The spiral field galaxies sample of
Mathewson et al. 1992 is a composite sample of
spiral galaxies lying in the field
(galaxies identified as cluster members have
been excluded of this catalog). It covers the south hemisphere
and extends in redshift up to km s-1 with
an effective depth of about km s-1.
Selection effects in observation are not trivial to model
since sampled galaxies have been
firstly selected in apparent diameter before Mathewson et al.
apply their own selection criteria (minimum limits in
inclination and velocity rotation). Some
galaxies inherited from others observational programs
have been also included in the sample.
The data used hereafter in the analysis are:
- the apparent magnitude where is the total I-band apparent magnitude corrected for internal and external extinction and K-dimming (Col. (6), second row in Mathewson et al. 1992).
- the log line-width distance indicator where is the maximum velocity of rotation of the spiral galaxy (Col. (9) in Mathewson et al. 1992).
- the redshift in km s-1 unit expressed in the CMB velocity frame (Col. (11), first row in Mathewson et al. 1992).
The application is presented as follows. In Sect. D.1 (click here) is explained how numerical simulations, used throughout the analysis for quantifying the amplitude of errors bars and velocity biases, are performed. Section D.2 (click here) is devoted to test on the calibration parameters proposed by Mathewson et al. 1992. Finally, NCA calibration of the Mathewson field galaxies sample is performed Sect. D.3 (click here).
Accuracies of the NCA calibration parameter estimators are herein calculated by using numerical simulations. Reliability of the values of such standard deviations is of course closely related to the way simulated samples succeed in reproducing the characteristics of the real sample under consideration. A preliminar study of the Mathewson field galaxies (MAT) sample characteristics is thus required.
On Fig. 4 (top left) is shown the decimal logarithm of the cumulative count in function of the apparent magnitude m for the MAT sample. It turns out that the completeness in magnitude of the MAT sample is violated beyond , corresponding to about 1/5 of the total number of sampled galaxies. Observational selection effects are then more complex than a mere cut-off in apparent magnitude. In order to mimic the real selection effects in observation affecting the MAT sample, simulations are built in two steps.
A virtual sample, complete in apparent magnitude up
to , is firstly
generated assuming the following characteristics:
- Variable p is generated according to a gaussian distribution function .
- Variable , accounting for the intrinsic scatter of the DTF relation, is generated according to a gaussian distribution function .
- The absolute magnitude is then formed, with aD and bD the slope and the zero-point of the DTF relation.
- Variable is generated according to an exponential distribution function and such that (i.e. distances are thus uniformly distributed in space).
- The redshift with H0 the Hubble constant in km s-1 Mpc-1 is finally formed according to the pure Hubble flow hypothesis.
Adopted values for the parameters p0, , , aD, bD and H0, as well as their corresponding "zero-points" B and B* introduced Eqs. (9 (click here), 20 (click here)), are given Table 1.
The next step is to extract from this virtual sample a subsamble of which has the same distribution in m and p than the observed distribution of the MAT sample. In effect this selection is achieved as follows. The m-p plane is divided in boxes box(i) of equal size and the number n(i) of MAT galaxies belonging to each box box(i) is memorized. Boxes box(i) are afterwards filled with galaxies belonging to the virtual sample while the observed n(i) are not reached. This selection procedure ensures that the m-p distribution of the simulated samples is approximately identical to the observed one. It corresponds to introduce a complex selection function in m and p directly derived from the data.
