def C_l_mu_single_ell(ell): Kp = np.geomspace(1e-4, 100, n_points) Mu = np.linspace(-0.9999, 0.9999, n_points) Z = np.geomspace(1e-4, z_r, n_points) mu, kp, z = np.meshgrid(Mu, Kp, Z) H_z = uf.H(Z) chi_z = uf.chi(Z) #chi_Z=chi(Z) tau_z = uf.tau_inst(Z) f_z = uf.f(Z) D_z = uf.D_1(Z) const = 1e12 * T_rad**2 * x**2 / cc.c_light_Mpc_s * ( sigma_T * rho_g0 / (mu_e * m_p))**2 / 8 / np.pi**2 #kp_norm=np.linalg.norm(Kp) #k_norm=np.linalg.norm(ell/chi_Z) k = ell / chi_z K = np.sqrt(k**2 + kp**2 - 2 * k * kp * mu) integral = sp.integrate.trapz(sp.integrate.trapz( sp.integrate.trapz(Mps_interpf(kp) * np.exp(-2 * tau_z) * f_z**2 * D_z**4 * H_z * (1 - mu**2) * (1 + z)**2 / chi_z**2, Mu, axis=0), Kp, axis=0), Z, axis=0) return integral
def powerspec_final(ell): z1, z2 = np.meshgrid(z, z) chi_z1 = chi(z1) chi_z2 = chi(z2) f_z1 = uf.f(z1) f_z2 = uf.f(z2) D_z1 = uf.D_1(z1) D_z2 = uf.D_1(z2) tau_z1 = tau(z1) tau_z2 = tau(z2) H_z = uf.H(z) chi_z = chi(z) tau_z = uf.tau_inst(z) f_z = uf.f(z) D_z = uf.D_1(z) Mps = np.array([]) for i in ell: Integrand = [ sp.integrate.trapz(powerspec(np.sqrt(j**2 + i**2 / chi_z**2)), z) for j in kpar ] Integral = sp.integrate.trapz(Integrand, kpar) #Integral=sp.integrate.trapz((1+z)**2/chi_z**2*powerspec(i/chi_z)*D_z**4*H_z*f_z**2*np.exp(-2*tau_z),z) #Integral=[sp.integrate.trapz(sp.integrate.trapz((1+z1)*(1+z2)*np.exp(-tau_z1) #*np.exp(-tau_z2)*2*f_z1*f_z2*D_z1**2*D_z2**2/chi_z1**2*powerspec(i/chi_z1),z,axis=0),z,axis=0)] Mps = np.append(Mps, Integral) return Mps
def C_l_integrand(ell, z): Kp = np.geomspace(1e-4, k_max, n_points) Mu = np.linspace(-0.9999, 0.9999, n_points) mu, kp = np.meshgrid(Mu, Kp) H_z = uf.H(z) chi_z = uf.chi(z) #chi_Z=chi(Z) tau_z = uf.tau_inst(z) f_z = uf.f(z) D_z = uf.D_1(z) const = 1e12 * T_rad**2 * x**2 / cc.c_light_Mpc_s * ( sigma_T * rho_g0 / (mu_e * m_p))**2 / 8 / np.pi**2 #kp_norm=np.linalg.norm(Kp) #k_norm=np.linalg.norm(ell/chi_Z) C_l = np.array([]) for i in ell: k = i / chi_z K = np.sqrt(np.abs(k**2 + kp**2 - 2 * k * kp * mu)) integral = [ const * sp.integrate.trapz(sp.integrate.trapz( k * (k - 2 * kp * mu) * (1 - mu**2) / K**2 * Mps_interpf(kp) * Mps_interpf(K) * (1 + z)**2 * np.exp(-2 * tau_z) * f_z**2 * D_z**4 / chi_z**2 * H_z, Mu, axis=0), Kp, axis=0) ] C_l = np.append(C_l, integral) return C_l
def vis_function(z): delta_z = 0.5 delta_y = 1.5 * np.sqrt(1 + z_r) * delta_z x = (1 / 2) * (1 + np.tanh((y(z_r) - y(z)) / delta_y)) #x_H=1/(1+np.exp(-(z-z_r)/delta_z)) vis_function = cc.c_light_Mpc_s * sigma_T * rho_g0 / mu_e / m_p * ( 1 + z)**2 * np.exp(-uf.tau_inst(z)) * (x) / H(z) return vis_function
