radial_comoving_distance: number of steps too low.

[jecampagne]

Hi,
concerning radial_comoving_distance at low z (< 5e-2)

def radial_comoving_distance(cosmo, a, log10_amin=-3, steps=256)

the number of steps needs to be increased to 1024.

If one sets

cosmo_jax = Cosmology(
    Omega_c=0.2545,
    Omega_b=0.0485,
    h=0.682,
    Omega_k=0.0,
    w0=-1.0,
    wa=0.0  
###  other params if needed
)
z = jnp.logspace(-3, 3,100)
grad_lum = vmap(grad(luminosity_distance), in_axes=(None,0))(cosmo_jax,z2a(z))

# direct integration d(d_L)/dOmega_k(cosmo_jax)
func = lambda a: -0.5*const.rh * (-1.+1./a**2) / (a**2 * Esqr(cosmo_jax, a)**1.5)
dLdOmega_k = quadIntegral(func,z2a(z),1.0,ClenshawCurtisQuad(100))*(1+z) 

plt.figure(figsize=(10,10))
plt.plot(z,grad_lum.Omega_c/cosmo_jax.h,ls='-',lw=3,label=r"$\frac{\partial d_L}{\partial \Omega_c}$")
plt.plot(z,grad_lum.Omega_b/cosmo_jax.h,ls='--',lw=3, label=r"$\frac{\partial d_L}{\partial \Omega_b}$")
plt.plot(z,grad_lum.h/cosmo_jax.h,ls='--',lw=3, label=r"$\frac{\partial d_L}{\partial h}$")
plt.plot(z,grad_lum.Omega_k/cosmo_jax.h,ls='-',lw=3, label=r"$\frac{\partial d_L}{\partial \Omega_k}$")
plt.plot(z,grad_lum.w0/cosmo_jax.h,ls='--',lw=3, label=r"$\frac{\partial d_L}{\partial w_0}$")
plt.plot(z,grad_lum.wa/cosmo_jax.h,ls='--',lw=3, label=r"$\frac{\partial d_L}{\partial w_a}$")
plt.plot(z,dLdOmega_k/cosmo_jax.h, ls='--',lw=3,c='k',label=r"Direct Integ: $\frac{\partial d_L}{\partial \Omega_k}$")
plt.yscale("symlog")
plt.xscale("log")
plt.legend()
plt.grid();

leads to a perfect matching (black vs red curves)

While using 256 steps leads to a discrepancy (black vs red curves):