Evolución de las componentes del Universo

El modelo solo contempla las componentes

Energía Oscura 68.50% del total en el Universo

Materia 31.50% del total en el Universo*

Se tienen las siguientes Ecuaciones

$$H^2=\dfrac{8\pi G}{3}\rho_m + \dfrac{8\pi G}{3}\dfrac{\rho_m^2}{2\lambda} + \dfrac{8\pi G}{3}\rho_p + \dfrac{8\pi G}{3}\dfrac{\rho_p^2}{2\lambda}$$

donde $\rho_p,\: \rho_m$ son las densidades de energía para la ecuación de estado del phantom y la materia, respectivamente; $\lambda$ esta relaciondado con la tensión de la brana.

Definiendo nuevamente

$$1=\frac{8\pi G}{3H^{2}}\sum_{i}\left(\rho_{i}+\bar{\rho}_{i}\right),$$ y con la constricción de Friedmann de la forma $$1=\sum\left(x_{i}^{2}+y_{i}^{2}\right),$$ se definen $$x_{i}^{2}=\frac{8\pi G}{3H^{2}}\rho_{i},\\y_{i}^{2}=\frac{8\pi G}{3H^{2}}\bar{\rho}_{i}.$$ Estas dos ecuciones se derivan con respecto a los e-folding, es decir, respecto al tiempo conforme $N$; y después se integran con respecto a N, se obtiene

$$x_{i}=x_{i0}\dfrac{e^{-\frac{3}{2}\left(1+w_{i}\right)N}}{H}\\ y_{i}=y_{i0}\dfrac{e^{-\frac{3}{2}\left(1+\bar{w}_{i}\right)N}}{H}.$$

Se hace los mismo para la ecuación de Friedmann $$\left(H^{2}\right)^{\prime}=\frac{8\pi G}{3}\sum\left(\rho_{i}^{\prime}+\bar{\rho}_{i}^{\prime}\right),$$ Finalmente se obtiene la ecuación para $H$ $$H=\sqrt{\sum\left(x_{i0}^{2}e^{-3\left(1+w_{i}\right)N}+y_{i0}^{2}e^{-3\left(1+\bar{w}_{i}\right)N}+C\right)}.$$

Las ecuaciones para las componentes que no dependen de la brana, son

$$x_1=x_M=x_{0M}\dfrac{e^{-\frac{3}{2}N}}{H}\\ x_2=x_p=x_{0p}\dfrac{e^{\frac{3}{4}N}}{H}$$

Las ecuaciones para las componentes que dependen de la brana, son

$$y_{1}=y_{\bar{M}}=y_{\bar{M}0}\dfrac{e^{-3N}}{H}\\ y_{2}=y_{\bar{p}}=y_{\bar{p}0}\dfrac{e^{\frac{3}{2}N}}{H}$$

Lo cual debería de ser

$$x_1^2=x_M^2=\Omega_m=x_{M0}^2\dfrac{e^{-3N}}{H^2}=\Omega_{0M}\dfrac{e^{-3N}}{H^2}\\ x_2^2=x_p^2=\Omega_p=x_{0p}^2\dfrac{e^{\frac{3}{2}N}}{H^2}=\Omega_{0p}\dfrac{e^{\frac{3}{2}N}}{H^2}$$ $$y_{1}^2=y_{\bar{M}}^2=\bar{\Omega}_{M}=y_{0\bar{M}}^2\dfrac{e^{-6N}}{H^2}=\bar{\Omega}_{0M}\dfrac{e^{-6N}}{H^2}\\ y_{2}^2=y_{\bar{p}}^2=\bar{\Omega}_{p}=y_{0\bar{p}}\dfrac{e^{3N}}{H^2}=\bar{\Omega}_{0p}\dfrac{e^{3N}}{H^2}$$