Figure 5: Accuracies of NCA estimators and amplitudes of velocity biases: (Top left) Ratio R(r*) of selected galaxies within the simulated samples function of the extra cut-off in distance estimate r*(aD,B*). (Top right) Standard deviation for the correlation coefficient and velocity biases created by GA, Bulk and Maxwellian flows. (Bottom left) Standard deviation for NCA slope estimator and velocity biases created by GA, Bulk and Maxwellian flows. (Bottom right) Standard deviation for NCA "zero-point" estimator and velocity biases created by GA, Bulk and Maxwellian flows
Figure 4 (top right) shows the cumulative count in apparent magnitude for the MAT sample and for a simulated sample. The cumulative count expected for a sample complete up to is also shown for comparison. Figures 4 (center left) and (center right) show respectively the distribution in the m-p plane of the MAT sample and a simulated sample. The difference between these two distributions are due to discretization effects which appear when applying the boxes algorithm. Figures 4 (bottom left) and (bottom right) show respectively the m-z distributions for the MAT sample and for a simulated sample. The fact that simulated sample distribution approximately reproduces the observed one is encouraging. It means that the working hypotheses assumed when generating simulated samples, such as the uniform spatial distribution of galaxies, are close to be verified by the Mathewson field galaxies sample. The likeness to data can certainly be improved, by choosing a more realistic shape for the luminosity function for example (i.e. the p distibution fp(p)), but is out of the scope of this study. Hereafter, these simulated samples will be considered as fair representatives of the observed catalog.
Figure 5 visualizes results obtained on these simulated samples. At the top left is plotted the ratio R(r*) of selected galaxies within a simulated sample when the subsampling in distance estimate (i.e. ) is applied. Standard statistical deviation of the correlation coefficient between p and the velocity estimates in function of the cut-off in distance estimate r* is shown in Fig. 5 (top right). Standard deviation (i.e. accuracy) of the NCA slope estimator and of the NCA "zero-point" are respectively presented Figs. 5 (bottom left) and (bottom right). If no peculiar velocity field is present (i.e. pure Hubble flow hypothesis, as it is the case for simulated samples), we see that and estimators are not biased, as it was previously proven appendices B and C (averaged over 1000 simulations, their values coincide with the input slope and "zero-point" of the simulated sample). It illustrates one of the potentialities of the null-correlation approach for calibrating TF like relations, i.e. its insentivity to observational selection effects in apparent magnitude m and log line-width distance indicator p. For these simulated samples supposed to mimic the Mathewson field galaxies sample (), accuracy of the NCA slope estimator sounds clearly good: or of aD if all the sample is selected, if half of the nearby galaxies of the sample are discarded (i.e. R(r*)=0.5 or km s-1) and at km s-1 (i.e. R(r*)=0.25) . The same remark holds for the NCA "zero-point" accuracy , or of B* at r*=0 and at km s-1.
Influence of peculiar velocity field on the NCA estimators is analysed using three examples: "Great Attractor" (GA) flow, constant or bulk flow and gaussian random or Maxwellian flow. In order to take into account the peculiarity of the 3D spatial distribution of the Mathewson field galaxies sample, biases created by these flows have been calculated by comparing the estimates of , and when one of these velocity field is added to the observed redshifts of the MAT sample, with these estimates for the real MAT sample. The Maxwellian flow has a velocity agitation of km s-1 and the bulk flow has been chosen to point toward direction l=310 and b=20 in galactic coordinates with an amplitude of 500 km s-1. The GA flow is the one of Bertschinger et al. (1988), centered at a distance of km s-1 toward l=310 and b=20 and creating an infall velocity for our Local group of 535 km s-1.
Figure 5 (bottom left) illustrates particularly well the discussion of Sect. 3.1 (click here) which was based on analytical results. Biases on the NCA slope estimate created by the presence of Maxwellian and Bulk flows become negligible when nearby galaxies of the MAT sample are discarded using the subsampling procedure in distance estimate (say herein for r* > 3000 km s-1). Influence of GA flow can be controled likewise. In practice, large values of the cut-off in distance estimate r* have to be preferred, of course with regards to the accuracy and to the r*-dependent statistical fluctuations affecting the NCA slope estimate at this distance r*.
Influences of velocity fields on and are shown respectively in Figs. 5 (bottom right) and (top right). As expected, Maxwellian flow does not bias these two quantities. Since the calibration of the simulated samples was not performed line-of-sight by line-of-sight, presence of Bulk flow slightly biases and . On the other hand, we can see that GA flow creates a significative bias on the two quantities (i.e. greater than their standard deviations). The fact that these biases vanish for large values of the cut-off in distance estimate r* is due to the specific form of the GA flow and to the characteristics of the 3D spatial distribution of the MAT sample. It cannot be interpreted as a general feature since the correlation between p and is not expected to vanish when the subsampling in distance estimate is applied, as it is the case for the p-X(aD) correlation. Some reasons may however be advocated for favouring large values of the cut-off in distance estimate r*. Since such a subsampling selects preferencially far away galaxies, a slighter coherence of their peculiar velocities is expected, consequently to their mutual distances. Finally, one remarks that amplitude of the bias created by huge flows such as the "Great Attractor" is not greater than , or of the value of B*. Same remark for the bias on the correlation coefficient which is less than whatever the value of the cut-off in distance estimate r*.
Figure 6: Correlation between p and Mathewson velocity estimates
Mathewson et al.1992 have proposed some values for the DTF calibration parameters. The authors calibrate the DTF slope aD in the Fornax cluster (14 galaxies). An estimate of is obtained by performing a linear regression in magnitude. Assuming that the true distance in km s-1 units of Fornax cluster is km s-1, authors derived a value of for DTF relative zero-point B. If H0=85 km s-1 Mpc-1, it implies a value of for the DTF zero-point bD. Averaging over the seven richest clusters of their catalog, a value of is estimated for the intrinsic scatter of the DTF relation, which gives a value of for the DTF relative "zero-point" B*. No error bars are proposed for these estimates.
Mathewson et al. DTF calibration parameters are tested in Fig. 6. Figure 6 (left) shows the correlation between p and Mathewson et al. peculiar velocity estimate for the of the MAT sample. Figure 6 (right) shows variations of with respect to the cut-off in distance estimate r* and the deviations from 0 expected when a flow such as the "Great Attractor" is present. It looks very unlikely that the strong correlation found between p and can be explained by the presence of large scale coherent peculiar velocity field. The same feature is observed for the p- correlation (not shown). This test leads to question in the validity of the calibration techniques used by Mathewson et al. when deriving slope and relative zero-point of the DTF relation.
As a matter of fact, it is known that the estimator of the DTF slope aD, obtained in a cluster by a linear regression on m, is biased by the presence of observational selection effects on m and p (see Lynden-Bell et al. 1988 for example). This bias can be corrected on in theory but a full description of selection effects in observation is thus required (see for example Willick 1994), and so is difficult to realize in practice. Same kind of remark holds for the relative zero-point estimate B. Since existing peculiar velocity field models proposed in the literature are far to be perfect nor accurate, a room of uncertainty remains when attributing a true distance to the calibration cluster (an error of 250 km s-1 for the assumed true distance of Fornax cluster will imply a relative error of for the relative zero-point B).
Figure 7: NCA calibration of the Mathewson spiral field galaxies sample: (Top left) Ratio R(r*) of selected galaxies within the MAT sample function of the extra cut-off in distance estimate . (Top right) Correlation coefficient between p and NCA peculiar velocity estimate . (Bottom left) NCA slope estimate function of the extra cut-off in distance estimate . (Bottom right) NCA relative "zero-point" estimate function of the extra cut-off in distance estimate
Figure 8: NCA "zero-point" estimate for remaining galaxies of the MAT sample when excluding the GA region (defined as a conic area pointing toward l=310 and b=20 with an angular aperture of ): (Left) Standard deviation for NCA "zero-point" estimator and velocity biases created by GA and Bulk flows. (Right) NCA "zero-point" estimate function of the extra cut-off in distance estimate
Figure 9: Top: Redshifts z versus NCA distance estimates in km s-1 for the MAT sample (left) Outside the GA region (right) Inside the GA region. Bottom: Peculiar velocity difference between Mathewson velocity estimates and NCA velocity estimates
NCA calibration is herein performed by following the discussions of Sect. D.1 (click here). The NCA estimate of the DTF slope aD is obtained from a subsample selected in distance estimate beyond km s-1, which corresponds to discard about of nearby galaxies of the MAT sample. NCA estimate of the DTF "zero-point" B* was achieved using galaxies beyond km s-1. Figure 7 shows the results of the NCA calibration of the Mathewson field galaxies catalog.
