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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 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 11a Cosmic Expansion |
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. |
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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. |
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(dR/dt)2/R2 = H(R)2 = (H0)2(![]() ![]() ![]() ![]() where ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Figure 11e Cosmological Models [view large image] |
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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 ![]() ![]() ![]() ![]() ![]() 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 ![]() |
Figure 11e1 Cosmic Distance |
Figure 11e1 is another attempt to explain the subtlety of Cosmic Distances. See also "Types of Cosmic Distance". |
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Figure 11f Measurements of H0 |
Figure 11g Modern Hubble Diagram [view large image] |
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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 11h CMBR Analyzer |
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. |
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(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. |
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Figure 11k BAO in Comoving Coordinates [view large image] |
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Figure 11l BAO Evolution of ONE Primordial Wavelet |
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). |
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Figure 11m2 Hubble Parameter [view large image] |
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This is the rationale to replace ![]() ![]() ![]() ![]() |
Figure 11n Continuity Equation |
It reveals that : |
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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. |
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Figure 11p Cosmic Timeline [view large image] |
Figure 11q [view large image] Cosmic Parameters |
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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 11r Hubble Constant in 21st Century |
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"). |
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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"). |
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Figure 11t Hubble Constant, 2019 [view large image] |
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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 ![]() ![]() ![]() |
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. |
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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 ![]() ![]() |
Figure 11w Atacama Cosmology Telescope [view large image] |
Figure 11x ACT-CMB, 2020 |
there's something wrong with the ![]() 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. |