## Relativity, Cosmology, and Time

### Hubble Constant and 2018, 2020, 2024 Updates

#### 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,
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 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.

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 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 :
(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.
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:
• In early 20th Century, determination of the Hubble constant depends on measuring the red shift z (and thus the expansion velocity) and the distance derived from the Cepheid Variable of nearly galaxies (up to ~ 2 Mpc at z ~ 0.001). The 1929 value of H0 = 500 (km/sec)/Mpc yields 2 Gyr for the age of the universe T ~ 1/H0. This value is certainly incorrect as it is more than 10 times lower than the age of various globular clusters. Since then the estimated value of H0 has been steadily improved over the years to cover off the problem (Figure 11f). Figure 11g shows the latest effort using Type 1a Supernova as standard candle. The method extents the distance to 10 Gpc at z ~ 1. It presents the comparison of some theoretical CDM models (in term of comoving distance Dc vs z) with observational data. The supposedly linear relationship between velocity and red shift is distorted by :

• The effect of General Relativity as shown by Eq.(4) below.
• The original form of Hubble's Law is recovered by equating DH = c/H0, then from Eq.(3) we obtain
H0Dc = v = c dz' / [M(1+z')3 + k(1+z')2 + ]1/2 ---------- Eq.(4)
• The linear Hubble's Law of v ~ cz = H0Dc is apparent in this graph only for z << 1, and k = 0 for which case i(i) = 1.
• Distortion from linear relationship with z in the Hubble diagram is even more pronounced if one of the variable is plotted in logarithmic scale as shown in Figure 11d.

#### Figure 11g Modern Hubble Diagram [view large image]

• Since the cosmological model depends on the various density parameters which is derived from the CMBR power spectrum, and the Hubble constant in turn has some influence on the CMBR; they form a coupled system as shown in Figures 11h,i. These parameters and a few more extras have to be adjusted to fit the observational data. See "Theoretical Models (on CMBR Spectrum)" for further explanation.
• Figure 11h illustrates how to adjust the input parameters to match a theoretical model to the observational data, which are measured by WMAP. The demonstration software is supplied by NASA's "Build a Universe". The Blue curve is the result of the "best fit". The official value for H0 is actually ~ 71 (km/sec)/Mpc (instead of 73 in the demonstration run, see "WMAP parameters"). The ESA/Planck team has its own "Cosmological parameters", which quotes a H0 value of 67.7 (km/sec)/Mpc - a rather substantial difference from the WMAP's estimate (see Tension). The latest estimate of H0 ~ 70 (km/sec)/Mpc in 2019 is derived from the TRGB measurement.

#### Figure 11i CMBR, H0 Dependence

Figure 11i shows the effect of varying the Hubble constant on the CMBR theoretical curve. The most affected portion is related to the time-dependent perturbations of the gravitational field at large scale.

Cosmic Ruler "Baryon Acoustic Oscillations (BAO)" is referred to the outward bound sound wave originated at recombination
(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.

• (a) Initially soon after the Big Bang, dark matter, baryonic matter, and photons etc. all mixed together in the cosmic plasma. Left over quantum fluctuations produced over-density spots everywhere.
• (b) The high pressure in those spots would generate 3-D spherical sound waves. There would be lot of such wavelets in the surrounding fluid.
• (c) At recombination time, the photons can travel freely through space, forming the CMBR; while acoustic wave propagation stops abruptly (since there is no more radiation to support the generation of waves). Systems of the wavelets originated in the last scattering shell (a shell between the last scattering of photon and the beginning of recombination ~ 20 Mpc thickness, see Figure 11k), leaving an imprint in the matter distribution called sound horizon as the wave is stalled at a distance of about 153 Mpc.
• #### Figure 11k BAO in Comoving Coordinates [view large image]

• (d) Hundreds of million years later, the over-density sound horizon should produce higher concentration of galaxies. The BAO measurements have used correlation function to look for such feature out of the jumbled galaxy distribution successfully. As shown in Figure 11m, there is high correlation of galaxies at small separation distances (due to the clumpy nature of galaxy formation) and a low correlation at large separation distances until at a point corresponding to the distance traveled by the initial over-density in a spherical shell, where averaging in the correlation is not random anymore and shows up as

#### Figure 11m Density Correlation

a peak in the diagram. There would be a lot of such spherical structures known as standard rulers mixing with other astronomical objects in certain part of the universe as the cosmos expanded (see Figure 11m2,a).
See "Cosmological Constraints from Baryonic Acoustic Oscillation Measurements" for a review of cosmic BAO.

• (e) BAO structures among group of galaxies can be located from the correlation function (Figure 11m2,a). The apparent size or the subtended angle (see Figure 11m2,c) is related to its distance to Earth. Mathematical derivation below also shows that the Hubble parameter at such location can be calculated accordingly. Theoretically, the value of dark energy is determined by the Hubble parameter. It prompts more galaxy survey in 2019 by the Dark Energy Spectroscopic Instrument (DESI) to investigate the nature of dark energy (see "Sky Map to Plot Dark Energy").

