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GaussianQuad.py

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+from __future__ import division
+import numpy as np
+import matplotlib.pyplot as plt
+import scipy.integrate as integrate
+from scipy.integrate import ode
+from scipy.interpolate import UnivariateSpline
+from scipy.interpolate import interp1d
+
+c=299792458*100  # cm/s
+
+def nt(z): # cm^-3
+    return 56*(1+z)**3
+
+def L0(energy_nu, z, k=1, gamma=2, E_max=1.0e7):
+    return k*np.power(energy_nu,-gamma)*np.exp(-energy_nu/E_max)
+
+
+def W(z, a=3.4 , b=-0.3 , c1=-3.5 , B=5000 , C=9 , eta=-10):
+    return ((1+z)**(a*eta)+((1+z)/B)**(b*eta)+((1+z)/C)**(c1*eta))**(1/eta)
+
+def L(z, energy_nu):
+    return W(z)*L0(energy_nu, z)
+
+def H(z, H0=0.678/(9.777752*3.16*1e16), OM=0.308, OL=0.692):  # s^-1
+    return H0*np.sqrt(OM*(1.+z)**3. + OL)
+
+
+def sigma(energy_nu, g, M, m=1.e-10):  # cm^2
+    return (g**4/(16*np.pi))*(2*energy_nu*m)/((2*energy_nu*m-M**2)**2+((M**4*g**4)/(16*np.pi**2)))* 0.389379e-27
+
+def Adiabatic_Energy_Losses(z, energy_nu, nu_density):
+    return H(z)*nu_density
+
+def Attenuation(z, energy_nu, nu_density, g, M, m=1.e-10):
+    return -c*nt(z)*sigma(energy_nu, g, M, m)*nu_density
+
+
+
+def Regeneration(z, energy_nu, interp_nu_density, g, M, m=1.e-10):
+
+    def Integrand(z, energy_nu_in, energy_nu, interp_nu_density, g, M, m=1.e-10):
+        return interp_nu_density(energy_nu_in)*sigma(energy_nu_in, g, M, m=1.e-10)
+
+    regen = 0   
+    regen = integrate.fixed_quad(lambda energy_nu_in: Integrand(z, energy_nu_in, energy_nu, interp_nu_density, g, M, m=1.e-10), energy_nu, 1e14, n=5)[0]
+
+    regen=c*nt(z)*regen/(energy_nu)
+
+    return regen
+
+def Propagation_Eq(z, nu_density, energy_nu, lst_energy_nu, lst_nu_density, interp_nu_density, g, M, m=1.e-10):
+    rhs = 0
+
+    rhs += Adiabatic_Energy_Losses(z, energy_nu, nu_density)
+    rhs += L(z, energy_nu)
+    rhs += Attenuation(z, energy_nu, nu_density, g, M, m=1.e-10)
+    rhs += Regeneration(z, energy_nu, interp_nu_density, g, M, m=1.e-10)
+
+    print(Attenuation(z, energy_nu, nu_density, g, M, m=1.e-10)/Regeneration(z, energy_nu, interp_nu_density, g, M, m=1.e-10))
+
+    rhs = rhs/(-(1+z)*H(z))
+
+
+    return rhs
+
+
+def Neutrino_Flux(z_min, z_max, lst_energy_nu, g, M, m=1.e-10):
+
+    def Integrand(z, nu_density, energy_nu, interp_nu_density):
+        return Propagation_Eq(z, nu_density, energy_nu, lst_energy_nu, lst_nu_density, interp_nu_density, g, M, m=1.e-10)
+
+    solver = ode(Integrand, jac=None).set_integrator('dop853', atol=1.e-4, rtol=1.e-4, nsteps=500, max_step=1.e-3, verbosity=1)
+
+    lst_nu_density = [0.0]*len(lst_energy_nu)
+
+
+    dz = 1.e-1 
+
+    z = z_max
+
+    while (z > z_min):
+        lst_nu_density_new = np.zeros(lst_energy_nu.size)
+        #interp_nu_density = interp1d(lst_energy_nu, lst_nu_density, kind='linear', bounds_error=False, fill_value='extrapolate')
+        interp_nu_density = UnivariateSpline(lst_energy_nu, lst_nu_density, k=3, ext=0)
+       # print(interp_nu_density(lst_energy_nu))
+        for i in range(len(lst_energy_nu)):
+
+            solver.set_initial_value(lst_nu_density[i], z)
+            solver.set_f_params(lst_energy_nu[i], interp_nu_density)
+            sol = solver.integrate(solver.t-dz)
+
+            lst_nu_density_new[i]=sol
+            #print(lst_nu_density_new)
+
+        lst_nu_density = [x for x in lst_nu_density_new]
+
+        #print(lst_nu_density)
+        z = z-dz
+
+    return lst_nu_density
+
+
+log10_E_min = 2
+log10_E_max = 8
+E_npts = 100
+log10_E=np.power(10, np.linspace(log10_E_min, log10_E_max, E_npts))
+Flux = Neutrino_Flux(0, 4, log10_E, 0.03 , 0.01, 1e-10)
+Flux = [log10_E[i]*log10_E[i]*Flux[i] for i in range(len(log10_E))]
+norm = 1/Flux[0]
+Flux = [norm*nu_flux for nu_flux in Flux]
+
+plt.figure()
+plt.plot(log10_E,Flux)
+plt.xlabel('Neutrino energy $E[GeV]$')
+plt.ylabel('$ E^2 J \ [10^{-8}GeV \ cm^{-2} s^{-1} sr^{-1}]$')
+plt.xscale('log')
+#plt.ylim([0.0, 1.5])
+plt.xlim([10**3.0, 10**8.0])
+#plt.savefig('test.png')

SimpsonsRule.py

@@ -0,0 +1,126 @@
+from __future__ import division
+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.integrate import ode
+from scipy.interpolate import interp1d
+from scipy.interpolate import UnivariateSpline
+
+c=299792458*100  # cm/s
+
+def nt(z): # cm^-3
+    return 56*(1+z)**3
+
+def L0(energy_nu, z, k=1, gamma=2, E_max=1.0e7):
+    return k*np.power(energy_nu,-gamma)*np.exp(-energy_nu/E_max)
+
+
+def W(z, a=3.4 , b=-0.3 , c1=-3.5 , B=5000 , C=9 , eta=-10):
+    return ((1+z)**(a*eta)+((1+z)/B)**(b*eta)+((1+z)/C)**(c1*eta))**(1/eta)
+
+def L(z, energy_nu):
+    return W(z)*L0(energy_nu, z)
+
+def H(z, H0=0.678/(9.777752*3.16*1e16), OM=0.308, OL=0.692):  # s^-1
+    return H0*np.sqrt(OM*(1.+z)**3. + OL)
+
+def sigma(energy_nu, g, M, m=1.e-10):  # cm^2
+    return (g**4/(16*np.pi))*(2*energy_nu*m)/((2*energy_nu*m-M**2)**2+((M**4*g**4)/(16*np.pi**2)))* 0.389379e-27    
+
+def Adiabatic_Energy_Losses(z, energy_nu, nu_density):
+    return H(z)*nu_density
+
+def Attenuation(z, energy_nu, nu_density, g, M, m=1.e-10):
+    return -c*nt(z)*sigma(energy_nu, g, M, m)*nu_density
+
+def Regeneration(z, energy_nu, interp_nu_density, g, M, m=1.e-10):
+    regen = 0
+    energy_x_min = energy_nu
+    energy_x_max = 8
+    energy_x_npoints = 20
+    energy_x = np.linspace(energy_x_min, energy_x_max, energy_x_npoints)
+
+    delta_x = (energy_x_max - energy_x_min)/energy_x_npoints
+
+    regen += 10**energy_x[0]*interp_nu_density(energy_x[0])*sigma(10**energy_x[0], g, M, m)
+    for j in range(1, energy_x_npoints-2, 2):
+        regen += 4*10**energy_x[j]*interp_nu_density(energy_x[j])*sigma(10**energy_x[j], g, M, m)
+
+    for k in range(2, energy_x_npoints-2, 2):
+        regen += 2*10**energy_x[k]*interp_nu_density(energy_x[k])*sigma(10**energy_x[k], g, M, m)
+
+    regen += 10**energy_x[-1]*interp_nu_density(energy_x[-1])*sigma(10**energy_x[-1], g, M, m)
+
