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A Simple Mathematical Model

Cosmic Evolution of Entropy

Magnetic Monopoles

According to the latest cosmological model, the universe sprang into being about 14 billion years ago. At birth, the space was likely to have been curved and warped due to quantum effect within the tiny speck and time may be meaningless. After about 10^{-35} seconds, there began a brief period of exponentially fast expansion, known as inflation, that ironed out any curves or warps in space and made the universe flat (because it becomes so large). Inflation also predicts a much smaller initial region, which is required for smoothing out the distribution of matter and radiation, only leaving behind tiny quantum fluctuations that match the observed spatial variations in the cosmic microwave background radiation and provide the seeds for galaxy formation. It also explains the mystery of the absence of magnetic monopole, which had been diluted during the phase of rapid inflation.
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## Figure 01 Cosmic Inflation [large image] |
See more in "Theory of Cosmic Inflation". |

The orange curve in Figure 01 shows the period of inflation from 10^{-35} sec to 10^{-32} sec after the initial expansion. Figure 02 shows the actual size of the universe after the inflation. Our observable unvierse is only part of the whole thing. The mechanism to drive the inflation is related to a "yet-to-be-discovered" inflaton field, which is thought to be similar to the Higgs fields responsible for the mass of the elementary particles. When the temperature fell below a certain value, a phase transition (similar to the transition of water to ice at 0^{o}C with the release of latent heat) of the inflaton field occurred. The phase transition released energy, which was conversed to hot matter and radiation. It also developed repulsive force to drive the inflation. The inflation stopped when the inflaton field settled down into lower energy state.
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## Figure 02 Unobservable Universe |

fundamental physics will eventually address these issues. Figure 03a depicts two "energy landscapes" for the process (at a certain point in space). In the old theory, bubbles of true vacuum form by tunneling. It results in a haphazard pattern of bubbles that never merge, so the decay process is never complete leaving an empty universe. In the new theory, the downhill rolling is very slow such that the energy density is almost at a constant value to sustain the exponential expansion. When the field gets to the steeper part of the energy slope, it rolls down faster, and when it finally reaches the minimum, it oscillates and dumps its energy into a hot fireball of particles. At this point we have an enormous, hot, expanding universe, which is also homogeneous and nearly flat - it solves the graceful exit problem on how to stop the inflation sensibly. | |

## Figure 03a Theories of Inflation [view large image] |
^{§} There are at least four different scenarios about the pre-bang universe: 1. Brane collision, 2. Universe inside blackhole, 3. Multiverse, and 4. Tunneling from nothing. (see also the "mathematics of Inflation") |

In spite of producing predictions in exquisite accord with observations, the theory of inflation is now faulted on theoretical ground. In the April 2011 issue of Scientific American, an article by the theoretical physicist Paul Steinhardt listed three or four shortcomings (as illustrated in Figure 03b) with the theory of inflation and proposed a cyclic universe to resolve the problems. Followings is a summary of his case against the theory of inflation : | |

## Figure 03b Problems with Inflation [view large image] |

- Scarcity of Good Inflation - There are literally an infinite number of models for the theory of inflation. Each of these model is controlled by a parameter called . An inflation model that matches closely with cosmological observations, is restricted to a narrow range around 10
^{-15}(Diagram a, Figure 03b). Otherwise the model would produce large temperature variation resulting in more stars and galaxies - a more habitable universe - in contrary to the one we live in. - Scarcity of Inflation - By extrapolating backward from the current conditions (e.g., flat and smooth) to the initial stage, it is found that only a tiny fraction of the initial states would lead to a uniform, flat universe. A majority of these do not require inflation, and only an infinitesimal fraction would do so by going through a period of inflation. A similar conclusion was reached in the 1980's by Roger Penrose using method in the Kinetic Theory of Gas to arrive at a probability of 1/10
^{100}for inflation (ends up as flat and smooth universe) to occur (Diagram b, Figure 03b). - Eternal Inflation - In the quantum version of inflation, there will be delay in reaching the end point (of inflation) due to quantum fluctuations (in time). If the delay happens to be of a relatively large time interval, the delayed expansion will engulf the well-behaved region, which becomes an island (bubble) surrounded by the (delayed) inflating space as shown in Diagram c, Figure 03b. Such process repeated an infinite times producing an unbounded number of island surrounded by ever more inflating space. In such scenario, all kinds of universes are possible. But if anything that can happen will happen an infinite number of times, then the theory is not a useful tool in generating testable predictions.

The discovery of B-mode polarization in 2014 has ruled out this cyclic model.

There is yet one more scenario to produce a flat and homogeneous (and isotropic) universe (see "Quantization of the Friedmann Equation").

The concept of cosmic inflation can be illustrated by simple mathematics using only elementary calculus. Suppose the universe is uniform and isotropic as demonstrated by observations. This means that every point in the universe is similar to every other point and can be considered as the "centre" (Figure 04). Now consider a "test particle" of mass m and at distance R from the centre. Since only mass inside the sphere has a net effect on the particle, and if the total mass M insider the sphere of radius R is constant, then the | |

## Figure 04 Repulsion vs Attraction [view large image] |
potential energy of the particle is: |

where G is the gravitational constant.

The gravitational force on the particle is the negative derivative of V(R):

F(r) = -dV(r)/dr = -GMm/r

where the minus sign signifies attraction, which tends to pull the particle towards the centre. The kinetic energy K.E. =

Next consider the case where the total mass M insider the sphere of radius R is not constant, instead the matter density is fixed (see Figure 04). The the gravitational energy of the particle is:

V(R) = -Gm x (4/3) x R

The force on the particle is now:

F(r) = -dV(r)/dr = 2Gm x (4/3) x r ---------- (4)

where the plus sign indicates that the force is repulsive.

