Survey Completeness

This page collects the key references for the completeness scan estimator implemented in completeness_scan().

The Johnston–Teodoro–Hendry series

Three companion papers develop the Tc/Tv completeness scan:

Paper

Citation

Key contribution

I

Johnston, Teodoro & Hendry (2007), MNRAS 376, 1757 (arXiv:astro-ph/0703040)

Extended Rauzy (2001) Tc/Tv to account for a bright magnitude limit in addition to the faint limit; introduced the full scan of m* so that Tc(m*) and Tv(m*) are computed as functions of the completeness threshold rather than as single numbers. Demonstrated on MGC, SDSS, and 2dFGRS — the original Rauzy formulation (faint limit only) gives spurious results for surveys with a non-negligible bright cut.

II

Teodoro, Johnston & Hendry (2010), MNRAS 405, 1187 (arXiv:1005.3888)

Adaptive S/N smoothing: instead of a fixed magnitude-redshift grid, the bin widths are adjusted dynamically so that each scan point contains the same information content (constant S/N = sqrt(N_bin)). Prevents shot-noise artefacts in sparse regions and false completeness detections. Survey-specific calibration is required; SDSS, MGC, and 2dFGRS thresholds are provided.

III

Johnston, Teodoro & Hendry (2012), MNRAS 421, 270 (arXiv:1107.0858)

Full error propagation for Tc and Tv; catalogue of characteristic incompleteness signatures (a drop in Tc/Tv followed by a peak at fainter magnitudes indicates missing objects in the affected range; over-completeness shows the opposite pattern). Framework extended to constrain pure luminosity evolution without assuming a parametric LF shape.

Background: Rauzy (2001)

Rauzy (2001), MNRAS 324, 51, introduced the ζ rank variable and the original Tc statistic (single faint-limit only). The Johnston–Teodoro–Hendry series builds directly on this foundation. The original single-threshold test is available as rauzy_completeness().

Method summary

For a flux-limited survey with faint limit m_lim and optional bright limit m_bright, the completeness scan proceeds as follows.

For each test threshold m*:

  1. Select the subsample S(m*) = {galaxies with m_bright ≤ m_app ≤ m*}.

  2. For each galaxy i in S(m*), form the associated set:

    \[J_i = \{j \mid z_j \le z_i \;\text{and}\; M_j \le m^* - \mu(z_i) \;\text{and}\; M_j \ge m_\mathrm{bright} - \mu(z_i)\}\]

    where μ(z) is the distance modulus. The bright-limit condition is omitted when m_bright is not specified.

  3. Compute the rank variable and ζ statistic (Rauzy 2001):

    \[R_i = \bigl|\{j \in J_i : M_j \le M_i\}\bigr|, \qquad \zeta_i = \frac{R_i}{|J_i| + 1} \in (0, 1)\]

    Under the null hypothesis of a complete sample, ζ_i ~ Uniform(0, 1).

  4. Compute the standardised statistics:

    \[T_c(m^*) = \frac{\bar{\zeta} - 0.5}{\sqrt{1/(12n)}}, \qquad T_v(m^*) = \frac{s^2_\zeta - 1/12}{\sqrt{1/(80n)}}\]

    where \(n = |S(m^*)|\) and both statistics are asymptotically N(0, 1) under H₀.

Significant negative Tc or Tv at threshold m* indicates that the sample is missing galaxies fainter than m* (incompleteness). The adaptive S/N smoothing (Paper II) replaces the cumulative scan with a sliding window of N_target galaxies to maintain constant detection sensitivity.

See also