Appendix 1: Measured Values for the Hubble Constant
219
Table 13.1 Values for the hubble constant
Method
Value
Error (approx.)
Reference
Ladder, sn
74.03
1.4
Reiss (2019)
Ladder, rg
69.8
2.5
Freedman (2019)
Lensing
67.4
4.1
Birrer (2020)
CMB
67.36
0.5
Planck (2018)
GW
70
10
Holz (2019)
The theoretical CMB spectrum obviously depends on the scale factor at the time of
emission as well as other physical properties such as the energy density over time
of the radiation, and the velocity of sound and standing waves in the cosmic fluid
(Knox 2019). (We will say more in Sect. 17.4.) By comparing the theoretical CMB
spectrum with the observed spectrum we can thus make a best fit that gives values
for the various cosmological parameters, in particular H 0 . The value obtained in this
way is about H 0 = 67 (km/s)/Mpc (Chalinor 2012; Planck 2018; NASA 2019)
Figure 13.7, along with Table 13.1, shows some of the interesting observational
results. It includes ladder method results using supernovae, red giants and gravitational lensing (Reiss 2019; Freedman 2019; Chen 2019). The CMB result using the
Planck satellite is shown, as well as the standard siren result based on gravitational
wave observations by LIGO (Planck 2018; Holz 2018). The values are clearly not in
violent disagreement, but the error estimates do not overlap; this causes concern for
many cosmologists (Crane 2019; Reiss 2019; Planck 2018).
The supernova ladder and CMB values differ by roughly 4 times more than the
observers error estimates (Reiss 2019). The red giant value lies between the supernova
value and the CMB value. The lensing value depends on the method of data analysis.
From this it appears that either the observers are overly optimistic concerning their
error estimates, or there is a problem with the basic theory and we must go beyond
the LCDM model. One example is that the number of neutrino types assumed may
be incorrect. Another is that the exponent for the dark energy term in (14.19) is not
correct. There are many possibilities (Knox 2019).
A historical note is in order. As we noted in Sect. 13.1, for decades the value of H 0
was unknown to about a factor of 2; the values favored by different observer groups
were about 50 and 100. No basic theoretical ideas emerged from this disagreement,
and it was resolved by later observations. For our pedagogical purposes we have used
the rough error estimate of about 5 (km/s)/Mp that pessimistically encompasses all
the individual error estimates.
Exercises
13.1 Take the radius of the observable universe to be roughly 10 billion light years,
and the separation between galaxies to be roughly 1 million light years. Very
roughly, how many galaxies are there in the observable universe? How many
stars? How many planets?
219
Table 13.1 Values for the hubble constant
Method
Value
Error (approx.)
Reference
Ladder, sn
74.03
1.4
Reiss (2019)
Ladder, rg
69.8
2.5
Freedman (2019)
Lensing
67.4
4.1
Birrer (2020)
CMB
67.36
0.5
Planck (2018)
GW
70
10
Holz (2019)
The theoretical CMB spectrum obviously depends on the scale factor at the time of
emission as well as other physical properties such as the energy density over time
of the radiation, and the velocity of sound and standing waves in the cosmic fluid
(Knox 2019). (We will say more in Sect. 17.4.) By comparing the theoretical CMB
spectrum with the observed spectrum we can thus make a best fit that gives values
for the various cosmological parameters, in particular H 0 . The value obtained in this
way is about H 0 = 67 (km/s)/Mpc (Chalinor 2012; Planck 2018; NASA 2019)
Figure 13.7, along with Table 13.1, shows some of the interesting observational
results. It includes ladder method results using supernovae, red giants and gravitational lensing (Reiss 2019; Freedman 2019; Chen 2019). The CMB result using the
Planck satellite is shown, as well as the standard siren result based on gravitational
wave observations by LIGO (Planck 2018; Holz 2018). The values are clearly not in
violent disagreement, but the error estimates do not overlap; this causes concern for
many cosmologists (Crane 2019; Reiss 2019; Planck 2018).
The supernova ladder and CMB values differ by roughly 4 times more than the
observers error estimates (Reiss 2019). The red giant value lies between the supernova
value and the CMB value. The lensing value depends on the method of data analysis.
From this it appears that either the observers are overly optimistic concerning their
error estimates, or there is a problem with the basic theory and we must go beyond
the LCDM model. One example is that the number of neutrino types assumed may
be incorrect. Another is that the exponent for the dark energy term in (14.19) is not
correct. There are many possibilities (Knox 2019).
A historical note is in order. As we noted in Sect. 13.1, for decades the value of H 0
was unknown to about a factor of 2; the values favored by different observer groups
were about 50 and 100. No basic theoretical ideas emerged from this disagreement,
and it was resolved by later observations. For our pedagogical purposes we have used
the rough error estimate of about 5 (km/s)/Mp that pessimistically encompasses all
the individual error estimates.
Exercises
13.1 Take the radius of the observable universe to be roughly 10 billion light years,
and the separation between galaxies to be roughly 1 million light years. Very
roughly, how many galaxies are there in the observable universe? How many
stars? How many planets?