So, what is the function for the redshift z as a function of distance r?

[jfy4]

This may seem elementary, however I have been unable to find a declarative statement that seems reliable. What is the equation for the red-shift $z$ as a function of distance $r$ to the best of our knowledge? That is

$$z(r)=?$$

Thanks in advance.

 

[IsometricPion]

These http://en.wikipedia.org/wiki/Comoving_distance" may be of use.

 

[jfy4]

These http://en.wikipedia.org/wiki/Comoving_distance" may be of use.

Thanks,

However, and this may seem odd, but I would like an expression that does not use the scaling parameter. I imagine data has been fit to the redshift without using $a(t)$. That is, what is the explicit function that describes the correlation between the redshift and the distance between the objects?

 

[Chalnoth]

Thanks,

However, and this may seem odd, but I would like an expression that does not use the scaling parameter. I imagine data has been fit to the redshift without using $a(t)$. That is, what is the explicit function that describes the correlation between the redshift and the distance between the objects?

If you don't like the scale factor, then just perform a change of variables:

$$a = {1 \over 1+z}$$

Or, equivalently,

$$z = {1 \over a} - 1$$

In this way, any equation that relates the scale factor to distance can be used to relate the redshift to distance.

 

[jfy4]

If you don't like the scale factor, then just perform a change of variables:

$$a = {1 \over 1+z}$$

Or, equivalently,

$$z = {1 \over a} - 1$$

In this way, any equation that relates the scale factor to distance can be used to relate the redshift to distance.

Yes, and thanks again for the quick response. However, I need to be more clear...

I imagine at some point, recently, there has been a data plot between the redshift of distance objects, $z$, and the distance to those objects, $r$. And using that data a best fit line, or a function, was generated to match the data as best as possible. I would like to know where to find, or what that best fit function is, because, I have been unable to find a reliable source that contains it. However, I'm almost positive an accurate report of this data exists and must be accessible.

 

[Chalnoth]

Yes, and thanks again for the quick response. However, I need to be more clear...

I imagine at some point, recently, there has been a data plot between the redshift of distance objects, $z$, and the distance to those objects, $r$. And using that data a best fit line, or a function, was generated to match the data as best as possible. I would like to know where to find, or what that best fit function is, because, I have been unable to find a reliable source that contains it. However, I'm almost positive an accurate report of this data exists and must be accessible.

Well, here are the distance measures in cosmology:
http://arxiv.org/abs/astro-ph/9905116

In particular, the one you're probably thinking of is called the comoving distance. This can be written as:
$$D_c = c \int_0^z {dz' \over H(z')}$$

Here you use the first Friedmann equation for $H(z)$:

$$H(z) = H_0 \sqrt(\Omega_m(1+z)^3 + \Omega_\Lambda)$$

(note: this is for flat space)

Then you just plug in the best-fit parameters for $H_0$, $\Omega_m$, and $\Omega_\Lambda$. For example, you can use the parameters from this combination of WMAP, baryon acoustic oscillation observations, and supernovae:
http://lambda.gsfc.nasa.gov/product/map/current/params/lcdm_sz_lens_wmap7_bao_snsalt.cfm

A short note on notation: $\Omega_m = \Omega_c + \Omega_b$, since $\Omega_m$ is the total matter density, and $\Omega_c$ and $\Omega_b$ are the cold dark matter and normal (baryonic) matter densities, respectively. The parameter $h$ at the above link is a different parameterization of the Hubble constant as follows:

$$h = {H_0 \over 100 km/sec/Mpc}$$

Unfortunately, yes, this is a bit complicated. But the real universe is complicated.

 

[IsometricPion]

http://arxiv.org/pdf/astro-ph/0311622" paper).

 

[jfy4]

Thank you for the responses.

The papers look helpful. I also found this equation from
http://hyperphysics.phy-astr.gsu.edu/hbase/astro/hubble.html#c3"

$$r=\left[\frac{(z+1)^2-1}{(z+1)^2+1}\right]\frac{c}{H_0}$$

which I rearranged to get

$$z(r)=\frac{H_0r-c\pm \sqrt{c^2-(H_0r)^2}}{c-H_0r}=-1\pm \sqrt{\frac{1+\frac{H_0r}{c}}{1-\frac{H_0r}{c}}}$$

I think the only physical solution to this is the plus case.

How strong is the validity of this equation with respect to what is observed?

 

[Chalnoth]

Thank you for the responses.

The papers look helpful. I also found this equation from
http://hyperphysics.phy-astr.gsu.edu/hbase/astro/hubble.html#c3"

$$r=\left[\frac{(z+1)^2-1}{(z+1)^2+1}\right]\frac{c}{H_0}$$

which I rearranged to get

$$z(r)=\frac{H_0r-c\pm \sqrt{c^2-(H_0r)^2}}{c-H_0r}=-1\pm \sqrt{\frac{1+\frac{H_0r}{c}}{1-\frac{H_0r}{c}}}$$

I think the only physical solution to this is the plus case.

How strong is the validity of this equation with respect to what is observed?

Well, this is the relativistic doppler effect in flat space-time. So it's definitely ruled out by our rather detailed observations today, but I seem to remember it's actually a pretty good approximation to surprisingly far out. I couldn't give you numbers on that off the top of my head, though.

 

[Jorrie]

Well, this is the relativistic doppler effect in flat space-time. So it's definitely ruled out by our rather detailed observations today, but I seem to remember it's actually a pretty good approximation to surprisingly far out.

Even more surprising is that if the distance r is taken to be the look-back distance (r=c(t_now-t)) rather than comoving distance, then the quoted relativistic Doppler equation holds within a few percent over the whole observable universe. I guess that is why the referenced page: "http://hyperphysics.phy-astr.gsu.edu/hbase/astro/hubble.html#c3"" states it so 'confidently'?