def crosscorr_rec_coords( ell, z_i, y, delta_z ): #with the assumption that zi=z so no cos factor, and we have dz=redshift bin width=2*delta_z defined above n = n_points #number of points over which to integrate Kp_par = np.geomspace(1.e-6, .15, n) Kp_perp = np.geomspace(1.e-6, 1., n) Z_min = z_i - delta_z Z_max = z_i + delta_z Z = np.geomspace(Z_min, Z_max, n) kp_perp, kp_par, z = np.meshgrid(Kp_perp, Kp_par, Z) chi_z = chi(z) T_mean_zi = uf.T_mean(z_i) chi_zi = chi(z_i) chi_z = chi(z) f_zi = uf.f(z_i) f_z = uf.f(z) D_zi = uf.D_1(z_i) r_zi = uf.r(z_i) D_z = uf.D_1(z) H_zi = uf.H(z_i) tau_z = uf.tau_inst(z) kpar = y / uf.r(z) const = 1.e6 / ( 4. * np.pi**2 ) * T_rad * T_mean_zi**2 / cc.c_light_Mpc_s * f_zi * D_zi**2 * H_zi / ( chi_zi**2 * r_zi) / (1. + z_i) * x * (sigma_T * rho_g0 / (mu_e * m_p)) kperp = ell / chi_z k = np.sqrt(kperp**2 + kpar**2) kp = np.sqrt(kp_perp**2 + kp_par**2) cos_theta = kp_par / kp #cos_theta=u, theta is azimuthal angle between k' and z (line of sight) axis sin_theta = kp_perp / kp sin_gamma = kpar / k #gamma is measured between k and xy plane, i.e. elevation of k cos_gamma = kperp / k zeta = sin_gamma * cos_theta + cos_gamma * sin_theta k_dot_kp = kperp * kp_perp + kpar * kp_par * zeta K = np.sqrt(k**2 + kp**2 - 2 * k_dot_kp) theta_K = kpar / K / k**2 * ( k**2 - k_dot_kp) - kp * kperp * np.sqrt(1 - zeta**2) / k / K theta_kp = cos_theta rsd = 1. + f_zi * kpar**2 / k**2 I = theta_kp * (theta_kp / kp**2 + theta_K / K / kp) integrand_1 = f_z * D_z**2 * ( 1 + z) * np.exp(-tau_z) * rsd**2 * Mps_interpf(kp) * Mps_interpf( K) * theta_kp**2 / kp**2 integrand_2 = f_z * D_z**2 * ( 1 + z) * np.exp(-tau_z) * rsd**2 * Mps_interpf(kp) * Mps_interpf( K) * theta_kp * theta_K / kp / K integral_1 = const * sp.integrate.trapz(sp.integrate.trapz( sp.integrate.trapz(integrand_1, Kp_perp, axis=0), Kp_par, axis=0), Z, axis=0) integral_2 = const * sp.integrate.trapz(sp.integrate.trapz( sp.integrate.trapz(integrand_2, Kp_perp, axis=0), Kp_par, axis=0), Z, axis=0) return integral_1 + integral_2
