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Origin of the Universe


Quantization of the Friedmann Equation (Quantum Origin)
Theory of Cosmic Inflation, Classical
Quantum Fluctuations and Cosmic Structures
Negative Cosmological Constant

Quantization of the Friedmann Equation (Quantum Origin)

There are many theories for the origin of the universe. Some of them are classical, while others take the quantum effect into account. The path to quantum gravity is usually via quantization of the gravitational field which endows the field with particle identity. Such treatment is not very
Energy Level and Cosmic Expansion 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 1.6x10-33cm; 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 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)2 - 2GM / Rr3 = -kc2 ----- (1)

where R is the dimension-less scale factor, M is the mass, r is the radius, and k is the spatial curvature.

Quantization of a particle is prescribed by the commutative relation : [px, x] = pxx - xpx = -i, where px = m dx/dt is the momentum along the x axis. This equation is satisfied if px = -i d/dx and operates on a (wave) function . In order to follow this quantization rule for the case of cosmic expansion, we can assign R to take the place of x and construct an entity having the dimension of ergs-sec = gm-cm2/sec (the dimension of ) such as Mr2dR/dt. Thus the commutative rule becomes:

[Mr2dR/dt, R] = -i, which can be satisfied for Mr2dR/dt = -i d/dR    or    dR/dt = (-i/Mr2) d/dR.

Then the quantized Friedmann equation can be written as :

d2/dR2 + (M2r4c2/2)[(2GM/c2r3R) - k] = 0 ----- (2)

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

d2/dR2 + (2/R)d/dR + (M2r4c2/2)[(2GM/c2r3R) - k] = 0 ----- (3)

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

d2/dr2 + (2/r)d/dr + (2m/2)[(e2/r) + E] = 0 ----- (4)

the solution of which yields the energy :

E = - (me4/22)(1/n2) ----- (5)

and the normalized wave function = [2/(nr0)½] (1/n2r0) e-r/nr0 Ln-1(1)(2r/nr0)  ----- (6),

where the generalized Laguerre polynomial (of differentiating once) Ln-1(1)(x) = (-1)j[n!/(n-j-1)!(j+1)!j!]xj
and the innermost Bohr orbit r0 = 2/me2 .

Wave Function 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 07).

Figure 07 Wave Function

n Ln-1(1)(x) n(R)
1 1 [2(R/R0)/(R0)½] e-R/R0
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

Table 01 Laguerre polynomial and Wave Function

The view of Loop Quantum Gravity is adopted by assuming the universe to begin with minimum length r = the Planck length Lp = (G/c3)½ = 1.6x10-33cm. However for the Planck mass Mp = (c/G)½ = 2.17x10-5gm, its corresponding Schwarzschild Radius rs is equal to twice the Planck length, i.e., rs = 2GMp/c2 = 2Lp implying the initial radius r is inside the event horizon, i.e., the particle is a black hole. It has been shown that such black hole would evaporate rapidly in 10-40 sec (see Black Hole Evaporation).

3-D White Hole 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".
Probability Spreading Thus, 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 08b), and R0/ = nLp would be the radius of curvature as depicted in Figure 06.

The commutative rule becomes : [t0(dR/dt), R] = -i,
where t0 = (G/c5)½ = 5.4x10-44 sec and dR/dt = (-i/t0)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 2x1034 folds expansion of the scale factor, n 2x1034 for R0 = 1; accordingly the spatial curvature lowers drastically to k 10-3 cm-2. Thus, the problem with the flatness of the universe can be explained as quantum jump to highly excited state. The transition to classic theory occurred naturally since such highly excited levels are crowded together like a continuum, where the discreteness characterized by quantum theory is lost. This theory has another virtue, e.g., the explosive expansion automatically stops with the end of the transition - unlike the original inflation theory, which requires a specific mechanism to have it terminated or it inflates forever creating endless universes.