#### Figure 11m2 Hubble Parameter [view large image]

1. Thus, the Hubble parameter (or the rate of cosmic expansion dR/dt) can reveal the nature of the dark energy by evaluating the w-parameter at various values of the red-shift z. Here's a few comments :
2. The w-parameter characterizes the Equation Of State : p = w c2 , where is the matter/energy density and p the pressure.
3. Various knind of matter/energy density has different value of w. For example :
w = 1/3 for ultra-relativistic matter (such as photon and neutrino).
w = 0 for non-relativistic matter moving with v << c.
w < -1/3 for matter/energy denstiy accelerating the expansion of the universe.
w = -1 for repulsive energy density ascribed to the cosmological constant.
w < -1 for very repulsive energy density leading to the "Big Rip" of the universe.
4. The value of w cannot be evaluated at z = 0 by Eq.(5) as it becomes indeterminate.
5. From the eBOSS measurement in Figure 11m2,d dR/dt = 56, z = 0.7, the CDM parameters M = 0.25, = 0.75, and
H0 = 70, the w-parameter can be calculated by Eq.(5) giving a value of -1.05.
6. The w-parameter as described above in Eq.(5) is supposed to be a constant that has to be verified by a lot of very accurate observations over very large range of the red-shift z.
7. The rate of cosmic expansion reaches its minimum dR/dt = 57 (km/s)/Mpac at z = 0.8 for H0 = 70 (km/s)/Mpac, w = -1.
8. It is conceivable that the whole universe can be considered as a fluid of different components having variable w at different time, i.e., at different red-shift z as described below.
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).
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 :
• The cosmic fluid in the universe doesn't move, it is the spatial expansion which creates the impression of motion.
• The continuity equation can be equated to the density parameters such as :

1/R3(1+w) = --------- Eq.(9a), or
3(1+w)log(1+z) = log(), or
w = {(1/3)[log()/log(1+z)]} - 1 ---------- Eq.(9b),
where = [(1+z)4 + M(1+z)3 + ] .

• for w > 0, the density decreases from small R to large R. The radiation era ends at z ~ 4000 (R ~ 10-4) with w = 1/3, and 1/R4.
• The matter era starts from w = 0, and 1/R3. It has been over taken by dark energy at z ~ 0.8 (R ~ 0.56).
• The dark energy era has w = -1 with = 0 = constant. Matter-energy is somehow infused into the universe to keep unchanged during the accelerated expansion - probably as "Vacuum Energy Density" from the newly created space in the expansion.
• Thus, w is actually a function of red shift z depending on the dominant component at that time. For k = 0, if the total density is known at z, then it can be calculated by Eq.(9b).

For the two extreme cases :
• When z , the radiation term is the dominant component, thus
w (1/3)[log()/log(1+z)] + 1/3 1/3.
• For z 0, w = (1/3)[log()/log(1+z)] - 1 , Eq.(9b) is not applicable.

• For intermediate value of z, all terms in have to be taken into account to determine w as shown in Figure 11o, which is calculated by a Basic program
• #### Figure 11o w-parameter and cosmic density vs log(1+z) [view large image]

running in lap-top computer. The computation is cut off at log(1+z) ~ 4, beyond which it returns the "out of range" error message.

• In the faraway future R >> 1, all terms in vanishes except the dark energy, which then becomes the critical density and thus = 1. The continuity equation could be satisfied only if w = -1 as shown by Eq.(9a).
• It seems that the cosmic density as function of z (see Figure 11o and conversion of z to various scales in Figures 11p and 11q) can be determined by using just one w-parameter without regard to the detailed composition as shown by Eq.(8). Actually, the w-parameter is computed directly from such composition (see Eq.(9b)) - providing a different point of view on the same thing. There's no free lunch in this world. Actually, the continuity equation provides an extra condition to determine an extra unknown, i.e., the w parameter as a function of z, which dictates ultimately the equation of state over the entire cosmic history since w=p/c2.

#### 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.

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 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).

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]

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 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.

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 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.
See the September 2021 H0 = 71 (km/sec)/Mpc via EDE.

[2024 Update]

In 2024, there were 2 publications (see "Could JWST solve cosmology’s big mystery?" + "The biggest mysteries") on the Hubble constant using the same standard candles (Cepheid, TRGB, and Carbon Stars) from JWST observations. The two results are at odd with 73 vs 69.1 km/sec-Mpc respectively (see Figure 11y). The discrepancy could cause by the fewer data (than by Hubble Space Telescope), choice of Density Parameters, or theoretical modeling. Anyway, the tension is still around at this point.

#### Figure 11y Hubble Constant, 2024 [view large image]

[End of 2024 Update]