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The existence of asymmetric solutions to a symmetric theory is common to many branches of physics. The reason lies in the fact that the symmetric state is not the state of minimum energy, i.e., the ground state, and that in the process of evolving towards the ground state, the intrinsic symmetry of the system has been broken. Figure 01 shows that the initial position of the marble on top of the bump is symmetric but not in a  
Figure 01 Symmetry Breaking _{} 
state of minimum energy. A small perturbation will cause the rotational symmetry to be broken and the system to assume a stable state configuration. When the symmetry of a physical system is broken in this way, it is often referred to as "spontaneous symmetry breaking (SSB)". 
process of SSB. Such process is similar to the phase transition (see Figure 02,a) between the gasliquidsolid phases, in which energy is released to the environment and the rotational symmetry changes to translational with the change of gas/liquid phase to solid. By such analogy, the SSB of electroweak at 10^{12} sec after the Big Bang can be considered as the transition of restmassless particles moving at the speed of light to motion constrained by having restmass (see Figure 03 of the Standard Model in symmetric and Higgs phases).  
Figure 02 Phase Transition _{} 
Figure 03 Standard Model _{} 
Figure 02,b portrays another interpretation of phase transition via lowering the potential energy V by some kind of catalyzing agent. 
The obvious omission in this model is the absence of the reheating process which is responsible for the generation of matter at the end of inflation (see "Inflation Era"). According to another model developed in 2014 (see "Inflation, LHC and the Higgs boson"), it is indeed possible to produce a potential V with an inflationary plateau at large values of the scalar field, and two minima (Figure 04,b).  
Figure 04 Higgs as Inflaton 
A much simplified version of the mathematic derivation is sketched below to illustrate the theoretical formulation. 
 
