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Fig. 1: Schematic illustration of the origin of the difference Δqbias be-tween the average IR/radio ratio mea-sured using an IR- or a radio-selected sample. Within each group of curves with a similar observed flux density, the central one represents the SED of an object with an average IR/radio ratio. The dashed (dotted) curve indi-cate +3σ (-3σ) outliers to the IR-radio relation. Arbitrary observational limits in the radio (left) and MIR (right) window are marked with horizontal bars.





Fig. 2: Cumulative/probability distribution functions of q24,0 in a radio- (light grey) and an IR-selected (dark grey) sample of star forming ga-laxies, as well as in the union of the two (black curves). The intersection of the black curve with the 50% pro-bability line (dotted horizontal line) de-fines the median value of q24,0 in the three different samples. Red lines are best-fitting Gaussian distributions for the jointly selected samples.



The selection effects that are the topic of this page arise in flux limited samples when flux information from one of the selection bands is directly used in the computation of the quantity being studied. In the present case the critical quantity is the logarithmic IR/radio ratio q, but analogous effects need to be considered in the context of studies of the distribution of spectral indices at different radio frequencies (e.g. Kellermann 1964; Condon 1984), of X-ray to optical continuum slopes of AGN (Francis 1993) or of the M•-σ and M•-L relationships (Lauer et al. 2007).
Fig. 1 illustrates the origin of the selection effect: consider the left hand panel in which the IR-to-radio SEDs of three sources with different observed bolometric flux are distributed along the vertical axis. Each of these three SEDs splits into three branches at the peak of the SED, thereby schematically reflecting the range of observed IR/radio ratios (from top to bottom: 3σ radio-excess outlier — dashed line; average source — solid line; and 3σ IR-excess outlier — dotted line). If we impose the indicated selection threshold at 1.4 GHz (red line) the resulting sample will contain (i) all sources of the brightest class, regardless of their IR/radio ratio; (ii) the source with an average IR/radio ratio and the radio-excess source from objects of the intermediate class and; (iii) in the faintest bin only the radio-excess sources. Since the fainter sources are more abundant (as parametrized by the slope of the differential source counts dN/dS ~ Sβ, with β&thinsp> 0) this results in a surplus of radio-excess sources and consequently a low average IR/radio ratio in a radio-selected sample. The right hand side of Fig. 1 shows that an IR-selected sample is biased in the opposite sense, i.e. towards high IR/radio ratios.

The analytical expression for the difference between the average IR/radio ratio of IR- and radio-selected samples is (Kellermann 1964; Condon 1984; Francis 1993; Lauer et al. 2007):

Δqbias = ln(10) (β - 1) σq2

It thus depends on β, the power law index of the source counts, and on σq, which is the dispersion of the IR/radio relation. Note that this offset will occur regardless of the relative depth of the two involved bands. An estimate of the 'intrinsic' (i.e. unbiased) IR/radio ratio can be obtained by constructing the sample using an unrelated selection criterion like optical luminosity, mass or morphological type (Lauer et al. 2007).
Under the simplified assumption of Euclidian source counts (β = 2.5) the previous equation predicts the offset Δqbias ≈ 0.35 dex, found in our lowest redshift bins (cf. Fig. 2). It also makes a fair prediction of a shift of ~0.7 dex between the IR- and radio-selected sample at z ~ 1 if one accounts for the larger scatter and the finding that at faint fluxes IR (Chary et al. 2004; Papovich et al. 2004) and radio (Richards 2000; Fomalont et al. 2006; Bondi et al. 2008) source counts are sub-Euclidean.
Fig. 2 shows that apart from biasing the average IR/radio ratio, selection effects can also produce spurious evolution. Based on the radio-selected sample alone we would infer a decrease of the mean ⟨q⟩ out to z ~ 1.

As described in Sargent et al. 2010a, there is ample evidence hat the offset Δqbias not only occurs is present in our COSMOS data but also that it can reconcile most apparently discrepant measurements of mean IR/radio ratios in the literature. The one exception to this generally encouraging agreement are the highly inconsistent radio stacking results of Boyle et al. (2007), Beswick et al. (2008) and Garn & Alexander (2009) who have all studied the mean q24 as a function of IR flux. Garn & Alexander (2009) in particular pointed out that the field-to-field variation of the mean IR/radio ratio can be considerable. The prospects are good that the issue will soon be resolved with the aid of significantly deeper EVLA observations at the μJy level that will even directly detect the radio emission of the faintest 24 μm sources.


More information on selection biases affecting average IR/radio properties:
  • 'The VLA-COSMOS Perspective on the IR-Radio Relation. I. New Constraints on Selection Biases and the Non-Evolution of the IR/Radio Properties of Star Forming and AGN Galaxies at Intermediate and High Redshift'
    Sargent, M. T., et al. 2010, ApJS, 186, 341

  • 'No Evolution in the IR-Radio Relation for IR-Luminous Galaxies at z < 2 in the COSMOS Field'
    Sargent, M. T., et al. 2010, ApJL, 714, 190


Related Links & Literature:

  • Collaborators: Gianni Zamorani, Eva Schinnerer, and Eric Murphy.
  • 'The Spectra of Non-Thermal Radio Sources'
    Kellermann, K. I. 1964, ApJ, 140, 969
  • 'The continuum slopes and evolution of active galactic nuclei'
    Francis, P. J. 1993, ApJ, 407, 519
  • 'Selection Bias in Observing the Cosmological Evolution of the M•-σ and M•-L Relationships'
    Lauer, T. R., et al. 2007, ApJ, 670, 249

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M. Sargent :: April 2010