## Relativity, Cosmology, and Time

### Hubble Constant and 2018 Update

#### Continuity Equation and the w Parameter,   Hubble Constant in 21st Century)

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 such 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].

Observationally for nearby astronomical objects, v can be measured from the red shift of spectral line, i.e., z = (0 / e) - 1 = v/c, where 0 is the red shifted wavelength, e 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 with dt = 0 :

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

For Line Of Sight (LOS) distance, d = 0, d = 0, it follows that : ds = R(t)dr.
The proper distance D(t) = ds = R(t)dr = R(t)r, where r is the initial separation (distance) of the two objects and does not change over time,

then v = dD/dt = (dR/dt)r = [(dR/dt)/R]D ---------- Eq.(1).

Finally, 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.
If there is no force acting on the pair of objects, the corresponding v(t0) is a constant, and the elapsed time is just t0 = D(t0)/v(t0) = D/(H0D) = 1/H0 which is called Hubble Time and is equated to the "age of the universe" by definition.
By expressing v = D/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).

The dynamic of the scale factor R(t) is governed by the Friedmann equation :
(dR/dt)2/R2 = (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]

Eq.(2) can be rewritten in the form : [(dR/dt)/R]DHdz(M(1+z)3 + k(1+z)2 + )-1/2 = cdz = dv.
Comparison to Eq.(1) shows that dD = DHdz(M(1+z)3 + k(1+z)2 + )-1/2 can be interpreted as the infinitesimal change in the proper distance corresponding to the infinitesimal change in velocity dv. The comoving distance Dc = D(t0) is just the sum of dD such as :

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

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

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 11f,g,h. These parameters and a few more extras have to be adjusted to fit the observational data.

#### Figure 11h CMBR-Hubble Dependence

See "Theoretical Models (on CMBR Spectrum)" for further explanation.

• Figure 11f 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. The latest estimate of H0 = 70 (km/sec)/Mpc in 2017 is derived from the GW170817 gravitational wave event.

• Figure 11g 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 distance is distorted by :
• The velocity is replaced by the red shift z.
• 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 in the Hubble diagram is even more pronounced if one of the variable is plotted in logarithmic scale (see Figure 11d).

• Figure 11h 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. The symbol h = [H0(km/sec)/Mpc]/[100(km/sec)/Mpc] is used there. The independent parameters in Eq.(2) are actually the (H0)2x's or h2x's, which is called physical density parameter. The value of h is obtained from :     h2 = i(h2i)/i(i), i.e., from the (CDM data)/(CMBR data).
Here's the procedure to determine the density parameters m and by the CMBR data and supernovae red shift (fit to the Hubble's Law) :

The CMBR spectrum was generated at the time of recombination when electrons and protons combined to form hydrogen atoms, and the universe became transparent to light. This event occurred about 378,000 years after the Big Bang at redshift of zCMBR ~ 1100. According to Eq.(1) and Eq.(2), The length scale at that time would be D = c/[(dR/dt)/R] = c/{H0[M(1+zCMBR)3]-1/2} =

#### Figure 11k Density Parm's Plot

0.25 Mpc, while the angular distance from Earth is DA ~ 14.5 Gpc/(1+zCMBR) = 14.5 Mpc. Therefore, the angular size SW corresponds to the size scale of the recombination epoch is
SW = 0.25/14.5 = 0.017 rad = 1o (Figure 11i). Angular fluctuation of temperature of such size and larger arise from the gravitational effect of primordial density fluctuations in the distribution of dark matter as shown by the following chain of effects :
• The dark matter energy density is dominant at the CMBR epoch (Figure 11j) and can be expressed as EDM(zCMBR) = DM0c2(1 + zCMBR)3, where 0 = 0.9x10-29 gm/cm3 is the critical density at the current epoch.
• At the time of last scattering the dark matter energy density varied as a function of position such that EDM(r) = EDM0 + E(r), where EDM0 is the spatial average.
• This varying component gives rise to a varying gravitational potential , which can be determined by the Poisson's equation in Newtonian mechanics : 2() = 4G(E)/c2.
• Finally, the variation of the temperature in angular size 1o or more in the CMBR spectrum was calculated by Sachs and Wolfe in 1967 :
T/T = ()/3c2 (Figure 11i).
The other CMBR features related to angular size < 1o was originated from Baryon Acoustic Oscillations (BAO) causing photons to lost and gain energy (hence the red and blue shifts) when they travel through the gravitational potential well (dark matter does not interact with photons). Detailed analysis of the T/T profile yield the value for the baryon density parameter B so that M = DM + B.

The dark energy density parameter is negligible in determining the CMBR power spectrum. It is obtained from the consistency relation :

(M + ) - 1 = kc2/(H0)2 ---------- Eq.(5).

It turns out that the position of the first peak is rather sensitive to the curvature of the universe. It would shift to the left or right (smaller or large than 1o) according to whether k < 0 (open universe) or k > 0 (closed). The match is right on k = 0, thus = 1 - M.

Figure 11k is a plot of vs M. Beside the CMBR fits with various confidence, there are the supernovae fits from the Hubble diagram (Figure 11g) and the BAO observations (as described below).
Cosmic "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 installed as the photons stream ahead faster. The sound horizon 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 150 Mpc from the place 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).

• (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 such spot 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, leaving an imprint in the matter distribution.
• (d) Hundreds of million years later, the over-density sound horizon should have higher concentration of galaxies. The BAO measurements have used correlation function to look for such feature out of the jumbled galaxy distribution successfully.

#### Figure 11m k-w Plot [view large image]

See "Cosmological Constraints from Baryonic Acoustic Oscillation Measurements" for a review of cosmic BAO.
The link between observational data and theory is embodied in these two formulas :

As shown in Figure 11m, the BAO measurements favors the case of k = 0, and w = -1. It also yields an estimate for B ~ 0.04.

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 support 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. 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. It is the radiation era ending at z ~ 4000 (R ~ 10-4) with w = 1/3, and 1/R4.
• The matter era corresponds to 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 1.
• For z 0, w = (1/3)[log()/log(1+z)] - 1 -1 as the numerator approaches zero faster than the denumerator when z 0.
• #### Figure 11o w-parameter and cosmic density vs log(1+z) [view large image]

• 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 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 Cosmic Parameters [view large image]

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 substantial difference from the Planck's evaluation (Figure 11s). It demands an

#### Figure 11s KiDS Survey [view large image]

explanation. 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).

See 2018 Update on the "tension" - "Star Map Adds to Cosmic Confusion"

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 11t) corresponding to 13.8 billion years for the age of the universe (see formula in Figure 11u) 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 rate

#### Figure 11u Age of the Universe, 2018

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.

N.B. See a possible resolution for this so-called "Hubble Tension".