{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 2097 "#Get the series so lutions in tight coupling\n#Start with the vector perturbation equatio ns, in A=0 gauge\n#See astro-ph/0403583\n#AL March 04\n\n# Use conform al time. \n# Includes cdm, baryons, lambda\n\nrestart;\nno_quint:=\{ps i(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(no_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*rhopi(t) + 4 * H(t) \+ * rhoq(t));\n\ndrag_t:=opac(t)*(4/3*v(t)-qg(t));\nphotbar_t:=rhog(t)/r hob(t);\n\ndqr:=-1/2*k*pir(t);\ndqg:=-1/2*k*pig(t)+drag(t);\ndv:=-(1-3 *c2(t))*H(t)*v(t)-photbar(t)*drag(t) - k/2*B0*rhog(t)/rhob(t);\n\n\n#d sigma:=simplify(-H(t)*sigma(t)+k*phi(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\n rho_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)*drh ob;\n\n\ndpig:=-opac(t)*(pig(t)-polter(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*Kf[4]*G_5(t));\n \nrhopi_t:=rhog(t)*pig(t)+rhor(t)*pir(t) + rhonu(t)*pinu(t) + rhog(t)* B0;\nrhoq_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:=\{r hopi(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);\ndB2:=-opac(t)*B 2(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(e xptau(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" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)no_quintG<(/-%\$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#%\"t G,\$*&*(%&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(#!\"\"\"\"#-%%d ragGF+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dvG,(*(,&\"\"\"F(*&\"\" \$F(-%#c2G6#%\"tGF(!\"\"F(-%\"HGF-F(-%\"vGF-F(F/*&-%(photbarGF-F(-%%dra gGF-F(F/*&#F(\"\"#F(*&*(%\"kGF(%#B0GF(-%%rhogGF-F(F(-%%rhobGF-F/F(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dsigmaG,\$*&,&*(-%\"HG6#%\"tG\"\" \"-%&sigmaGF+F-%\"kGF-\"\"#*(%&kappaGF-)-%\"SGF+F1F--%&rhopiGF+F-F-F-F 0!\"\"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+-%%rhogG F(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(-%#qrG F+F(F(*(#\"\")F/F(F'F(-%&sigmaGF+F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dG_3G*&%\"kG\"\"\",&-%\$pirG6#%\"tG#\"\"\$\"\"(*&#\"#:\"#GF'-%\$G_4 GF+F'!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%dG_4G*&%\"kG\"\"\", &-%\$G_3G6#%\"tG#\"\"%\"\"**&#\"\")\"#:F'*&&%#KfG6#F.F'-%\$G_5GF+F'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,(*&-%%rhogG6#%\"tG\"\"\"-%# qgGF)F+F+*&-%%rhorGF)F+-%#qrGF)F+F+*&,&-%%rhobGF)F+-%#pbGF)F+F+-%\"vGF )F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%&sigmaG6#%\"tG,\$*&*(%&kapp aG\"\"\")-%\"SGF&\"\"#F,,(*&-%%rhogGF&F,-%#qgGF&F,F,*&-%%rhorGF&F,-%#q rGF&F,F,*&,&-%%rhobGF&F,-%#pbGF&F,F,-%\"vGF&F,F,F,F,*\$)%\"kGF0F,!\"\"F 0" }}{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),*F 3#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#>%)polte r_tG,&-%\$pigG6#%\"tG#\"\"\"\"#5*&#\"\"\$\"\"&F+-%#E2GF(F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$dE2G,(*&-%%opacG6#%\"tG\"\"\",&-%#E2GF)F+ -%'polterGF)!\"\"F+F1*&#\"\")\"#FF+*&%\"kGF+-%#E3GF)F+F+F1*(#F+\"\"\$F+ F7F+-%#B2GF)F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$dB2G,(*&-%%opac G6#%\"tG\"\"\"-%#B2GF)F+!\"\"*&#\"\")\"#FF+*&%\"kGF+-%#B3GF)F+F+F.