Approaching a final comoving co-ordinate

1
Joining the Hubble Flow
M.J. Francis1, L.A. Barnes1,2, J.B. James1,3,
G.F. Lewis1 1 IOA, Sydney University 2 IOA,
Cambridge 3 IFA, Edinburgh
Modern cosmology was born out of Edwin Hubble's
discovery that distant galaxies were all
receeding with a velocity proportional to their
distance. This motion is known as the 'Hubble
Flow'. In practice the velocities of galaxies
have an additional 'peculiar velocity' that
deviates from the Hubble Flow. It has long been
supposed that the expansion of the Universe
'washes out' these perturbations and that after
sufficient time all galaxies will join the Hubble
Flow. We have found that joining the Hubble Flow
is in fact not a generic property of expanding
universes and have determined the conditions for
which all objects will join the Hubble Flow
asmyptotically.
Co-ordinates, Distances and Velocites A number
of important measures needed to understand this
work are detailed Proper Distance, Dp Tthe
distance between any two points at a constant
time Scale Factor, a(t) The dimensionless
factor by which the expansion of the Universe is
described Comoving Co-ordinate, c A Galaxy in
the Hubble Flow is said to be co-moving, and has
a constant co-moving co-ordinate c The above
three measures are connected through the
relation This indicates that the proper
distance between two comoving galaxies will
simply increase in proportion with the scale
factor. Differentiating this simple relation with
time shows that which we can now use to
define where vrec is the reccession velocity
due to the Hubble Flow and vpec is the peculiar
velocity that indicates the perturbation to the
Hubble Flow for a given galaxy. Decceleration
Parameter, q(t) The evolution of the expansion
a(t) is governed by the properties of the energy
content of the Universe, such as how much matter,
radiation and other forms of energy, such as a
cosmological constant, are present. We can
describe the sum of their effect via the
Decceleration Parameter q(t). A postive value
indicates decceleration while a negative value
indicates acceleration.
Why does anyone care? Despite General Relativity
being proposed just over 80 years ago, there is
surpringly many things about the theory that are
poorly understood on both a technical and popular
level. A heated debate in recent years has
surrounded the question of whether the expansion
of the Universe can sensibly be described as the
expansion of space and whether, regardless of the
answer on a technical level, this is a good way
to describe cosmology to students. An important
thought experiment in this debate is the question
of whether the expansion of space causes all
objects to eventually join the Hubble Flow. In
this study we found that the very definition of
joining the Hubble Flow, while never accurately
defined previous, is implicitly different in
different papers, adding to the confusion. We
have identified 7 different definitions that have
been used to describe whether an object has
joined the Hubble Flow and determined that for
several of these definitions, joining the Hubble
Flow is not a generic property of expanding
Universes, contrary to previous assertions.
What does the real Hubble Flow look like?
Shown here is the local Hubble Flow as found by
the Hubble Space Telescope key project ( Freedman
et al 2001). The distance-velocity law is clear
to see, however there are perturbations to this
law due to peculiar velocites of individual
galaxies.
Approaching a final comoving co-ordinate Since
the Hubble Flow is defined by the motion of
galaxies at fixed comoving co-ordinates a
reasonable definition of joining the Hubble Flow
might be that a galaxy approaches some final
resting comoving co-ordinate, i.e that We find
that this does not occur in all expanding
universes. For rapidly decclerating universes
there is no asymptotic final resting comoving
co-ordinate and hence by this definition does not
join the Hubble Flow If q lt1
then we can define the final comoving co-ordinate
that a galaxy is approaching. Given this, we can
then define the qauntity as the difference
betwen the particles current proper distance from
the origin and the proper distance to a galaxy
residing at the asymptotic comoving co-ordinate.
It might be expected that this quantity would go
to zero as t goes to infinity. However we have
found that this only occurs for deccelerating or
coasting universes and that for accelerating
cases this qauntity does not go to zero.
Velocities as t goes to infinity One reasonable
definition for joining the Hubble Flow might be
that the peculiar velocity of a galaxy should go
to zero at t goes to infinity. We have verified
that this does indeed hold for all expanding
universes. Since we know that it has been
assumed that the proper velocity approaches the
reccession velocity since vpec goes to zero, and
hence the proper velocity of all galaxies will
approach the Hubble Law recession velocity.
However we have found this in not true in all
cases. In a rapidly deccelerating universe vrec
goes to zero 'faster' than vpec and the velocity
of the galaxy does not approach the expected
reccession velocity even after infinite time. By
this definition galaxies will not join the Hubble
Flow.
For a rapidly deccelerating universe (q gt1) the
comoving co-ordinate of test galaxy with an
intial peculiar veloctity in unbounded with time
and does not approach a final value. In this
Universe galaxies do not join the Hubble Flow. In
more slowly deccelerating or accelerating
universes galaxies do approach a final comoving
co-ordinate.
In a moderately decelerating Universe (0 lt q
1), the peculiar velocity decays more rapidly
than the recession velocity at the Universe
expands such that the proper velocity of a test
galaxy approaches the Hubble Law recession
velocity. Therefore the velocity of the galaxy
approaches that of comoving galaxies and hence
can be said to join the Hubble Flow by this
definition.
In a rapidly decelerating universe (qgt1), the
recession velocity decays more rapidly than the
peculiar velocity and therefore the proper
velocity of the test galaxy does not approach the
Hubble Law recession velocity. Therefore the
velocity of the galaxy does not approach that of
comoving galaxies and hence does not join the
Hubble Flow by this definition.
What is different about our study? Previous
attempts to address this problem have suffered a
number of shortcomings. Since General Relativity
can present difficult mathematical challenges
there has been an overwhelming desire to
approximate GR by something else. Some studies
have realised that pure Newtonian analysis gives
similiar results locally but have failed by
extending the Newtonian equations into the
relativistic realm and finding erroneous results
(Whiting 2004). Other studies have used GR but
have only solved the equations numerically
(Davis, Lineweaver Webb 2003). Since the
question is of asymptotic behaviour at t goes to
infinity numerical integration can easily mislead
since it can only be used for finite times. We
have solved the full General Relativistic
equations in the limit as t goes to infinity
analytically in all cases, revealing previously
unknown properties of test particle motion in
expanding space.
For those Universes where we can define a final
comoving co-ordinate for initially non comoving
test galaxies, the difference in proper distance
between the origin and the galaxy and the origin
and a galaxy at the final comoving co-ordinate
actually increases unboundedly with time. Only
for an accelerating Universe does this go to zero
and hence by this definition can be said to join
the Hubble Flow.
For more detail see Barnes et al (2006) MNRAS
Vol. 373 pp. 382-390 Francis et al (2007)
(Submitted to PASA) Lewis et al (2007) (in
prep) Whiting (2004) The Observatory, Vol. 124,
pp 174-189 Davis, Lineweaver Webb (2003) Am. J.
Phys. Vol. 71 pp 358-365