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Quantization of the Empty Universe (-only)

Theory of Cosmic Inflation

Quantum Fluctuations and Cosmic Structures

successful as infinities keep coming up in the formulation. A less ambitious way is to quantize the Friedmann equation for a homogeneous and isotropic universe when the size of the universe was at the Planck scale of 1.6x10^{-33}cm; 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. | |

## Figure 01a Energy Level and Cosmic Expansion |
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 |

See also the conclusion in "Realm of Planck Scale" about not to generalize too much of the Friedmann equation beyond the period of inflation.

Following is the mathematical derivation (for the matter only "Quantization of the Friedmann Equation") :

The classical Friedmann equation for a matter-only homogeneous and isotropic universe is in the form :

(dR/dt)

where the various mathematical symbols are now re-interpreted such that M is the mass of a particle at the moment of the Big Bang,

Quantization of a particle is prescribed by the commutative relation : [p

[M

Then the quantized Friedmann equation can be written as :

d

By substituting = R , this equation can be re-cast to :

d

which is in a form similar to the Schrodinger Equation for hydrogen atom :

d

the solution of which yields the energy :

E = - (me

and the normalized wave function = [2/(nr

where the generalized Laguerre polynomial (of differentiating once) L

and the innermost Bohr orbit r

By substituting k to -E, R to r, M^{2}r^{4}c^{2}/^{2} to 2m/^{2}, and 2GM/c^{2}r^{3} to e^{2}, the quantized Friedmann equation yields :k = (2GM/c ^{2}r^{3})^{2} (M^{2}r^{4}c^{2}/4^{2})(1/n^{2}) = G^{2}M^{4}/r^{2}c^{2}^{2}n^{2} ----- (7),= 2[(R/nR _{0})/(nR_{0})^{½}](1/n) e^{-R/nR0} L_{n-1}^{(1)}(2R/nR_{0}) ----- (8),where R _{0} = (c^{2}r^{3}/2GM) (2^{2}/M^{2}r^{4}c^{2}) = ^{2}/GM^{3} . The Laguerre polynomial and wave function for a few low laying states are listed in Table 01 below (also see Figure 01b).
r | |

## Figure 01b Wave Function |

n | L_{n-1}^{(1)}(x) |
_{n}(R) |
---|---|---|

1 | 1 | [2(R/R_{0})/(R_{0})^{½}] e^{-R/R0} |

2 | 2(1 - x/2) | [2(R/2R_{0})/(2R_{0})^{½}] (1 - R/2R_{0}) e^{-R/2R0} |

3 | 3(1 - x + x^{2}/6) |
[2(R/3R_{0})/(3R_{0})^{½}] [(1 - 2(R/3R_{0}) + (2/3)(R/3R_{0})^{2}] e^{-R/3R0} |

n-1 | n[1 - (n-1)(x/2) + ... (-1) ^{n-1}(x^{n-1}/n!)] |
[2(R/nR_{0})/(nR_{0})^{½}][(1 - 2(R/nR_{0}) + ... (-1)^{n-1} (2^{n-1}/n!)(R/nR_{0})^{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. This idea is similar to the black hole evaporation. Therefore from such perspectives, the event horizon is not a barrier for expansion. | |

## Figure 01c 3-D White Hole |
See earlier work on "LQG Cosmology". |

Anyway, expressing in term of the Planck length, k and R_{0} can be written in a very simple form :k = 1/(n ^{2}L_{p}^{2}) = 40x10^{64}/n^{2} cm^{-2}, and R_{0} = 1 ----- (9).In analogy to the Bohr radius for hydrogen atom, R _{0} = Lr_{p} for n = 1 can be identified as the classical radius of the particle at ground state (Figure 01d), and R_{0}/_{} = nL_{p} would be the radius of curvature as depicted in Figure 01a.
The commutative rule becomes : [t _{0}(dR/dt), R] = -i,where t _{0} = t_{Pl} = (G/c^{5})^{½} = 5.4x10^{-44} sec and dR/dt = (-i/t_{0})d/dR so that[-i(d/dR), R] = -i. | |

## Figure 01d Probability Distribution |

The "Theory of Inflation" was introduced to address the astronomical puzzle of a flat and homogeneous (and isotropic) universe. Although homogeneous and isotropic conditions have been built into the Friddmann equation explicitly, it is the small size which ensures whatever inside had enough time to mix. The quantization is able to replicate the expansion without additional assumption by considering excitation to nearly continuum of the principle quantum n. Thus, for the transition from n = 1 to n 2x10