Figure 05 Higgs/Inflaton, Evolution with Time _{} 
Figure 05a Semiclassical Model [view large image] 
According to the consensus view on cosmic inflation, the duration and expanse during that period are : T ~ c(10^{32}/10^{44}) ~ 10^{22}, R ~ (10^{6}/10^{33}) ~ 10^{27} and T = R =1, ds^{2} = 0 timelike in the beginning. Thus, R >> T and ds^{2} < 0 thereafter, such events are qualified as "Inflation"  QED. 
Figure 06 H Atom Energy Level & Cosmic Radius 
Now the hydrogenic wave function _{n}(R) can be identified to the inflaton field (or the Higgs field _{}, or whatever designation). Inflation corresponds to the transition from the level of n 1 to the continuum (n ). Energy should be infused to pump the transition as shown in Figure 07. After inflation, the 
Effort to formulate theory in time or size smaller than the Planck scale (~10^{44} sec, Figure 08) is doomed to fail as any system below such limit violates the uncertainty principle of t t_{PL} (for the corresponding Planck mass/energy, i.e., E_{PL}t = E_{PL}t_{PL}) and x L_{PL} rendering it meaningless (see "What Is The Smallest Possible Distance In The Universe?").  
Figure 07 Cosmic Energy Density [view large image] 
Figure 08 _{} Cosmic Evolution 
BTW, the cosmic evolution in Figure 08 shows that the structures in quantum domain had completed within a very short time interval about 10^{32} sec. Then it went through a period of making the atoms for building the large structures, 
Figure 08b Dark Energy [view large image] 
Meanwhile the quantum fields are also homogeneous, isotropic, and constant over the eon; it is tempting to identify each of the 24 quantum fields (Figure 08b2) to the eigenfunction _{} with different value of H (labeled by _{}_{n}, H_{n} and _{n} for n = 1, 2, 3, ... 24). It is estimated that they would have energy density _{} _{} 10^{8} erg/cm^{3}, with the corresponding vacuum (zero point) energy E_{n} = h_{n}/2, where _{n} = (k/m)^{1/2}, m is the mass, k the spring constant for the n^{th} particle. The elementary particles in this world are just in the excited state of the quantum fields. As the Universe expands, the quantum fields have to be replenished similarly to the dark energy.  
Figure 08b2 Quantum Fields _{} 
As for where does the infusing energy come from ? The quick answer would be from the 4^{th} spatial dimension (Figure 08b3). The corollary for contraction to the beginning is to remove such energy back to the bulk. Thus, such reversal process would not end up with infinite matterenergy as oftquoted in the past. Unfortunately, the "Superstring Theory" is not viable anymore due to failure of finding any experimental evidence. Or, is the bulk just another 3D environment beyond this Universe ?  
Figure 08b3 _{} 
This is part of the progress in science  more problems are created following the resolution of an old one. 
The pattern on Cosmic Microwave Background (CMB) has shown conclusively that it is the relic of density fluctuation. The theory of cosmic inflation is rather convincingly (not 100%) ascribed as the origin. Mathematically, the CMB (the earliest observable) is expressed by the Power Spectrum : P_{s}(k) = A_{s}k^{(ns1)}  (9), where k is the Fourier transform of the spatial coordinate x, n_{s} 1 the scalar index, A_{s} the scalar amplitude, the subscript "s" denotes scalar type perturbation (quantum fluctuation), P_{s}(k) represents the density fluctuation in kspace corresponding to random fluctuation in xspace (see Figure 08c).  
Figure 08c Primordial Fluctuations _{} 
Some versions of the "Inflation Theory" also include the fluctuation in gravity (or metric tensor in General Relativity paradigm). It is often referred as primordial gravitational wave which is expressed as P_{t}(k) = A_{t}k^{nt}  (10), 
The gravitational attraction in the density enhanced regions, and the radiation repulsion acted together to produce the incoherent acoustic oscillations (noise). Compressing a gas heats it up; letting it expand cools it down  this is the origin of the temperature variation which eventually shows up as CMB. As shown in Figures 08d and 08e, the main peak is the oscillatory mode that went through 1/4 of a period (reaching maximal compression) at the time of recombination (between electrons and protons to form neutral atoms). The lower peaks correspond to the harmonic series of the main peak frequency.  
Figure 08d acoustic oscillations [view large image] 
Figure 08e Power Spectrum _{} 
In angular coordinates, the temperature fluctuation is expressed mathematically by : where Y_{m}(,_{}), is the Spherical Harmonics, and = 0 denotes the monopole, = 1 the dipole, = 2 the quadrupole, ..., and m can be any integer between  and . Each a_{lm} constitutes a multipole mode. The lowest multipoles are the largestarea, continent and oceansize undulations on the temperature map. Higher multipoles are like successively smallerarea plateaus, mountains and hills (trenches and valleys) inserted on top of the larger features. The entire complicated topography is the sum of the individual multipoles (Figure 08f).  
Figure 08f Multipole _{} 
Eq.(11) is the Inverse Fourier Transform back to the xspace with integration replaced by summation over the multipole moment , Y_{lm} playing the role of e^{ikx}, and a_{lm} the power spectrum P_{s}(k) in Eq.(9). 
There are two distinct gravitational fluctuations corresponding to two kinds of polarizations as shown in Figure 08g. The Emode is similar to the scalar fluctuation with g_{} for = = 1,2,3 producing compression/expansion perturbation. While the Bmode is for producing shear perturbation.  
Figure 08g Tensor Polarization [view large image] 
Thus, the scalar fluctuation generates only Emode polarization; while the tensor fluctuation (gravitational wave) generates both Emode and Bmode polarizations. 
Polarization in CMB can be detected by using a polarizer (by the antenna in astronomical instrument). The corresponding formulas to Eq.(11) for CMB in Emode and Bmode are : where E(n) and B(n) are CMB for the Emode, and Bmode respectively, and n denotes the direction on the sky (see "Probing Inflation with CMB Polarization").  
Figure 08h CMB Spectra, and Polarization Patterns _{} 
Figure 08h,a shows the unpolarized, Emode and Bmode CMB from both scalar and tensor perturbations (dotted curves). The Bmode by lensing is detected from only a small patch of the sky via gravitational lensing. Figure 08h,b shows the patterns around hot and cold spots. 
BICEP2's (Figure 08i) first attempt was a failure because it has not taken into account the intergalactic dust (see "Bmode Detection, 2010's"). Since then it has created a precise intergalactic model to subtract out the masking. The 2021 BICEP2 measurement provides the tightest constraint to gravitational perurbation with r < 0.036. The BICEP2 team have arrived at this limit by combining data at three frequencies (96 GHz, 150 GHz, 220 GHz) from their own experiment, complemented by archival data from the WMAP and Planck.  
Figure 08i BICEP2 Experiment [view large image] 
Figure 08j rRatio [view large image] 
Figure 08j plots the rratio r = P_{t}/P_{s} vs the scalar index n_{s}. It encompasses the prediction of some Inflation Models, and experimental measurement (current and future). It is related to the efold expansion of space during inflation and is 
 
Figure 08k Quantum Field _{} 
 
Figure 08l The 3 Problems _{} 
Furthermore, Entropy is defined as the degree of randomness, which can be expressed alternatively as the degree of freedom in a system, i.e., the number of different parameters or arrangements needed to specify completely the state of a system. The evolution of entropy in the universe can be separated into four phases as described in Figure 08m, which shows lot of (???) for the very low entropy at the beginning of the universe. It has been a puzzling problem since a hot Big Bang should involve a lot of entropy (randomness/disorder). The mathematics in the above quantum cosmological model is specified by only "one" parameter, i.e., the Planck length L_{PL}, which is the absolute minimum entropy for any system. Thus, low entropy came naturally in the model resolving the paradox.  
Figure 08m Evolution of Entropy [view large image] 
The universe starts with gravity (which goes on by itself) and a quantum wave function that is identified as the Higgs field. This Higgs field went on to undergo many phase changes and finally settled down with 3 distinct quantum fields plus the Higgs field itself as shown in Figure 09. Such phenomena is similar to the phase transition of water (see Figure 02), which can exist in the form of gas, liquid, and solid in macroscopic states (similar to the strong, weak, and em quantum fields in Figure 09).  
Figure 09 Quantum Field History (modified) [view large image] 
NB : The gravitational field is a classical field that does not follow the rules for the quantum fields. It only coupled to the Higgs field at the very early cosmic time in the form of quantum gravity. It detaches from the Higgs after inflation when the size of the universe is beyond the quantum domain. 
 
