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|for a homogeneous and isotropic universe when the size of the universe was at the Planck scale of 1.6x10-33cm; then the whole universe can be considered as a small particle (with certain mass and size). By quantizing the dimensionless scale factor R and its time derivative dR/dt, this particle is endowed with a wave property; the corresponding wave function can be interpreted as the probability amplitude at certain value of the scale factor R. It turns out that the resulting wave equation is similar to that for the electron in the hydrogen atom (with different interpretations of the parameters). In particular, the energy in the case of hydrogen atom is now replaced by the spatial curvature k. The transition from its extremely large value to small number corresponding to a nearly flat space can be interpreted as quantum jump to near continuum during the period of inflation. Figure 01a compares the energy levels of the hydrogen atom and the inverse of the spatial curvature (~ radius of curvature) in cosmic expansion. The appearance seems to be in reverse since the correspondence is between E (energy level) and k (spatial curvature), and the plot is for 1/k to make it looking like the expansion of the universe. This formulation is similar to the Wheeler-DeWitt equation, which is also derived from the Friedmann equation with an un-specified potential.|
|This treatment adopts a potential corresponding to a matter-only universe. The other alternative with "cosmological constant -only" is explored in the section on "Quantization of the Empty Universe".|
By substituting k to -E, R to r, M2r4c2/2 to 2m/2, and 2GM/c2r3 to e2, the quantized Friedmann equation yields :|
k = (2GM/c2r3)2 (M2r4c2/42)(1/n2) = G2M4/r2c22n2 ----- (7),
= 2[(R/nR0)/(nR0)½](1/n) e-R/nR0 Ln-1(1)(2R/nR0) ----- (8),
where R0 = (c2r3/2GM) (22/M2r4c2) = 2/GM3r . The Laguerre polynomial and wave function for a few low laying states are listed in Table 01 below (also see Figure 01b).
|2||2(1 - x/2)||[2(R/2R0)/(2R0)½] (1 - R/2R0) e-R/2R0|
|3||3(1 - x + x2/6)||[2(R/3R0)/(3R0)½] [(1 - 2(R/3R0) + (2/3)(R/3R0)2] e-R/3R0|
|n-1||n[1 - (n-1)(x/2) + ...
|[2(R/nR0)/(nR0)½][(1 - 2(R/nR0) + ... (-1)n-1 (2n-1/n!)(R/nR0)n-1] e-R/nR0|
|An article in "New Scientist, January 2-8 2016" with the title "Quantum Bounce" postulates that the collapse to black hole can be reversed to ejection from white hole by the space-time loop in the "Theory of Loop Quantum Gravity" (Figure 01c). Calculations show that bigger black hole would take longer to bounce. Thus, the authors are looking for evidences from small black holes more common in earlier history of universe; and that's why we don't see such event nearby. However, nobody would perceive the very slow process that turns the whole universe from its previous collapsing cycle back to the expansion phase today. Such idea is similar to the black hole evaporation (particularly at its very last stage as considering here). Therefore from such perspectives, the event horizon is not a barrier for expansion.|
|See earlier work on "LQG Cosmology".|
Anyway, expressing in term of the Planck length, k and R0 can be written in a very simple form :|
k = 1/(n2Lp2) = 40x1064/n2 cm-2, and R0 = 1 ----- (9).
In analogy to the Bohr radius for hydrogen atom, R0r = Lp for n = 1 can be identified as the classical radius of the particle at ground state (Figure 01d), and R0/ = nLp would be the radius of curvature as depicted in Figure 01a.
The commutative rule becomes : [t0(dR/dt), R] = -i,
where t0 = tPl = (G/c5)½ = 5.4x10-44 sec and dR/dt = (-i/t0)d/dR so that
[-i(d/dR), R] = -i.
Figure 01h Quantum Harmonic Oscillator [view large image]
Figure 01i Hermite Polynomials
|The eigenvalue and eigenfunction for the quantum harmonic oscillator is respectively :|
En = (n + 1/2) ---------- (13a),
n = (m/)1/4(2nn!)-1/2Hn(y)e-y2 ---------- (13b),
where y = (m/)1/2, 2 = /m, and Hn(y) is the Hermit Polynomials (see Figures 01h, 01i).