The particles are now being pushed away, the whole universe will expand (inflate). Since the kinetic energy acquired by the test particle is equal to the sum over the distance with the force acting on it:

K.E. =

The total energy = K.E. + V = 0 (for the test particle) is conserved like all other dynamical systems. The increase in K.E. is cancelled exactly by the decrease in the potential energy V cumulating in a huge explosion called the Big Bang. Meanwhile, the mass inside the sphere increases with R like:

M = (4/3) x x R

Thus in this scenario, there is a repulsive force to drive the inflation (Eqs.(4) and (5)); matter and radiation are being created according to Eq.(6). Under normal circumstance, the matter density would decrease with the expansion. It is thought that the decay of the "inflaton field" to the lower energy state is responsible for keeping the matter density constant. Figure 05 shows the variation of the energy density since the Big Bang. It maintained a constant value during the inflationary era as described in the simple | |

## Figure 05 Energy Density |
mathematical model. The energy density of water (10^{21} ergs/cm^{3}) and of an atomic nucleus (10^{36} ergs/cm^{3}) are included in the graph for comparison. A mathematical treatment based on general relativity can be found in the appendix on "Relativity". |

Entropy is defined as the degree of randomness^{§}, which can be expressed alternatively as the degree of freedom in a system (the degree of freedom is the number of different parameters or arrangements needed to specify completely the state of a particle or system). The evolution of entropy in the universe as a whole can be separated into four phases as described briefly below (also see Figure 06, which supplies more details):
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## Figure 06 Evolution of Entropy [view large image] |
## Figure 07 Initial Entropy |

- The inflaton field is a coherent system changing rapidly until the end of the inflationary era. Such system has very few degrees of freedom, so it has a very low entropy. (Figure 07, it also shows a brief portray of leptogenesis, which is a theory to explain the asymmetry between matter and anti-matter.)
- At the end of inflation the energy density of the inflaton field decays to zero (see Figure 03a), thereby releasing lots of energy to produce particle anti-particle pairs, and to heat up the universe. It is this "reheating" that produce lots of degree of freedom, and thus lots of entropy.
- The infusion of energy dU ceased once the inflaton energy density vanished, i.e., dU = 0. According to the thermodynamics relation dU = TdS - pdV (where p = pressure, V = volume, T = temperature), the entropy now varies as dS = (p/T)dV. The universe was dominated by radiation up to 10
^{4}years after the Big Bang. During this era p T^{4}; since in term of the size of the universe R, T 1/R, dV R^{2}dR, thus dS dR/R and the entropy S log (R). - In a matter dominated universe p = 0, thus dS = 0; the entropy is conserved as a whole for the rest of the cosmic expansion.
- If acceleration of the cosmic expansion is taken into account, then there is infusion of energy by an amount dU. The entropy dS dU/T will increase until space is nearly empty in attaining the highest entropy state.

The scenario above assumes a quasi-equilibrium approach, which is not entirely correct with the cosmos expanding rapidly. Diagram (a) in Figure 08 is a more realistic sequence according to Boltzmann's general definition of entropy in which the evolution of entropy is described by the phase point moving in phase space without any assumption concerning the equilibrium of the system. Diagram (b) shows another sequence of entropy evolution when the gravitational degrees of freedom is introduced to the system. | |

## Figure 08 Entropy and Gravity |

It is pointed out that the above model fails to explain what set up the initial low entropy state. An alternate scenario adds a period of prehistory (see Figure 09) in which space was nearly empty (thus avoiding the necessity of setting up the low entropy state), and inflation was brought about by fluctuation of quantum fields. It implies the existence of multiverse where the arrow of time may run backward. The appearance of stars and galaxies is a temporary deviation from the equilibrium of empty space. | |

## Figure 09 Entropy Evolution with Prehistory [view large image] |

- Near the center ( about 10
^{-29}cm ) there is a GUT symmetric vacuum. - At about 10
^{-16}cm, its content is the electroweak gauge fields of the standard model. - At 10
^{-15}cm, it is made up of photons and gluons. - At the edge to the distance of 10
^{-13}cm, there are fermion-antifermion pairs. - Far beyond nuclear distances it behaves as a magnetically-charged pole of the Dirac type.
In effect, the sequence of events during the earliest moment of the universe had been fossilized inside the magnetic monopole. | |

## Figure 08 Monopole Structure in GUT [view large image] |

Meanwhile in 1995 the MACRO (Monopole, Astrophysics, and Cosmic Ray Observatory) detector located at the Gran Sasso National Laboratories in Italy had been designed to look for supermassive magnetic monopoles among other exotic particles. It had stopped collecting data by December 2000 and represents another failed attempt for the search of magnetic monopoles.

Figure 09 shows the flux upper limit of monopole search in cosmic ray. It indicates that the detectors are sensitive only up to that level and failed to detect any monopole; in fact the flux could be even lower all the way down to zero. The data are in unit of cm^{-2}s^{-1}sr^{-1} (sr = steradian, is the SI unit of solid angle). The flux upper limit is in the range of 10^{-15} - 10^{-16} with this unit. Such flux level is much lower than that for the cosmic ray bombarding the Earth regularly in the range of 1 - 10. The MACRO experiment comprised three different types of detector : liquid scintillator, limited stream tubes, and NTDs (Nuclear Track Detector for detecting cosmic ray tracks
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## Figure 09 Monopole Flux Upper Limit |
inside solid material). The OHYA experiment (using array of NTDs) is located inside a mine in Japan. SLIM is a high-altitude experiment. |