 

[Jorrie]

http://hyperphysics.phy-astr.gsu.edu/hbase/astro/hubble.html#c3"

$$r=\left[\frac{(z+1)^2-1}{(z+1)^2+1}\right]\frac{c}{H_0}$$

which I rearranged to get

$$z(r)=\frac{H_0r-c\pm \sqrt{c^2-(H_0r)^2}}{c-H_0r}=-1\pm \sqrt{\frac{1+\frac{H_0r}{c}}{1-\frac{H_0r}{c}}}$$

The z(r) equation is not as useful as the r(z) one, because if you put independently determined r (lookback or light travel) distance in there, then at Ho r >= c, it diverges and blows up. This happens around r = 4000 Mpc, where z ~ 8 (using http://lambda.gsfc.nasa.gov/product/map/current/params/lcdm_sz_lens_wmap7_bao_snsalt.cfm" values.

The first (r(z)) equation tracks the integration for LCDM light travel distance within 8%, with the max error around z=1.3.

 

[jfy4]

The z(r) equation is not as useful as the r(z) one, because if you put independently determined r (lookback or light travel) distance in there, then at Ho r >= c, it diverges and blows up. This happens around r = 4000 Mpc, where z ~ 8 (using http://lambda.gsfc.nasa.gov/product/map/current/params/lcdm_sz_lens_wmap7_bao_snsalt.cfm" values.

The first (r(z)) equation tracks the integration for LCDM light travel distance within 8%, with the max error around z=1.3.

Okay. With all this in mind, along with the data in the papers posted here, is there a data matching function $z(r)$ instead of various functions which each correspond to different proposed models? In my previous post I gave an equation which was model based, not derived from data. Is there a function that is a best fit for our current redshift vs distance data in the form $z(r)$ ?

I know I appear to be repeating myself, so if this has already been answered, I'm sorry I haven't been able to identify it...

 

[jfy4]

Okay. Everything I have said may seem a bit vauge without some context or motivation. I am working on an exercise (not homework) where I am attempting to model the cosmological red-shift not using a scaling parameter, but rather by the use of constructing a cosmological solution to the Einstein field equations where the $g_{00}$ is responsible for the observed red-shift. In this way the universe is static, and the structure of the gravitational field causes the observed red-shift.

Hopefully no one here will scream blasphemy after I say this and horrible repercussions come my way...

I would like to assume a static, spherically-symmetric, perfect fluid cosmology. I would like to start using canonocal coordinates

$$ds^2=-e^{2\nu(r)}dt^2+e^{2\lambda(r)}dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2.$$

Now here is where my curiosity for the red-shift equation comes into play.

The energy of the light observed is modeled by

$$E=-g_{\alpha\beta}p^{\alpha}u_{pbs}^{\beta}=\hbar\omega.$$

Assume a stationary observer. Then the 4-velocity constraint on the observer is

$$g_{\alpha\beta}u_{obs}^\alpha u_{obs}^\beta=-1$$

however for a stationary observer only the temporal component remains resulting in

$$g_{00}(u_{obs}^0)^2=-1\rightarrow u_{obs}=\sqrt{-g^{00}}=e^{-\nu(r)}$$

since the metric is diagonal. There is a time-like killing vector too so

$$u_{obs}^{\alpha}=e^{-\nu(r)}\xi^{\alpha}$$

Then the energy formula from above takes the form

$$\hbar\omega=e^{-\nu(r)}(-\mathbf{p\cdot\xi}).$$

However the item in parenthesis is a constant regardless of radius. So I evaluate the previous equation at some distance $R$, and at "here" where I assume the exponential function reduces to 1. Then I have the following

$$\hbar\omega_{R}=e^{-\nu(R)}(-\mathbf{p\cdot\xi})_{R}$$

and

$$\hbar\omega_{\ast}=(-\mathbf{p\cdot\xi})_{\ast}$$

Then upon substitution

$$\frac{\omega_{R}}{\omega_{\ast}}=e^{-\nu(R)}.$$

However, by definition the redshift is defined as

$$\frac{\omega_{R}}{\omega_{\ast}}=z(R)+1$$

then

$$e^{-\nu(R)}=z+1\implies e^{2\nu(R)}=(z(R)+1)^{-2}.$$

This is the part where in my exercise I would like to know what the current accepted function is for the redshift as a function of distance, based on the observation data that we have.

I am not putting forth in any way that this is the way the universe is, and I don't think people should think this way. The current model has been remarkably successful. This is simply an exercise for myself in cosmology and a curiosity about expressing the redshift solely in terms of a static gravitational field.

 

[Jorrie]

However, by definition the redshift is defined as

$$\frac{\omega_{R}}{\omega_{\ast}}=z(R)+1$$

then

$$e^{-\nu(R)}=z+1\implies e^{2\nu(R)}=(z(R)+1)^{-2}.$$

This is the part where in my exercise I would like to know what the current accepted function is for the redshift as a function of distance, based on the observation data that we have.

I'm not aware of an 'accepted function' compatible with current observations, other than integrating for comoving distance or lookback distance using the LCDM model. Before accelerating expansion has been observed, it was relatively easy to obtain a function, but not any more, I think...

That said, it is interesting to note that a linear function can be fitted reasonably well to the standard expansion factor vs. lookback distance curve, up to around 12 billion years light travel time:

$$a = \frac{1}{z+1} = 1-\frac{r H_0}{ c}$$ , where r is lookback distance.

I attach a graph showing 'best-fit' values of this linear function and also the Doppler shift function (of the prior posts) to the standard LCDM curve, with H_0 as shown.