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Relativity, Cosmology, and Time

Hubble Constant and 2018 Update

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 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 (in order of billion years = Gyr).

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].

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.

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 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 :
Cosmological Models (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.

CMBR Analyzer Modern Hubble Diagram CMBR-Hubble Dependence Since the cosmological model depends on the various density parameters which is derived from the CMBR 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 11f CMBR Analyzer
[view large image]

Figure 11g Modern Hubble Diagram
[view large image]

Figure 11h CMBR-Hubble Dependence

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

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) :

Power Spectrum Cosmic Energy Density Density Parameter's Plot 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 11i Power Specturm

Figure 11j Cosmic Energy Density [view large image]

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 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).
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, cosmic acceleration for all cases with w < -1/3, and in particular w = -1 for the case with cosmological constant. Since all observations converges to w = -1, that's why data from 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 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 :

Figure 11p 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. Meanwhile, the SHOES Program (Supernovae and H0 for the Equation of State) have made precision measurement (<5%) of H0
Hubble Constant in 21st Century KIDS Survey 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 11q). It demands an explanation.

Figure 11p Hubble Constant in 21st Century [view large image]

Figure 11q KiDS Survey [view large image]

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).

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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