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from an undefined
place,
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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 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.
Convolutional neural networks
Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry. – Richard Feynman
Following up on our Neural network primer column, this month we explore a different kind of network architecture: a convolutional network.
The convolutional network replaces the hidden layer of a fully connected network (FCN) with one or more filters (a kind of neuron that looks at the input within a narrow window).
Even through convolutional networks have far fewer neurons that an FCN, they can perform substantially better for certain kinds of problems, such as sequence motif detection.
Derry, A., Krzywinski, M & Altman, N. (2023) Points of significance: Convolutional neural networks. Nature Methods 20:.
Background reading
Derry, A., Krzywinski, M. & Altman, N. (2023) Points of significance: Neural network primer. Nature Methods 20:165–167.
Lever, J., Krzywinski, M. & Altman, N. (2016) Points of significance: Logistic regression. Nature Methods 13:541–542.
Neural network primer
Nature is often hidden, sometimes overcome, seldom extinguished. —Francis Bacon
In the first of a series of columns about neural networks, we introduce them with an intuitive approach that draws from our discussion about logistic regression.
Simple neural networks are just a chain of linear regressions. And, although neural network models can get very complicated, their essence can be understood in terms of relatively basic principles.
We show how neural network components (neurons) can be arranged in the network and discuss the ideas of hidden layers. Using a simple data set we show how even a 3-neuron neural network can already model relatively complicated data patterns.
Derry, A., Krzywinski, M & Altman, N. (2023) Points of significance: Neural network primer. Nature Methods 20:165–167.
Background reading
Lever, J., Krzywinski, M. & Altman, N. (2016) Points of significance: Logistic regression. Nature Methods 13:541–542.
Cell Genomics cover
Our cover on the 11 January 2023 Cell Genomics issue depicts the process of determining the parent-of-origin using differential methylation of alleles at imprinted regions (iDMRs) is imagined as a circuit.
Designed in collaboration with with Carlos Urzua.
Akbari, V. et al. Parent-of-origin detection and chromosome-scale haplotyping using long-read DNA methylation sequencing and Strand-seq (2023) Cell Genomics 3(1).
Browse my gallery of cover designs.
Science Advances cover
My cover design on the 6 January 2023 Science Advances issue depicts DNA sequencing read translation in high-dimensional space. The image showss 672 bases of sequencing barcodes generated by three different single-cell RNA sequencing platforms were encoded as oriented triangles on the faces of three 7-dimensional cubes.
More details about the design.
Kijima, Y. et al. A universal sequencing read interpreter (2023) Science Advances 9.
Browse my gallery of cover designs.
Regression modeling of time-to-event data with censoring
If you sit on the sofa for your entire life, you’re running a higher risk of getting heart disease and cancer. —Alex Honnold, American rock climber
In a follow-up to our Survival analysis — time-to-event data and censoring article, we look at how regression can be used to account for additional risk factors in survival analysis.
We explore accelerated failure time regression (AFTR) and the Cox Proportional Hazards model (Cox PH).
Dey, T., Lipsitz, S.R., Cooper, Z., Trinh, Q., Krzywinski, M & Altman, N. (2022) Points of significance: Regression modeling of time-to-event data with censoring. Nature Methods 19:1513–1515.
Music video for Max Cooper's Ascent
My 5-dimensional animation sets the visual stage for Max Cooper's Ascent from the album Unspoken Words. I have previously collaborated with Max on telling a story about infinity for his Yearning for the Infinite album.
I provide a walkthrough the video, describe the animation system I created to generate the frames, and show you all the keyframes
The video recently premiered on YouTube.
Renders of the full scene are available as NFTs.