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from an undefined
place,
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create (a place)
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— Viorica Hrincu
Sometimes when you stare at the void, the void sends you a poem.
Universe—Superclusters and Voids
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.
Propensity score weighting
The needs of the many outweigh the needs of the few. —Mr. Spock (Star Trek II)
This month, we explore a related and powerful technique to address bias: propensity score weighting (PSW), which applies weights to each subject instead of matching (or discarding) them.
Kurz, C.F., Krzywinski, M. & Altman, N. (2025) Points of significance: Propensity score weighting. Nat. Methods 22:1–3.
Happy 2025 π Day—
TTCAGT: a sequence of digits
Celebrate π Day (March 14th) and sequence digits like its 1999. Let's call some peaks.
Crafting 10 Years of Statistics Explanations: Points of Significance
I don’t have good luck in the match points. —Rafael Nadal, Spanish tennis player
Points of Significance is an ongoing series of short articles about statistics in Nature Methods that started in 2013. Its aim is to provide clear explanations of essential concepts in statistics for a nonspecialist audience. The articles favor heuristic explanations and make extensive use of simulated examples and graphical explanations, while maintaining mathematical rigor.
Topics range from basic, but often misunderstood, such as uncertainty and P-values, to relatively advanced, but often neglected, such as the error-in-variables problem and the curse of dimensionality. More recent articles have focused on timely topics such as modeling of epidemics, machine learning, and neural networks.
In this article, we discuss the evolution of topics and details behind some of the story arcs, our approach to crafting statistical explanations and narratives, and our use of figures and numerical simulations as props for building understanding.
Altman, N. & Krzywinski, M. (2025) Crafting 10 Years of Statistics Explanations: Points of Significance. Annual Review of Statistics and Its Application 12:69–87.
Propensity score matching
I don’t have good luck in the match points. —Rafael Nadal, Spanish tennis player
In many experimental designs, we need to keep in mind the possibility of confounding variables, which may give rise to bias in the estimate of the treatment effect.
If the control and experimental groups aren't matched (or, roughly, similar enough), this bias can arise.
Sometimes this can be dealt with by randomizing, which on average can balance this effect out. When randomization is not possible, propensity score matching is an excellent strategy to match control and experimental groups.
Kurz, C.F., Krzywinski, M. & Altman, N. (2024) Points of significance: Propensity score matching. Nat. Methods 21:1770–1772.
Understanding p-values and significance
P-values combined with estimates of effect size are used to assess the importance of experimental results. However, their interpretation can be invalidated by selection bias when testing multiple hypotheses, fitting multiple models or even informally selecting results that seem interesting after observing the data.
We offer an introduction to principled uses of p-values (targeted at the non-specialist) and identify questionable practices to be avoided.
Altman, N. & Krzywinski, M. (2024) Understanding p-values and significance. Laboratory Animals 58:443–446.
Depicting variability and uncertainty using intervals and error bars
Variability is inherent in most biological systems due to differences among members of the population. Two types of variation are commonly observed in studies: differences among samples and the “error” in estimating a population parameter (e.g. mean) from a sample. While these concepts are fundamentally very different, the associated variation is often expressed using similar notation—an interval that represents a range of values with a lower and upper bound.
In this article we discuss how common intervals are used (and misused).
Altman, N. & Krzywinski, M. (2024) Depicting variability and uncertainty using intervals and error bars. Laboratory Animals 58:453–456.