The corresponding solutions to the quantum empty universe's are :
kn = k = -|k| = -(n + 1/2)(2/a0r0) ---------- (14a),
n = (1/a0r0)1/4(2nn!)-1/2Hn[(r0/a0)1/2 R]e-(r0/a0)R2 ---------- (14b)
Figure 01j Pseudo-sphere
|If the initial size of the universe L and (3/||)1/2 are taken to be the Planck length LP = (G/c3)1/2 = 1.62x10-33 cm, then r0 ~ a0 ~ LP, the expressions assume the simple forms :|
Figure 01k -only Potential [view large image]
Figure 01l -only Wavefunction [view large image]
|Actaully, the formulation has a serious conceptual problem, here's the comments :|
|For the case of k = 0 cm-2, b0 = , P = 1, i.e., models of early universe with small curvature is free-range.|
|Eternal inflation is a peculiar property arising from the concept of inflation. The above-mentioned scenario is for just one point in space. It has been suggested that every point in the space can be initially in a false vacuum state. The decay to true vacuum happens randomly similar to the decay of radioactive substance. The difference is the continuously growing of the false vacuum (FV). As depicted in Figure 04b, the FV itself inflates continuously getting bigger and bigger (cannot be shown in the drawing). Those decayed regions are the "pocket universe" like ours, which keeps on expanding but not in exponential rate.|
Figure 04b Eternal Inflation [view large image]
|The chaotic inflation proposes that false vacuum with sufficient high energy density can pop up randomly within the the normally expanding universe. Exploration of variations on the inflationary theme has produced many speculative "theories" such as extra-dimension, multiverse, many worlds, pre-BigBang, ...|
|About a year after the introduction of the theory of inflation without considering any quantum effect, it was realized that quantum fluctuations in the initial region could have a profound consequence as tiny imperfection would become huge defect with the cosmic expansion (Figure 05a). Since then theoretical calculations yield close agreements with observations in the forms of CMBR power spectrum and super-clusters structures. The basic assumptions, the different kinds of spectrum, the link of gravitational potential fluctuations to the other types, and the evolution of the fluctuations to astronomical objects will be explored in the followings. In order to understand what it is all about, it is crucial to define the relationship between the fluctuation and power spectrum.|
Figure 05a Exponential Expansion [view large image]
|In Figure 05g the frozen waves are shown as if they are trapped under the "mountain" (it may take a moment to appreciate the contents as it plots everything inverse to length, i.e., inside out; contrary to the normal perception in the insert). As expansion of the universe continues, new particle waves are frozen on top of the previously ones. The process goes on until the end of the inflation when the universe already becomes populated by inhomogeneous fluctuations (see insert in Diagram a, Figure 05d). They become the source of mass-energy density|
Figure 05g Cosmic Horizon
|fluctuations (Diagram e) leading to gravitational fluctuations (Diagram d) and eventually the wall/void in super-galactic cluster and the temperature variations in the CMBR.|
Figure 05h Formation of CMBR Spectrum and Cosmic Large Structures [view large image]
|See "Power Spectrum" and "Hubble Constant and 2018 Update" for more info about the formation of CMBR.|
|measurement related to the left of the "mountain" in Figure 05g. In order to prevent runaway gravitational fluctuation, it is believed that P(k) k for the mass density spectrum (Figure 05i). Subsequent interaction with the cold dark matter bends the curve as shown in Figure 05i. The large scale structures are in simpler form since there is no radiation to counteract the gravity. Although the CMBR spectrum also follows this scenario (Figure 05d, diagram a), its more complicated shape can be explained by further modifications (see "CMBR Power Spectrum").|
Figure 05i Power Spectrum, Evolution [view large image]
|A rough estimate of the thickness of the "Last Scattering Shell" (Figure 05h) : For ne ~ 103cm-3 (from Figure 01f), scatter = 6.65x10-25cm2; ~ 1021cm - about 10-2 times the observable cosmic size of 1023cm at recombination (see "History of the Universe").|
|produce negligible amounts of gravitational waves. No primordial or any other forms of gravitational wave has been detected so far (up until 2014). However, indirect evidence has been observed in the B-mode polarization imprinted on the CMB (to be confirmed by further evaluations). As shown in Figure 05j, the B-mode polarization is subjected to contamination by many effects in the intervening space before its detection on Earth making its confirmation very difficult.|