Figure 09b Convergence of Quantum Gravity [view large image] 
Figure 09b shows all kinds of objects in this world with different size and mass. Ultimately, they all emerge from a tiny speck at the Planck scale via the action of Quantum Gravity (see the corresponding Table). 
Figure 10 QuarkGluon Plasma [view large image] 
Higgs and hence the quantum particles (excited from the fields) are restmassless moving at the velocity of light (Figure 12). Most of them acquired restmass only after SSB. The event separated the electroweak field into the electromagnetic and weak; it also made the left/righthanded versions of the quarks and leptons acting differently in weak interaction.  
Figure 11 SM, Parameters _{} 
Figure 12 Mass, Origin of 
Another phase change occurs about 10^{5} sec after BB. This one transforms the QuarkGluon Plasma to the bound state of Hadons (see "Chiral Symmetry Breaking"). 
Figure 13 Stability of the Universe [view large image] 
and the sum is over all SM particles acquiring a Higgsdependent mass M_{i}. The precise form of V_{1} is not important in the present context, it just shows that the Higgs potential also depends on the particles it acts upon. Furthermore, only the heaviest top quark in the sum is retained in the following consideration. 
In Gauge theory, the Largangian or field equation is invariant under the gauge transformation : ^{'} = e^{i(at)} , where a is the phase angle associates with the "axis", t the generator in the form of nxn matrix, n the dimension of the Gauge space, the number of phase angles is determined by the formula N_{a} = n^{2}  1 (except for n = 1, N_{a} = 1, see a little bit more in "Unitary Groups"). Figure 14 is an example of simplest Gauge transformation with n = 1. It shows that the SO(2) rotational transformation of 2component field (interpreted as two +/ charge states) is equivalent to a single complex field in U(1)= e^{iI} (I = 1) Gauge group. In general SO(n+1) ~ SU(n), where "S" means that the transformation is "Special" with det(U) = +1, which implies continuous rotations without reflection.  
Figure 14 [view large image] 
This is "Global Gauge Transformation" applicable to all space. It is the spatial dependence "Local Gauge Transformation" that generate the Gauge bosons responsible for interaction of particles. 
why, and when, but in certain circumstance (probably via decreasing temperature caused by cosmic expansion), the Higgs field would develop to a false vacuum state and then decays to the state of true vacuum H (as depicted in Figure 15). Then the particles would interact with the Higgs field to slow down by exchange of energy and appear as acquiring mass.  
Figure 15 Higgs Field 
It is not known whether the quantum fields are inherent to the initial environment or developed gradually. Regardless of either possibility, one Gauge theory assumes that it started with a more encompassed Gauge group namely the SU(5) (also see "Group"). 
about 10^{35} sec after the Big Bang. These additional bosons carry new forces which can transform quarks into leptons and vice versa (hence the proton decay, which failed to be confirmed experimentally leading to the desertion of the theory).  
Figure 16 GUT, SU(5) 
See the more encampassing SO(10) Group. 
transformation of electroweak bosons to (W^{+}, W^{}, Z) bosons involves the combination (eating as shown in Figue 17) with the three Goldstone bosons from the Higgs to acquire the longitudinal polarization (equivalent to endowing mass). The transformation also breaks the left/right handed symmetry. The left handed one still has SU(2) gauge symmetry running both weak and electromagnetic interactions; while the right handed version is relegated to have U(1) symmetry, (a singlet without neutrino) interacting via electromagnetic only.  
Figure 17 SSB + ESB 
Without further ado, here's the mathematics to show the consequence of SSB using a simplified example of one boson field (see "Spontaneous Symmetry Breaking" for further details of the mathematics) : 
 
Figure 18a SU(3) Gauge Theory [view large image] 
 
Figure 18b SSB + CSB 

Figure 19 QGP/Hadron Gas Transition _{} 
The main way of processing crude oil is to separate it using fractional distillation. Oil is made up of many different hydrocarbons of different boiling points (Quantum Fields). The first step in distillation is to boil the oil to a very high temperature (Big Bang), usually around 400^{o}C. When the solution boils, it forms vapors entering the fractional distillation column (GUT), which has many trays to collect the liquid (Quantum Fields). As the height in the column increases, the temperature gets cooler, so as the vapors rise, they cool. Since every hydrocarbon chain has a different boiling point, as a chain reaches a height where the temperature is lower than the boiling point, it will cool into a liquid and be collected by a tray (Symmetry Breaking Processes). This method allows for all parts of the oil to be separated and collected so that they can be further processed and refined (Standard Model). A very good animation of the distillation process can be found in Figure 20.  
Figure 20 Q Fields Evolve and Crude Oil Distillation _{} 
According to this scenario, all the quantum fields were unified (mixed together) in the beginning. Each one emerges from the whole and manifests its characteristics only at suitable temperature by the process of Symmetry Breaking. Ultimately, it is the cosmic expansion (which keeps lowering the temperature) that causes all these cosmic phase changes (Figure 09). 