The "cosmological" proper distance from the origin, [itex]D(t)[/itex], to an object at radial co-ordinate [itex]r[/itex] at cosmological time [itex]t[/itex] is given by(adsbygoogle = window.adsbygoogle || []).push({});

[itex]D(t) = a(t) r(t) [/itex]

The corresponding "cosmological" proper velocity [itex]v[/itex] of the object is given by

[itex] v = \frac{dD}{dt} = \frac{da}{dt} r(t) + a(t) \frac{dr}{dt} [/itex]

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

[itex] v(t) = H(t) D(t) + a(t) \frac{dr}{dt} [/itex]

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

[itex] d\tau = \frac{dt}{a(t)}[/itex]

to rewrite the peculiar velocity term so that we have

[itex] v(t) = H(t) D(t) + \frac{dr}{d\tau} [/itex]

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 [itex](r,\tau)[/itex].

Is this the right interpretation of the above equation?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Correct interpretation of terms in proper velocity expression?

**Physics Forums | Science Articles, Homework Help, Discussion**