Quantum Cosmology

Contents

Quantization of the Empty Universe (-only)

Quantization of the Friedmann Equation (Matter-only)

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.

Figure 01a Energy Level and Cosmic Expansion

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).

Figure 01b 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) + ... (-1)n-1(xn-1/n!)]	[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

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.

Figure 01c 3-D White Hole

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 01d Probability Distribution

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_bR_b = r_aR_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 ~ 10²⁸ cm respectively, R_a 4.6x10^-27 for R_b 2x10³⁴. 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 ta = t0Ra2, where t0=1/2H0=2x1017sec yielding ta 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.

Figure 01e Scale Factor Inflation



• 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 10⁹² erg (from the energy density at inflation and the expanded volume). Consequently, the velocity of expansion v r_bR_b/t 10¹²⁰ cm/sec >> c.



Checkpoint #4 - The energy density associated with the spatial curvature is Ek = kc4/(8G) 10113 ergs/cm3 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 01f Energy Density Evolution

Quantization of the Empty Universe (-only)

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

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 [view large image]

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 :

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) = R₀ sin(Ht) ---------- (16),
  where R₀ = 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.

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 = R₀ cosh(Ht) ---------- (18),
  which approaches infinity as R (R₀/2) 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 :
  d²/dR² + v₀R⁴ = 0 ---------- (19a),
  where v₀ = (r₀/a₀)². Solution of this equation can be obtained readily by a change of variable x = R³, so that dx = 3R²dR,
  it follows : = A cos{[(v₀)^1/2/3]R³} ---------- (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, b0 = , P = 1, i.e., models of early universe with small curvature is free-range.

Theory of Cosmic Inflation

Figure 02a Different Types of Solution	Figure 02b Damped and Forced Oscillation

Figure 03a Inflaton Potential	Figure 03b Inflaton Field

Initially, the force drives the inflaton field toward the true vacuum, then it blocks further change and the field oscillates to complete stop (Figures 3a,b). In particular, V can take the form V() = (1/2)m²² + (/4!)⁴, in which m² is negative for endowing mass to particles (see "Spontaneous Symmetry Breaking").

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, ...

Quantum Fluctuations and Cosmic Structures

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]

Figure 05b Fourier Transform	Figure 05c Gaussian Distribution

The power spectrum has the general form with index n : P(k) = Akⁿ, where A is a constant equal to the amplitude of the wave as shown in Figure 05e. For n = 1, P(k)/k = d(²)/dk = A, the variance receives equal contributions from a given range of k. The gravitational potential fluctuation is said to be scale invariance as shown in Figure 05d. It is estimated that (x) is of the order 10^-4 at kc = H and t = 1/H.

Figure 05d Power Spectrum and Cosmic Fluctuations	Figure 05e

		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 05f Quantum Fluctuation	Figure 05g Cosmic Horizon [view large image]	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.

Figure 05j B-mode from Gravitational Wave

" In quantum field theory (QFT), forces are described using particles (e.g., photons for electromagnetism) with interactions defined by perturbation theory.
" When we apply perturbative techniques to General Relativity (GR), the resulting theory produces infinite divergences at higher orders in the perturbative expansion.
" Unlike the Standard Model forces (electromagnetic, weak, and strong), these infinities cannot be systematically removed (renormalized) because GR has no natural dimensionless coupling constant. The gravitational coupling constant, GNG_NGN, has dimensions of length squared, making the theory non-renormalizable.
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2. Conceptual Clash Between GR and QFT
" Background Independence: General Relativity is background-independent, meaning spacetime geometry is dynamic and shaped by matter and energy. In contrast, QFT relies on a fixed spacetime background.
o Reconciling a dynamic spacetime with quantum fluctuations is profoundly challenging.
" Quantum Superpositions of Spacetime: In GR, spacetime itself is a geometric entity. In quantum theory, states exist in superpositions. What does it mean for spacetime to be in a superposition, e.g., a "superposition of different curvatures"? This remains poorly understood.
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3. Lack of Experimental Guidance
" Unlike quantum electrodynamics or other quantum theories, there is no direct experimental data to guide the construction of a quantum theory of gravity.
" The effects of quantum gravity are expected to be significant only near the Planck scale (?10?35\sim 10^{-35}?10?35 meters), far beyond the reach of current experiments.
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4. Problem of Time
" In quantum mechanics, time is an external parameter that evolves the system. In GR, time is part of the spacetime fabric, which is dynamic and relative.
" Reconciling these two views leads to the "problem of time", where the standard notion of time evolution in quantum mechanics breaks down in the context of a quantum gravitational framework.
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5. Uniqueness and Ambiguity
" There are multiple approaches to quantum gravity (e.g., string theory, loop quantum gravity), each with distinct assumptions, methodologies, and predictions.
" The lack of empirical evidence means there is no consensus on which, if any, of these approaches is correct.
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6. Singularities
" General Relativity predicts singularities (e.g., inside black holes or at the Big Bang) where spacetime curvature becomes infinite, and classical physics breaks down.
" A quantum theory of gravity should resolve these singularities, but how this happens is not yet understood.
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Possible Resolutions
Efforts to overcome these challenges have led to several approaches:
" String Theory: Proposes that fundamental particles are 1D "strings" rather than point-like particles, with gravity emerging as a natural consequence.

" Loop Quantum Gravity (LQG): Seeks to quantize spacetime directly, suggesting that space has a discrete, "quantized" structure.

" Asymptotic Safety: Suggests that a quantum field theory of gravity might be renormalizable at a non-perturbative fixed point.


None of these approaches is yet complete or experimentally confirmed, leaving the problem of quantizing gravity as one of the most profound challenges in modern physics.

see this illustration on separation of various theories. AND :