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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 H_{0} 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 H_{0} is equated to the age of the universe ~ 13.8 Gyr (see more accurate formula).
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## Figure 11a Cosmic Expansion |
## Figure 11b Proper Distance [view large image] |
A recent practice often expresses H_{0} = 100h (km/sec)/Mpc with the dimensionless h 100 times smaller than H_{0}, i.e., h = H_{0}/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 = (

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 |
## 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 :

ds

For Line Of Sight (LOS) distance, d = 0, d = 0, the proper distance is defined by : dD = R(t)dr.

Then D(t) =

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 = t

where H

- By expressing v = D(t
- Object further away has higher velocity.
- The cosmic horizon D
_{H}= ct_{0}= c/H_{0}. - For D > D
_{H}, the velocity can excess c, that's OK since cosmic expansion is not restricted by Special Relativity.

1 + z = (

BTW, dR = d[1/(1+z)] = -dz/(1+z)

The dynamic of the scale factor R(t) is governed by the Friedmann equation :

(dR/dt)^{2}/R^{2} = H(R)^{2} = (H_{0})^{2}(_{M}/R^{3} + _{k}/R^{2} + _{}) ---------- Eq.(2),where _{M} = 8G_{M}/3 H_{0}^{2}, _{k} = -kc^{2}/ H_{0}^{2}, _{} = c^{2}/3 H_{0}^{2} 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.
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## Figure 11e Cosmological Models [view large image] |

D = -D

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

The Light Travel or (Look Back) distance D

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 dD _{T} = cdt = dD/(1+z) = R(t)dD ---------- Eq.(3a), D _{T} = -D_{H} _{}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., D _{T} ~ D_{c} = D(t_{0}) ~ cz/H_{0} or V(t_{0}) ~ H_{0}D(t_{0}).The age of the universe is computed from D _{T} integrating from z to 0, it is not exactly equal to 1/H_{0} (see Figure 11u).
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## Figure 11e1 Cosmic Distance |
Figure 11e1 is another attempt to explain the subtlety of Cosmic Distances. See also "Types of Cosmic Distance". |

- 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 H
_{0}= 500 (km/sec)/Mpc yields 2 Gyr for the age of the universe T ~ 1/H_{0}. 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 H_{0}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 D_{c}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 D
_{H}= c/H_{0}, then from Eq.(3) we obtain

H_{0}D_{c}= v = c_{}dz' / [_{M}(1+z')^{3}+_{k}(1+z')^{2}+_{}]^{1/2}---------- Eq.(4) - The linear Hubble's Law of v ~ cz = H
_{0}D_{c}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 11f Measurements of H

_{0}

[view large image]#### 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 H_{0} 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 H_{0} value of 67.7 (km/sec)/Mpc - a rather substantial difference from the WMAP's estimate (see Tension). The latest estimate of H_{0} ~ 70 (km/sec)/Mpc in 2019 is derived from the TRGB measurement.
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## Figure 11h CMBR Analyzer |
## Figure 11i CMBR, H |
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. |

(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.
- (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
- (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]

- The w-parameter characterizes the Equation Of State : p = w c
^{2}, where is the matter/energy density and p the pressure. - 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. - The value of w cannot be evaluated at z = 0 by Eq.(5) as it becomes indeterminate.
- From the eBOSS measurement in Figure 11m2,d dR/dt = 56, z = 0.7, the CDM parameters
_{M}= 0.25,_{}= 0.75, and

H_{0}= 70, the w-parameter can be calculated by Eq.(5) giving a value of -1.05. - 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.
- The rate of cosmic expansion reaches its minimum dR/dt = 57 (km/s)/Mpac at z = 0.8 for H
_{0}= 70 (km/s)/Mpac, w = -1. - 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.

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## Figure 11k BAO in Comoving Coordinates [view large image] |

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## Figure 11l BAO Evolution of |
## 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). |

- 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 :

From the Friedann equation :

[(dR/dt)/R]

and the equation of continuity (see footnote for derivation) :

the equation for the acceleration of scale factor R(t) can be derived as :

(d

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

The solution of Eq.(6) yields : =

This is the rationale to replace _{} by _{}/R^{3(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.
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## Figure 11n Continuity Equation |
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/R^{3(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/R^{4}. - The matter era starts from w = 0, and 1/R
^{3}. 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.

- When z , the radiation term is the dominant component, thus
- 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
- 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/
_{}c^{2}.

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## 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. |

## Figure 11p Cosmic Timeline [view large image] |
## Figure 11q [view large image] Cosmic Parameters |

Anyway, Figure 11r lists 14 H

Meanwhile, the SHOES Program (Supernovae and H0 for the Equation of State) have made precision measurement (<5%) of H_{0} 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 S_{8} from the "KiDS-450 Weak Lensing Power Spectrum" has contributed additional tension with another
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## Figure 11r Hubble Constant in 21st Century |
## Figure 11s Shear Power S |
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"). |

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 H _{0} ~ 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"). |
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## 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
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## 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. |

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 H_{0} = 67.6 1.1 km/s-Mpc in very close agreement with Planck's H_{0} = 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
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## Figure 11w Atacama Cosmology Telescope [view large image] |
## Figure 11x ACT-CMB, 2020 |
there's something wrong with the CDM cosmological model. See "Mystery over Universe’s expansion deepens with fresh data" by Nature, 23 July 2020. |

By definition

but from thermodynamics dE = -pdV, while from cosmic expansion V R

combining these equations yields :

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