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Relativity, Cosmology, and Time

Hubble Constant and 2018, 2020 Updates

( Hubble's Law in Modern Cosmology,   Hubble Constant Over the Last 100 Years,   The Role of BAO,

Continuity Equation and the w Parameter,   Tension with Hubble Constant)

The Hubble's Law was first published in 1929 in the form of : v = H0D,
Cosmic Expansion Proper Distance where v is the recessional velocity (in km/sec) of the astronomical object due to the expansion of the universe (Figure 11a), D is the proper distance (in mega parsecs = Mpc) to the same object including the effect due to the expansion (Figure 11b), and H0 is called the Hubble constant, which is in unit of (km/sec)/Mpc and can be interpreted as the increase in expanding velocity for every Mpc from the observer. The inverse of H0 is equated to the age of the universe ~ 13.8 Gyr (see more accurate formula).

Figure 11a Cosmic Expansion
[view large image]

Figure 11b Proper Distance [view large image]

A recent practice often expresses H0 = 100h (km/sec)/Mpc with the dimensionless h 100 times smaller than H0, i.e., h = H0/100[(km/sec)/Mpc], h = 0.7 is the popular choice.

Observationally for nearby astronomical objects, v can be measured from the red shift of spectral line, i.e., z = (e / 0) - 1 = v/c, where e is the red shifted wavelength, 0 is the wavelength of the original spectral line; while the proper distance D is obtained by Parallax or Cepheid Variables of those "Standard Candles" closer to Earth. In term of z, the Hubble's Law can be expressed as z = (H0/c)D = D/DH, where DH = c/H0 is the cosmic horizon. This formula is valid only for D << DH.

Standard Candles lambdaCDM Model For objects further away, the standard candles (Figure 11c) usually invokes the magnitude-distance relation to calculate D :
m = M - 97.5 + 5xlog(D),
where m is the apparent magnitude which can be measured directly, M is the absolute magnitude, which is unique for a special class of astronomical obejcts. The distance D can be calculated from the above formula once M is known.

Figure 11c Standard Candles
[view large image]

Figure 11d lambdaCDM Model [view large image]

As shown in Figure 11d, neither the naive v = cz nor the special relativistic expression : v = c[(1 + z)2 - 1] / [(1 + z)2 + 1] is suitable as they have not taken the cosmic expansion into account.

General Relativity is required for the formulation as prescribed by Eq.(4).

Starting from the k = 0 (flat space) Friedmann-Lemaitre-Robertson-Walker (FLRW) metric at time t :

ds2 = - c2dt2 + R(t)2 (dr2 + r2 (d2 + sin2 d2)).

For Line Of Sight (LOS) distance, d = 0, d = 0, the proper distance is defined by : dD = R(t)dr.
Then D(t) = dD = R(t)dr = R(t)r, where r is the initial separation (distance) of the two objects and does not change over time.

The rate of change for the proper distance due to cosmic expansion is v = dD/dt = (dR/dt)r = [(dR/dt)/R]D ---------- Eq.(1).

We obtain the Hubble's Law for the current epoch t = t0 :
v(t0) = H0D(t0),
where H0 = [(dR/dt)/R]t=t0 is the Hubble constant, the inverse of which is called Hubble Time t0 = "age of the universe" by definition, and D(t0) is the comoving distance.
    By expressing v = D(t0)/t0, it shows that :
  1. Object further away has higher velocity.
  2. The cosmic horizon DH = ct0 = c/H0.
  3. For D > DH, the velocity can excess c, that's OK since cosmic expansion is not restricted by Special Relativity.
The relationship between the red shift z and the scale factor R(t) can be derived by considering the propagation of light between the astronomical object and the observer on Earth. For this case, ds = 0, the FLRW metric is reduced to : dr = cdt/R(t), which indicates the ratio is independent of t. If the galaxy emits one cycle of light wave with wavelength e at time te, then dt = [(te + e/c) - te] = e/c; similarly the observer would receive the red shifted wave with dt = 0/c . Equating the ratio yields e/R(te) = 0/R(t0), thus :
1 + z = (0 / e) = R(t0)/R(te), or 1 + z = 1/R(t) with the usual convention of equating R(t0) = 1 and re-labeling te to t. It is through this relation the cosmological models are linked to the red shift z (Figure 11d).
BTW, dR = d[1/(1+z)] = -dz/(1+z)2

The dynamic of the scale factor R(t) is governed by the Friedmann equation :
Cosmological Models (dR/dt)2/R2 = H(R)2 = (H0)2(M/R3 + k/R2 + ) ---------- Eq.(2),
where M = 8GM/3 H02, k = -kc2/ H02, = c2/3 H02 are the density parameters, and M is the baryonic B + dark matter DM denstiy, k the spatial curvature, the cosmological constant as dark energy (all of them for the current epoch). This combination consists the modern Standard Cosmological Model called CDM model. The red curve in Figure 11e is the solution with M = 0.3, = 0.7 and k = 0.

