The presence of the radial peculiar
velocities v is included in the statistical modelization
by rewriting the density probability of Eq.
(5 (click here))
where is the distribution function of radial peculiar velocities depending in general on the spatial position with (l,b) the direction of the line-of-sight in galactic coordinates.
The aim is herein to estimate the calibration parameters of the DTF relation entering into the radial peculiar velocity estimator of Eq (9 (click here)) (i.e. the DTF slope aD and DTF "zero-point" defined as ). The calibration sample is constituted of field galaxies selected along the same line-of-sight of direction (l,b) for which apparent magnitude m, log line-width distance indicator p and redshift z are measured. It is assumed hereafter that the sample is described by the density probability of Eq. (11 (click here)) and satisfies the following hypotheses:
Assumption implies that the radial peculiar velocity field is not correlated with the distance r. is thus ruled out if flows such like the Great Attractor are present along the line-of-sight. On the other hand, is less restrictive than the pure Hubble flow hypothesis (i.e. v=0 everywhere). First, galaxies may have a Maxwellian agitation of velocity dispersion . Second, the mean radial peculiar velocity along the line-of-sight u is not forced to zero. It allows to mimic the following situation. Suppose that a whole-sky sample is calibrated in a velocity frame of reference where sampled galaxies are not globally at rest (say that the sample has a bulk flow in cartesian galactic coordinates with respect to the velocity frame of reference). The radial peculiar velocity of galaxies belonging to the same line-of-sight of direction (l,b) will be shifted by . Assumption in fact tolerates this kind of situation.
Calibration of the DTF slope aD using null-correlation
approach (NCA) is based on the following remark.
For a calibration sample satisfying assumptions
(i.e. pure Hubble flow hypothesis),
the variable X=X(aD) defined as:
is not correlated with p (see Appendix C for proof):
where is the covariance of variables p and X. On the other hand, a wrong value of the DTF slope (i.e. ) correlates p and the random variable :
where is the square of the standard deviation of p. The null-correlation approach consists in adopting the value of the parameter a verifying as the correct estimate of the DTF slope aD:
Figure 1: Variation of the ratio R(r*) of selected object within the observed sample with respect to the extra cut-off in distance estimate r*
Figure 2: Variation of the bias inferred by the presence of a Maxwellian velocity agitation of dispersion with respect to the extra cut-off in distance estimate r* ()
which gives in practice:
Note that in the case of pure Hubble flow, the NCA is a quite robust technique for calibrating the slope of the DTF relation. It furnishes indeed an unbiased estimate of aD whatever the selection effects on m and p which affect the observed sample, and whatever the specific shape of the theoretical distribution function fp(p) of the variable p (see Appendix C).
the presence of radial peculiar velocities such as
the ones assumed in with or
biases the NCA estimate of the slope
aD since Eq. (13 (click here)) rewrites in this case
(see Appendix C):
However, the magnitude of this bias can be attenuated by selecting only distant galaxies of the observed sample. It is not surprising since the term in Eq. (17 (click here)) becomes negligible for distances r large enough. Such a subsampling can be performed by discarding galaxies of the observed sample which have a distance estimate smaller than a given r*. It corresponds to introducing an extra selection function defined as follows:
where is given Eq. (6 (click here)). Introducing this extra selection effect does not alterate the result obtained Eq. (13 (click here)) since this property is insensitive to the specific shape adopted for the selection function on m and p.
In order to evaluate amplitude of the bias appearing Eq. (17 (click here)) and its variation with respect to the cut-off in distance estimate r*, calculations have been performed on a synthetic sample characterized as follows:
Figure 1 shows variation of R(r*), i.e. the ratio of selected objects within the observed sample, with respect to the cut-off in distance estimate r* (analytical expression of R(r*) is given Eq. (C13)). On one hand, high values of r* are required in order to minimize as far as possible the amplitude of the bias created by radial peculiar velocities. On the other hand, accuracy of the NCA slope estimate depends on the size of the selected subsample (i.e. due to the intrinsic statistical fluctuations affecting the sample). Since theses two features are indeed competitive (i.e. R(r*) decreases when r* increases, compromise on the optimal value of r* has to be chosen with regard to the specific characteristics of the data sample under consideration.
The variations of term , i.e. the contribution
of a Maxwellian agitation of velocity dispersion
to the bias of Eq.
(19 (click here)), are illustrated Fig. 2.
Influence of on the NCA estimate
of the DTF slope aD given Eq.
(15 (click here)) is obtained by comparing
with the contribution
of to the covariance
of Eq. (14 (click here)).
For example, if the mean radial peculiar velocity
along the line-of-sight u is zero,
the magnitude of the bias on the NCA estimate of aD
created by a velocity field of dispersion
km s-1 is greater than
for (i.e. ),
for (i.e. )
and finally falls below
for (i.e. ).
Figure 2 reveals two important features.
- If nearby galaxies are not discarded from the observed sample, the presence of a Maxwellian velocity agitation for galaxies contaminates strongly the NCA estimate of the DTF slope aD (a velocity dispersion of 500 km s-1 induces a bias on aD of magnitude ).
- This bias can be rendered arbitrarily small by selecting only distant galaxies by means of the extra cut-off in distance estimate r*.
Figure 3: Variation of the bias inferred by the presence of a constant velocity u along the line-of-sight with respect to the extra cut-off in distance estimate r*
Figure 3 shows the variations of , i.e. the dominant term of the bias entering Eq. (19 (click here)) due to the presence of a mean radial peculiar velocity along the line-of-sight u, with respect to r*. A careful analysis of Fig. 3 leads to the three following remarks.
The term decreases a function of r* less rapidly as than . For comparable values of and at r*=0, say for u = -750 km s-1 and km s-1, bias falls to at r* =10 (to be compared with for bias), to at r* =20 ( for ) and finally to at r* =60 ( for ). It thus turns out that a particular attention has to be paid in priority to the presence of constant velocity fields.
As the existence of constant velocities strongly biases the NCA estimate of aD (at r*=0, for u = -500 km s-1), discarding nearby galaxies by means of distance estimate selection appears as a quite crucial step. Note however that the situation is not so stringent for samples affected by bulk flow. Since the bias is antisymmetric with respect to u (see Fig. 3), the bias on the NCA estimate of aD cancels in average if the opposite line-of-sight direction is also considered. By scanning the sky by line-of-sight directions, this interesting symmetry allows indeed to detect bulk flows already at the level of the calibration step.
Finally, some upper bounds on the aD bias created by large-scale coherent velocity fields can be extracted from analysis of Fig. 3. Suppose that the line-of-sight of the calibration sample points toward the direction of a Great Attractor, located at a distance estimate of Mpc and creating back-side infall velocities, say of amplitude u = -500 km s-1 at Mpc and slowly decreasing at larger distances. The bias on the NCA estimate of aD will be necessarily smaller than the bias for u = -500 km s-1 (i.e. at r*=50, ). This property is closely related to the subsampling in distance estimate allowed by the null-correlation approach. The peculiar velocity field, whatever its specific form, becomes negligible compared to the mean Hubble flow as long as the cut-off in distance estimate r* is large enough. In this case the null-correlation approach furnishes unbiased estimate of the DTF slope aD.
Assuming that the DTF slope aD has been correctly calibrated
by means of the technique presented in Sect.
3.1 (click here) or others calibration procedures,
the null-correlation approach is herein used for
calibrating the remaining calibration parameters entering
into the radial peculiar velocity estimator
of Eq. (9 (click here)).
For this purpose,
this equation is
rewritten as follows:
where depends on H0, bD and . For a calibration sample satisfying assumptions , , and , the radial peculiar velocity estimator is not correlated with p (see Appendix B for proof):
On the other hand, a wrong value of the B* parameter (i.e. ) correlates p and the random variable :
which does not vanish since selection effects on apparent magnitude m correlate variables p and (i.e. for selected galaxies, observable p increases in average with the distance estimate ). Assuming that the DTF slope aD has been correctly calibrated, the NCA estimate of the "zero-point" B* is then defined as:
which gives in practice:
NCA estimate of B* is clearly robust. It is insensitive to observational selection effects on m and p, specific shape of the luminosity function fp(p), constant velocity field and Maxwellian agitation of galaxies (see Appendix B). Note however that presence of non-constant large scale velocity fields (such GA flow for example) biases NCA estimate of B*. Unfortunatly the subsampling procedure in distance estimate previously proposed is not efficient for dealing with this kind of biases since the contribution of the peculiar velocity v to entering Eq. (24 (click here)) does not decrease with the distance r.
If one want to express the distance estimator
of Eq. (6 (click here)) in true distance units (Mpc),
the Hubble constant H0 has to be estimated (i.e.
For this purpose, estimates of the parameters B*
--by using NCA calibration for example--,
DTF zero-point bD
--using primary distance indicators--
and DTF intrinsic dispersion
--in galaxies clusters for example--
are required (i.e. ).
If B* is estimated with an error of ,
bD and with errors of
the dominant term of the relative error on the H0 estimate