Are Redshifts Used to Specify Locations in Cosmology?

[center o bass]

I have heard cosmologists use the phrase "at redshift", presumably indicating the location of something. Are redsifts used to specify locations in cosmology, and if so, how is that done?

 

[PeterDonis]

The redshift basically indicates location, yes, if you interpret that as "location in spacetime". The reason cosmologists give the redshift instead of a distance (e.g., "1 billion light-years away") or a time (e.g., "1 billion years ago") is that the redshift is what we actually observe, and how it translates into a distance or a time depends on other cosmological parameters whose values we haven't necessarily pinned down. So rather than have to specify which particular values of all those parameters are being assumed when a distance or a time is given based on an observed redshift, cosmologists just give the observed redshift directly.

 

[center o bass]

The redshift basically indicates location, yes, if you interpret that as "location in spacetime". The reason cosmologists give the redshift instead of a distance (e.g., "1 billion light-years away") or a time (e.g., "1 billion years ago") is that the redshift is what we actually observe, and how it translates into a distance or a time depends on other cosmological parameters whose values we haven't necessarily pinned down. So rather than have to specify which particular values of all those parameters are being assumed when a distance or a time is given based on an observed redshift, cosmologists just give the observed redshift directly.

Is this idea based on Hubbles law? I.e. that $v = H d$ where v is the velocity of a galaxy and d is it's distance from us? Using that
$$v/c :=z = (\lambda - \lambda_0)/\lambda_0 = d/H$$
we see that given the redshift $z$, we can determine $d$. Is this the basic idea?

 

[Chalnoth]

Is this idea based on Hubbles law? I.e. that $v = H d$ where v is the velocity of a galaxy and d is it's distance from us? Using that
$$v/c :=z = (\lambda - \lambda_0)/\lambda_0 = d/H$$
we see that given the redshift $z$, we can determine $d$. Is this the basic idea?

Essentially, yes.

It's worth noting that the local motions of galaxies can cause their redshifts to vary by as much as about $\pm$0.003 from this value. For far-away galaxies, this is inconsequential. But for nearby galaxies, the redshift can't reasonably be used as a distance measure due to this uncertainty.

 

[PeterDonis]

Is this the basic idea?

Yes, but the expansion rate of the universe (which is what $H$ refers to) changes with time, and we don't know how, exactly, it changes with time.

 

[center o bass]

Yes, but the expansion rate of the universe (which is what $H$ refers to) changes with time, and we don't know how, exactly, it changes with time.

Indeed, but it does not change as fast that $d = H z$ will be significantly tomorrow (or next year) from what it was today?

 

[Jorrie]

Is this idea based on Hubbles law? I.e. that $v = H d$ where v is the velocity of a galaxy and d is it's distance from us? Using that
$$v/c :=z = (\lambda - \lambda_0)/\lambda_0 = d/H$$
we see that given the redshift $z$, we can determine $d$. Is this the basic idea?

Note that the equation that you have used is approximate and only holds for low z (z << 1). Using H₀ = 67.9 km s^-1 Mpc^-1, z=1 represents a recession speed of 0.77c (231,000 km/s) and a proper distance of 11 billion light years (3.8 Mpc), which you can see do not quite fit the equation.

 

[PeterDonis]

Indeed, but it does not change as fast that $d = H z$ will be significantly tomorrow (or next year) from what it was today?

It's not a question of how fast $H$ is changing right now. It's a question of how much $H$ changed during all the time that the light we are seeing now from an object with a given redshift $z$ was traveling. The larger the redshift $z$, the more $H$ will have changed during the light's travel, so the worse an approximation the formula $d = H z$, which assumes that $H$ is constant, will be. (Alternatively, instead of using the value of $H$ right now in the formula, you could use some sort of average value of $H$ over the travel time of the light, but then the value of $H$ you used in the formula would depend on $z$.)