The presence of the radial peculiar
velocities v is included in the statistical modelization
by rewriting the density probability of Eq.
(5 (click here))
as follows:
where
is the distribution function of radial
peculiar velocities depending in general
on the spatial position
with
(l,b) the direction of the lineofsight 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 a^{D} and DTF "zeropoint" defined as ). The calibration sample is constituted of field galaxies selected along the same lineofsight of direction (l,b) for which apparent magnitude m, log linewidth 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 lineofsight. 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 lineofsight u is not forced to zero. It allows to mimic the following situation. Suppose that a wholesky 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 lineofsight of direction (l,b) will be shifted by . Assumption in fact tolerates this kind of situation.
Calibration of the DTF slope a^{D} using nullcorrelation
approach (NCA) is based on the following remark.
For a calibration sample satisfying assumptions
,
,
and
with
(i.e. pure Hubble flow hypothesis),
the variable X=X(a^{D}) 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 nullcorrelation approach consists in adopting
the value of the parameter a verifying
as the correct estimate
of the DTF slope a^{D}:
Figure 1:
Variation of the ratio R(r^{*}) of selected object within the observed
sample with respect to the extra cutoff 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 cutoff 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 a^{D} whatever the selection effects
on m and p which affect the observed sample,
and whatever the specific shape of the theoretical
distribution function f_{p}(p) of the variable p
(see Appendix C).
Unfortunately,
the presence of radial peculiar velocities such as
the ones assumed in with or
biases the NCA estimate of the slope
a^{D} 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 cutoff 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 cutoff 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 a^{D} 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 lineofsight u is zero,
the magnitude of the bias on the NCA estimate of a^{D}
created by a velocity field of dispersion
km s^{1} is greater than
for r^{*}=0,
falls to
for (i.e. ),
equalizes
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 a^{D} (a velocity dispersion of
500 km s^{1}
induces a bias on a^{D} of magnitude
).

This bias can be rendered arbitrarily small by selecting
only distant galaxies by means of the extra cutoff in
distance estimate r^{*}.
Figure 3:
Variation of the bias inferred by the presence
of a constant velocity u along the lineofsight
with respect to the extra cutoff 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 lineofsight 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 a^{D} (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 a^{D} cancels in average if the opposite lineofsight direction is also considered. By scanning the sky by lineofsight directions, this interesting symmetry allows indeed to detect bulk flows already at the level of the calibration step.
Finally, some upper bounds on the a^{D} bias created by largescale coherent velocity fields can be extracted from analysis of Fig. 3. Suppose that the lineofsight of the calibration sample points toward the direction of a Great Attractor, located at a distance estimate of Mpc and creating backside 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 a^{D} 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 nullcorrelation approach. The peculiar velocity field, whatever its specific form, becomes negligible compared to the mean Hubble flow as long as the cutoff in distance estimate r^{*} is large enough. In this case the nullcorrelation approach furnishes unbiased estimate of the DTF slope a^{D}.
Assuming that the DTF slope a^{D} has been correctly calibrated
by means of the technique presented in Sect.
3.1 (click here) or others calibration procedures,
the nullcorrelation 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
H_{0},
b^{D} 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 a^{D} has been correctly
calibrated, the NCA estimate of the "zeropoint" 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 f_{p}(p), constant velocity
field and Maxwellian agitation of galaxies (see Appendix B).
Note however that presence of
nonconstant 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 H_{0} has to be estimated (i.e.
).
For this purpose, estimates of the parameters B^{*}
by using NCA calibration for example,
DTF zeropoint b^{D}
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 ,
b^{D} and with errors of
and
respectively,
the dominant term of the relative error on the H_{0} estimate
reads: