Section 4
4. DEVIATIONS FROM HOMOGENEITY: THE PECULIAR VELOCITY
FIELD
4.1 INTRODUCTION
In addition to the effects on the propagation of light rays and the gravitational lensing
effects (Sect. 2), the cosmological Large Scale Structure is supposed to leave imprints
on the velocity fields of galaxies and other tracers. Indeed, the general homogeneity
and isotropy of the Universe, and the Hubble law itself have only validity if
considered on the largest spatial scales. On smaller scales, the matter distribution
shows fluctuations that we have quantified in Sect. 1. And the Hubble flow itself
reveals perturbations that go under the name of peculiar velocities. Systematic
measurements of the latter offer a fundamental cosmological observable.
If we consider the spatial scales over which we expect the Ω Λ parameter operates,
they are only of the order or larger than many hundreds or thousands Mps, the
extremely large scales. As we know from Sect.1, these are not the scales where large
matter fluctuations are present. In conclusion it is unlikely that the Ω Λ parameter
strongly affects the cosmic peculiar velocity field on the scales of interest in a direct
way (it does it indirectly by modifying the structure formation history, as we
discussed in Sect. 3.4). On the other hand, it is very likely that this is influenced by
the the Ω m parameter, and studies of peculiar velocities can constrain it. Notice that
the information that we will get from peculiar motions will be weakly affected by the
cosmological bias problem that instead influences other analyses based on the
observed galaxy distributions (Sect. 1).
The knowledge and measurements of the velocity fields are also important for many
other reasons, like for example correcting the distance determinations based on
redshift and the Hubble law with a large-scale model for the peculiar velocity flow.
4.2 PECULIAR MOTIONS FROM THE HUBBLE LAW AND A 3D
MAPPING OF THE UNIVERSE
As it is well known, the spectral lines observed in the spectrum of the Sun show
changes in wavelength due to the combined motion of the observer relative to solar
photosphere, and due to the motion of rotation of the Earth around its axis and the
The Peculiar Velocity Field
4. 1
rotation of the Sun itself. Observing other stars or galaxies, the spectral lines will
reflect their relative motions to us. These motions are e.g. well measured
components in the Local Group of galaxies, where they appear with a few exceptions
shifts toward the blue (approaching motions). All these line shifts are due to the
Doppler special relativistic effect, due to the relative motion of source and observer,
ruled by the Lorentz transform to be:
[4.1]
where λ and λ o are the emitted and observed wavelength. For v sufficiently small
compared to the velocity of light, eq. [4.1] becomes
[4.2]
Figure 4.1 Left: the graph published by Hubble with the expansion’s
discovery (by Hubble E. 1929). Right: the modern version of the Hubble
law from observations with the Hubble Space Telescope, with the different
used distance indicators listed in the insert (by Freeman K. et al. 2001).
This relation defines the redshift z in the local Universe. For negative v (sources
approaching the observer, like in the Local Group) this is a blue-shift, while
becoming a redshift for positive v. The great discovery by Hubble in the 1929 was
that of the existence of a relation between the distance of a galaxy and the
spectroscopically measured redshift
The Peculiar Velocity Field
4. 2
cz  H 0 ⋅ d .
[4.3]
The costant H 0 is the Hubble costant, expressed in Km/sec/Mpc. Current
measurements (mainly performed measuring distances of galaxies in the Virgo
cluster with the cefeids method, with an error <10%, thanks to the systematic use of
HST) indicate
H 0  70 ± 6 Κm/sec/Μpc
[4.3b]
H 0 is known today with much higher precision (∼1%) thanks to observations of the
Cosmic Microwave Background, see Sect. 6).
The source distances in [4.3] are measured from distance indicators, whose sequence
as a function of the distance scale is reported in the cosmic distance ladder in Fig.
4.2.
Figure 4.2
Representation of the
so-called cosmic
distance ladder. At each
step of this,
corresponding to a
certain range of
distances, different
cosmic sources are used
as indicators. Each of
these different
categories are
characterized by
different luminosity
classes. [Chart taken
from the book The
cosmological distance
ladder (RowanRobinson, 1985].
The Hubble law applies on large scales (it is not valid, for example, in the Local
Group, where we measure only blue-shifts). On smaller scales, where the LSS is
prominent, we have to consider the effects of peculiar motions (or proper motions) of
galaxies, according to the relation:
z  costant ⋅ d + v pec / c
The Peculiar Velocity Field
[4.4]
4. 3
such that the peculiar velocity field can be directly estimated from the two
independent measurements of the recession velocity v redshift and distance d:
=
v pec v redshift − H 0 d
[4.5]
Vice-versa, a correct 3D mapping of the Universe that we might wish to obtain from
application of the Hubble law thanks to the relative simplicity to obtain redshift
measurements today, requires some good modelling and correction for the
systematics inherent the peculiar velocity flow
=
d (v redshift − v pec ) / H 0
[4.5b]
Peculiar motions and the cosmological Large Scale Structure
When cosmologists have started to study the cosmological large-scale structure, they
have also considered the possibility that the large voids, walls and sheets (see e.g.
Figs. 1.1 and 1.2 of Sect. 1) could be simply due to the accumulation effects of the
peculiar motions on large scales.
We have seen that large voids, for example, may extend up to scales of order of a
hundred Mpc. The question then is: what velocity fields might generate this structure
during a Hubble time? The simple answer is
v pec ≈
100 Mpc
100 Mpc

 7000 Km/s .
1 / 70( Km / s / Mpc)
tH
Enormous velocities would then be required. The observations of the peculiar
velocities, discussed later in this Sect., appear to be inconsistent with such large
values and indicate instead much lower velocities, of order of a few to several
hundred Km/sec.
The consequence is that the large scale structure should be rather interpreted as
regions in which galaxies have formed either more (in walls, sheets, and filaments) or
less (in voids) efficiently.
Peculiar motions, on their side, are interpreted as the integrated effect of gravitation
by large scale in-homogeneities in the matter distribution, operating during a time
comparable to the Hubble time t H .
The Cosmic Virial Theorem
We may start having a semi-quantitative assessment of the role of the large-scale
structure in inducing perturbations in the galaxy velocity field by application of the
virial theorem to the matter distribution. The theorem's validity requires a kind of
dynamical situation with some level of relaxation in the energy components at play
(which strictly speaking we cannot really assume, but let us having it as a first
The Peculiar Velocity Field
4. 4
approximation). These components are the self-gravitation in the matter distribution
and the kinetic energy of test particles (as galaxies are):
v 2pec ≈
2GM
.
r
As for the matter distribution, the effective term about peculiar velocities is that
corresponding to matter enhancements above the average matter density of the
Universe, enhancements that we can represent in terms of the 2-point spatial
correlation function (see Sect. 1.3). The effective “excess” mass producing the
peculiar velocity field can then be estimated as (the following is a particularly good
assumption on small enough scales that ξ (r ) > 1 , but holds approximately true also
for larger r, ξ (r ) measures the excess density with respect to the underlying
homogeneous density field):
 r 
4p
4p
r0 r 3ξ (r )
r0 r 3 
M (r ) ..
−1 
3
3
 8.3 h 
v
2
pec
v pec
 r 
8p G
≈
r0 r 2 
−1 
3
 8.3 h 
−1.8
−1.8
such that
 r 
8p G
.
Ω m rC r 2 
−1 
3
 8.3 h 


r
≈ (1000 Km / sec) Ω m1/2  −1

 h Mpc 
−1.8
;
[4.7]
0.1
where we get a very weak dependence on the spatial scale to which the peculiar
velocity is referred. This relation can be effectively inverted to get a constraint on the
density parameter from the measured velocities:
Ω m ≈ ( v pec
 r 
1000 Km / sec ) 

 Mpc 
2
−0.2
[4.8]
If we put the typical observed values, that are velocities of about 500 Km/sec
(approximately the value of the Local Group measured against the dipole component
of the CMB, see Sect. 6) on scales of about 10 Mpc, we get
Ω m ≈ 0.16
In spite of the crudeness of the approach (the virialisation assumption is essentially
incorrect except to first order), this is an interesting reference value, not so far from
the correct accepted figure for the density parameter. This result, although qualitative
at the moment, illustrates the fact that peculiar velocities and the density parameter
are indeed tightly related. The advantage of this crude approach is its independence
from the bias parameter.
The Peculiar Velocity Field
4. 5
4.3 MODELLING THE PECULIAR VELOCITY FIELD
We have found in the previous Sect. 3.1 an important result about the peculiar
velocity field: in the absence of forces inducing perturbations in the velocity field,
any such perturbations quickly decay in time (proportionally to a (t ) −1 for the proper
velocities), as a result of the expanding medium (eq. [3.22]). The existence of
peculiar motions is direct consequence and evidence for a perturbation field in
gravitating matter continuously generating and maintaining at the same time these
motions.
In particular, eqs. [3.20], [3.38] and [3.39] are the most general expressions about the
evolution of structures in its various components, both the relativistic and nonrelativistic ones (in particular these will be needed for treating perturbations before
recombination in Sect. 6).
Rotational motions.
If we are interested, as we are now, in a physical modelling of the relationship
between the structure in the matter distribution and the velocity field, things can be
simplified significantly. Let us first of all consider eq. [3.20], where we can neglect
the pressure term at all, considering that galaxies and dark matter are pressure-less
components (except in case the peculiar motions might become relativistic, which is
obviously not the case as we have seen). That eq. then becomes
d
du
1
 a  d
+ 2 u =
− 2 ∇ cdφ
dt
a
a
[4.10]

u being the peculiar velocity component in comoving units. Now let us come to the
important point that these velocity perturbations, the peculiar velocities, correspond
essentially to potential motions, and are irrotational. Indeed, consider splitting the

velocity into a component u parallel to the gradient of the perturbed gravitational

potential ∇ cδφ , and one perpendicular to it, u⊥ .
This latter is mostly independent on the gravitational potential, and corresponds to a
field of rotational motions, like vortices. By definition, these motions obey our eqs.
[3.21] and [3.22] and feel the Hubble drag, because (4.10) becomes, in the absence of
torque forces,
d
d u⊥
 a  d
=
−2   u⊥ ;
dt
a
⇒
d
u⊥ ∝ a (t ) −2 .
The Peculiar Velocity Field
4. 6
These turbulent motions are not supported by any force, and then quickly decay
according to the rule of the Hubble drag [in proper units: v ⊥ pec ∝ a (t ) −1 ] 1 .
We mention just in passing that turbulence of primordial origin has been sometimes
considered as a mechanism driving galaxy formation. However, because these
motions decay quickly, one would need to find some mechanisms maintaining them
if they have to survive. Since there are no candidates for such forces, cosmological
turbulence is not supposed to drive peculiar motions in the Universe.
Indeed, forces potentially inducing shear fields and rotation might be the tidal ones;
however in our situation of small assumed perturbations these terms are negligible.
These instead become significant during the non-linear phase of the collapse and are
expected to be the source of rotational momentum in cosmic objects (like galaxies).
From an observational perspective, important components of rotational motions and
vorticity on large scales can be excluded from the following consideration. Let us
suppose that the velocity field consisted of rotational motion components with
random orientation of the rotational axes. Then one would expect strong
discontinuities in the velocity field here and there to happen where two eddies with
opposite rotation come in touch. No such discontinuities are observed in fact.
Potential motions.
Motions along the direction of the gravitational field gradient are maintained and
developed by the field itself. To start with, we can give up using the whole eqs.
[3.20], [3.38] and [3.39], and we can give up the Euler equation, and just consider the
Poisson and mass conservation conditions. However, in addition, we make use of our
previous results about the evolution of the density contrast (the growth factor). This is
an important element of our analysis, because the velocity field is not merely due to
perturbations to the density field at the present time, but it also integrates the
operation of the gravitational potential during the whole Hubble time, hence it is
sensitive on how the LSS has evolved with time. Since we now have from Sect. 3.3
solutions for the evolving field of fluctuations in the matter distribution, we can
easily calculate from this the perturbed gravitational potential that they generate.
Let us start from the perturbed Poisson eq. [3.16] either in proper or in comoving
units:
( ) 4π G r ⋅ ∆;
=
∇ 2δφ 4πδr
G=
or
2
4πδr
∇=
Ga 2
c δφ
3Ω m H 2
Because the average matter density reads ρρ
, we get
=
Ωm C =
8π G
1
As remarked by Longair (Galaxy Formation) , this can be seen as a simple rule of conservation of the rotational
angular momentum mvr = const .
The Peculiar Velocity Field
4. 7
 3Ω m H 2  2
3
∇ c δφ = 4π G 
Ωm H 2 a 2 ∆
 a ∆=
2
 8π G 
2
[4.13]
It may also be useful to refer to the peculiar gravitational acceleration as
δδ
1
δ
[4.14]
g = −∇δφ = −
∇ cδφ
a (t )
 
Now, since these motions are irrotational ( ∇ × v pec ≈ 0 ), we can express the
gravitational pull in terms of a new potential Φ v inducing a peculiar velocity field,
like defined as:


∇c Φ v

v pec = −
= −∇Φ v
a (t )
whose divergence gives:
[4.15]


∇ c2Φ v

2
∇ ⋅ v pec = −∇ Φ v = − 2
a (t )
[4.16]
Next, let us include the continuity equation for the perturbed mass distribution, eq.
(3.13):
d d
d  dρ  d ∆
1d d
= −∇ ⋅ v pec = − ∇ c ⋅ v pec
 =
dt  ρ 0  dt
a
[4.17]
and putting together [4.16] and [4.17],
d
d d
d∆
[4.18]
∇ c2Φ v = − a (t )∇ c ⋅ v pec = a 2
= a 2 ∆
dt
Let us manipulate this by multiplying and dividing by ∆ and H =
a / a
2
a 2 ∆
[4.19]
∇
=
Φ
H
a=
∆ H (t ) f (Ω m )a 2 ∆
v
c
a ∆
where we have introduced the function f (Ω m ) expressing the evolution of the
matter density contrast (the growth factor discussed in Sect. 3.3), defined as
a ∆ a d ∆ d ln ∆
f (Ω m ) ≡ =
=
a ∆ ∆ da d ln a
.
[4.20]
We can borrow from the analysis made in Sect. 3.3 all details about the function
f (Ω m ) . In particular, we have seen in [3.52] that for an Einstein-de Sitter Universe
this evolution is simply
The Peculiar Velocity Field
4. 8
∆=
δρ
−1
∝ (1 + z ) ∝ a (t ), such that f (Ω m = 1)= 1 .
ρ
[3.52]
More in general, we have also seen that the situation is more complex for Ω m ≠ 1 , in
which however a fair representation of the evolution is given by (Peebles 1980)
f (Ω m ) . Ω m 0.6
[4.21]
Note that this result is almost independent on Ω Λ , so it is surprisingly very general.
From [4.19]

∇ c2Φ v= a 2 ∆= Hf (Ω)a 2 ∆
and putting together this and the Poisson [4.13], by substituting the product a 2 ∆ we
get
−1
δ2
3
2
2
∇ c Φ=
∆ Hf (Ω)  Ω m H  ∇ c 2δf
Hf (Ω)a =
v
2

−1
δ2
 3Ω m H 
2 f (Ω m )
2
,
for
which
one
solution
is
,
=
∇c Φ v 
∇
=
Φ
δfδf
v
c

3Ω m H
 2 f (Ω ) 
[4.23]
and taking the gradient of this we get the important relation:
δδ
2 f (Ω m )
2 f (Ω m )
δδ
∇δf =
g
v pec = ∇Φ v =
3Ω m H 0
3Ω m H 0
.
[4.24]


Note that a first integral of [4.23] may be formally obtained as ∇ c Φ=
v pec
=
v
δ

= (2 f 3ΩH )∇δf + const = (2 f 3ΩH ) g + const , but the constant velocity term
can be considered to quickly decay due to Hubble drag, hence [4.24] is obtained.
We see in [4.24] that the action on the velocity fields is done by a (factorized)

combination of the peculiar acceleration g and the growth factor f : because of the
long term action of gravity, this latter factor is relevant.
We can also obtain a second useful expression by considering the divergence of the
velocity field, through [4.18] and using [4.19]:


∇ ⋅ v pec = − a −2∇ c2Φ v = − a −2 Ha 2 f (Ω)∆ = − H (t ) f (Ω)∆
[4.25]
where the divergence on the velocity is here expressed in proper coordinates.
Equations [4.24] and [4.25] are the important ones relating the peculiar velocity field
to the present-time gravitational pull and its time evolution. These will be our workhorses in our later applications.
The Peculiar Velocity Field
4. 9
4.4 OBSERVATIONS OF PECULIAR MOTIONS
As we will see below, peculiar motions reflect in-homogeneities in the Universe on
large scales (<300 Mpc). The corresponding collapse phase is linear, so can be treated
with good precision with the linear theory discussed in Sect. 3. The analysis of
peculiar velocity fields not only offers a solid confirmation of the gravitational
instability scenario for the formation of the cosmological large scale structure, but
also a robust method to measure the average matter content of the Universe through
eqs. [4.24] and [4.25]. 2
We will consider in the following two applications of our previous analysis to two
schematic situations, one in which the peculiar velocity field feels a large mass
concentration that might be assumed as point-like. The second to associate to a largescale extended mass distribution and the associated cosmological dipoles.
4.4.1 Infall towards large point-like mass concentrations. Virgo infall.
The situation we consider here is that of a circularly symmetric matter concentration,
like depicted in the scheme below, whose gravitational effects on the velocity fields
might be described as those of a point mass concentration.

So with reference to eq. [4.24], we get for the gravitational acceleration g :
 4π G r 3 ∆ ⋅ r 0
g=
3
r2
and for the peculiar induced velocity
2
This Section refers partly to the Chapter 16.10 of the book "Physical Cosmology" of J. Peacock, and takes material
from there.
The Peculiar Velocity Field
4. 10
2 f (Ω m )
2 f (Ω m ) 4p G r 0
8p G 3H 02 f (Ω m )

=
=
=
∆
=
g
r
r ⋅∆
v pec
Ωm
3H 0 Ω m
3H 0
3
9 8p G 3H 0
f (Ω m )
H0r ⋅ ∆
3
2
having considered that ρρπ
0 Ωm =
c = 3H 0 8 G is the critical density, and noting
that H 0 r = v Hubble is the Hubble recession velocity at distance r, we have

v pec
v Hubble
=

v-v Hubble
v Hubble
f (Ω m )
Ω0.6
=
∆. m ∆
3
3
[4.27]
This is a remarkable result: the percentage deviation of peculiar velocities with
respect to the Hubble flow does not depend on the spatial scale but only (linearly) on
the percentage density fluctuation and on the density parameter Ω m through the
growth function f .
The most obvious application concerning observations from our reference frame is
that about the perturbation of the Hubble flow induced by the Local Super-cluster
centred on the Virgo cluster, as shown in Fig. 4.3.
This mass concentration reduces the expansion velocity of our reference system with
respect to it, and consequently the recession velocity of Virgo is smaller than that
expected for its distance, given the unperturbed Hubble flow.
From the 3D distribution of galaxies in the Virgo environment we can measure the
average density contrast in the galaxy distribution compared to the average density,
and this is approximately ∆ ≈ 2.5 . Figure 4.4 also shows the distribution of
measured recession velocities of galaxies in the Virgo direction. From this, the
observed redshift is 1000 Km/sec, while the distance (precisely measured by distance
indicators) is 18.5 Mpc, that translates onto an unperturbed recession velocity for
H 0 = 70 of 1300 Km/sec, or an infall peculiar velocity of 300 Km/sec. All this put
together into eq. [4.27] brings to an estimate of the matter density of Ω m if galaxies
trace the underline dark matter field. This brings to an estimate of Ω m . 0.18 (this
includes a small correction factor to account for some non-linearity effect due to the
relatively large value of ∆ ), which in any case points towards an open Universe and a
matter density quite insufficient to closure.
The Peculiar Velocity Field
4. 11
Figure 4.4
Measured
velocities of
galaxies within 6
degrees of the
Virgo cluster
center.
The Peculiar Velocity Field
4. 12
4.4.2 Cosmological dipoles towards large scale structures.
The situation here is assumed to represent the effects of mass concentration in a large
cosmic wall, as we see in 3D maps of the galaxy distribution and schematically
illustrated in the graph below.
Again from [4.24] we have:
d 3r
θ 2 r 2 dr
d
g G
=
=
r0 ⋅ ∆ G 2 r0 ⋅ ∆
r2
r
because θ 2 r 2 dr = d 3 r and integrating over the sheet
3H 02
dr 3
d
=
g G ∫ 2 ∆ ⋅ Ωm
r
8π G
2 f (Ω m ) d
3H 02 2 f 3H 0
2 fG d 3 r
d 3r
d
v pec
g
rˆ
=
=
∆⋅Ω
=
=
∆
m
3H 0 Ω m
3H 0 Ω m ∫ r 2
8p G
3 8p ∫ r 2
[4.29]
d
H f (Ω m ) ∆ ( r ) 3
= 0
∫ r 2 rˆ d r
4p

Note that the mass fluctuation field ∆ (r ) appearing here, and responsible for
inducing peculiar motions, includes all non-relativistic matter (baryons and dark
matter), ∆ mass . Of course, to evaluate this mass reconstruction we need to use
luminous matter, like galaxies, whose fluctuations are ∆ light . The relation between
the two is given by the bias parameter b that we introduced in Sect. 1.3.4:
The Peculiar Velocity Field
4. 13
∆ light
∆ mass =
b
[4.30]
such that
d
d
∆
∆
r
r
(
)
(
) 3
b
H 0 Ω0.6
H
d
light
light
3
m
0
ˆ
=
r
d
r
rˆ d r
v pec =
4p b ∫ r 2
4p ∫ r 2
[4.31]
where the parameter
Ω0.6
b≡ m .
b
[4.32]
is often used, and makes the direct observable from this analysis.
The local dipole. The flux-weighted dipole.
So, given some kind of mapping of the galaxy distribution ∆ ( r ) , we can predict the
induced velocity field. An interesting application concerns the one case in which we
know in good detail and precision the peculiar velocity in amplitude and direction:
the well-determined absolute motion of our reference frame with respect to the local
fundamental observer defined by the Cosmic Microwave Background (CMB)
photons. As we will see in detail in Sect. 6, observations of the angular distribution
of the CMB intensity allow us to define with high precision both the velocity and
direction of the absolute motion of the Earth in the sky. Once we subtract from this
the velocity components of the Sun in the Galaxy and of the Galaxy in the Local
Group, we get the absolute motion of the local fundamental observer (the local
frame). We defer to Sect. 6 for more details.

So, assuming that the vector v pec is known in [4.31], a model for the mass
distribution ∆ ( r ) can be used to infer estimates of the parameter b = Ω 0.6
m b , and
eventually for the density parameter itself.
In fact, a reasonable answer for the predicted dipole can be obtained without distance
or even redshift data, but just using the angular projected distribution of galaxies (like
for example discussed in Sect. 5.1 of the 3rd year Course). Any flux-limited survey
will have a radial distribution peaking at some typical distance depending on the
survey flux limit. So the anisotropies of the galaxy distributions calculated at
different magnitude limits allow us in principle to infer an integrated estimate of the
3D map. We can write that, in terms of the galaxy luminosity function Φ ( L) , the
number counts can be expressed as a function of the sky coordinates with integrals of
the density fluctuation field ∆ ( r ) weighted by Φ ( L) :
The Peculiar Velocity Field
4. 14
dN [ RA, DEC ]
=
dS
∫ (1 + D[r , RA, DEC ] ) Φ( Sr
2
1
d
r (r )Φ ( Sr
r∫
r
) d 3=
2
) d 3 r [4.33]
This flux-weighted dipole is therefore roughly proportional to the gravitating mass
integral. An important question when performing this analysis is at which radial
distance from us the integral [4.33] converges, to get the total gravitational pull. This
obviously depends on how fast the cosmic large-scale structure converges to
homogeneity. This then sets a requirement on the depth of the survey used to
determine the counts. From what we found in Sect. 1, it is clear that survey depths
approaching 300 Mpc are required for the peculiar acceleration to converge (Peacock
1992).
This approach has been followed by many authors, not always with consistent results.
For example, the use of the IRAS extragalactic source catalogue (about 20.000
galaxies, flux-limited sample, homogeneous selection, average distance of ∼400 Mpc;
see Sect. 5.1 of the 3rd year Course) has brought to rather high values of the
Ω0.6
parameter b =
m b IRAS . 0.85 ± 0.15 , where bIRAS is the bias of the IRAS
galaxies: that would mean an high value of the density parameter ( Ω m ∼0.7), unless
this bias factor is proportionally small. Indeed this might be the case if IRAS
galaxies tend to avoid large matter concentrations, like the galaxy clusters and
superclusters, because of lack of gas, dust and star-formation in the latter. Indeed this
seems to be the case, as also revealed by clustering analyses of the IRAS galaxies,
showing small values of the correlation length compared to optical galaxies. With
reference to Fig. 6 Sect.1, the IRAS galaxies have a 2-point correlation function like
blue optical galaxies. If we assume bIRAS . 0.6 , completely consistent with the IRAS
2-point correlation function, we get Ω m =
( 0.85bIRAS )
1/0.6
. 0.32 .
In conclusion, the high observed values for the β parameter from the IRAS galaxies
is not a problem anymore (while it has been so in the past).
Weighting the Universe. The POTENT programme.
As we have anticipated, studies of peculiar velocity fields allow us to robustly
constrain the average global matter density. We have discussed in the previous
paragraph that considerations of the matter distribution in the local Universe can be
used to predict the peculiar motion of the local group, to be compared with
measurements of the motion of our reference system within the CMB photon field.
We extend now this analysis to generalize to the velocity fields of galaxies on very
large scales. The present analysis has at least three motivations. One is to test the
gravitational instability model for the origin of the motions. The second is to achieve
The Peculiar Velocity Field
4. 15
more precise evaluations of the average matter density
. The third is to get a 3D
mapping of peculiar velocities in the local Universe to parallel the 3D matter
distribution: with such velocity fields we can estimate precise corrections to redshift
distance measurements to get the real distances from spectroscopic observations, with
good accuracy.
Our reference relation in this analysis is still eq. [4.31], but we attempt to use it here
to relate the general velocity field with the matter distribution in a large volume of
the Universe. So the vector in [4.31] is the vector field as a function of local
field is not known directly, except for a small
coordinates. Of course, the
subsample of galaxies having distance measurements from distance indicators. For
the bulk of galaxies, distances are still velocity distances (redshift distances and
Hubble law).
Figure 4.5 A comparison of peculiar velocities inferred from the Tully-Fisher distance
measurements (left), with predictions of the peculiar velocity field based on the IRAS galaxy
density map, for galaxies within 30 degrees of the supergalactic plane and within 6000 km s-1. The
coordinate system is in Supergalactic coordinates. The model assumes
[Davis et al. 1996].
In practice, eq. [4.31] is used in a classical iterative way. A first guess of the density
is obtained via Hubble law from a large spectroscopic survey of galaxies,
field
with further constraints on a subset of those based on distance indicators. With this
from eq.
first-guess density model we get a first-guess peculiar velocity model
[4.29], which is then used to get corrections to the recession velocities hence to the
Hubble distances, and so an improved model for the density field. And the procedure
The Peculiar Velocity Field
4. 16
continues so on until the variations from one step to the following in the iteration for
both the velocity and density fields become small enough. An example of such an
analysis is reported in Figure 4.5. On the left panel the peculiar velocity field after
convergence appears, with distance measurements including those based on the
Tully-Fisher method, in which the galaxy luminosity is related to its rotational
velocity (and, from luminosity, the luminosity distance is found). On the right panel
the velocity field from the iteration method is reported, with a best-fit value for the
beta parameter as:
.
Since most galaxies are concentrated towards the super-galactic plane defined by the
local super-cluster, it makes sense to plot the velocity vectors projected onto this
plane. Since thousands of galaxies are involved, the velocity field is also shown
averaged onto a grid.
Figure 4.6
The POTENT density field in the Super-galactic plane out to 80 h−1 Mpc, shown as landscape
maps observed from two different directions. The height of the surface is proportional to
. Note the
attractors GA (Great Attractor) and PP (Perseus-Pisces super-cluster), and the extended void in between.
The figure shows that, within 5000 km s-1, the velocity field appears dominated by a
few distinct regions in which the flow is coherent. The feature that has received some
particular attention is at coordinates SGX: -4000 km s-1, SGY : -3000 km s-1. This
suggests the existence of a single mass concentration, which has been named the
"Great Attractor" 3. This makes a large deviation from the Hubble flow. The
structure is explained with a large concentration of clusters and super-clusters at a
distance range of about 47 to 80 Mpc (the latter being the most recent estimate) from
the Milky Way, in the direction of the constellations Hydra and Centaurus. Objects
in that direction lie in the Zone of Avoidance (the part of the night sky obscured by
3
Popular reports in the journals of this subject have sometimes given the impression of some
mysterious concentration of mass that is detected only through its gravitational pull on
surrounding galaxies. This is no more a problem, the Great Attractor has been identified and
resolved.
The Peculiar Velocity Field
4. 17
the Milky Way galaxy) and are thus difficult to study with visible wavelengths, but
X-ray observations have revealed that the region of space includes many massive
clusters.
One of the most systematic applications of the above concepts have been achieved
with the POTENT programme (Dekel, Yahil, Burnstein, Faber, and others). Some
results about the density field reconstruction using the velocity fields and all other
available information are reported in Figure 4.6.
based on an
The bottom line of these analyses is about the value of Ω0.6
m . 0.5 b ± 0.07
optically-selected sample of galaxies (Peacock et al. 2001), corresponding to a
density parameter Ω m . 0.3 , unless the bias of optical galaxies is either very larger or
very smaller than unity, which is unlikely to be the case.
The Peculiar Velocity Field
4. 18