def crosscorr_integral_y_rec_coords( ell, z_i, delta_z ): #This is not working out-don't try this for now. Just leave it. n = n_points #number of points over which to integrate Kp_par = np.geomspace(1.e-6, .15, n) Kp_perp = np.geomspace(1.e-6, 1., n) Z_min = z_i - delta_z Z_max = z_i + delta_z z = np.geomspace(Z_min, Z_max, n) Kpar = np.geomspace(1.e-6, .15, n) kp_perp, kp_par, kpar = np.meshgrid(Kp_perp, Kp_par, Kpar) chi_z = chi(z) T_mean_zi = uf.T_mean(z_i) chi_zi = chi(z_i) f_zi = uf.f(z_i) f_z = uf.f(z) D_zi = uf.D_1(z_i) r_zi = uf.r(z_i) D_z = uf.D_1(z) H_zi = uf.H(z_i) tau_z = uf.tau_inst(z) kpar = y / uf.r(z) const = 1.e6 / ( 8. * np.pi**3 ) * T_rad * T_mean_zi**2 / cc.c_light_Mpc_s * f_zi * D_zi**2 * H_zi / ( chi_zi**2) / (1. + z_i) * x * (sigma_T * rho_g0 / (mu_e * m_p)) kperp = ell / chi_zi k = np.sqrt(kperp**2 + kpar**2) kp = np.sqrt(kp_perp**2 + kp_par**2) cos_theta = kp_par / kp #cos_theta=u, theta is azimuthal angle between k' and z (line of sight) axis sin_theta = kp_perp / kp sin_gamma = kpar / k #gamma is measured between k and xy plane, i.e. elevation of k cos_gamma = kperp / k zeta = sin_gamma * cos_theta + cos_gamma * sin_theta k_dot_kp = kperp * kp_perp + kpar * kp_par * zeta K = np.sqrt(k**2 + kp**2 - 2 * k_dot_kp) theta_K = kpar / K / k**2 * ( k**2 - k_dot_kp) - kp * kperp * np.sqrt(1 - zeta**2) / k / K theta_kp = cos_theta rsd = 1. + f_zi * kpar**2 / k**2 I = theta_kp * (theta_kp / kp**2 + theta_K / K / kp) z_dep_integ = sp.integrate.trapz(f_z * D_z**2 * (1 + z) * np.exp(-tau_z), z) integrand_1 = rsd**2 * Mps_interpf(kp) * Mps_interpf( K) * theta_kp**2 / kp**2 * kp_perp integrand_2 = rsd**2 * Mps_interpf(kp) * Mps_interpf( K) * theta_kp * theta_K / kp / K * kp_perp integral_1 = const * z_dep_integ * sp.integrate.trapz(sp.integrate.trapz( sp.integrate.trapz(integrand_1, Kp_perp, axis=0), Kp_par, axis=0), Kpar, axis=0) integral_2 = const * z_dep_integ * sp.integrate.trapz(sp.integrate.trapz( sp.integrate.trapz(integrand_2, Kp_perp, axis=0), Kp_par, axis=0), Kpar, axis=0) return integral_1 + integral_2
def crosscorr_squeezedlim( ell, z_i, y, delta_z ): #with the assumption that zi=z so no cos factor, and we have dz=redshift bin width=2*delta_z defined above n = n_points #number of points over which to integrate #y=np.geomspace(1.,3000.,n) U = np.linspace(-.9999, .9999, n) Kp = np.geomspace(1.e-6, .1, n) Z_min = z_i - delta_z Z_max = z_i + delta_z z = np.geomspace(Z_min, Z_max, n) u, kp = np.meshgrid(U, Kp) T_mean_zi = uf.T_mean(z_i) chi_zi = chi(z_i) chi_z = chi(z) f_zi = uf.f(z_i) f_z = uf.f(z) D_zi = uf.D_1(z_i) r_zi = uf.r(z_i) D_z = uf.D_1(z) H_zi = uf.H(z_i) tau_z = uf.tau_inst(z) kpar = y / uf.r(z) const = 1.e6 / ( 4. * np.pi**2 ) * T_rad * T_mean_zi**2 / cc.c_light_Mpc_s * f_zi * D_zi**2 * H_zi / ( chi_zi**2 * r_zi) / (1. + z_i) * x * (sigma_T * rho_g0 / (mu_e * m_p)) #Cl=np.array([]) kp_perp = kp * np.sqrt(1 - u**2) kp_par = kp * u k_perp = ell / chi_zi k = np.sqrt(k_perp**2 + kpar**2) rsd = 1. + f_zi * kpar**2 / k**2 zeta = (kpar / k * u + k_perp / k * np.sqrt(1 - u**2)) k_dot_kp = k_perp * kp_perp + kpar * kp_par * zeta K = np.sqrt(k**2 + kp**2 - 2 * k_dot_kp) #theta_kp=kpar*zeta/k+k_perp*np.sqrt(np.abs(1-zeta**2))/k theta_K = kpar / K / k**2 * ( k**2 - k_dot_kp) - kp * k_perp * np.sqrt(1 - zeta**2) / k / K #print (theta_K.min(),theta_K.max()) theta_kp = u #theta_K=np.where(theta_K > 0, theta_K, 0) I = theta_kp * (theta_kp / kp**2 + theta_K / K / kp) z_integral = sp.integrate.trapz(f_z * D_z**2 * (1 + z) * np.exp(-tau_z), z) integrand_1 = z_integral * Mps_interpf(kp) * Mps_interpf( K) * rsd**2 * theta_kp**2 #+theta_K/K/kp)#-mu*kp*np.gradient(Mps_interpf(k),axis=0)) integrand_2 = z_integral * Mps_interpf(kp) * Mps_interpf( k) * rsd**2 * kp**2 * theta_kp * theta_K / kp / K integral_sing_1 = const * sp.integrate.trapz( sp.integrate.trapz(integrand_1, U, axis=0), Kp, axis=0) integral_sing_2 = const * sp.integrate.trapz( sp.integrate.trapz(integrand_2, U, axis=0), Kp, axis=0) #Cl=np.append(Cl,integral) return integral_sing_1 + integral_sing_2
def crosscorr_squeezed_integral_y(ell, z_i, delta_z): n = n_points #number of points over which to integrate #y=np.geomspace(1.,3000.,n) #Kpar=y/uf.r(z) Kpar = np.geomspace(1.e-6, .15, n) U = np.linspace(-.9999, .9999, n) Kp = np.geomspace(1.e-6, .1, n) Z_min = z_i - delta_z Z_max = z_i + delta_z z = np.geomspace(Z_min, Z_max, n) u, kp, kpar = np.meshgrid(U, Kp, Kpar) T_mean_zi = uf.T_mean(z_i) chi_zi = chi(z_i) chi_z = chi(z) f_zi = uf.f(z_i) f_z = uf.f(z) D_zi = uf.D_1(z_i) r_zi = uf.r(z_i) D_z = uf.D_1(z) H_zi = uf.H(z_i) tau_z = uf.tau_inst(z) const = 1.e6 / ( 8. * np.pi**3 ) * T_rad * T_mean_zi**2 / cc.c_light_Mpc_s * f_zi * D_zi**2 * H_zi / ( chi_zi**2) / (1. + z_i) * x * (sigma_T * rho_g0 / (mu_e * m_p)) #Cl=np.array([]) kp_perp = kp * np.sqrt(1 - u**2) kp_par = kp * u k_perp = ell / chi_zi k = np.sqrt(k_perp**2 + kpar**2) rsd = 1. + f_zi * kpar**2 / k**2 zeta = (kpar / k * u + k_perp / k * np.sqrt(1 - u**2)) k_dot_kp = k_perp * kp_perp + kpar * kp_par * zeta K = np.sqrt(k**2 + kp**2 - 2 * k_dot_kp) #theta_kp=kpar*zeta/k+k_perp*np.sqrt(np.abs(1-zeta**2))/k theta_K = kpar / K / k**2 * ( k**2 - k_dot_kp) - kp * k_perp * np.sqrt(1 - zeta**2) / k / K theta_kp = u #theta_K=np.where(theta_K > 0, theta_K, 0) I = theta_kp * (theta_kp / kp**2 + theta_K / K / kp) z_dep_integ = sp.integrate.trapz(f_z * D_z**2 * (1 + z) * np.exp(-tau_z), z) integrand = Mps_interpf(kp) * Mps_interpf( k ) * rsd**2 * kp**2 * I #+theta_K/K/kp)#-mu*kp*np.gradient(Mps_interpf(k),axis=0)) integral_sing = const * z_dep_integ * sp.integrate.trapz( sp.integrate.trapz( sp.integrate.trapz(integrand, U, axis=0), Kp, axis=0), Kpar, axis=0) #Cl=np.append(Cl,integral) return integral_sing
def C_l_quad_integrand(Mu, Kp, Z, ell): mu, kp, z = np.meshgrid(Mu, Kp, Z) H_z = uf.H(z) chi_z = chi(z) #chi_Z=chi(Z) tau_z = uf.tau_inst(z) f_z = uf.f(z) D_z = uf.D_1(z) k = ell / chi_z K = np.sqrt(k**2 + kp**2 - 2 * k * kp * mu) integrand = k * (k - 2 * kp * mu) * ( 1 - mu**2) / K**2 * Mps_interpf(kp) * Mps_interpf(K) * ( 1 + z)**2 * H_z * D_z**4 * f_z**2 * np.exp(-2 * tau_z) / chi_z**2 return integrand
def C_l_integrand(ell, z): H_z = uf.H(z) chi_z = chi(z) tau_z = uf.tau_inst(z) f_z = uf.f(z) D_z = uf.D_1(z) const = 1e12 * 8 * np.pi**2 * T_rad**2 * x**2 * ( sigma_T * rho_g0 / (mu_e * m_p))**2 / cc.c_light_Mpc_s integrand = const * (1 + z)**2 * np.exp( -2 * tau_z) * chi_z * f_z**2 * H_z * D_z**4 * Delta_b_noredshift( ell / chi_z) / (2 * ell + 1)**3 ##print (np.shape(integrand)) #integrand=np.resize(integrand,(n_points+1)) ##print (np.shape(integrand)) return integrand
def C_l_mu_integral_threeints(ell, z_min): Kp = np.geomspace(1.e-10, 10., n_points) Mu = np.linspace(-0.9999, 0.9999, n_points) Phi = np.geomspace(1.e-4, 2 * np.pi, n_points) Z = np.geomspace(z_min, z_r, n_points) mu, phi, kp, z = np.meshgrid(Mu, Phi, Kp, Z) H_z = uf.H(z) chi_z = uf.chi(z) #chi_Z=chi(Z) tau_z = uf.tau_inst(z) f_z = uf.f(z) D_z = uf.D_1(z) const = 1e12 * T_rad**2 * x**2 / cc.c_light_Mpc_s * ( sigma_T * rho_g0 / (mu_e * m_p))**2 / 16 / np.pi**3 #kp_norm=np.linalg.norm(Kp) #k_norm=np.linalg.norm(ell/chi_Z) theta_kp_arr = np.array([]) theta_K_arr = np.array([]) I_arr = np.array([]) C_l = np.array([]) for i in ell: k = i / chi_z K = np.sqrt(k**2 + kp**2 - 2 * k * kp * mu * np.cos(phi)) theta_kp = np.sqrt(1 - mu**2) theta_K = -np.sqrt(1 - mu**2) * kp / K theta_kp_arr = np.append(theta_kp_arr, theta_kp) theta_K_arr = np.append(theta_K_arr, theta_K) I = theta_kp**2 / kp**2 + theta_K * theta_kp / K / kp I_arr = np.append(I_arr, I) #I=k*(k-2*kp*mu)*(1-mu**2)/K**2 integral = [ const * sp.integrate.trapz(sp.integrate.trapz(sp.integrate.trapz( sp.integrate.trapz( kp**2 * I * Mps_interpf(kp) * Mps_interpf(K) * (1 + z)**2 * np.exp(-2 * tau_z) * f_z**2 * D_z**4 / chi_z**2 * H_z, Mu, axis=0), Phi, axis=0), Kp, axis=0), Z, axis=0) ] C_l = np.append(C_l, integral) return C_l, theta_kp_arr, theta_K_arr, I_arr