+    regen=delta_x*c*nt(z)*regen*np.log(10)/(3*10**energy_x_min)
+
+
+    return regen
+
+
+def Propagation_Eq(z, nu_density, energy_nu, lst_energy_nu, lst_nu_density, interp_nu_density, g, M, m=1.e-10):
+    rhs = 0
+
+    rhs += Adiabatic_Energy_Losses(z, 10**energy_nu, nu_density)
+    rhs += L(z, 10**energy_nu)
+    rhs += Attenuation(z, 10**energy_nu, nu_density, g, M, m=1.e-10)
+    rhs += Regeneration(z, energy_nu, interp_nu_density, g, M, m=1.e-10)
+
+    rhs = rhs/(-(1+z)*H(z))
+
+
+    return rhs
+
+def Neutrino_Flux(z_min, z_max, lst_energy_nu, g, M, m=1.e-10):
+
+    def Integrand(z, nu_density, energy_nu, interp_nu_density):
+        return Propagation_Eq(z, nu_density, energy_nu, lst_energy_nu, lst_nu_density, interp_nu_density, g, M, m=1.e-10)
+
+    solver = ode(Integrand, jac=None).set_integrator('dop853', atol=1.e-4, rtol=1.e-4, nsteps=500, max_step=1.e-3, verbosity=1)
+
+    lst_nu_density = [0.0]*len(lst_energy_nu)
+
+
+    dz = 1.e-1 
+
+    z = z_max
+
+    while (z > z_min):
+        lst_nu_density_new = np.zeros(lst_energy_nu.size)
+        #interp_nu_density = interp1d(lst_energy_nu, lst_nu_density, kind='linear', bounds_error=False, fill_value='extrapolate')
+        #print(interp_nu_density(lst_energy_nu))
+        interp_nu_density = UnivariateSpline(lst_energy_nu, lst_nu_density, k=3, ext=0)
+        for i in range(len(lst_energy_nu)):
+
+            solver.set_initial_value(lst_nu_density[i], z)
+            solver.set_f_params(lst_energy_nu[i], interp_nu_density)
+            sol = solver.integrate(solver.t-dz)
+
+            lst_nu_density_new[i]=sol
+            #print(lst_nu_density_new)
+
+        lst_nu_density = [x for x in lst_nu_density_new]
+
+        #print(lst_nu_density)
+        z = z-dz
+
+    return lst_nu_density
+
+log10_E_test_min = 2
+log10_E_test_max = 8
+E_test_npts = 150
+E_test=np.linspace(log10_E_test_min, log10_E_test_max, E_test_npts)
+#E_test=np.append(E_test, [5e3, 4.5e4, 5e5, 5e7 ])
+E_test=np.append(E_test, np.log10(5e5))
+E_test=np.sort(E_test)
+flux = Neutrino_Flux(0, 4, E_test, 0.03, 0.01, 1e-10)
+flux = [10**E_test[i]*10**E_test[i]*flux[i] for i in range(len(E_test))]
+norm = 1/flux[0]
+flux = [norm*nu_flux for nu_flux in flux]
+
+plt.figure()
+plt.plot(E_test,flux)
+plt.xlabel('Neutrino energy $E[GeV]$')
+plt.ylabel('$ E^2 J \ [10^{-8}GeV \ cm^{-2} s^{-1} sr^{-1}]$')
+#plt.ylim([0.0, 1.5])
+plt.xlim([3.0, 8.0])
+plt.savefig('test.png')
+#plt.xscale('log')

Trapz.py

@@ -0,0 +1,114 @@
+from __future__ import division
+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.integrate import ode
+from scipy.interpolate import interp1d
+
+c=299792458*100  # cm/s
+
+def nt(z): # cm^-3
+    return 56*(1+z)**3
+
+def L0(energy_nu, z, k=1, gamma=2, E_max=1.0e7):
+    return k*np.power(energy_nu,-gamma)*np.exp(-energy_nu/E_max)
+
+
+def W(z, a=3.4 , b=-0.3 , c1=-3.5 , B=5000 , C=9 , eta=-10):
+    return ((1+z)**(a*eta)+((1+z)/B)**(b*eta)+((1+z)/C)**(c1*eta))**(1/eta)
+
+def L(z, energy_nu):
+    return W(z)*L0(energy_nu, z)