Finalmente la ecuación para $H$

$$H=\sqrt{x_{0M}^2 e^{-3N}+x_{0p}^2 e^{\frac{3}{2}N}+ y_{0\bar{M}}^2 e^{-6N}+ y_{0\bar{p}}^2 e^{3N} +C}$$

donde $H=H(N)$ entonces $$H(0)=\sqrt{x_{0M}^2+x_{0p}^2+ y_{0\bar{M}}^2 + y_{0\bar{p}}^2 +C}$$ $$H(0)=\sqrt{1 + C}\;\rightarrow\;C=H(0)^2-1$$



In [14]:

    
function H(N::Float64,x0,z0)
    w0=0.315-x0
    y0=0.685-z0
    H0=1.0
    Hache = w0*exp(-3.0*N) + x0*exp(-6.0*N) + y0*exp(3.0*N/2.) + z0*exp(3.0*N)
    return Hache
end









    Out[14]:





H (generic function with 1 method)

Ahora una función para cada una de mis componentes



In [15]:

    
function f1(fH,w0,x0,z0,a,b,p)
    x=zeros(0)
    
    for N in a:p:b
        x_n = (w0-x0)*exp(-3.0*N)
        x_nn = x_n/fH(N,x0,z0)
        push!(x,x_nn)
#        push!(Nt,[i])
    end
    x
end









    Out[15]:





f1 (generic function with 1 method)



In [16]:

    
function f2(fH,x0,z0,a,b,p)
    x=zeros(0)
    
    for N in a:p:b
        x_n = x0*exp(-6.0*N)
        x_nn = x_n/H(N,x0,z0)
        push!(x,x_nn)
    end
    x
end









    Out[16]:





f2 (generic function with 1 method)



In [17]:

    
function f3(fH,y0,x0,z0,a,b,p)
    x=zeros(0)
    
    for N in a:p:b
        x_n = (y0-z0)*exp(3.0*N/2.0)
        x_nn = x_n/fH(N,x0,z0)
        push!(x,x_nn)
    end
    x
end









    Out[17]:





f3 (generic function with 1 method)



In [18]:

    
function f4(fH,z0,x0,a,b,p)
    x=zeros(0)
    
    for N in a:p:b
        x_n = z0*exp(3.0*N)
        x_nn = x_n/fH(N,x0,z0)
        push!(x,x_nn)
    end
    x
end









    Out[18]:





f4 (generic function with 1 method)

Se gráfican



In [19]:

    
## using Winston
using PyPlot
## using Gadfly
## using RDatasets



In [20]:

    
x0 = 0.0;
z0 =0.0;
a = -10.0;
b = 5.0;
p = 0.01;

ws = f1(H,0.315,x0,z0,a,b,p);
xs = f2(H,x0,z0,a,b,p);
ys = f3(H,0.685,x0,z0,a,b,p);
zs = f4(H,z0,x0,a,b,p);
ts = collect(a:p:b);

Función graficadora



In [71]:

    
function graficador(x₀::Float64,z₀::Float64,a::Float64,b::Float64,p::Float64,xlim1,xlim2,ylim1,ylim2,location::Int,name::ASCIIString)
    ws = f1(H,0.315,x₀,z₀,a,b,p);
    xs = f2(H,x₀,z₀,a,b,p);
    ys = f3(H,0.685,x₀,z₀,a,b,p);
    zs = f4(H,z₀,x₀,a,b,p);
    ts = collect(a:p:b);
    
    w₀ = abs(0.315-x₀)
    y₀ = abs(0.685-z₀)
    
    xlim(xlim1,xlim2)
    ylim(ylim1,ylim2)
    Ome1 = plot(ts,ws,"--",label=L"$\Omega_m$");
    Ome2 = plot(ts,xs,label=L"$\bar{\Omega}_m$");
    Ome3 = plot(ts,ys,"--",label=L"$\Omega_p$");
    Ome4 = plot(ts,zs,label=L"$\bar{\Omega}_p$");
    grid("on");
    legend(handles=[Ome1,Ome2,Ome3,Ome4],loc=location);
    xlabel(L"$N=log(a)$");
    ylabel(L"$\Omega$");
    axvline(0,color="gray")
    title(L"$\Omega_{0m}=$""$w₀"L"$,\;\bar{\Omega}_{0m}=$""$x₀"L"$,\;\Omega_{0p}=$""$y₀"L"$,\;\bar{\Omega}_{0p}=$""$z₀")
    savefig(name)
    return show()
end