def C_l_mu_integral_squeezed(ell, z_min): Kp = np.geomspace(1.e-10, .1, n_points) Mu = np.linspace(-0.9999, 0.9999, n_points) Z = np.geomspace(z_min, z_r, n_points) mu, kp, z = np.meshgrid(Mu, Kp, Z) H_z = uf.H(z) chi_z = uf.chi(z) #chi_Z=chi(Z) tau_z = uf.tau_inst(z) f_z = uf.f(z) D_z = uf.D_1(z) const = 1e12 * T_rad**2 * x**2 / cc.c_light_Mpc_s * ( sigma_T * rho_g0 / (mu_e * m_p))**2 / 8 / np.pi**2 #kp_norm=np.linalg.norm(Kp) #k_norm=np.linalg.norm(ell/chi_Z) C_l = np.array([]) for i in ell: k = i / chi_z #K=np.sqrt(k**2+kp**2-2*k*kp*mu) #original K K = np.sqrt(k**2 + kp**2 - 2 * k * kp * (np.sqrt(1 - mu**2))) #changed K, correct one #I=(1-mu**2) #in squeezed with original #I=k*(k-2*kp*mu)*(1-mu**2)/K**2 #original I, without squeezed #I=k*(k-2*kp*np.sqrt(1-mu**2))*(mu**2)/K**2 #I, without squeezed, changed, correct one I = mu**2 #I, with squeezed, changed, correct one integral = [ const * sp.integrate.trapz(sp.integrate.trapz(sp.integrate.trapz( I * Mps_interpf(kp) * Mps_interpf(k) * (1 + z)**2 * np.exp(-2 * tau_z) * f_z**2 * D_z**4 / chi_z**2 * H_z, Mu, axis=0), Kp, axis=0), Z, axis=0) ] C_l = np.append(C_l, integral) return C_l
def crosscorr_squeezedlim(ell, y): #with the assumption that zi=z so no cos factor z_i = 1. delta_z = 0.3 n = 100 Mu = np.linspace(-0.9999, 0.9999, n) Kp = np.linspace(1.e-4, 10., n) mu, kp = np.meshgrid(Mu, Kp) z = z_i T_mean_zi = uf.T_mean(z_i) chi_zi = chi(z_i) chi_z = chi(z) f_zi = uf.f(z_i) f_z = uf.f(z) D_zi = uf.D_1(z_i) r_zi = uf.r(z_i) D_z = uf.D_1(z) H_zi = uf.H(z_i) tau_z = uf.tau_inst(z) const = 1.e6 / ( 2. * np.pi )**2 * T_rad * T_mean_zi / cc.c_light_Mpc_s * f_zi * D_zi**2 * H_zi / ( chi_zi**2 * r_zi) / (1. + z_i) * x * ( sigma_T * rho_g0 / (mu_e * m_p)) * delta_z * f_z * D_z**2 * (1 + z) * np.exp(-tau_z) #Cl=np.array([]) kpar = y / r_zi k_perp = ell / chi_zi k = np.sqrt(k_perp**2 + kpar**2) rsd = 1. + f_zi * kpar**2 / k**2 theta_kp = kpar * mu / k + k_perp * np.sqrt(1. - mu**2) / k integrand = Mps_interpf(kp) * rsd * theta_kp**2 * ( Mps_interpf(k)) #-mu*kp*np.gradient(Mps_interpf(k),axis=0)) integral_sing = const * sp.integrate.trapz( sp.integrate.trapz(integrand, Mu, axis=0), Kp, axis=0) #Cl=np.append(Cl,integral) return integral_sing