2.1.6 Cosmological coverage and visualizing tables ¶

Having found the angular coverage of the dataset in Angular coverage on the sky, we can now use Gnuastro to answer a more physically motivated question: “How large is this area at different redshifts?”. To get a feeling of the tangential area that this field covers at redshift 2, you can use Gnuastro’s CosmicCalcular program (CosmicCalculator). In particular, you need the tangential distance covered by 1 arc-second as raw output. Combined with the field’s area that was measured before, we can calculate the tangential distance in Mega Parsecs squared (\(Mpc^2\)).

## If your system language uses ',' (not '.') as decimal separator.
$ export LC_NUMERIC=C

## Print general cosmological properties at redshift 2 (for example).
$ astcosmiccal -z2

## When given a "Specific calculation" option, CosmicCalculator
## will just print that particular calculation. To see all such
## calculations, add a `--help' token to the previous command
## (under the same title). Note that with `--help', no processing
## is done, so you can always simply append it to remember
## something without modifying the command you want to run.
$ astcosmiccal -z2 --help

## Only print the "Tangential dist. for 1arcsec at z (physical kpc)".
## in units of kpc/arc-seconds.
$ astcosmiccal -z2 --arcsectandist

## It is easier to use the short (single character) version of
## this option when typing (but this is hard to read, so use
## the long version in scripts or notes you plan to archive).
$ astcosmiccal -z2 -s

## Short options can be merged (they are only a single character!)
$ astcosmiccal -sz2

## Convert this distance to kpc^2/arcmin^2 and save in `k'.
$ k=$(astcosmiccal -sz2 | awk '{print ($1*60)^2}')

## Calculate the area of the dataset in arcmin^2.
$ n=$(aststatistics flat-ir/xdf-f160w.fits --number)
$ r=$(astfits flat-ir/xdf-f160w.fits -h1 --keyvalue=CDELT1 -q)
$ a=$(echo $n $r | awk '{print $1 * ($2*60)^2 }')

## Multiply `k' and `a' and divide by 10^6 for value in Mpc^2.
$ echo $k $a | awk '{print $1 * $2 / 1e6}'

At redshift 2, this field therefore covers approximately 1.07 \(Mpc^2\). If you would like to see how this tangential area changes with redshift, you can use a shell loop like below.

$ for z in 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0; do        \
    k=$(astcosmiccal -sz$z);                                  \
    echo $z $k $a | awk '{print $1, ($2*60)^2 * $3 / 1e6}';   \
  done

Fortunately, the shell has a useful tool/program to print a sequence of numbers that is nicely called seq (short for “sequence”). You can use it instead of typing all the different redshifts in the loop above. For example, the loop below will calculate and print the tangential coverage of this field across a larger range of redshifts (0.1 to 5) and with finer increments of 0.1. For more on the LC_NUMERIC command, see Numeric locale.

## If your system language uses ',' (not '.') as decimal separator.
$ export LC_NUMERIC=C

## The loop over the redshifts
$ for z in $(seq 0.1 0.1 5); do                                  \
    k=$(astcosmiccal -z$z --arcsectandist);                      \
    echo $z $k $a | awk '{print $1, ($2*60)^2 * $3 / 1e6}';   \
  done

Have a look at the two printed columns. The first is the redshift, and the second is the area of this image at that redshift (in mega-parsecs squared). Redshift (\(z\)) is often used as a proxy for distance in galaxy evolution and cosmology: a higher redshift corresponds to larger line-of-sight comoving distance.

Now, have a look at the first few values. At \(z=0.1\) and \(z=0.5\), this image covers \(0.05 Mpc^2\) and \(0.57 Mpc^2\) respectively. This increase of coverage with redshift is expected because a fixed angle will cover a larger tangential area at larger distances. However, as you come down the list (to higher redshifts) you will notice that this relation does not hold! The largest coverage is at \(z=1.6\): at higher redshifts, the area decreases, and continues decreasing!!! In flat FLRW cosmology (including \(\Lambda{}\)CDM), the only factor contributing to this is the \((1+z)\)$ factor from the expansion of the universe, see the Wikipedia page, with no curvature effect.

In case you have TOPCAT, you can visualize this as a plot (if you do not have TOPCAT, see TOPCAT). To do so, first you need to save the output of the loop above into a FITS table by piping the output to Gnuastro’s Table program and giving an output name:

$ for z in $(seq 0.1 0.1 5); do                                  \
    k=$(astcosmiccal -z$z --arcsectandist);                      \
    echo $z $k $a | awk '{print $1, ($2*60)^2 * $3 / 1e6}';   \
  done | asttable --output=z-vs-tandist.fits

You can now use Gnuastro’s astscript-fits-view to open this table in TOPCAT with the command below. Do you remember this script from Dataset inspection and cropping? There, we used it to view a FITS image with DS9! This script will see if the first dataset in the image is a table or an image and will call TOPCAT or DS9 accordingly: making it a very convenient tool to inspect the contents of all types of FITS data.

$ astscript-fits-view z-vs-tandist.fits

After TOPCAT opens, you will see the name of the table z-vs-tandist.fits in the left panel. On the top menu bar, select the “Graphics” menu, then select “Plain plot” to visualize the two columns printed above as a plot and get a better impression of the turn over point of the image cosmological coverage.