Figure 11e Cosmological Models [view large image]

Since [(dR/dt)/R] = dv/dD = c(dz/dD) from Eq.(1), Eq.(2) can be rewritten as : dD = -[DH / (M(1+z)3 + k(1+z)2 + )1/2] dz, which can be interpreted as the infinitesimal change in the proper distance corresponding to the infinitesimal change in expansion speed dv or dz. The comoving distance Dc = D(t0) is just the sum of dD to z = to 0, while the proper distance is :

D = -DH dz' / [M(1+z')3 + k(1+z')2 + ]1/2 ---------- Eq.(3).

The relationship between D and z is plotted in Figure 11d for the CDM model.

The Light Travel or (Look Back) distance DT or DLB is defined by the time taken by the photon from source to observation.
Cosmic Distance Since ds = 0 for the photon cdt = dD, but it has to be modified by the Doppler effect due the cosmic expansion as well. Thus, an additional term is introduced into the equation, i.e.,
cdt + vdt = (1+z)cdt, or

dDT = cdt = dD/(1+z) = R(t)dD ---------- Eq.(3a),
DT = -DH dz' / {(1+z')[M(1+z')3 + k(1+z')2 + ]1/2} ---------- Eq.(3b).

The original Hubble's Law is recovered for z ~ 0, i.e., DT ~ Dc = D(t0) ~ cz/H0 or V(t0) ~ H0D(t0).
The age of the universe is computed from DT integrating from z to 0, it is not exactly equal to 1/H0 (see Figure 11u).

Figure 11e1 Cosmic Distance
[view large image]

Figure 11e1 is another attempt to explain the subtlety of Cosmic Distances. See also "Types of Cosmic Distance".

Hubble Constant:
Cosmic Ruler "Baryon Acoustic Oscillations (BAO)" is referred to the outward bound sound wave originated at recombination
BAO Horizon (decoupling) time. Initially, it traveled with the CMBR together. It lost energy and stalled as the photons stream ahead faster. The sound horizon with a fixed radius (in comoving coordinates) then became a relic with higher matter density to produce more astronomical objects such as galaxies at that location which is estimated to be about 153 Mpc ~ 500 Mlyr from the point of origin. Detection of such feature would yield information on some of the cosmic parameters as shown below after a brief summary on its evolution (see visual aid in Figure 11l).

Figure 11j BAO Horizon in Light Travel Distance

Figure 11j shows the BAO horizon expanding with the cosmic expansion.

W-Parameter: The effect of w is apparent if we consider the composition of the universe to be a fluid with matter-energy density and equation of state p/c2 = w.

From the Friedann equation :
[(dR/dt)/R]2 = 8G/3 ,
and the equation of continuity (see footnote for derivation) :
+ 3[(dR/dt)/R]( + p/c2) = 0 ---------- Eq.(6),
the equation for the acceleration of scale factor R(t) can be derived as :
(d2R/dt2) / R = - (4G/3) (1 + 3w) ---------- Eq.(7).

For relativistic matter w = 1/3, non-relativistic matter w = 0, w < -1/3 for cosmic acceleration. All observations are in favor of w = -1 for the current epoch in supporting the case of cosmological constant as the cause of cosmic acceleration. Early universe are not sensitive to until recently when the dark energy acceleration took over at z ~ 0.8 (see "Vacuum Energy Density").

The solution of Eq.(6) yields : = 0/R3(1+w) = 0(1+z)3(1+w) ---------- Eq.(8).
Continuity Equation This is the rationale to replace by /R3(1+w) as mentioned previously in BAO. Comparison with the equation of continuity in fluid dynamics shows that it is really not an equation for the conservation of mass (Figure 11n). It actually dictates the variation of the matter-energy density as the result of cosmic expansion.

