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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-ordinatec

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 <1 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 >1) 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 < 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 (q>1), 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

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