The value of NCA estimate of the DTF slope aD is given in Fig. 7 (bottom left): with an accuracy at km s-1 given by the numerical simulations of (or in other words ). Since this estimate is obtained with a large cut-off in distance estimate, it is in principle free of biases created by large scale peculiar velocity field. Adding to this point that NCA calibration technique is insentive to observational selection effects on apparent magnitude m and log line-width distance indicator p, NCA estimate of DTF slope aD appears as fairly secure.
NCA estimate of the relative "zero-point" B* is shown in Fig. 7 (bottom right). The value of has been arrested accounting for the more or less stable trend of beyond km s-1. To give a value of the accuracy is a little bit more tricky. It was previously mentioned that the subsampling procedure in distance estimate does not remove bias on estimator created by the presence of large scale peculiar velocity field (excepting the case of bulk flows). It turns out that the value of this bias depends on the specific geometry and amplitude of the real cosmic velocity field, and so cannot be estimated without modeling this velocity field. The amplitude of this bias for the "Great Attractor" flow model is less than (or about of B*) for the MAT sample. It means that if the GA flow is real, error on estimator is dominated by velocity bias rather than statistical fluctuations (accounting for the value of the accuracy on obtained from numerical simulations). At this stage, other criterion for calibrating DTF relative "zero-point" B* may be preferred, for example in constraining the average of radial peculiar velocity estimates over the sample to vanish. Unfortunatly such property is theoretically expected for fair samples (i.e. samples large enough to be representative at any scales of the kinematical fluctuations of the Universe), but not expected for calalogs such as the MAT sample.
Amplitude of velocity bias on the estimator may however be attenuated by removing of the sample areas presumed to have a strong kinematical activity. Such a procedure was achieved by discarding galaxies of the MAT sample belonging to a cone pointing toward l=310 and b=20 in galactic coordinates with an angular aperture of (the GA region). This subsample (i.e. MAT sample, GA region excluded) contains galaxies. Figure 8 (left) shows amplitudes of biases created by bulk and GA flow when the GA region is excluded of the MAT sample. Compared to the biases shown in Fig. 5 (bottom left) for the whole MAT sample, the discarding procedure looks efficient (if of course, the "Great Attractor" model of Bertschinger et al. 1988 succeeds in mimicking the real cosmic peculiar velocity field). Figure 8 (right) shows the NCA estimate for the MAT sample, GA region excluded. If the situation improves for km s-1 (compared to Fig. 6 (bottom left) showing for the whole MAT sample), a residual bias persists between km s-1 and km s-1. The value of the NCA relative "zero-point" has then to be read cautiously, keeping in mind that it can be affected by a velocity bias (anyway presumed not to be greater than ).
Finally, distance estimate and velocity estimate given respectively Eq. (6 (click here)) and Eq. (20 (click here)) can be inferred using the NCA calibration parameters and previously derived. Figure 9 (left) and (right) show respectively the redshift z versus the NCA predicted distance for galaxies of the MAT sample outside and inside the GA region. Figure 9 (bottom) illustrates the difference between the peculiar velocity estimates of Mathewson et al. 1992 and the ones derived in this present appendix. The averaged velocity difference by bins of redshift z is plotted function of the redshift. It turns out that an erroneous input value of the calibration parameters can interpret, especially at large redshifts, as fictitious large scale and coherent peculiar velocity flows. The preliminar calibration step of Tully-Fisher like relation is thus of crucial importance for kinematical studies.