Talk:2853: Redshift

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Does it make any sense to try and relate the two Z values to a more "normal" time? MAP (talk) 21:08, 10 November 2023 (UTC)

I have added exactly that, using `astropy.cosmology` for the calculations Juandesant (talk) 21:16, 10 November 2023 (UTC)

There is a big error in the current explanation, saying it means they were interested in this since the early universe, but that would only be true if the z value was much closer to 1. I'm not exactly sure as I haven't done the math, but with that many decimal places of zero's it is probably near enough history to be during their lifetime. So the joke isn't being interested in the early universe since the early universe, but rather so interested in it that they talk about other things with the same terminology.--172.70.126.142 21:18, 10 November 2023 (UTC)

Why aren't these values expressed in scientific notation? Separately, do they make sense logarithmically? JohnHawkinson (talk) 07:12, 11 November 2023 (UTC)

Part of the joke is that the values are unwieldy to use compared to everyday language and units. The scientific notation would have lessened this. However they are given in a universal standard time instead of some solar system or human related units, so they generally should be clearly preferred. Sebastian --172.68.110.188 15:37, 11 November 2023 (UTC)

I feel like the Earth's gravitational field would affect the answer as z goes to zero? 172.69.22.210 06:57, 12 November 2023 (UTC)

Calculation

Thanks to ChatGPT-4 and the Fortran-90 code from arxiv:1303.5961 (which please see, it's really cool) here's how to get the look-back time from redshift in Python: 

from scipy.special import hyp2f1  # hypergeometric 2F1 function is in integral solution
from numpy import format_float_positional

# Cosmological parameters from the Fortran params.f90 header
#H0 = 67.15       # Hubble constant in km/s/Mpc (or, 73.5: the "crisis in cosmology")
H0 = 69.32        # from Explainxkcd for 2853: Redshift; seems a consensus compromise
#OL = 0.683       # Cosmological constant for dark energy density, Omega_Lambda or _vac
#Om = 0.317       # Density parameter for matter, Omega_mass
Om = 0.286        # From https://arxiv.org/pdf/1406.1718.pdf page 8
OL = 1.0 - Om - 0.4165/(H0**2)  # flat curvature, from https://www.astro.ucla.edu/~wright/CosmoCalc.html
#print(f"{OL=:.3F}")  # 0.714

# Age of universe at redshift z as a closed-form solution to its integral definition, ...
def age_at_z(z):  # ...which is 27 times faster than the original numeric integration.
    hypergeom = hyp2f1(0.5, 0.5, 1.5, -OL / (Om * (z + 1)**3))
    return (2/3) * hypergeom / (Om**0.5 * (z + 1)**1.5) * (977.8 / H0)  # 977.8 for Gyr

# Current age of the universe at redshift 0 in Gyr
age0 = age_at_z(0)  # 13.78

# Function to calculate the look-back time at redshift z in Gyr
def zt(z):  # from the function name in the Fortran cosmonom.f90 code
    return age0 - age_at_z(z)

# For z = 0.00000000038
z1 = 0.00000000038
look_back_time_years_z1 = zt(z1) * 1e9 # from Gyr to years
print("Look-back time for z=" + 
      f"{format_float_positional(z1)}: {look_back_time_years_z1:.1f} years")

# For z = -0.000000000000045
z2 = -0.000000000000045
look_back_time_hours_z2 = zt(z2) * 1e9 * 365.25 * 24
print("Look-back time for z=" +
      f"{format_float_positional(z2)}: {look_back_time_hours_z2:.1f} hours")

The output being: 

Look-back time for z=0.00000000038: 5.4 years
Look-back time for z=-0.000000000000045: -5.5 hours

And should you wish to make this quick reference chart for interpreting observations in the JWST era:

Supplemental code for Look-back Time by Redshift chart 
import matplotlib.pyplot as plt
import matplotlib.ticker as ticker

redshifts = [z * 20 / 299 for z in range(300)]  # 0 to 20 in 300 steps
look_back_times = [zt(z) for z in redshifts]

plt.plot(redshifts, look_back_times)
plt.title('Look-back Time by Redshift')
plt.xlabel('z')
plt.ylabel('Billion Years Ago')
plt.xticks(range(21))
plt.yticks(list(range(14)) + [age0])
plt.gca().yaxis.set_major_formatter(ticker.FormatStrFormatter('%.1f'))
plt.grid(True, color='lightgray')
plt.gca().spines['right'].set_visible(False)
plt.gca().spines['top'].set_visible(False)

for t in range(0, 13):
    z = redshifts[min(range(len(look_back_times)),
                      key=lambda i: abs(look_back_times[i]-t))]
    plt.text(z, t, f"   z = {z:.2f}", ha='left', va='center', fontsize='small')

for z in range(7, 20, 2):
    t = zt(z)
    plt.text(z, t - 0.2, f"{t:.2f}", ha='center', va='top', fontsize='small')

for z in range(6, 21, 2):
    t = zt(z)
    plt.text(z, t + 0.1, f"{t:.2f}", ha='center', va='bottom', fontsize='small')

plt.savefig('time_by_redshift.png', bbox_inches='tight')
#plt.show()  # https://i.ibb.co/v4nt6GM/time-by-redshift.png

Or this one showing the age of the universe from redshift:

Supplemental code for Age of Universe by Redshift chart 
plt.clf()  # reset plot

redshifts = [z * 15 / 299 + 5 for z in range(300)]  # 5 to 20 in 300 steps
ages = [age_at_z(z) * 1000 for z in redshifts]  # Gyr to million years 

plt.plot(redshifts, ages)
plt.title('Age of Universe by Redshift')
plt.xlabel('z')
plt.ylabel('Million Years')
plt.xticks(range(5, 21))
plt.yticks(range(0, 1300, 100))
plt.grid(True, color='lightgray')
plt.gca().spines['right'].set_visible(False)
plt.gca().spines['top'].set_visible(False)

plt.savefig('age_by_redshift.png', bbox_inches='tight')
#plt.show()  # https://i.ibb.co/yybQpjL/age-by-redshift.png

Enjoy! 172.69.22.12 08:50, 12 November 2023 (UTC)

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