distance between them remains constant at all times i.e., fixed as the universe ... observer in a comoving frame and is a measure of the cosmological time.
Comoving frame: Comoving frame is a coordinate system where an observer moves with the Hubble’s flow of space expansion and, perceives the universe and the cosmic microwave background radiation (CMBR) to be isotropic. So, for a given pair of comoving objects, while the proper (physical) distance between them grows with the expansion of space, the comoving distance between them remains constant at all times i.e., fixed as the universe expands. However due to the gravitational attraction, a peculiar velocity of an object (galaxies etc.) may arise relative to an observer in the comoving frame. The comoving time is the elapsed time since the Big Bang according to the clock of an observer in a comoving frame and is a measure of the cosmological time. Scale factor: The Robertson-Walker scale factor a (t) is a function of time and represents the relative expansion of the universe. It relates the proper distance between a pair of objects, e.g. two galaxy clusters, moving with the Hubble flow in an expanding or contracting FLRW universe at any arbitrary time t to their distance at some reference time t0 . d (t) = d0 a (t) ⇒ d0 =
d (t) a (t)
Here, d (t) and d0 are the proper distances at epoch t and t0 respectively. The scale factor is dimensionless, with t counted from the birth of the universe and t0 set to the present age of the universe. So, by definition, a (t0 ) = 1. The evolution of the scale factor is determined by the Friedman equations of general relativity for a locally isotropic, locally homogeneous universe. d˙ (t) = d0 a˙ (t) d (t) d˙ (t) = a˙ (t) ⇒ d˙ (t) = Hd (t) , which is the Hubble’s law. a (t) Here, H ≡ a1 da . dt Hubble parameter: Hubble parameter (prefer not to call it a constant), H measures the rate of change of the scale factor. H≡
1 1 da and Hubble time = a dt H0 −1
s The SI unit of H is s−1 , but it is mostly expressed in kmM pc . H0 is the value of the Hubble parameter today. It is a constant only in space, not in time. A parsec is a unit of length used to measure large astronomically distances. One parsec is the distance at which one astronomical unit subtends an angle of one arcsecond.
1 pc ≈ 3.0856776 × 1016 m ≈ 3.2615638 light years 1 M pc ≈ 3.0856776 × 1022 m Conformal time and particle horizon: We know that the comoving distance is the distance light could have traveled (in the absence of interactions) since t = 0 i.e., Big Bang. In a time dt, light travels a comoving distance dx = dta (setting, c = 1). So, the total comoving distance light could have traveled is ˆt η=
0
dt . a (t0 )
0
This distance is very important, as no information could have propagated further (on the comoving grid) than η since the beginning of time. Therefore, distances greater than η are not 1
causally connected. If they appear similar, we should be suspicious. We can think of η as the comoving horizon or, even better as the conformal time. 1 η ∝ a 2 , in a matter-dominated universe. η ∝ a, in a radiation-dominated universe. In terms of comoving distance, the particle horizon is equal to the conformal time η that has passed since the Big Bang, times the speed of light c. Conformal time today is, η (t0 ) = η = 1.48 × 1018 s. Note that the conformal time is not the age of the universe. Rather, the conformal time is the amount of time it would take a photon to travel from where we are located to the furthest observable distance provided the universe ceased expanding. The particle horizon Hp recedes constantly as time passes and the conformal time grows. As such, the observed fraction of the universe always increases. Since proper distance at a given time is just comoving distance times the scale factor, the proper distance to the particle horizon at time t is given by ˆt a (t) Hp (t) = a (t)
0
dt and for today t = t0 . c a (t0 )
0
And, Hp (t0 ) = cη0 = 14.4 Gpc = 46.9 billion light years.
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