#Get second order action for scalar perturbations #in terms of user-defined variable, following astro-ph/9902265 #I paste into Maple what I want to run #AML 13 Nov 00 #Note Delta here = box+3K = -k^2+3K restart; phidot:=diff(phi(t),t); phiddot:=diff(diff(phi(t),t),t); phidddot:=diff(phiddot,t); phidot_sub:=sqrt(2)*z(t)*H(t)/sqrt(kappa); zdot:=diff(z(t),t); zddot:=diff(diff(z(t),t),t); nudot:=diff(nu(t),t); #Newtonian potential PsiN_sub:=psi(t)+H(t)*kappa/2/a(t)^2*Pipsi(t)/Delta; #Curvature perturbation chi_sub:=(H(t)*dphi(t)/phidot + psi(t)); chi_sub:=(dphi(t) + psi(t)*phidot/H(t)); Gamma_sub:=1+(K-diff(H(t),t))/H(t)^2; #Define variable nu(t) we want to use NuDef:=nu(t)=chi_sub; #nu(t)=(dphi(t) + phidot/H(t)*psi(t))/sqrt(1-K/Delta*kappa/2*phidot^2/H(t)^2); #nu(t)=a(t)*(dphi(t) + phidot/H(t)*psi(t)); #nu(t)=((H(t)*dphi(t)/phidot + psi(t))-2*K/kappa/phidot^2*PsiN_sub)); #nu(t)=Delta*dphi(t)-kappa*phidot*Pipsi(t)/2/a(t)^2; #nu(t)=alpha*H(t)/phidot*(dphi(t)-kappa*phidot*Pipsi(t)/2/a(t)^2/Delta)+(psi(t)+H(t)*kappa/2/a(t)^2*Pipsi(t)/Delta); #nu(t)=a(t)*(dphi(t)-kappa*phidot*Pipsi(t)/2/a(t)^2/Delta); #nu(t)=simplify(-(K*PsiN_sub+H(t)^2*chi_sub)/(H(t)^2+K)+Delta/3/Gamma_sub/H(t)^2*PsiN_sub); #V3 comaptible coeff coef:= proc(X,Y) collect(X,Y); coeff(%,Y); RETURN(%); end; #background eqs: background:={diff(H(t),t)=-kappa*diff(phi(t),t)^2/2+H(t)^2 + K,diff(a(t),t)=a(t)*H(t)}; Ebackground:={diff(H(t),t)=-kappa*diff(phi(t),t)^2/2+H(t)^2 - K,diff(a(t),t)=a(t)*H(t)}; ##End useful definitions NuDef:=simplify(subs(background,NuDef)); #Do full thing in maple S2:=Pipsi(t)*diff(psi(t),t)+Piphi(t)*diff(dphi(t),t) - kappa/4/a(t)^2/Delta*(-K*Pipsi(t)^2+2*Delta*Piphi(t)^2/kappa) -kappa/2*phidot*Pipsi(t)*dphi(t)- a(t)^2/kappa*(Delta*psi(t)^2 - kappa*dphi(t)^2/2*(Delta-H(t)^2-diff(H(t),t) + phidddot/phidot)); #A integral: -H(t)*Pipsi(t) + phidot*Piphi(t) + 2*a(t)^2/kappa*(-Delta*psi(t) + kappa/2*(H(t)*phidot - phiddot)*dphi(t)); solve(%,psi(t)); psi(t):=simplify(%); solve(NuDef,dphi(t)); dphi(t):=simplify(%); simplify(S2); subs(background,%); subs(background,%); simplify(%); S2:=%; collect(S2,diff(Pipsi(t),t)); coeff(%,diff(Pipsi(t),t)); tmp:=simplify(%); collect(%,Pipsi(t)); coeff(%,Pipsi(t)); T1:=%; subs(Pipsi(t)=0,tmp); -diff(%,t)*Pipsi(t)-1/2*diff(T1,t)*Pipsi(t)^2; subs(diff(Pipsi(t),t)=0,S2)+%; subs(background,%); subs(background,%); simplify(%); S2:=%; collect(S2,diff(Piphi(t),t)); simplify(coeff(%,diff(Piphi(t),t))); subs(Piphi(t)=1,%); -1/2*diff(%,t)*Piphi(t)^2+subs(diff(Piphi(t),t)=0,S2); simplify(%); subs(background,%); subs(background,%); simplify(%); S2:=%; collect(S2,diff(nu(t),t)); coeff(%,diff(nu(t),t)); Pinu(t)=%; solve(%,Piphi(t)); Piphi(t):=simplify(%); simplify(S2); subs(background,%); subs(background,%); simplify(%); S2:=%; #Got rid of Pipsi OK! collect(S2,Pinu(t)); coeff(%,Pinu(t)^2); Pinu2:=simplify(%); collect(S2,Pinu(t)); coeff(%,Pinu(t)); Pinu1:=simplify(%); tmp:=S2; #Do gaussian integral subs(Pinu(t)=0,tmp) - Pinu1^2/4/Pinu2; tmp:=simplify(%); collect(tmp,diff(nu(t),t)); coeff(%,diff(nu(t),t)); subs(nu(t)=1,%); nudotnusurface:=simplify(%); nudotterm:=-1/2*diff(%,t); collect(tmp,diff(nu(t),t)); coeff(%,diff(nu(t),t)^2); %*diff(nu(t),t)^2 + nudotterm*nu(t)^2 + subs(diff(nu(t),t)=0,tmp); subs(background,%); subs(background,%); simplify(%); res:=%; #####res is the result for the action##### z_sub:= proc(X) subs(phidot=phidot_sub,X): simplify(%): subs(background,%): subs(background,%): simplify(%): subs(background,%): subs(background,%): simplify(%): subs(phidot=phidot_sub,%): simplify(%): subs(background,%): subs(phidot=phidot_sub,%): simplify(%): RETURN(%): end; zres:=z_sub(res); #put in terms of z #See what individual terms are collect(zres,nudot); coeff(%,nudot^2); simplify(%); subs(nu(t)=1,zres); zres2:=simplify(%); collect(%,zddot); coeff(%,zddot); simplify(%); subs(zddot=0,zres2); collect(%,zdot); simplify(coeff(%,zdot)); subs(zdot=0,zres2); simplify(%); #collect(zres,nudot); #coeff(%,nudot^2); #nudotnusurface*nu(t)^2/2+%*nudot*nu; ############End of general stuff######################### #Put in schodinger form, for Mukhanov nu subs(nu(t)=sqrt(1-K*z(t)^2/Delta)*Q(t),zres); Qres:=simplify(%); collect(%,diff(Q(t),t)); coeff(%,diff(Q(t),t)); subs(Q(t)=1,%); -1/2*diff(%,t); term:=%; collect(Qres,diff(Q(t),t)); coeff(%,diff(Q(t),t)^2); %*diff(Q(t),t)^2 + subs(diff(Q(t),t)=0,Qres) + term*Q(t)^2; Qres:=simplify(%); collect(Qres,diff(Q(t),t)); coeff(%,diff(Q(t),t)^2); simplify(%); subs(diff(Q(t),t)=0,Qres); subs(Q(t)=1,%); Q2terms:=%; collect(Q2terms,diff(diff(z(t),t),t)); simplify(coeff(%,diff(diff(z(t),t),t))); Z2:=%; subs(diff(diff(z(t),t),t)=0,Q2terms); collect(%,diff(z(t),t)); coeff(%,diff(z(t),t)); simplify(%); Z1:=%; subs(diff(diff(z(t),t),t)=0,Q2terms); collect(%,diff(z(t),t)); coeff(%,diff(z(t),t)^2); simplify(%); Z12:=%; subs(diff(z(t),t)=0,Q2terms); Z0:=simplify(%); #Got final answer in terms of z #Get in terms of phi for computation #U(t):=simplify(subs(z(t)=sqrt(kappa/2)*phidot/H(t),Q2terms)*2-diff(H(t),t)-H(t)^2); U(t)=simplify(subs(z(t)=sqrt(kappa/2)*phidot/H(t),Q2terms)*2); subs(background,%); subs(background,%); simplify(%); U(t):=%; ########### U(t); #Go Euclidean subs(K=-K,%); subs(Delta=-Delta,%); UEuc(t):=%; subs(phiddot=a(t)^2*Vdash -2*H(t)*phidot,UEuc(t)); subs(Ebackground,%); simplify(%); subs(H(t)=H,%); subs(phiddot=dotdotphi,%); subs(phidot=dotphi,%); subs(a(t)=a,%); subs(phi(t)=phi,%); collect(%,{dotphi,Delta,K,kappa,H}); fortran(%); #Get value at t=0 exp_subs:={phi(t)=phi0 + phi2*t^2/2 + phi4*t^4 + phi6*t^6,H(t)=-K*t + H3*t^3+H5*t^5,a(t)=1 - K*t^2/2 + a4*t^4}; V := V0+V2*t^2+V4*t^4; phiddot-a(t)^2*diff(V,t)/phidot + 2*H(t)*phidot; subs(exp_subs,%); collect(%,t); series(%,t,5); ser1:=%; coeff(%,t,0); solve(%,phi2); phi2:=%; phi2:=sqrt(2*V2); ser1:=collect(simplify(ser1),t); K-H(t)^2 - kappa/3*(-phidot^2/2 + a(t)^2*V); subs(exp_subs,%); collect(%,t); series(%,t,5); ser2:=%; coeff(collect(ser1,t),t^2); solve(%,phi4); phi4:=%; diff(a(t),t)/a(t)-H(t); subs(exp_subs,%); series(%,t); coeff(%,t^3); solve(%,a4); a4:=%; collect(ser2,t); coeff(%,t^4); simplify(%); solve(%,H3); H3:=%; diff(V,t)/phidot; subs(exp_subs,%); simplify(%); series(%,t); coeff(%,t,0); Vdash0=%; solve(%,V2); V2:=%; diff(diff(V,t)/phidot,t)/phidot; simplify(%); subs(exp_subs,%); simplify(%); series(%,t); coeff(%,t,0); Vddash0=%; solve(%,V4); V4:=simplify(%); #Get z sqrt(kappa/2)*phidot/H(t); subs(exp_subs,%); z:=collect(simplify(%),t); series(z,t); subs(K=kappa/3*V0,%); simplify(%); coeff(%,t,2); diff(z,t)/z/H(t); subs(exp_subs,%); subs(K=kappa/3*V0,%); series(%,t); tmp2:=%; coeff(%,t,0); simplify(%); #Get classical solution near origin p:=subs(exp_subs,-z^2*a(t)^2/(1-z^2*K/Delta)); U:=p*(2*K/(1-z^2*K/Delta)*tmp2+Delta+K); G:=G0+G1*t+G2*t^2; diff(p*diff(G,t),t) + U*G; series(%,t); Eq:=%; coeff(%,t,0); subs(K=kappa/3*V0,%); simplify(%); solve(%,G2); G2:=simplify(%); coeff(Eq,t,1); subs(K=kappa/3*V0,%); simplify(%); subs(exp_subs,UEuc(t)); simplify(%); series(%,t,5); coeff(%,t,0); simplify(%); tmp:=%; simplify(tmp); simplify(%,{K=kappa*m*phi0^2/3}); subs(phi0^2=3*K/kappa/m,%); simplify(%); fortran(%); ###################### ##relation between nu and psi_N (nu=bigQ) #Restart to do this... nu(t):=a(t)*(dphi(t) + phidot/H(t)*psi(t)); subs(dphi(t)=2/kappa*(diff(psi(t),t)+H(t)*psi(t))/phidot,%); nu_sub=%; solve(%,diff(psi(t),t)); psidot_sub:=%; diff(nu(t)/a(t)/phidot*H(t),t)*phidot^2/H(t); subs(dphi(t)=2/kappa*(diff(psi(t),t)+H(t)*psi(t))/phidot,%); subs(background,%); subs(diff(diff(psi(t),t),t)=-2*(H(t)-phiddot/phidot)*diff(psi(t),t)-2*(diff(H(t),t)-H(t)*phiddot/phidot)*psi(t) - (-Delta - K)*psi(t),%); subs(background,%); simplify(%); subs(diff(psi(t),t)=psidot_sub,%); simplify(%); X=%; solve(%,psi(t)); subs(X=diff(n(t)*H(t)/a(t)/phidot,t)*phidot^2/H(t),%); subs(nu_sub=n(t),%); subs(background,%); subs(n(t)=Q(t)*sqrt(1-K*kappa/2*phidot^2/H(t)^2/Delta),%); psi:=simplify(%); phidot_sub:=sqrt(2)*z(t)*H(t)/sqrt(kappa); subs(phidot=phidot_sub,psi); subs(background,%); subs(phidot=phidot_sub,%); subs(background,%); simplify(%); tmp:=%; collect(%,diff(Q(t),t)); coeff(%,diff(Q(t),t)); simplify(%); subs(diff(Q(t),t)=0,tmp); simplify(%) #check sqrt(kappa/2)*H(t)/a(t)/Delta/sqrt(1-K*z(t)^2/Delta)*(z(t)*diff(Q(t),t) - diff(z(t),t)*Q(t)/(1-K*z(t)^2/Delta) - z(t)*Q(t)*(H(t)+K/H(t)))-tmp; simplify(%); #OK ##For N... nu(t):=sqrt(2/kappa)*(H(t)*dphi(t)/phidot + psi(t)); subs(dphi(t)=2/kappa*(diff(psi(t),t)+H(t)*psi(t))/phidot,%); nu_sub=%; solve(%,diff(psi(t),t)); psidot_sub:=%; diff(nu(t),t); subs(dphi(t)=2/kappa*(diff(psi(t),t)+H(t)*psi(t))/phidot,%); subs(background,%); subs(diff(diff(psi(t),t),t)=-2*(H(t)-phiddot/phidot)*diff(psi(t),t)-2*(diff(H(t),t)-H(t)*phiddot/phidot)*psi(t) - (-Delta - K)*psi(t),%); subs(background,%); simplify(%); subs(diff(psi(t),t)=psidot_sub,%); simplify(%); dnu(t)=%; solve(%,psi(t)); simplify(%); subs(phidot=phidot_sub,%); simplify(%); psi_sub:=%; #Get Chi 1/(1+K/H(t)^2)*(psidot_sub/H(t) + psi_sub) + psi_sub*(1/(1+K/H(t)^2) - 1); simplify(%); subs(psi(t)=psi_sub,%); subs(phidot=phidot_sub,%); simplify(%); tmp:=%; psi_sub+2/3*tmp; simplify(%); subs(H(t)=H,%); subs(dnu(t)=yprime[5],%); subs(nu_sub=y[5],%); subs(z(t)=z,%); fortran(%); ###################End subs(phidot=2*z(t)*H(t),phidddot/phidot); subs(phidot=sqrt(2*Y(t)*H(t)^2),%); subs(diff(phi(t),t)=H(t)*z(t)/a(t),%); subs(background,%); subs(background,%); simplify(%); subs(diff(phi(t),t)=H(t)*z(t)/a(t),%); subs(background,%); simplify(%); %check K=0; subs(phidddot=phidddot_sub,%); subs(K=0,%): simplify(%); ############################################### #tests #See if get correct term for X bterm:=H(t)*diff(phi(t),t) - diff(diff(phi(t),t),t); -diff(beta(t),t)/2*a(t)^2*Gamma(t) + a(t)^2/2*beta(t)*diff(Gamma(t),t) - Gamma(t)^2/2 - a(t)^2/4/Delta*( bterm^2*beta(t)^2*a(t)^2 - 2*bterm*beta(t)*Gamma(t)*diff(phi(t),t) + Gamma(t)^2*diff(phi(t),t)^2/a(t)^2) + a(t)^4/2*beta(t)^2*(Delta-H(t)^2-diff(H(t),t)+diff(diff(diff(phi(t),t),t),t)/diff(phi(t),t)); simplify(%); subs(diff(a(t),t)=H(t)*a(t),%); subs(diff(H(t),t)=-diff(phi(t),t)^2/2+H(t)^2 + K,%); simplify(%); #Yes #coeff of v after int -H(t) - diff(Gamma(t),t)/Gamma(t) - a(t)^2/4/Delta*(-2*(H(t)*diff(phi(t),t) - diff(diff(phi(t),t),t))^2/Gamma(t)*beta(t) + 2*(H(t)*diff(phi(t),t)-diff(diff(phi(t),t),t))*diff(phi(t),t)/a(t)^2); ##Old stuff (junk) #Work #coeff of v before int: bterm:=H(t)*diff(phi(t),t) - diff(diff(phi(t),t),t); -H(t) - diff(Gamma(t),t)/Gamma(t) - a(t)^2/4/Delta*(-2*bterm^2/Gamma(t)*beta(t) + 2*bterm*diff(phi(t),t)/a(t)^2) -beta(t)/Gamma(t)*a(t)^2*(Delta-H(t)^2-diff(H(t),t)+phidddot/phidot); simplify(%); subs(background,%); simplify(%); A(t):=%; #Do integral. nu^2 term is diff(A(t),t)-A(t)*diff(X(t),t)/X(t) - A(t)^2; subs(background,%); simplify(%); %/4/X(t) - bterm^2/Gamma(t)^2/4/Delta + 1/Gamma(t)^2/2*(Delta-H(t)^2 - diff(H(t),t)+phidddot/phidot); subs(background,%); simplify(%); tmp:=%; subs(phidddot=phidddot_sub,tmp); simplify(%); \$check K=0 subs(K=0,%): simplify(%); subs(diff(Pinu(t),t)=0,tmp); collect(%,Pinu(t)); coeff(%,Pinu(t)); subs(diff(nu(t),t)=0,%); subs(nu(t)=1,%): simplify(%); A2:=%; tmp; subs(Pinu(t)=0,%); subs(diff(nu(t),t)=0,%); collect(%,nu(t)); coeff(%,nu(t)^2); nu2:=%; -(diff(A2,t)-A2*diff(X(t),t)/X(t) - A2^2)/4/X(t) + nu2: subs(background,%); subs(background,%); simplify(%); res:=%; #Change vars to q = nu/2/sqrt(X) -res* 4*X(t) - diff(diff(X(t),t)/2/X(t),t) + 1/4*diff(X(t),t)^2/X(t)^2; simplify(%); subs(background,%); subs(background,%); simplify(%); U(t):=%; subs(phidot=phidot_sub,U(t)); simplify(%): subs(background,%); subs(background,%); simplify(%); subs(background,%); subs(background,%); simplify(%); subs(phidot=phidot_sub,%); simplify(%); subs(background,%); subs(phidot=phidot_sub,%); simplify(%); Uz(t):=%; collect(Uz(t),diff(diff(z(t),t),t)); coeff(%,diff(diff(z(t),t),t)); Z2:=simplify(%); subs(diff(z(t),t)=0,Uz(t)); simplify(%); Z0:=%; collect(Uz(t),diff(z(t),t)); subs(diff(diff(z(t),t),t)=0,%); coeff(%,diff(z(t),t)); Z1:=simplify(%); collect(Uz(t),diff(z(t),t)); subs(diff(diff(z(t),t),t)=0,%); coeff(%,diff(z(t),t)^2); Z12:=simplify(%); Uz(t) - Z2*diff(diff(z(t),t),t)- Z1*diff(z(t),t) - Z0 - Z12*diff(z(t),t)^2; simplify(%); #so we can write it down moderately neatly