+
+def H(z, H0=0.678/(9.777752*3.16*1e16), OM=0.308, OL=0.692):  # s^-1
+    return H0*np.sqrt(OM*(1.+z)**3. + OL)
+
+
+def sigma(energy_nu, g, M, m=1.e-10):  # cm^2
+    return (g**4/(16*np.pi))*(2*energy_nu*m)/((2*energy_nu*m-M**2)**2+((M**4*g**4)/(16*np.pi**2)))* 0.389379e-27
+
+
+def Adiabatic_Energy_Losses(z, energy_nu, nu_density):
+    return H(z)*nu_density
+
+def Attenuation(z, energy_nu, nu_density, g, M, m=1.e-10):
+    return -c*nt(z)*sigma(energy_nu, g, M, m)*nu_density
+
+
+
+def Regeneration(z, energy_nu, lst_energy_nu, lst_nu_density, g, M, m=1.e-10):
+    regen = 0
+    index = list(lst_energy_nu).index(energy_nu)
+    #print(index)
+    for j in range (index, len(lst_energy_nu)-1):
+        regen += sigma(lst_energy_nu[j], g, M, m)*lst_nu_density[j]*(lst_energy_nu[j+1]-lst_energy_nu[j])
+
+    regen=c*nt(z)*regen/(energy_nu)
+
+
+    return regen
+
+def Propagation_Eq(z, nu_density, energy_nu, lst_energy_nu, lst_nu_density, interp_nu_density, g, M, m=1.e-10):
+    rhs = 0
+
+    rhs += Adiabatic_Energy_Losses(z, energy_nu, nu_density)
+    rhs += L(z, energy_nu)
+    rhs += Attenuation(z, energy_nu, nu_density, g, M, m=1.e-10)
+    rhs += Regeneration(z, energy_nu, lst_energy_nu, lst_nu_density, g, M, m=1.e-10)
+
+    rhs = rhs/(-(1+z)*H(z))
+
+    return rhs
+
+def Neutrino_Flux(z_min, z_max, lst_energy_nu, g, M, m=1.e-10):
+
+    def Integrand(z, nu_density, energy_nu, interp_nu_density):
+        return Propagation_Eq(z, nu_density, energy_nu, lst_energy_nu, lst_nu_density, interp_nu_density, g, M, m=1.e-10)
+
+    solver = ode(Integrand, jac=None).set_integrator('dop853', atol=1.e-4, rtol=1.e-4, nsteps=500, max_step=1.e-3, verbosity=1)
+
+    lst_nu_density = [0.0]*len(lst_energy_nu)
+
+
+    dz = 1.e-1 
+
+    z = z_max
+
+    while (z > z_min):
+        lst_nu_density_new = np.zeros(lst_energy_nu.size)
+
+        for i in range(len(lst_energy_nu)):
+
+            solver.set_initial_value(lst_nu_density[i], z)
+            solver.set_f_params(lst_energy_nu[i], lst_nu_density)
+            sol = solver.integrate(solver.t-dz)
+
+            lst_nu_density_new[i]=sol
+            #print(lst_nu_density_new)
+
+        lst_nu_density = [x for x in lst_nu_density_new]
+
+        #print(lst_nu_density)
+        z = z-dz
+
+    return lst_nu_density
+
+log10_E_test_min = 2
+log10_E_test_max = 8
+E_test_npts = 100
+E_test=np.power(10, np.linspace(log10_E_test_min, log10_E_test_max, E_test_npts))
+#E_test=np.append(E_test, [5e3, 4.5e4, 5e5, 5e7 ])
+E_test=np.append(E_test, np.log10(5e5))
+E_test=np.sort(E_test)
+flux = Neutrino_Flux(0, 4, E_test, 0.01, 0.001, 1e-10)
+flux = [E_test[i]*E_test[i]*flux[i] for i in range(len(E_test))]
+norm = 1/flux[0]
+flux = [norm*nu_flux for nu_flux in flux]
+
+plt.figure()
+plt.plot(E_test,flux)
+plt.xlabel('Neutrino energy $E[GeV]$')
+plt.ylabel('$ E^2 J \ [10^{-8}GeV \ cm^{-2} s^{-1} sr^{-1}]$')
+plt.xscale('log')
+#plt.ylim([0.0, 1.5])
+plt.xlim([10**3.0, 10**8.0])
+plt.savefig('TestSimpleD.png')