The streaming model

The integral

The streaming model maps the real-space correlation function and the pairwise velocity distribution to the redshift-space correlation function:

\[ 1 + \xi^s(s_\perp, s_\parallel) = aH \int_{-\infty}^{\infty} \bigl[1 + \xi(r)\bigr]\, \mathcal{P}\!\left(v_\text{los} \mid r_\perp, r_\parallel\right)\, \mathrm{d}y \]

It is formally exact given the true pair-conditional velocity PDF \(\mathcal{P}\): all the modelling approximation lives in \(\mathcal{P}\), which is where liulu puts its effort (see Velocity PDFs).

Geometry and parametrisation

liulu integrates over \(y\), the real-space line-of-sight separation, rather than over \(v_\text{los}\). With the line of sight along \(\hat z\) and a transverse separation \(s_\perp = r_\perp\):

quantity	definition
\(y\)	real-space parallel separation (integration variable)
\(r\)	\(\sqrt{s_\perp^2 + y^2}\)
\(v_\text{los}\)	\(aH\,(s_\parallel - y)\,\operatorname{sign}(y)\)
PDF call	\(\mathcal{P}(v_\text{los},\, r_\perp,\, r_\parallel{=}\lvert y\rvert)\)

The PDF is always evaluated with non-negative \(r_\parallel = |y|\); the model uses the symmetry \(\mathcal{P}(v\mid\mu) = \mathcal{P}(-v\mid-\mu)\) to fold the sign into \(v_\text{los}\). Callers passing \(r_\parallel\) to a VelocityPDF directly must therefore pass \(|y|\).

The integration range is \(y \in [-y_\text{max}, y_\text{max}]\); choose \(y_\text{max}\) large enough that the integrand has decayed (it must exceed the largest \(r\) the integrand samples, i.e. \(\sqrt{s_{\perp,\max}^2 + y_\text{max}^2}\) should cover where \(\xi\) and the velocity dispersions are non-negligible).

Multipoles

The redshift-space anisotropy is summarised by Legendre multipoles

\[ \xi_\ell(s) = \frac{2\ell+1}{2}\int_{-1}^{1} \xi^s(s,\mu)\, L_\ell(\mu)\,\mathrm{d}\mu , \]

computed by StreamingModel.multipoles with Gauss–Legendre quadrature in \(\mu\) (n_mu nodes). The monopole \(\xi_0\), quadrupole \(\xi_2\) and hexadecapole \(\xi_4\) are returned.

Resolution knobs

n_y (LOS quadrature points per half-interval) and n_mu (Legendre nodes) control accuracy vs cost. Defaults are n_y=600, n_mu=100. The Fingers-of-God dip in \(\xi_2\) at \(s\sim3\)–\(5\,\mathrm{Mpc}/h\) is sensitive to the LOS resolution: do not drop n_y below ~300 or the dip is under-resolved and any downstream fit/objective is biased.

What you supply

• A real-space \(\xi(r)\) as a RealSpaceCorrelation (e.g. TabulatedXi from a measurement, or LinearTheoryXi/HalofitXi from a \(P(k)\)).
• Pairwise velocity moments as a PairwiseVelocityMoments (typically TabulatedMoments).
• A VelocityPDF built from those moments.
• The constant \(aH = a\,H(z)\) in km/s/(Mpc/\(h\)).

See the Quickstart to put these together.