*&#F +\"\"\$F+*&F4F+-%#E2GF)F+F+F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(sub listG<)/-%%diffG6\$-%'exptauG6#%\"tGF--%\"gGF,/-F(6\$-%\"HGF,F-,\$*()-%\" SGF,\"\"#\"\"\"%&kappaGF;,&-%\$rhoGF,F;*&\"\"\$F;-%\"pGF,F;F;F;#!\"\"\" \"'/-F(6\$-%\$pigGF,F-,**&-%%opacGF,F;,&FJF;-%'polterGF,FEF;FE*&#\"\")\" #:F;*&%\"kGF;-%\$J_3GF,F;F;FE*(#F:\"\"&F;FXF;-%#qgGF,F;F;*&**#\"#;FVF;F %-subtotderivsG<\$/-%%diffG6\$-%\"HG6#%\"tGF-,\$*()-%\"SGF,\"\"#\"\"\"% &kappaGF4,&-%\$rhoGF,F4*&\"\"\$F4-%\"pGF,F4F4F4#!\"\"\"\"'/-F(6\$F1F-*&F1 F4F*F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1952 "#Get series sol ution using tight coupling result:\nOrd:=4;\nDoQuint = \"F\";\n\nOrder :=Ord+5;\n\nmakeseries:=proc(v,x,ord)\n local 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)=s eries(Res,x,ord); \nend;\n \n\nexpandfully:=proc(x)\n eval(x);\n %; \n %;\nend;\n\n\nRb:=1-Rc;\nv(t):=3/4*qg(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;\npig(t):=0;\n\n\nrhog(t):=3*omg*H0^2/kappa/S(t)^4;\nrh or(t):=3*omr*H0^2/kappa/S(t)^4;\nrhob(t):=3*omb*H0^2/kappa/S(t)^3;\nrh oc(t):=3*omc*H0^2/kappa/S(t)^3;\nrhov(t):=3*omv*H0^2/kappa;\nH(t):=H_t ;\n\nassign(subtots);\n\nK:=0;\nKfac:=beta[2];\n\n\n\n#Start series so lving\ndsolve(\{expandfully(Friedmann),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*omega^2*t^4);\n\n\nRb:=1-Rc;\n\nv(t):= 3/4*qg(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;\npig(t):=0;\n\n#U se tight coupling result\nR:=4/3*rhog(t)/rhob(t);\ndrag(t):=1/(1+R)*(- 4/3*H(t)*v(t) - k/2*B0*R);\n\n#Some frame invariant variables\n\nsolv evars:= \{qr,qg,pir,G_3,G_4\};\n\neqs:=\{simplify(dqr-diff(qr(t),t)),s implify(dqg-diff(qg(t),t)),simplify(diff(pir(t),t)-dpir),simplify(diff (G_3(t),t)-dG_3),simplify(diff(G_4(t),t)-dG_4)\};\n\n\nknownvars:=\{H0 ,omm,omv,k,beta[2],Kf[2],Kf[3],Kf[4],omega,Rv,Rc,B0,sqg[0]\};\n\nserie s_subs:=map(makeseries,solvevars,t,Ord+3);\nassign(%);\n\n#spir[0]:=0; \n#sG_3[1]:=0;\n#sG_4[2]:=0;\n\nG_5(t):=sG_5[3]*t^3+sG_5[4]*t^4+sG_ 5*t^5;\nsG_3[0]:=0;\nsG_4[0]:=0;\nsG_4[1]:=0;\n\n\nseries_eqs:=map(exp andfully,eqs);\n\nmap(series,series_eqs,t):\nmap(simplify,%):\nseries_ eqs:=%;\n\ndosolve:=proc(ineqs)\n local tmp,vars;\n map(series,ineqs,t );\n map(simplify,%);\n map(convert,%,polynom);\n map(tcoeff,%,t):\n t mp:=map(simplify,%);\n vars:=indets(tmp) minus knownvars;\n print(vars );\n solve(tmp,vars);\n assign(%);\nend;\n\n\n\nfor i from 1 to Ord+1 \+ do\n dosolve(series_eqs);\nend do;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\$OrdG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(DoQuintGQ\"F6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&OrderG\"\"*" }}{PARA 11 "" 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#>-%\"vG6#%\"tG,\$-%#qgGF&#\"\"\$\"\"%" }}{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,&*&*&)%\$omm G\"\"#\"\"\")%#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#>-%\$pigG6#%\"tG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%%rhogG6#%\"tG,\$*&*&,&*&*&)%\$ommG\"\"#\"\"\")%#H0GF0F1F1*\$)%&omeg aGF0F1!\"\"F1*&*(%#RvGF1F.F1F2F1F1*\$F5F1F7F7F1F2F1F1*&%&kappaGF1)-%\"S GF&\"\"%F1F7\"\"\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%%rhorG6#%\"tG, \$*&*(%#RvG\"\"\")%\$ommG\"\"#F,)%#H0G\"\"%F,F,*()%&omegaGF/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#\"\"#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"SG6#%\" tG+1F'*&*&)%#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!