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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 initialize the Cosmological models
Step1:  initialize the ClusterAbundance object
Step2:  Binned approach
Compute the 2D integral
$$
N_{\alpha\beta}^{\rm predicted} = \Omega_s\int_{z_\alpha}^{z_{\alpha + 1}} dz\int_{\log_{10}M_\beta}^{\log_{10}M_{\beta + 1}}\frac{d^2V(z)}{dz d\Omega}\frac{dn(M,z)}{d\log_{10}M}d\log_{10}M
$$
Step3:  Un-binned approach
Compute $$\frac{d^2V(z_i)}{dz d\Omega}\frac{dn(M_i,z_i)}{d\log_{10}M_i}$$ for each masses and redshifts in a dark matter halo catalog
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Python Code:
#CCL cosmology
cosmo_ccl = ccl.Cosmology(Omega_c = 0.30711 - 0.048254, Omega_b = 0.048254, h = 0.677, sigma8 = 0.8822714165197718, n_s=0.96, Omega_k = 0, transfer_function='eisenstein_hu')
ccl_cosmo_set_high_prec (cosmo_ccl)
cosmo, dist, ps_lin, ps_nln, hmfunc = create_nc_obj (cosmo_ccl)
psf = hmfunc.peek_psf ()
Explanation: initialize the Cosmological models
End of explanation
#Numcosmo Cluster Abundance
#First we need to define the multiplicity function here we will use the tinker
mulf = nc.MultiplicityFuncTinker.new()
mulf.set_linear_interp (True)
mulf.set_mdef(nc.MultiplicityFuncMassDef.CRITICAL)
mulf.set_Delta(200)
#Second we need to construct a filtered power spectrum 
hmf = nc.HaloMassFunction.new(dist,psf,mulf)
hmf.set_area((0.25)*4*np.pi)
#Here we define the mass proxy in the first analysis is used the true mass and redshift of the clusters
cluster_m = nc.ClusterMass.new_from_name("NcClusterMassNodist{'lnM-min':<%20.15e>, 'lnM-max':<%20.15e>}" % (math.log(10)*np.log10(1e14),math.log(10)*np.log10(1e16)))
cluster_z = nc.ClusterRedshift.new_from_name("NcClusterRedshiftNodist{'z-min': <%20.15e>, 'z-max':<%20.15e>}" % (0.25,2))
ca = nc.ClusterAbundance.new(hmf,None)
mset = ncm.MSet.new_array([cosmo,cluster_m,cluster_z])
mset.pretty_log()
print('sigma8 = '+str(cosmo.sigma8(psf)))
#CCL Cluster Abundance
clc = cl_count.ClusterAbundance()
#choose the halo mass function and mass definition
massdef = ccl.halos.massdef.MassDef(200, 'critical', c_m_relation=None)
hmd = ccl.halos.hmfunc.MassFuncTinker08(cosmo_ccl, mass_def=massdef)
clc.set_cosmology(cosmo = cosmo_ccl, hmd = hmd, massdef = massdef)
clc.sky_area = (0.25)*4*np.pi
Explanation: initialize the ClusterAbundance object
End of explanation
t0 = time.time()
#Bins definition
z_nodes = np.linspace(0.25, 2, 8)
log10M_nodes = np.linspace(14, 14.8, 11)
lnM_nodes = log10M_nodes * math.log(10)
#Numcosmo cluster counts in the bins
Numcosmo_Abundance = []
ca.prepare (cosmo, cluster_z, cluster_m)
for zl, zu in zip (z_nodes, z_nodes[1:]):
    nc_bin_mass = []
    for lnMl, lnMu in zip (lnM_nodes, lnM_nodes[1:]):
        Pbin = ca.intp_bin_d2n(cosmo, cluster_z, cluster_m, [lnMl], [lnMu], None, [zl], [zu], None)
        nc_bin_mass.append(Pbin)
    Numcosmo_Abundance.append(nc_bin_mass)
#Numcosmo_Abundance[i][j] is the number of clusters in i-th z_bin and j-th mass_bin
t1 = time.time()
print('time (seconds) = ' + str(t1-t0))
#CCL cluster count in the bins
t0 = time.time()
CCL_Abundance_exact = clc.Cluster_Abundance_MZ(zbin_edges = z_nodes, proxybin_edges = log10M_nodes, method = 'exact')
t1 = time.time()
print('time (seconds) = ' + str(t1-t0))
