Projected Density Profiles

Weak lensing never sees \(\rho(r)\) directly — it sees mass projected along the line of sight. Both NfwProfile and EinastoProfile implement the same two projected quantities from Halo Density Profiles’s \(\rho(r)\): the surface density and the excess surface density that a lensing survey actually measures.

\(\Sigma(R)\) and \(\Delta\Sigma(R)\) for the same mass-matched pair as Halo Density Profiles. Both track closely inside \(r_{200}\); Einasto’s extra mass beyond \(r_{200}\) shows up as a shallower \(\Sigma(R)\) tail and a \(\Delta\Sigma(R)\) that has not yet turned over at \(r_{200}\), unlike the NFW’s sharp truncation there.

The projection

The surface density is the line-of-sight integral of \(\rho\), and the excess surface density compares it to its own interior mean — \(\gamma_t = \Delta\Sigma/\Sigma_{\rm crit}\) is what a lensing survey actually fits:

\[ \Sigma(R) = \int_{-\infty}^{\infty} \rho\!\left(\sqrt{R^2+\chi^2}\right) d\chi, \qquad \Delta\Sigma(R) \equiv \bar\Sigma(<R) - \Sigma(R), \qquad \bar\Sigma(<R) = \frac{2}{R^2}\int_0^R \Sigma(R')\,R'\,dR'. \]

Substituting \(r=\sqrt{R^2+\chi^2}\) turns the \(\chi\)-integral above into the radial form it actually is — the Abel transform of \(\rho(r)\),

\[ \Sigma(R) = 2\int_R^\infty \frac{\rho(r)\,r}{\sqrt{r^2-R^2}}\,dr. \]

\(\Sigma(R)\) is a single, local integral. \(\bar\Sigma(<R)\) is not: it is a cumulative integral over \(\Sigma\) from \(0\) to \(R\), so getting \(\Delta\Sigma(R)\) at one radius numerically means having already integrated \(\Sigma\) over every smaller radius first — and doing that accurately on a tabulated grid is genuinely delicate. clenspy.utils.sigma_to_deltasigma_cumtrapz (the fallback used where no closed form exists, e.g. Einasto’s \(n\le3/2\) below) does this by cumulative trapezoidal quadrature in \(\ln R\), which implicitly assumes \(\Sigma\) is constant inward of the grid’s first point — accurate only once that first point is small enough relative to the profile’s scale, and capable of giving exactly \(0\) for a cored profile if it isn’t.

NFW and Einasto sidestep all of this: both classes have full analytical \(\Sigma\) and \(\Delta\Sigma\) (Wright & Brainerd 2000 for NFW; the Catalan series below for Einasto), so the cumulative numerical route above is never on the hot path for either — it exists only as the last-resort fallback. Both classes also expose mean_sigma, \(\bar\Sigma(<R)\), from its own closed form rather than reconstructed as \(\Sigma+\Delta\Sigma\) — the two routes agree to machine precision, checked in the example below.

NFW: the Wright & Brainerd (2000) closed forms

Both the truncated and untruncated Abel transform above have a closed form for NFW, so NfwProfile.sigma/deltasigma never touch the numerical fallback described above:

\[ \Sigma(R) = 2r_s\rho_s\,f(x), \qquad \Delta\Sigma(R) = r_s\rho_s\,g(x), \qquad x = \frac{R}{r_s}, \]

with \(f\), \(g\) the piecewise (inside/outside \(r_s\)) closed forms of Wright & Brainerd (2000). Both kernels are 0/0 at the \(x=1\) kink in their direct form; NfwProfile switches to a Taylor series in \(|x-1|\) there, and to a small-\(x\) series below \(x\sim3\times10^{-3}\) where the direct branches cancel catastrophically (both traps documented inline in _fNfw/_gNfw).

