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Theory of Cosmic Inflation, Classical

Quantum Fluctuations and Cosmic Structures

Negative Cosmological Constant

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 06 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 06 Energy Level and Cosmic Expansion |
BTW, this formulation is entirely different from the Wheeler-DeWitt equation, which is also derived from the Friedmann equation. See also the conclusion at the end in "Realm of Planck Scale". |

Following is the mathematical derivation :

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

(dR/dt)

where R is the dimension-less scale factor, M is the mass,

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 07).
r | |

## Figure 07 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 08a). 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 08a 3-D White Hole [view large image] |
See earlier work on "LQG Cosmology" also "Realm of Planck Scale". |

Thus, 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 08b), and R_{0}/_{} = nL_{p} would be the radius of curvature as depicted in Figure 06.
The commutative rule becomes : [t _{0}(dR/dt), R] = -i,where t _{0} = (G/c^{5})^{½} = 5.4x10^{-44} sec and dR/dt = (-i/t_{0})d/dR so that[-i(d/dR), R] = -i. | |

## Figure 08b Probability Spreading |

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 a 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 09a). 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 09b). 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 universe. See a 2015 update in "Vacuum Energy Density", and also "Composition of Early Universe".

## Figure 09a Scale Factor Inflation |

## Figure 09b Energy Density Evolution |

## Figure 01 Different Types of Solution [view large image] |
## Figure 02 Damped and Forced Oscillation [view large image] |

Suppose the dawn of the universe contains a homogeneous scalar field

## Figure 03a Inflaton Potential |
## Figure 03b Inflaton Field [view large image] |
In particular, V can take the form V(_{}) = V_{o} + (1/2)m^{2}_{}^{2} + (/4!)_{}^{4}, in which m^{2} is negative for endowing mass to particles (see "Spontaneous Symmetry Breaking"). |

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

Figure 03b also plots the energy density

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

where M

The duration of inflation can be estimated by H

Another seemingly discrepancy is about the magnitude of _{o}, which is equal to about 10^{71} gm/cm^{3}. Since inflation stopped when the universe had the size of about 10 cm and now expends to 10^{28} cm, the mass density at the current epoch should be equal to 10^{-10} gm/cm^{3}; but the value deduced from the WMAP observation is about 10^{-29} gm/cm^{3}, where had all the matter gone? The puzzle can be resolved by realizing that we can perceive only a small portion of the whole universe, there is an event horizon (10^{28} cm) to prevent us from looking beyond to the very large unobservable part as shown in Figure 04a (not to scale). There is much more matter beyond the horizon.
| |

## Figure 04a Universe, Unobservable |

Eternal inflation is one more peculiar property arising from the concept of inflation. The above-mentioned scenario is for just one point in the 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 |
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 inflaton field 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] |

## Figure 05b Fourier Transform [view large image] |
## Figure 05c Gaussian Distribution [view large image] |

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

The square of the inflaton field fluctuation in k-space is plotted in Figure 05d for t = H

Ultimately, the quantum fluctuation during the inflation period is created out of the virtual inflaton anti-inflaton pair. Similar to the Hawking radiation generated at the event horizon of the black hole, the inflaton 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) H

uncertainty principle is t ~ /c, i.e., 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 inflaton 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 scalar field (see insert in Diagram a, Figure 05e). They become the source of mass-energy density fluctuations (Diagram e) leading to | ||

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

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] |

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 mass density spectrum. Subsequent interaction with the cold dark matter bends the curve as shown in Figure 05i. The CMBR spectrum is in agreement with this estimate (Figure 05e), the detailed shape of which can be explained roughly by the "Photon Fluid Approximation", while the large scale structures are in simpler form since there is no radiation to counteract the gravity. | |

## 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 09b), _{scatter} = 6.65x10^{-25}cm^{2}; ~ 10^{21}cm - about 10^{-4} times the cosmic size of 10^{25}cm (from r = 10^{28}/(z+1) cm for z = 1100 at recombination time). |

In general relativity, distortions of the geometry (metric tensor) would induce quantum fluctuations similar to that in the scalar (inflaton) field in the form of gravitational wave. There are two components H

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. | |

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

Recently in 2012, a paper by Stephen Hawking etal indicates that our universe with its accelerated expansion can best be described by a negative cosmology constant contrary to the usual wisdom. This 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 Super-string 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. | ||

## Figure 11 Pseudosphere |
## Figure 12 Pseudosphere Generation [view large image] |
Such theory is incompatible to the "Quantization of the Friedmann Equation". Beside the weird shape of a pseudosphere (for negative curvature k, see Figure 11), the probability amplitude at large R approaches Acos(R/2n) contrary to the boundary condition of vanishing exponentially. |

- Since the tangent of the w curve dw/dz = -(r
^{2}- z^{2})^{1/2}/z, therefore by Pythagoras' relation the length (squared) of the tangential line to the w axis PF^{2}= z^{2}+ (r^{2}- z^{2}) = r^{2}, or PF = r = constant (see the PF lines in Figure 12e). - The Gaussian curvature k is defined as the product of the curvatures for the two osculating circles (Figure 12c) k
_{1}=1/r_{1}, k_{2}=1/r_{2}on the two principle planes corresponding to the maximum and minimum curvatures respective (Figure 12d, the red circle has been rotated 90^{o}). - There are three Pythagoras' relations for the three right-angled triangle in Figure 12d : (O
_{1}F)^{2}= (r_{1})^{2}+ r^{2}, (O_{2}F)^{2}= (r_{2})^{2}+ r^{2}, and (O_{1}F)^{2}+ (O_{2}F)^{2}= (r_{1}+ r_{2})^{2}. - Combination of these relations yields k = k
_{1}k_{2}= -1/(r_{1}r_{2}) = -1/r^{2}= constant. The negative sign denotes the two principle curves warped toward opposite directions. - By defining = angle(r
_{1}O_{1}F) and using the trigonometric relations : tan() = r/r_{1}, and sin() = z/r_{1}, it can be shown that r_{1}= rz/(r^{2}- z^{2})^{1/2}, r_{2}= (r/z)(r^{2}- z^{2})^{1/2}. Actually r_{2}can be calculated directly from the definition of curvature k_{2}= |d(dw/dz)/dz| / [1+(dw/dz)^{2}]^{3/2}= 1/r_{2}. BTW, the vuvuzela from South Africa should be the object with a shape closest to the pseudosphere - manmade or otherwise (Figure 12f).