Figure 11n Continuity Equation
[view large image]

It reveals that :
Cosmic Timeline Cosmic Parameters

Figure 11p Cosmic Timeline [view large image]

Figure 11q [view large image] Cosmic Parameters

Hubble Constant (2018):

Anyway, Figure 11r lists 14 H0 measurements in the 21st century. Neglecting the error bar in each of these measurements, the averaged value is 70.8 (km/sec)/Mpc corresponding to a cosmic age of 13.8 billion years.

Hubble Constant in 21st Century KIDS Survey Meanwhile, the SHOES Program (Supernovae and H0 for the Equation of State) have made precision measurement (<5%) of H0 by refining the SN and Cepheids observations. Their results are plotted in the insert of the same graph. There is a considerable discrepancy with the Planck's values after 2013 when the respective error bar would not overlap anymore. The measurement of the so called shear power S8 from the "KiDS-450 Weak Lensing Power Spectrum" has contributed additional tension with another

Figure 11r Hubble Constant in 21st Century

Figure 11s Shear Power S8 [view large image]

substantial difference from the Planck's evaluation (Figure 11s). It demands an explanation (see "How Heavy is the Universe? Conflicting Answers Hint at New Physics").
Such tension is revealed by an anecdote of an encounter between Catherine Heymans of KiDS and George Efstathiou, a senior figure on the Planck team :

Heymans recalls that when she presented the conflicting results in 2012, it seemed that George Efstathiou was always sitting in the front row, hand raised. "He would say, 'Catherine, can you tell the audience what you've done wrong?'," she says. "I didn't have the balls to say 'George, can you tell the audience what your team's done wrong?'" (as reported by New Scientist in the article "Dark energy is mutating, with grave consequences for the cosmos", December 9-15, 2017).

Hubble Constant 2019 2018 Update on the "tension" - "Star Map Adds to Cosmic Confusion"

2019 Update on the "tension" - The 2019 TRGB data produce a Hubble constant H0 ~ 70 (km/sec)/Mpc which sits right at the middle between the contentious measurements from local Cepheids and the faraway CMB (see Figure 11t, and New Measure of Hubble Constant").

Figure 11t Hubble Constant, 2019 [view large image]

Hubble Constant 2018 Age of the Universe, 2018 The grand finale for CMB observation occurred on July 17, 2018 when the Planck Collaboration released to the public a new and improved version of the data acquired by the Planck satellite. Among other cosmic parameters (see "Planck Data, 2018"), the Hubble constant remains little change at a value of 67.4 km/sec-Mpc (Figure 11u) corresponding to 13.8 billion years for the age of the universe (see formula in Figure 11v) with M = 0.317, and = 0.683. Thus the tension is un-resolved by this final measurement (funding for another CMB satellite is unlikely). It remains the biggest controversy in the modern view of the Universe. Note that although the expansion

Figure 11u Hubble Constant, 2018 + 2020 [view large image]

Figure 11v Age of the Universe, 2018

rate has a problem, the age of the universe as derived by WMAP and Planck turns out to be rather consistent, thank to the slightly different values of the density parameters.

Atacama Cosmology Telescope ACT-CMB In July 2020, the Atacama Cosmology Telescope (Figure 11w) collaboration submitted 2 CMB research papers for publication. From the observed CMB map (Figure 11x), it calculates a value of H0 = 67.6 1.1 km/s-Mpc in very close agreement with Planck's H0 = 67.9 1.5 km/s-Mpc. Indeed its derived cosmic parameters are in agreements with CMB measurements by WMAP and Planck as a group; but the tension with the "local" measurements remains un-resolved. It is suggested that perhaps

Figure 11w Atacama Cosmology Telescope [view large image]

Figure 11x ACT-CMB, 2020
[view large image]

there's something wrong with the CDM cosmological model.
See "Mystery over Universe’s expansion deepens with fresh data" by Nature, 23 July 2020.
Return to the "pre-21st Century Standard Cosmology".

§§ Footnote for derivation of the equation of continuity :
By definition c2 = E/V, and c2d = dE/V - (c2/V)dV;
but from thermodynamics dE = -pdV, while from cosmic expansion V R3 and dV = (3V/R)dR,
combining these equations yields : + 3[(dR/dt)/R]( + p/c2) = 0, or
= -3H(1+w), where H = (dR/dt)/R.

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