- There is no test to verify or falsify the above-mentioned idea. The most that can be provided, is to check out its consistency. The theory has to pass at least some checkpoints.
- Checkpoint #1 - The theory should not contradict with the current observation of temperature fluctuation of 1 part in 100000 in CMBR. Considering the proper distance at the time of transition from quantum to classical Friedmann universe to be : r
_{b}R_{b}= r_{a}R_{a}, where the subscripts "b" and "a" refer to "before" and "after" the transition. Since the scale factors R_{b}and R_{a}refer to the length scale of r_{b}= 2.3x10^{-33}cm and r_{a}~ 10^{28}cm respectively, R_{a}4.6x10^{-27}for R_{b}2x10^{34}. The averaged quantum fluctuation (carried over from the quantum era) of the scale factor |R_{a}| would not be larger than R_{a}, thus |R_{a}| R_{a}4.6x10^{-27}< T/T 10^{-5}, which is at least consistent with the observation. - Checkpoint #2 - It is related to the time scale of the inflationary period. Since the early universe was dominated by radiation, therefore the time at transition t
_{a}= t_{0}R_{a}^{2}, where t_{0}=1/2H_{0}=2x10^{17}sec yielding t_{a}4.2x10^{-36}sec, which is at the threshold of the inflationary epoch estimated by the theory of inflation (Figure 01e). It is close but not exactly equal to the value commonly quoted in the literatures. - Checkpoint #3 - This one is about the speed of inflation. The duration of the quantum jump is estimated to be about 10
^{-119}sec according to the uncertainty relationship E t = with E 10^{92}erg (from the energy density at inflation and the expanded volume). Consequently, the velocity of expansion v r_{b}R_{b}/t 10^{120}cm/sec >> c. - Checkpoint #4 - The energy density associated with the spatial curvature is E
_{k}= kc^{4}/(8G) 10^{113}ergs/cm^{3}in agreement with the estimated energy infusion at the beginning of the universe (Figure 01f). While the cosmological constant term in the Friedmann Equation is considered to be negligible as the current estimate for the value of 10^{-56}cm^{-2}is very small, and it is supposed to be constant throughout the history of the post-inflation universe. See a 2015 update on "Vacuum Energy Density", and also "Composition of Early Universe".

## Figure 01e Scale Factor Inflation |

## Figure 01f Energy Density Evolution |

(dR/dt)

where is the cosmological constant, k is the curvature of space, t the comoving time, and R the dimension-less scale factor.

In order to follow the conventional quantization rule with the commutative relation : [p

[(c

Then the quantized Friedmann equation can be written as :

## Figure 01h Quantum Harmonic Oscillator [view large image] |
## Figure 01i Hermite Polynomials |

The eigenvalue and eigenfunction for the quantum harmonic oscillator is respectively : E _{n} = (n + 1/2) ---------- (13a),_{n} = (m/)^{1/4}(2^{n}n!)^{-1/2}H_{n}(y)e^{-y2} ---------- (13b),where y = (m/) ^{1/2}, ^{2} = /m, and H_{n}(y) is the Hermit Polynomials (see Figures 01h, 01i).The corresponding solutions to the quantum empty universe's are : k _{n} = k = -|k| = -(n + 1/2)(2/a_{0}r_{0}) ---------- (14a),_{n} = (1/a_{0}r_{0})^{1/4}(2^{n}n!)^{-1/2}H_{n}[(r_{0}/a_{0})^{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 L_{P} = (G/c^{3})^{1/2} = 1.62x10^{-33} cm, then r_{0} ~ a_{0} ~ L_{P}, the expressions assume the simple forms : |

and the corresponding energy density

- Some Comments :
- The negative cosmological constant as specified here manifesting an attractive force, which tends to contract the universe instead of expanding.
- On the other hand, the negative curvature -|k| represents an open universe, which tends to expand it.
- ||
^{2}can be interpreted as the probability at certain value of the scale factor R for the universe. For example at ground state n = 0, R = 0 is the most probable configuration, i.e., the universe is most probable to be nothing. - Transition to higher states of n > 0 would increase the curvature instead of flattening it - contrary to the observation of a flat universe by CMB.
- The shape of subject with negative curvature (such as the pseudo-sphere in Figure 01j) is rather alien to normal perception. It doesn't look like the appearance of the universe around.
- The classical empty cosmological model is called Anti-de Sitter (AdS) universe. The solution of Eq.(10) with = -|| and k = -1 is :

R(t) = R_{0}sin(Ht) ---------- (16),

where R_{0}= 1, H = c(||/3)^{1/2}. For the current epoch ~ 10^{-56}, the oscillating period T = 2/H ~ 100 Gyr or t_{max}= T/4 ~ 25 Gyr, i.e., it is compatible with the cosmic age even though the other parameters do not seem to be quite right. - A paper by Stephen Hawking et al in 2012 indicates that our universe with its accelerated expansion can best be described by a negative cosmology constant contrary to the usual wisdom. They claimed that the idea comes from running the Wheeler-DeWitt equation, which can produce a viable accelerating universe only with negative cosmology constant. The new finding receives a boost from the Superstring Theory, which can accommodate a stable universe only if both the cosmology constant and the curvature of space are negative. Since the observed universe has a flat geometry, it is argued that the hyperbolic shape applies only to the early epoch of the universe.