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"vG6#%\"tG,\$-%#qgGF&#\"\"\$\"\"%" }}{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,&*&*&)%\$omm G\"\"#\"\"\")%#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#>-%\$pigG6#%\"tG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG,\$*&,&*&*&)%\$ommG\"\"#\"\"\")%#H0GF,F-F-*\$)%&omegaGF,F-!\"\"F -*&*(%#RvGF-F*F-F.F-F-*\$F1F-F3F3F-*(-%\"SG6#%\"tGF-,&F-F-%#RcGF3F-F+F- F3#\"\"%\"\"\$" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-%%dragG6#%\"tG*&,&* 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\"\"F.%#RvGF.F.F.,&F1\"\"%\"\"&F.F0!\"\$F-,\$*&,,*(%\"kGF.%#B0GF.)F1\"\" #F.F3**\"\"(F.%&omegaGF.F*F.F1F.F.*(F:F.F;F.F1F.F.*(F4F.F:F.F;F.F0*(F? F.F@F.F*F.F0F.,(*\$F&%\$sqgG6#\"\"!,\$*&*&%%sig0G\"\"\",&%#RvG\"\"%\"\"&F,F,F,,&!\"\" F,F.F,F2#F2\"\"\$" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#++%\"tG%%sig0G\"\" !,\$*&,(*(%\"kG\"\"\"%#B0GF,%#RvGF,\"\"\$*(F/F,F+F,F-F,!\"\"*(\"\"(F,%&o megaGF,F%F,F1F,,&F.\"\"%\"#:F,F1#F7\"#9F,,\$*&,.*()F+\"\"#F,F%F,F.F,\"# ;**\"#GF,)F4F?F,F%F,F.F,F,*,\"\$!=F,F4F,F+F,F-F,F.F,F,*(\"#gF,F>F,F%F,F ,*(\"\$:\$F,FCF,F%F,F1**FEF,F4F,F+F,F-F,F1F,*&F5F,,&F.F?F7F,F,F1#!#:\"\$7 \"F?-%\"OG6#F,F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "#This \+ is the solution for sigma\n#velocities\n#photon\nvg(t):=simplify(serie s(expandfully(3/4*qg(t)),t,3));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>-% #vgG6#%\"tG++F',\$*&*&%%sig0G\"\"\",&%#RvG\"\"%\"\"&F-F-F-,&!\"\"F-F/F- F3#F3F0\"\"!,\$*&,0*(%&omegaGF-F,F-F/F-!\"%*(F1F-F:F-F,F-F3*,F0F-F:F-F, F-%#RcGF-F/F-F-**F1F-F:F-F,F-F>F-F-*(\"\"#F-%\"kGF-%#B0GF-F3**F0F-FBF- FCF-F/F-F-**FAF-FBF-FCF-)F/FAF-F3F-*\$)F2FAF-F3#\"\"\$\"#;F-,\$*&*&F:F-,> **FBF-FCF-F/F-F>F-\"#7*,\"\"'F-FBF-FCF-FFF-F>F-F3*,F0F-F:F-F,F-F>F-FFF -F3*,\"#DF-F:F-F,F-F>F-F/F-F3**F0F-F:F-F,F-FFF-F-**\"#8F-F:F-F,F-F/F-F -**FQF-FBF-FCF-F/F-F3**FSF-FBF-FCF-FFF-F-*(\"#5F-F:F-F,F-F-*(FSF-FBF-F CF-F-**FSF-FBF-FCF-F>F-F3**FVF-F:F-F,F-F>F-F3*,FQF-F:F-F,F-)F>FAF-F/F- F-**\"#:F-F:F-F,F-F\\oF-F-F-F-*&,(F-F-*&FAF-F/F-F3*\$FFF-F-F-F2F-F3#!\" \$\"#kFA-%\"OG6#F-FJ" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "#neu trino\n\nvr(t):=simplify(series(expandfully(3/4*qr(t)),t,3));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>-%#vrG6#%\"tG++F',\$*&*&%%sig0G\"\"\", &%#RvG\"\"%\"\"&F-F-F-F/!\"\"#F2F0\"\"!,\$*&*(%\"kGF-%#B0GF-,&F2F-F/F-F -F-F/F2#!\"\$\"\")F-,\$*&*&)F8\"\"#F-F,F-F-F/F2#F-F=FB-%\"OG6#F-\"\"\$" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "#neutrino anisotropy stres s\nsimplify(series(expandfully(pir(t)),t,3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#++%\"tG*&*&%#B0G\"\"\",&!\"\"F(%#RvGF(F(F(F+F*\"\"!,\$*& *&%\"kGF(%%sig0GF(F(F+F*#!\"#\"\"\$F(,\$*&*&F0F(,(*(F0F(F'F(F+F(\"#X**\" #GF(%&omegaGF(F1F(F+F(F(*(F:F(F0F(F'F(F*F(F(*&,&F+\"\"%\"#:F(F(F+F(F*# F*\"#9\"\"#-%\"OG6#F(F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "#Note that the above results have corrections from the full evolution , though the leading terms in B0 and sig0 are OK" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "10 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }