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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 such 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). | ||
Figure 11a Cosmic Expansion [view large image] |
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]. |
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 |
(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. | |
Figure 11e Cosmological Models [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 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. |
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 z_{CMBR} ~ 1100. According to Eq.(1) and Eq.(2), The length scale at that time would be D = c/[(dR/dt)/R] = c/{H_{0}[_{M}(1+z_{CMBR})^{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 D_{A} ~ 14.5 Gpc/(1+z_{CMBR}) = 14.5 Mpc. Therefore, the angular size _{SW} corresponds to the size scale of the recombination epoch is |
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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. |
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. | |
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] |
Figure 11p Cosmic Timeline [view large image] |
Figure 11q Cosmic Parameters |
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 | ||
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 : |
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 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. |