#CCL_Abundance_exact[i][j] is the number of clusters in i-th z_bin and j-th mass_bin
for zl, zu, ccl_a_z, nc_a_z in zip(z_nodes, z_nodes[1:], CCL_Abundance_exact, Numcosmo_Abundance):
    for lnMl, lnMu, ccl_a, nc_a in zip (lnM_nodes, lnM_nodes[1:], ccl_a_z, nc_a_z):
        print("(% 16.9g % 16.9g) (% 16.9g % 16.9g) % 16.9g cmp %.2e" % (lnMl / math.log(10), lnMu / math.log(10), zl, zu, ccl_a, math.fabs (nc_a/ccl_a - 1.0)))
diff = []
for i in range(len(CCL_Abundance_exact)):
        diff.append(100*abs((Numcosmo_Abundance[i]-CCL_Abundance_exact[i])/Numcosmo_Abundance[i]))
plt.figure(figsize = (14,9))
plt.yscale('log')
z_ = [np.mean([z_nodes[i],z_nodes[i+1]]) for i in np.arange(len(z_nodes)-1)]
mass = [np.mean([10**log10M_nodes[i],10**log10M_nodes[i+1]]) for i in np.arange(len(log10M_nodes)-1)]
plt.tick_params(axis='both', which="both", labelsize= 15)
legend = []
for i in range(len(z_)):
        plt.scatter(mass, diff[:][i])
        legend.append(str(round(z_nodes[i],3))+'< z <'+str(round(z_nodes[i+1],3)))
plt.legend(legend,loc =4, fontsize = 10)
plt.grid(True, which="both" )
plt.ylabel('|bias| CCL exact method to Numcosmo $(\%)$', fontsize = 20)
plt.xlabel(r'$log_{10}(M)$', fontsize = 20)
plt.ylim(1e-9,1e-1)
plt.show()
#diff[i][j] is the percentual difference between Numcosmo and CCL cluster abundance in i-th z_bin and j-th mass_bin
Explanation: Binned approach
Compute the 2D integral
$$
N_{\alpha\beta}^{\rm predicted} = \Omega_s\int_{z_\alpha}^{z_{\alpha + 1}} dz\int_{\log_{10}M_\beta}^{\log_{10}M_{\beta + 1}}\frac{d^2V(z)}{dz d\Omega}\frac{dn(M,z)}{d\log_{10}M}d\log_{10}M
$$
End of explanation
#Generates cluster count catalog data
ncdata = nc.DataClusterNCount.new(ca,'NcClusterRedshiftNodist','NcClusterMassNodist')
rng = ncm.RNG.pool_get("example_ca_sampling");
ncdata.init_from_sampling(mset,(0.25)*4*np.pi,rng)
ncdata.catalog_save("ca_nodist_unbinned_numcosmo.fits", True)
ncdata_fits = fits.open('ca_nodist_unbinned_numcosmo.fits')
#ncdata_fits.info()
ncdata_data =  ncdata_fits[1].data
ncdata_Table = Table(ncdata_data)
ncdata_Table.columns
ncdata_Table.sort('Z_TRUE')
display(ncdata_Table)
#Using Numcosmo
t0 = time.time()
d2n = []
for i in ncdata_Table:
    d2n.append(ca.d2n(cosmo,cluster_z,cluster_m,i[1],i[0]))
d2n = np.array(d2n)
t1 = time.time()
print('time (seconds) = ' + str(t1-t0))
d2n
#Using CCL
t0 = time.time()
log10M = np.array(ncdata_Table[1][:]/np.log(10))
multiplicity_exact = clc.multiplicity_function_individual_MZ(z = np.array(ncdata_Table[0][:]), logm =log10M, method = 'exact')
multiplicity_exact *= clc.sky_area/math.log(10)
t1 = time.time()
print('time (seconds) = ' + str(t1-t0))
print(multiplicity_exact)
diff = abs(100*(multiplicity_exact-d2n)/d2n)
plt.figure(figsize = (14,7))
plt.yscale('log')
plt.tick_params(axis='both', which = 'major', labelsize= 15)
plt.scatter(ncdata_Table[0][:], diff, s = 0.4, c = log10M)
plt.colorbar()
plt.grid(True, which='both')
plt.xlabel(r'$redshift$', fontsize = 20)
plt.ylabel('|bias| CCL exact method to Numcosmo $(\%)$', fontsize = 20)
plt.ylim(10e-10,1)
Explanation: Un-binned approach
Compute $$\frac{d^2V(z_i)}{dz d\Omega}\frac{dn(M_i,z_i)}{d\log_{10}M_i}$$ for each masses and redshifts in a dark matter halo catalog
End of explanation
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