Theory of Cosmic Inflation, Classical

The theory of cosmic inflation has been introduced to resolve some observational astronomical features in 1980. Since then it has received ample supports from both WMAP and Planck. However, in the absence of a quantum-gravity theory its foundation is not solid, it often resorts to ad hac assumption which relies on special form of potential and is not quantum mechanical and/or relativistic. Anyway, a brief re-visit of the "forced and damped oscillations" would clarify somewhat the mystery shrouded behind the subject.

Damped Oscillation Damped and Force Wave

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 (t). As it will be shown in the followings, this is the inflaton that drives the inflation and later on becomes the Higgs field endowing mass to all particles (see "Could the Higgs Boson be the Inflaton?"). It can be shown (see for example "Early Universe") that the energy density and the pressure p associated with the scalar field can be expressed in the form (in natural units):
Inflaton Potential Inflaton

Figure 03a Inflaton Potential

Figure 03b Inflaton Field [view large image]

In particular, V can take the form V() = Vo + (1/2)m22 + (/4!)4, in which m2 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 = 0. The scalar field has been solved numerically using this form of potential with a little push in the beginning to start the process. The result shown in Figure 03b is not quite satisfactory as it reaches the steady state too soon. A more realistic scenario is shown in the inserts (Figure 03a), in which the potential falls off much more gently allowing a slow-roll to the true vacuum. Note that the general form of is similar to the damped oscillation with a constant force in Figure 02.

Figure 03b also plots the energy density (blue), and pressure p (black) as functions of the time t. It shows that the energy density and pressure vanish as the inflaton settles down to the true vacuum. This field at true vacuum can be identified as the Higgs field discovered on July, 2012. While the energy is converted to the particles in this world, the negative pressure requires some explanation. It has been interpreted as suction or negative gravity (hence the repulsion). Close examination reveals that the effect of negative pressure never shows up in the continuity equation nor the field equation. Only (d/dt)2 or dV/d appears in the process. The mechanism of inflation can be explained without invoking the concept of negative pressure. It is the coupling of the cosmic expansion to the inflaton - via the term 3H(d/dt)2 - that actually requires further explanation. The very early cosmo is represented here by the de Sitter universe with the scale factor R(t)=RoeHt. This model universe is derived from General Relativity without any consideration on the quantum effect. Thus, the theory presented here is a contrived scheme for illustrative purpose only.

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

where Mp = 1/(8G)1/2 is the reduced Planck mass, and o is the total mass density of the system in Planck scale (= 5x1093 gm/cm3), which is related to Vo = oc2 = 10114 erg/cm3. It remains to be a constant during the exchange of energy between the inflaton's and particles' until the end of the inflation. Thus according to this scenario the early exponential expansion depends on the value of H = c(/3)1/2 = (o/3Mp2)1/2 = constant 5.3x1043 sec-1. Ultimately, the inflation is caused by the constant energy density Vo during that period. (see the simple mathematics to demonstrate the effect of constant energy density on gravitational force). During this period, the cosmological constant = 1067 cm-2, i.e., a 10123 folds difference from the value in the current epoch. See a 2015 update on "Vacuum Energy Density".

The duration of inflation can be estimated by H-1 = 2x10-44 sec. It is about 1011 times shorter than the quotations of 10-33 - 10-32 sec in many scientific literatures. It turns out that all the estimates above are related to the Planck era before the inflation period. If we equate H-1 = 2x10-33 sec, then Vo = 1092 erg/cm3 (a good agreement to the value shown in Figure 09b); correspondingly, all the other parameters such as , and o would be changed by a factor of 1022 to = 1045 cm-2 and o = 1071 gm/cm3.

Unobservable Universe Another seemingly discrepancy is about the magnitude of o, which is equal to about 1071 gm/cm3. Since inflation stopped when the universe had the size of about 10 cm and now expends to 1028 cm, the mass density at the current epoch should be equal to 10-10 gm/cm3; but the value deduced from the WMAP observation is about 10-29 gm/cm3, 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 (1028 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 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, ...


