CMB Redshift: 13.7B Yrs Ago, 10.4x Speed of Light Today

[HarryWertM]

The CMB is believed to have started [or ended] 13.7 billion years ago. Recent estimates of Hubble constant are around 70 km/s/Mpc, or 228 km/s/Mlyr. When we look at CMB we are looking back 13700 million years into the past. Multiplying 228 by 13700 gives us 1.04 times the speed of light as the current [in our time frame] rate of recession of CMB. Yes?

Uhhh... Make that 10.4 times the speed of light.

 

[AleLucca]

This is not correct. You are using the expression v = Hd, where v is the recession velocity, H is the Hubble constant and d is the distance of the source, right?
This expression is only valid under the assumption that the Hubble constant is truly a constant during the light travel time between the source and the observer. This condition holds only for small source redshifts. For example, at the emission time of the CMB the Hubble constant was many orders of magnitude larger than it is today.

The redshfit of a source is connected with the scale length of the universe at the time of the emission of the photons observed today, a(t). In particular we have 1 + z = a(t_0)/a(t), where a(t_0) is the scale length of the universe today and z is the redshift.
So, if you want to calculate the redshift of a source by knowing the time of emission of the signal, you must know the value of the scale length at that time, a(t). This in turn is connected to the Hubble constant and how it changes with time.

a(t) goes approximately as t^(1/3), so in principle you could calculate the redshift of the CMB as z = (t_0/t)^(1/3) - 1, but this gives a value too low because a(t) departs significantly from t^(1/3) at recent times.

The redshift of the CMB is instead z~1100.

I hope this is not too confusing.

 

[HarryWertM]

So the Hubble constant changes with time. How are its previous values determined?

 

[marcus]

The CMB is believed to have started [or ended] 13.7 billion years ago. Recent estimates of Hubble constant are around 70 km/s/Mpc, or 228 km/s/Mlyr. When we look at CMB we are looking back 13700 million years into the past. Multiplying 228 by 13700 gives us 1.04 times the speed of light as the current [in our time frame] rate of recession of CMB. Yes?

Uhhh... Make that 10.4 times the speed of light.

Harry, the Hubble law does not use light travel time as a handle on the distance. In the Hubble law v = Hd, the distance d is the freezeframe distance you would get if you could stop the expansion process right now, and then measure the distance to the matter that emitted the Cmb by timing a light or radar signal.

That distance is about 45 billion lightyears (because of expansion it is large).

The usual estimate for the CMB redshift is z = 1090. Redshift has no simple relation to recession rate because it cannot be analyzed as a doppler shift in any simple straightforward manner. z essentially just tells the factor by which the universe has expanded while the light was in transit---actually that is z+1, so to be picky say the factor is 1091.

Your figure 70 km/s/Mpc is about right, but your figure 228 km/s/Mlyr is way off because you seem to have multiplied by 3.26 instead of dividing. That would put you off by a factor of the square of 3.26 or about a factor of 10.

As I recall the matter which emitted the CMB which we are now receiving is now around 45 billion lightyears from us and is receding at over 3 times the speed of light. That is the rate the Hubble law distance is growing. (Recession rates are not like ordinary motion, in General Rel, distances can grow faster than light.)

Anyway, you asked about the CMB redshift. That is easy to say. z = 1090.
Over a thousandfold expansion of distance has occurred while the CMB light was traveling to us.

If you want to know past values of the Hubble parameter (it has changed enormously over time!) then you should learn to use the Cosmos Calculator. It is very simple. Just google "cosmos calculator".

To start you have to type in the matter fraction and the cosmological constant fraction. Usual values are .27 and .73. the H is already set to 70. So you are ready and you just type in a value of z and it will tell the distance and light travel time and what the value of H was back when the light was emitted, and even what the recession rate was (whether it was smaller or larger than the speed of light etc.) Very simple to use.
If you try it and have any trouble, ask. Many people here have had experience with it.

I have the link in my sig. It is the uni.edu link. But it also works to just google "cosmos calculator".

 

[HarryWertM]

Hi Marcus. You are very correct - I multiplied with 3.26 instead of dividing. Totally brain dead with math. I have looked at Ned Wright's calculator. It does not provide anything I am looking for. I am looking for as much insight and understanding as I can manage. If I were a Java Script expert maybe I could tease out something from the JavaScript, but I am not.

What I am looking for is as simple a formula as possible for Hubble's constant. AleLucca clearly explained that it is not constant and depends on time and the scale factor. But the scale factor also depends on time. So is there a simple formula for Hubble that depends only on time and these coefficients for matter and cosmological constant?

 

[nicksauce]

The general formula is (ignoring radiation)

$$H^2 = (\frac{\dot{a}}{a})^2= H_0^2(\Omega_Ma^{-3} + \Omega_{\Lambda})$$

To the best of my knowledge, however, this ODE doesn't have closed form solution, and thus you can't write a formula for H(t)... you just have to solve this ODE numerically.

 

[AleLucca]

Yeah, if you neglect $$\Omega_\Lambda$$ you get $$a(t) \propto t^{1/3}$$, but you cannot do that for recent times, because $$\Omega_\Lambda = 0.7$$ and $$\Omega_{M} = 0.3$$