Einasto: a closed form, just not an elementary one

\(\Sigma\) and \(\Delta\Sigma\) do have a closed form for general \(n\) — it is just not the single elementary function Halo Density Profiles’s \(M_{\rm 3D}(r)\) gets from the incomplete gamma function. Closing a Mellin-Barnes contour integral of the line-of-sight projection instead gives an infinite residue series that is exact at every radius, for every \(n>0\) (proved by contour closure, checked against an independent numerical Mellin-Barnes integral and a brute-force Abel quadrature to \(10^{-40}\) — see Einasto Profile: Mathematics and Current Implementation) — not a truncated approximation standing in for a missing closed form:

\[ \Sigma(x) = \sqrt{\pi}\rho_0 h \Big[\sum_{k\ge1}A_k\,x^{k/n+1} +\sum_{j\ge0}S_j\,x^{2j}\Big], \qquad \Delta\Sigma(x) = \sqrt{\pi}\rho_0 h \Big[\sum_{k\ge1}D_k\,x^{k/n+1} +\sum_{j\ge1}T_j\,x^{2j}\Big], \qquad x=\frac{R}{h}, \]

\[ A_k = \frac{(-1)^k}{k!}\frac{\Gamma\!\big(-\tfrac12-\tfrac{k}{2n}\big)} {\Gamma\!\big(-\tfrac{k}{2n}\big)}, \quad D_k = -\frac{n+k}{3n+k}A_k, \qquad S_j = \frac{2n(-1)^j}{j!}\frac{\Gamma(n-2nj)}{\Gamma(\tfrac12-j)}, \quad T_j = -\frac{j}{j+1}S_j\;\;(T_0=0). \]

This is EinastoProfile.sigma/deltasigma’s dispatch for every non-anchor \(n\) (both above and below \(3/2\)), via EinastoLowN: the Retana-Montenegro et al. (2012) case-1 residue series above with resonance pairing where the two coefficient families collide, switching to an all-positive \(E_{\nu_k}\) representation only at \(z=(R/h)^{1/n}\) far beyond any physical radius for \(n>3/2\). \(n=1\) and \(n=1/2\) collapse this to genuinely elementary closed forms in modified Bessel functions.

See Einasto Profile: Mathematics and Current Implementation for the full per-regime dispatch and the small-\(R\) asymptotic used where the native series cancels catastrophically.

Example

# same nfw/einasto halos as above; the line-of-sight projection of rho(r).
R = np.array([0.1, 0.5, 1.0, 2.0])  # Mpc, projected radius
print("Sigma_NFW(R)          [Msun/Mpc^2] =", nfw.sigma(R))
print("Sigma_Einasto(R)      [Msun/Mpc^2] =", einasto.sigma(R))
print("DeltaSigma_NFW(R)     [Msun/Mpc^2] =", nfw.deltasigma(R))
print("DeltaSigma_Einasto(R) [Msun/Mpc^2] =", einasto.deltasigma(R))

# both profiles satisfy Sigmabar(<R) = Sigma(R) + DeltaSigma(R) identically,
# even though mean_sigma is evaluated from its own closed form, not this sum
nfw_check = nfw.mean_sigma(R) / (nfw.sigma(R) + nfw.deltasigma(R)) - 1.0
print("NFW Sigmabar consistency, max|rel| =", np.max(np.abs(nfw_check)))

Sigma_NFW(R)          [Msun/Mpc^2] = [1.91291224e+14 3.25021969e+13 1.11712030e+13 3.36400736e+12]
Sigma_Einasto(R)      [Msun/Mpc^2] = [1.97446111e+14 3.17987731e+13 9.66245761e+12 2.13438607e+12]
DeltaSigma_NFW(R)     [Msun/Mpc^2] = [8.51802198e+13 3.87203748e+13 2.00729059e+13 8.74034644e+12]
DeltaSigma_Einasto(R) [Msun/Mpc^2] = [8.89031956e+13 4.03108101e+13 2.08324780e+13 8.73234570e+12]
NFW Sigmabar consistency, max|rel| = 4.218847493575595e-15

See also: API Reference for the full clenspy.halo reference, Notation for the symbol table, Einasto Profile: Mathematics and Current Implementation for the Einasto profile’s derivations.