Another model of empty quantum universe under the title "Wave function of the Universe" had been developed in 1983 by Hartle and Hawking. It specifies a positive curvature

d

where V(R) = k(r

and shown graphically in Figure 01k, where

It is in the form of a potential barrier for R/(a

Solution of Eq.(16a) is published in "Quantum Cosmology for Pedestrians" and shown below in a slightly different format :

## 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 : |

- The classical solution of the empty Friedmann universe with k = +1 cm
^{-2}is :

R = R_{0}cosh(Ht) ---------- (18),

which approaches infinity as R (R_{0}/2) e^{Ht}for t , i.e., the universe expands forever leading to "Big Chill" which may not be a problem. - However,

as mentioned earlier, the curvature k becomes the eigenvalue for the quantum universe. It admits only negative k for the -only universe. The case for k = +1 cm^{-2}is a rather contrived scheme. It really doesn't represents the quantum universe as it implies a linear size of 1 cm, for which any subject would be outside the quantum domain. On the contrary, in classical cosmology, the value of curvature has been arbitrarily assigned a number without much problem especially when the universe (since the inflationary period) is observed to have a curvature ~ 0 (see Planck). - Anyway,

irrespective of the above criticism, the scheme can be carried further by taking a simpler case of k = 0 cm^{-2}for illstruction purpose.

The quantum empty Friedmann equation becomes :

d^{2}/dR^{2}+ v_{0}R^{4}= 0 ---------- (19a),

where v_{0}= (r_{0}/a_{0})^{2}. Solution of this equation can be obtained readily by a change of variable x = R^{3}, so that dx = 3R^{2}dR,

it follows : = A cos{[(v_{0})^{1/2}/3]R^{3}} ---------- (19b),

which is identical to Eq.(17c) if the k = +1 cm^{-2}is replaced by 0 cm^{-2}. There is no barrier to overcome. - BTW,

for the case of V(R) in Figure 01k with a barrier, the tunneling probability P can be estimated by :

#### Figure 01m Quantum Tunneling

_{}**For the case of k = 0 cm**^{-2}, b_{0}= , P = 1, i.e., models of early universe with small curvature is free-range.

## Figure 02a Different Types of Solution [view large image] |
## Figure 02b Damped and Forced Oscillation [view large image] |

Suppose at one spatial point in the dawn of the universe, it contained a homogeneous scalar field

## Figure 03a Inflaton Potential |
## Figure 03b Inflaton Field |

Figure 03a depicts this form of V graphically showing the false and true vacuum at dV/d

Figure 03b also plots the inflaton energy density

Now a word about the Hubble parameter H, which is derived from standard cosmology and can be written as (for k = 0) :

where M

Eternal inflation is a peculiar property arising from the concept of inflation. The above-mentioned scenario is for just . 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.
one point in space | |

## 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 |

## Figure 05b Fourier Transform |
## Figure 05c Gaussian Distribution |

## Figure 05d Power Spectrum and Cosmic Fluctuations |
## Figure 05e Quantum Fluctuation |

Ultimately, the quantum fluctuation during the inflation period is created out of the virtual particle pair. Similar to the Hawking radiation generated at the event horizon of the black hole, the pair can be separated by the Hubble horizon preventing them from recombination back to the vacuum and thus become frozen (real wave, not virtual anymore) as shown in Figure 05f. This condition is applicable for those long wavelength (small k) fluctuations outside the Hubble horizon (sphere) d

more free time for longer wavelength. 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 fluctuations (Diagram e) leading to gravitational fluctuations | ||

## Figure 05f Quantum Fluctuation |
## Figure 05g Cosmic Horizon |
(Diagram d) and eventually the wall/void in super-galactic cluster and the temperature variations in the CMBR. |

Followings is a short recap to trace the fluctuations from the invisible inflaton to something that is tangible (Figure 05h).

## 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. |

The observed fluctuation patterns are not the original ones generated in the inflationary era. Ordinary matter, dark matter, cosmic expansion, and gravity together have modified the appearance almost beyond recognition. It is from this jumbling mess that some of the cosmic parameters are extracted. In particular, the energy density in the inflation period is estimated to be about 10

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 n_{e} ~ 10^{3}cm^{-3} (from Figure 01f), _{scatter} = 6.65x10^{-25}cm^{2}; ~ 10^{21}cm - about 10^{-2} times the observable cosmic size of 10^{23}cm at recombination (see "History of the Universe"). |

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. | |

## Figure 05j B-mode from Gravitational Wave |