Correct interpretation of terms in proper velocity expression?

[johne1618]

The "cosmological" proper distance from the origin, $D(t)$, to an object at radial co-ordinate $r$ at cosmological time $t$ is given by

$D(t) = a(t) r(t)$

The corresponding "cosmological" proper velocity $v$ of the object is given by

$v = \frac{dD}{dt} = \frac{da}{dt} r(t) + a(t) \frac{dr}{dt}$

Using the definition of the Hubble parameter $H(t) = \dot{a} / a$ and the above equation $D = a r$ we find

$v(t) = H(t) D(t) + a(t) \frac{dr}{dt}$

The first term is Hubble's law for the recessional velocity of a co-moving object whereas the second is the peculiar velocity term.

I would like to further understand the meaning of the peculiar velocity term.

To do so I use the relationship between an interval of co-moving time τ and cosmic time t

$d\tau = \frac{dt}{a(t)}$

to rewrite the peculiar velocity term so that we have

$v(t) = H(t) D(t) + \frac{dr}{d\tau}$

Thus the cosmological proper velocity of an object is the recessional velocity of its co-moving inertial frame plus the velocity of the object within this inertial frame. The co-ordinates of the object within its inertial frame are $(r,\tau)$.

Is this the right interpretation of the above equation?

 

[marcus]

I would like to further understand the meaning of the peculiar velocity term.

To do so I use the relationship between an interval of co-moving time τ and cosmic time t

$d\tau = \frac{dt}{a(t)}$
...

Instead of "co-moving time" can I interpret that as "conformal time"?
I don't mean to be picky about terminology but sometimes I get slowed down just by unfamiliar words.

According to paragraph 6 of this essay in John Baez physics FAQ
http://math.ucr.edu/home/baez/physics/Relativity/GR/hubble.html
"comoving time" is actually just the SAME AS COSMIC TIME.
So it is not the same as conformal time.
An interval of conformal could differ by a factor of 1000 from an interval of cosmic (ie. "comoving").

==quote from Baez FAQ==
... However, this explanation glosses over one crucial point: the time coordinate. FRW spacetimes come fully equipped with a specially distinguished time coordinate (called the comoving or cosmological time). For example, a comoving observer could set her clock by the average density of surrounding speckles, or by the temperature of the Cosmic Background Radiation. (From a purely mathematical standpoint, the comoving time coordinate is singled out by a certain symmetry property.)...
==endquote==

I haven't heard "comoving time" used much--maybe others have and I just didn't notice. If it is as uncommon as I think, it could cause confusion.

IMHO better to say cosmic time t, or FRW time t.
the tau as you define it would be conformal time
=====================

I think your interpretation is perfectly fine, though. Good handling of the equations. Straightforward derivation. You clearly indicate that r is the CO-MOVING radial distance, so it doesn't change except due to the objects own peculiar motion.
The objects own peculiar radial velocity is then, as you say, dr/dτ

Maybe someone else will find something wrong. I don't

 

[johne1618]

Instead of "co-moving time" can I interpret that as "conformal time"?

Sorry - yes I meant conformal time $\tau$.

 

[marcus]

Hey great!
That sets my mind at rest. So AFAICS everything is OK.

 

[johne1618]

Hey great!
That sets my mind at rest. So AFAICS everything is OK.

So do inertial observers measure conformal time $\tau$ rather than cosmological time $t$?