# Help with velocity/redshift/distance law

1. Nov 19, 2008

### Dave Rutherfo

I've derived a velocity/redshift/distance law,

$$v = c \ln(1 + z) = H_0 d_0$$

where $v$ is the recession velocity, $c$ is the speed of light, $z$ is the cosmological redshift, $H_0$ is the present Hubble constant, and $d_0$ is the present distance of the source.

I would like to relate my law to the data, hopefully to show that the
expansion rate of the universe is not accelerating, thus eliminating the
need to invoke dark energy. Any help would be greatly appreciated.

For the derivation of this law and more, please click on the following link

http://www.softcom.net/users/der555/horizon.pdf

Thanks,
Dave Rutherford

2. Nov 19, 2008

### marcus

Your proposed equation looks wrong, Dave. Check it against the recession speeds given by Morgan's calculator. The link is in my sig. Be sure to enter the usual parameters 0.27, 0.73, and 71 for present matter fraction, lambda fraction, and hubble rate.

When I do that and put in z=10, I get that the recession speed is 2.28 c.
You would have the recession speed be ln(11) c. That is 2.40 instead of 2.28. Well that is not too bad!

Now when I put in z=1090, I get that the recession speed is 3.3.
But you would have it be be ln(1091) = 7.0. That is way off. Either Morgan's calculator, or your formula, or both must be being pushed too far.

Maybe your formula is all right as a rough approximation as long as you just apply it to small redshifts, and will therefore suit your purposes (depending on how you intend to use it.) But as a rule I don't think it works. The relation between recession speed and redshift is not that simple.

Part of your equation is right though. Hubble law does say v = Hd. Hubble law does not talk directly about redshift. It gives the recession speed.

Last edited: Nov 19, 2008
3. Nov 19, 2008

### Dave Rutherfo

Thanks for the reply, Marcus.

Using the current limits of observation for redshift, 0 to 6, my values for recession velocity (my v), for a given redshift (z), are less than Morgan's calculator's corresponding values for recession velocity (Morgan's v) using the values 0.27, 0.73, and 71 for the other quantities that you gave above.

Here are the comparisons I came up with (all velocities x c):

z ....... Morgan's v ...... my v
---------------------------
0 ........... 0.00 ........... 0.00
1 ........... 0.78 ........... 0.69
2 ........... 1.24 ........... 1.10
3 ........... 1.53 ........... 1.39
4 ........... 1.73 ........... 1.61
5 ........... 1.87 ........... 1.79
6 ........... 1.99 ........... 1.95

If we only go by these results (which are based on the current limits of observation for redshift), my values seem to indicate that the universal expansion rate is either not accelerating or accelerating more slowly than the currently accepted rate. Correct? If yes, how can I determine which it is? If no, why not?

Thanks,
Dave