Quantum Fluctuations and Cosmic Structures

Exponential Expansion 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]

Fourier Transform Gaussian Distribution

Figure 05b Fourier Transform [view large image]

Figure 05c Gaussian Distribution [view large image]

Inflaton Field Fluctuation Power Spectrum

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-1. This graph corresponds to the intersection of the green vertical line with all the k waves in Figure 05b. Since the power spectrum P(k) depends only on the amplitude k, it is a constant according to the graph, thus P(k) = constant in agreement with other investigations pointing to a scale invariance power spectrum, i.e., independent of k. The wavy curve (in blue) is interpreted as pattern of the fluctuations at a particular time. An observable example is the CMBR spectrum (Diagram b, Figure 05e) to be explained later.

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-1 since the fluctuating duration according to the
Quantum Fluctuation Cosmic Horizon 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
[view large image]

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).
Acoustic Materials

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 10102 erg/cm3 from the measurement related to the left
Power Spectrum, Evolution 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 ne ~ 103cm-3 (from Figure 09b), scatter = 6.65x10-25cm2; ~ 1021cm - about 10-4 times the cosmic size of 1025cm (from r = 1028/(z+1) cm for z = 1100 at recombination time).

N.B. Although the power spectrum P(k), the CMBR spectrum, and the cosmic density fluctuation are related, they are defined differently. While P(k) only concerns with the amplitude (magnitude) of the cosmic fluctuation, the CMBR spectrum involves displacements (the amount of deviation from equilibrium at certain point) of the temperature from the average, and the cosmic density fluctuation plots are similar to P(k) but often displayed with the length scale (instead of k).

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 generated in the early universe. It turns out that both of them obey the same equation for the k. The power spectrum is also a constant, i.e., P(k) = (8/MP2)(H/2)2 implying low-energy inflation would produce
Primordial Gravitational Wave 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


Negative Cosmological Constant

Pseudosphere Pseudosphere Generation 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
[view large image]

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.
N.B. - Figure 12a plots the generator w as a function of z for different values of r. The pseudospheres are generated by rotating the curves around the axis. It shows that the shapes of the pseudosphers do not change with r, just the same with the real spheres (Figure 12b). The latitude circles are similar in both cases, the only difference is the radius decreases from z = 0 to r for the real sphere (see insert in Figure 12) while it is the opposite for the pseudosphere. The following provides a proof that the curvature is constant for the pseudosphere (Figure 12c, d, e) :
  1. Since the tangent of the w curve dw/dz = -(r2 - z2)1/2/z, therefore by Pythagoras' relation the length (squared) of the tangential line to the w axis PF2 = z2 + (r2 - z2) = r2, or PF = r = constant (see the PF lines in Figure 12e).
  2. The Gaussian curvature k is defined as the product of the curvatures for the two osculating circles (Figure 12c) k1=1/r1, k2=1/r2 on the two principle planes corresponding to the maximum and minimum curvatures respective (Figure 12d, the red circle has been rotated 90o).
  3. There are three Pythagoras' relations for the three right-angled triangle in Figure 12d : (O1F)2 = (r1)2 + r2, (O2F)2 = (r2)2 + r2, and (O1F)2 + (O2F)2 = (r1 + r2)2.
  4. Combination of these relations yields k = k1k2 = -1/(r1r2) = -1/r2 = constant. The negative sign denotes the two principle curves warped toward opposite directions.
  5. By defining = angle(r1O1F) and using the trigonometric relations : tan() = r/r1, and sin() = z/r1, it can be shown that r1 = rz/(r2 - z2)1/2, r2 = (r/z)(r2 - z2)1/2. Actually r2 can be calculated directly from the definition of curvature k2 = |d(dw/dz)/dz| / [1+(dw/dz)2]3/2 = 1/r2. BTW, the vuvuzela from South Africa should be the object with a shape closest to the pseudosphere - manmade or otherwise (Figure 12f).