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▲ THE ENTIRE UNIVERSE | Put it on your wall. (buy artwork / see all my art)

π in the sky ·

If you like space, you will love this. The 2017 π Day art imagines the digits of π as a star catalogue with constellations of extinct animals and plants. The work is featured in the article Pi in the Sky at the Scientific American SA Visual blog.

Sky constellation figures ·

If you like space, you'll love my the 12,000 billion light-year map of clusters, superclusters and voids. Find the biggest nothings in Boötes and Eridanus.The largest map there is shows the location of voids and galaxy superclusters in our visible universe.

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from an undefined
place,
undefined
create (a place)
an account
of us
— Viorica Hrincu

Sometimes when you stare at the void, the void sends you a poem.

Universe—Superclusters and Voids

▲ The Universe — Superclustesr and Voids. The two supergalactic hemispheres showing Abell clusters (blue), superclusters (magenta) and voids (black) within a distance of 6,000 million light-years from the Milky Way.

The average density of the universe is about `10 \times 10^{-30} \text{ g/cm}^3` or about 6 protons per cubic meter. This should put some perspective in what we mean when we speak about voids as "underdense regions".

expressing distances in the universe

Distances in the universe can be expressed as either the light-travel distance or the comoving distance to the object. The first tells us how long light took to travel from the object to us.

For example, the furthest object observed is the galaxy GN-Z11 and its light-travel distance is 13 billion light-years (Gly).

But because space has expanded during the time the light from GN-Z11 has been travelling to us, the galaxy is now actually much further away. This is measured by its comoving distance which accounts for space expansion, which is 29.3 Gly for GN-Z11.

The redshift, `z`, is commonly used to specify distance, since it's a quantity that can be observed. For GN-Z11, `z = 11.09`.

All distances on the poster are expressed in terms of the light-travel distance.

To calculate these distances, the redshift `z` is used along with a few cosmological parameters.

The Hubble parameter, `H(z)`, is the function used for these calculations. It can be derived from the Friedmann equation. $$ H(z) = H_0 \sqrt { \Omega_r({1+z})^4 + \Omega_m({1+z})^3 + \Omega_k({1+z})^2 + \Omega_\Lambda } $$

The values of the parameters in `H(z)` are being continually refined and the values of some depend on various assumptions. I use the Hubble constant `H_0 = 69.6 \text{ km/s/Mpc}`, mass density of relativistic particles `\Omega_r = 8.6 \times 10^{-5}`, mass density `\Omega_m = 0.286`, curvature `\Omega_k = 0` and dark energy fraction `\Omega_\Lambda = 1 - \Omega_r - \Omega_m - \Omega_k = 0.713914`.

Bennett, C.L. et al The 1% Concordance Hubble Constant Astrophysical Journal 794 (2014)

Now given a redshift, `z` the light-travel distance is $$ d_T(z) = c \int_0^z \frac{dx}{({1+x})E(x)} $$

The age of the universe can be computed from this expression. The edge of the universe has an infinite redshift so w can calculate it using `\lim_{z \rightarrow \infty} d_T(z)`.

The comoving distance to the object with redshift `z` is $$ d_C(z) = c \int_0^z \frac{dx}{E(x)} $$

It's convenient to express the above integrals by making a variable substitution. Using the scale factor `a = 1/(1+z)`, $$E(a) = H_0 \sqrt { \frac{\Omega_r}{a^2} + \frac{\Omega_m}{a} + \Omega_k + a^4\Omega_\Lambda } $$

The light-travel distance is $$D_T(z) = c \int_a^1 \frac{dx}{E(x)}$$

The comoving distance is $$D_C(z) = c \int_a^1 \frac{dx}{xE(x)}$$

The light-travel distance to the edge of the universe is $$D_{T_U}(z) = c \int_0^1 \frac{dx}{E(x)}$$

and the light-travel distance from the edge of the universe to the object as we're observing it now is $$D_{T_0}(z) = c \int_0^a \frac{dx}{E(x)}$$

which can be interpreted as the age of the object when it emitted the light that we're seeing now.

The proper size of the universe is the comoving distance to its edge, $$D_{C_U}(z) = c \int_0^1 \frac{dx}{xE(x)}$$

Below you can You can download the full script.

### Cosmological distance calculator
### Martin Krzywinski, 2018
#
# The full script supports command-line parameters
# http://mkweb.bcgsc.ca/universe-voids-and-superclusters/cosmology_distance.py

z  = 1                     # redshift 
a  = 1/(1+z)               # scale factor
Wm = 0.286                 # mass density
Wr = 8.59798189985467e-05  # relativistic mass
Wk = 0                     # curvature
WV = 1 - Wm - Wr - Wk      # dark matter fraction
n  = 10000                 # integration steps

# Hubble parameter, as function of a = 1/(1+z)
def Ea(a,Wr,Wm,Wk,WV):
    return(math.sqrt(Wr/a**2 + Wm/a + Wk + WV*a**2))

H0   = 69.6            # Hubble constant
c    = 299792.458      # speed of light, km/s
pc   = 3.26156         # parsec to light-year conversion
mult = (c/H0)*pc/1e3   # integrals are in units of c/H0, converts to Gy or Gly
sum_comoving = 0
sum_light    = 0
sum_univage  = 0
sum_univsize = 0

for i in range(n):
    f   = (i+0.5)/n
    x   = a + (1-a) * f # a .. 1
    xx  = f             # 0 .. 1
    ex  = Ea(x,args.Wr,args.Wm,args.Wk,args.WV)
    exx = Ea(xx,args.Wr,args.Wm,args.Wk,args.WV)
    sum_comoving += (1-a)/(x*ex)
    sum_light    += (1-a)/(  ex)
    sum_univsize += 1/(xx*exx)
    sum_univage  += 1/(   exx)

results = [mult*i for i in [sum_univage,sum_univsize,sum_univage-sum_light, \
                            sum_light,sum_comoving]]
print("z {:.2f} U {:f} Gy {:f} Gly T0 {:f} Gy T {:f} Gly C {:f} Gly". \
      format(args.z,*results))

Use the script to generate distances for a given redshift, `z`. For example,

# For galaxy GN-Z11, furtest object ever observed
./cosmology_distance.py -z 11.09
z 11.09 U 13.720 Gy 46.441 Gly T0 0.414 Gy T 13.306 Gly C 32.216 Gly

The galaxy GN-Z11 has a light-travel distance of 13.3 Gly and a comoving distance of 32.2 Gly. We're seeing it now as it was only 0.4 Gy after the beginning of the universe, which is 13.7 Gy old and the distance to its edge is 46.4 Gly.

# For quasar J1342+0928, furthest quasar ever observed
./cosmology_distance.py -z 7.54
z 7.54 U 13.720 Gy 46.441 Gly T0 0.699 Gy T 13.021 Gly C 29.355 Gly

The values for U (age and size of universe), will always be the same for a given set of cosmological parameters for any value of `z`. I include them in the output of the script for convenience.

These values match those generated by Ned's online cosmological calculator for a flat universe.