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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 = 0.71 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 |

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

ds

For Line Of Sight (LOS) distance, d = 0, d = 0, it follows that : ds = R(t)dr.

The proper distance D(t) =

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 = t

If there is no force acting on the pair of objects, the corresponding v(t

- 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 = (

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

(dR/dt)^{2}/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] |

Eq.(2) can be rewritten in the form : [(dR/dt)/R]D

Comparison to Eq.(1) shows that dD = D

D

The relationship between D

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 11f CMBR Analyzer |
## Figure 11g Modern Hubble Diagram |
## 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 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. The latest estimate of H_{0}= 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 D
_{c}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 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 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 = [H
_{0}(km/sec)/Mpc]/[100(km/sec)/Mpc] is used there. The Hubble constant can be calculated from Eq.(4) with the cosmic parameters determined by CMB and the z-D_{c}measurements as shown in Figure 11g.

- (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.
- The link between observational data and theory is embodied in these two formulas :

(3) 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.

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## Figure 11l BAO Evolution |
## Figure 11m k-w Plot |
See "Cosmological Constraints from Baryonic Acoustic Oscillation Measurements" for a review of cosmic BAO. |

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 support 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. 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. It is the radiation era ending at z ~ 4000 (R ~ 10
^{-4}) with w = 1/3, and 1/R^{4}. - The matter era corresponds to 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 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.

- 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 running in lap-top computer. The
- 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] |

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

Anyway, Figure 11r lists 14 H

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 substantial difference from the Planck's evaluation (Figure 11s). It demands an
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## Figure 11r Hubble Constant in 21st Century |
## 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
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## Figure 11t Hubble Constant, 2018 [view large image] |
## 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. |

Return to the "pre-21st Century Standard Cosmology".

By definition

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

combining these equations yields :

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