2F,,&F+F3FCF,F,F-#\"\"\$\"#9F,,\$*&,N* ,)F+FNF,FAF,FFF,F=F,F:F,\"\$7&*.\"%7DF,FAF,FFF,F;F,F=F,F:F,F-**\"#!)F,) F:F " 0 "" {MPLTEXT 1 0 39 "sv[0]:=solve(sig0=coeff(%,t,0),sv[0]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#svG6#\"\"!,\$*&*&%%sig0G\"\"\",&%#RvG\"\"%\"\"&F,F,F ,,&F.F,F,!\"\"F2#F2F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "#S o solution for sigma is\nsimplify(sig_ser);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#++%\"tG%%sig0G\"\"!,\$*&,**(%#RvG\"\"\"%#B0GF,%\"kGF,!#: *(\"#:F,F-F,F.F,F,*(\"#NF,%&omegaGF,F%F,F,**\"\$7\"F,F4F,%,delta_tightG F,F%F,F,F,,&F+\"\"%F1F,!\"\"#!\"\$\"#9F,,\$*&,N*,)F+\"\"\$F,F4F,F7F,F-F,F .F,!\$7&*,\"%![%F,)F+\"\"#F,)F4FHF,F7F,F%F,F,*,\"\$+*F,F4F,FGF,F-F,F.F,F ,*.\"\$'*)F,FGF,FIF,F7F,F%F,%#RcGF,F:*.\"%kOF,F4F,F7F,FGF,F-F,F.F,F,** \"#!)F,)F.FHF,F%F,FGF,F,**\"\$S\"F,FIF,FGF,F%F,F,*,\"%)o#F,FIF,FGF,)F7F HF,F%F,F:*.FFF,FIF,F7F,F+F,F%F,FNF,F:*,\"%CIF,FIF,F7F,F%F,F+F,F:**\"%: " 0 "" {MPLTEXT 1 0 43 "simplify(subs(delta_tight=0, subs(B0=0,%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"tG%%sig0G\"\"! ,\$*&*&%&omegaG\"\"\"F%F+F+,&%#RvG\"\"%\"#:F+!\"\"#!#:\"\"#F+,\$*&*&,**& )%\"kGF3F+F-F+\"#;*(\"#GF+)F*F3F+F-F+F+*&\"\$:\$F+F>F+F0*&\"#gF+F9F+F+F+ F%F+F+*&F,F+,&F-F3F/F+F+F0#F2\"\$7\"F3-%\"OG6#F+\"\"\$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "#photon heat flux\nsimplify(series(expand fully(qg(t)),t,3));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#++%\"tG,\$*&*&%% sig0G\"\"\",&%#RvG\"\"%\"\"&F)F)F),&F+F)F)!\"\"F/#F/\"\"\$\"\"!,\$*&,0** F(F)%&omegaGF)%#RcGF)F+F)F,**F-F)F(F)F7F)F8F)F)**F,F)F7F)F+F)F(F)F/*(F -F)F7F)F(F)F/**\"\"#F)%\"kGF))F+F=F)%#B0GF)F/**F,F)F+F)F@F)F>F)F)*(F=F )F@F)F>F)F/F)*\$)F.F=F)F/#F)F,F),\$*&*&F7F),>*(F+F)F@F)F>F)!#7**\"\"'F)F 8F)F@F)F>F)F/*,\"#7F)F8F)F+F)F@F)F>F)F)*,FMF)F8F)F?F)F@F)F>F)F/**FMF)F >F)F?F)F@F)F)*(FMF)F@F)F>F)F)**\"#8F)F7F)F+F)F(F)F)*(\"#5F)F7F)F(F)F)* *F,F)F7F)F?F)F(F)F)*,\"#DF)F(F)F7F)F8F)F+F)F/*,FOF))F8F=F)F7F)F+F)F(F) F)*,F,F)F8F)F(F)F7F)F?F)F/**FYF)F(F)F7F)F8F)F/**\"#:F)FenF)F7F)F(F)F)F )F)*\$)F.F1F)F/#F/\"#;F=-%\"OG6#F)F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "simplify(subs(delta_tight=0,subs(B0=0,%)));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#++%\"tG,\$*&*&%%sig0G\"\"\",&%#RvG\"\"%\"\"&F )F)F),&F+F)F)!\"\"F/#F/\"\"\$\"\"!,\$*&*(%&omegaGF)F(F),**&%#RcGF)F+F)F, *&F-F)F9F)F)*&F,F)F+F)F/F-F/F)F)*\$)F.\"\"#F)F/#F)F,F),\$*&*()F6F>F)F(F) ,2F+\"#8\"#5F)*&F,F))F+F>F)F)*(\"#DF)F9F)F+F)F/*(\"#7F))F9F>F)F+F)F)*( F,F)F9F)FHF)F/*&FJF)F9F)F/*&\"#:F)FMF)F)F)F)*\$)F.F1F)F/#F/\"#;F>-%\"OG 6#F)F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "#neutrino heat fl ux\nsimplify(series(expandfully(qr(t)),t,3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"tG,\$*&*&%%sig0G\"\"\",&%#RvG\"\"%\"\"&F)F)F)F+!\" \"#F.\"\"\$\"\"!,\$*&*(%#B0GF)%\"kGF),&F+F)F)F.F)F)F+F.#F.\"\"#F),\$*&*&) F6F9F)F(F)F)F+F.#F)\"\"'F9-%\"OG6#F)F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "#neutrino anisotropic stress\nsimplify(series(expandf ully(pir(t)),t,3));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#++%\"tG*&*&%#B0 G\"\"\",&%#RvGF(F(!\"\"F(F(F*F+\"\"!,\$*&*&%\"kGF(%%sig0GF(F(F*F+#!\"# \"\"\$F(,\$*&*&,***%&omegaGF(%,delta_tightGF(F1F(F*F(\"\$[%**\"\$S\"F(F:F( F*F(F1F(F(**\"\$D#F(F*F(F'F(F0F(F(*(F@F(F'F(F0F(F+F(F0F(F(*&F*F(,&F*\" \"%\"#:F(F(F+#F+\"#q\"\"#-%\"OG6#F(F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "16 0 0" 7 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }