b[T_]:=((16 (G^2))/(π α))(2+((Cos[w])^2))((7(π^2)(T^5))/120);
Ω[M_,T_,θ_]:=(Sqrt[((θ^2)/((((1+((6T)/M)b[T]))^2)+(θ^2)))]);
α=(1/30);
w=(30/180)π;
G=(1/(((300000))^2));

Block[{M,θ},
M=500;θ=(10^(-5));
Plot[((Ω[M,T,θ])/θ),{T,0,(10^4)},AxesLabel->Automatic,PlotRange->All,PlotLabel->"((θ (T))/(Subscript[θ, 0]))",AxesLabel->{"Temperature, MeV","θ(T)/(Subscript[θ, 0])"}]
]

Plot3D[((Ω[m,T,θ])/θ)/.θ->(10^(-5)),{T,0,(10^4)},{m,100,500},PlotLabel->"((θ (T))/(Subscript[θ, 0]))",AxesLabel->{"Temperature, MeV","Mass, MeV","θ(T)/(Subscript[θ, 0])"}]

g[T_]:=Piecewise[{{10.75,T≤105},{14.25,T≤170},{61.75,T≤ 1000},{75,True}},T];
Plot[g[T],{T,1,2000},PlotRange->All,PlotLabel->"(Subscript[g, ×])"]
(Subscript[M, pl])[T_]:=((1.22 (10^22))/(1.66 (Sqrt[g[T]])));
γ[T_]:=(G^2)(θ^2)(Subscript[M, pl])[T];

Manipulate[Block[{θ},(
θ=(10^x);

RegionPlot[{
(m≥y T && (m^2)<(1/(γ[T]T)))/.T->(10^t),
((1/y)T<m<y T&&(m^2)<(1/(γ[T]T))&&(T^3)γ[T] <1)/.T->(10^t),
(m≤ (1/y)T&&(T^3) γ[T]<1)/.T->(10^t)
},{m,1,500},{t,4,-3},PlotLabel->"Decoupling particles",FrameLabel->{"Mass, MeV","(Subscript[log, 10])(T), MeV"},PlotLegends->{"Non-relativistic","Intermediate","Relativistic"},GridLines->{{135,490},{}}]
)],{{x,-4,"log(θ)"},-10,0,1,Appearance->"Labeled"},{{y,5,"regime factor"},1,10,1,Appearance->"Labeled"}]

SetDirectory[NotebookDirectory[]]
Export["decoupling_regimes_10^-2.png",DynamicModule[{x=-2,y=5},θ=(10^x);RegionPlot[{m≥y T&&(m^2)<(1/(γ[T] T))/. T->(10^t),(T/y)<m<y T&&(m^2)<(1/(γ[T] T))&&(T^3) (Subscript[M, pl])[T]<(1/((G^2) (θ^2)))/. T->(10^t),m≤(T/y)&&(T^3) (Subscript[M, pl])[T]<(1/((G^2) (θ^2)))/. T->(10^t)},{m,1,500},{t,4,-3},PlotLabel->"Decoupling particles",FrameLabel->{"Mass, MeV","\!\(\*SubscriptBox[\(log\), \(10\)]\)(T), MeV"},PlotLegends->{"Non-relativistic","Intermediate","Relativistic"},GridLines->{{135,490},{}}]]]

Plot[(1/200)((((m/T)))^(5/2))Γ[m](E^(-(m/T)))(E^(-((Γ[m](Subscript[M, pl])[T])/(2 (T^2)))))*(((Subscript[M, pl])[T])/(T^2))/.θs[3]/.{m->300,(Subscript[T, N])->1000},{T,0.01,300},PlotLabel->"Sterile neutrino correction to H",AxesLabel->{"T, MeV","((ΔH)/H)"},PlotRange->All]

(((1.2 (G^2)(Subscript[M, pl])[3]))^(-1/3))
((((G^2)(Subscript[M, pl])[3]))^(-1/3))

Manipulate[
RegionPlot[{
(m≥y T && (m^2)<(1/(γ[T]T)))/.T->(10^t),
((1/y)T<m<y T&&(m^2)<(1/(γ[T]T))&&(T^3) (Subscript[M, pl])[T]<(1/(G^2)))/.T->(10^t),
(m≤ (1/y)T&&(T^3) (Subscript[M, pl])[T]<(1/(G^2)))/.T->(10^t)
},{m,0,10},{t,2,-2},PlotLabel->"Decoupling particles",FrameLabel->{"Mass, MeV","(Subscript[log, 10])(T), MeV"},PlotLegends->{"Non-relativistic","Intermediate","Relativistic"},GridLines->{{135,490},{}}]
,{{y,5,"regime factor"},1,10,1,Appearance->"Labeled"}]