    Out[71]:





graficador (generic function with 1 method)



In [73]:

    
# graficador(x₀::Float64,z₀::Float64,a::Float64,b::Float64,p::Float64,xlim1,xlim2,ylim1,ylim2,location::Int)
graficador(0.0,0.0,-5.,2.0,0.01,-5,2,0,1,6,"AD_01.pdf")



In [74]:

    
# graficador(x₀::Float64,z₀::Float64,a::Float64,b::Float64,p::Float64,xlim1,xlim2,ylim1,ylim2,location::Int)
graficador(1e-4,0.0,-5.0,2.0,0.01,-5,2,0,1,6,"AD_02.pdf")



In [75]:

    
# graficador(x₀::Float64,z₀::Float64,a::Float64,b::Float64,p::Float64,xlim1,xlim2,ylim1,ylim2,location::Int)
graficador(1e-3,0.0,-5.,2.0,0.01,-5,2,0,1,6,"AD_03.pdf")



In [76]:

    
# graficador(x₀::Float64,z₀::Float64,a::Float64,b::Float64,p::Float64,xlim1,xlim2,ylim1,ylim2,location::Int)
graficador(1e-2,0.0,-5.,2.0,0.01,-5,2,0,1,6,"AD_04.pdf")



In [77]:

    
# graficador(x₀::Float64,z₀::Float64,a::Float64,b::Float64,p::Float64,xlim1,xlim2,ylim1,ylim2,location::Int)
graficador(0.0,1e-3,-2.5,6.0,0.01,-2.5,6,0,1,6,"AD_05.pdf")



In [78]:

    
# graficador(x₀::Float64,z₀::Float64,a::Float64,b::Float64,p::Float64,xlim1,xlim2,ylim1,ylim2,location::Int)
graficador(0.0,1e-2,-2.5,4.0,0.01,-2.5,4,0,1,6,"AD_06.pdf")



In [79]:

    
# graficador(x₀::Float64,z₀::Float64,a::Float64,b::Float64,p::Float64,xlim1,xlim2,ylim1,ylim2,location::Int)
graficador(0.0,0.05,-2.5,4.0,0.01,-2.5,4,0,1,6,"AD_07.pdf")



In [80]:

    
# graficador(x₀::Float64,z₀::Float64,a::Float64,b::Float64,p::Float64,xlim1,xlim2,ylim1,ylim2,location::Int)
graficador(1e-3,1e-3,-4.0,6.0,0.01,-4.0,6,0,1,6,"AD_08.pdf")



In [81]:

    
# graficador(x₀::Float64,z₀::Float64,a::Float64,b::Float64,p::Float64,xlim1,xlim2,ylim1,ylim2,location::Int)
graficador(1e-2,1e-2,-3.0,4.0,0.01,-3,4,0,1,6,"AD_09.pdf")



In [82]:

    
# graficador(x₀::Float64,z₀::Float64,a::Float64,b::Float64,p::Float64,xlim1,xlim2,ylim1,ylim2,location::Int)
graficador(0.05,0.05,-2.5,4.0,0.01,-2.5,4,0,1,6,"AD_10.pdf")



In [83]:

    
# graficador(x₀::Float64,z₀::Float64,a::Float64,b::Float64,p::Float64,xlim1,xlim2,ylim1,ylim2,location::Int)
graficador(0.315,0.685,-1.0,1.0,0.01,-1,1,0,1,6,"AD_11.pdf")



In [ ]: