A Brief Study of the Universe

In the present study, an oscillatory model of the universe is proposed wherein the universe undergoes a sequence of big bang, expansion, contraction, big bang—repeated ad infinitum. The universe comprises of a “World” and an “Antiworld” in both of which matter operates in the positive time zone and antimatter in the negative time zone. Big bang is predicted as the violent encounter between matter and antimatter. It is suggested that antimatter has negative mass and is hugely abundant. It is also shown why it is extremely rare in spite of its abundance. It is predicted that a built in transformation converts matter to antimatter and vice versa. Finally, it is established that symmetry between matter and antimatter in the universe is maintained throughout.


Introduction 1.Methodology
This paper presents a study of the universe using a new method, to be named as the "Method of Indices".It is named so, because it defines and evaluates the "Growth Indices" (briefly, "Indices") of scalars, at play in the universe, links these indices to the Hubble's Constant, and subjects various equations of Physics and Cosmology to unveil new cosmological truths.
The index of any scalar, X , appearing in an equation of physics or cosmology, is denoted by X  and given by where d d , t denoting time.X in the denominator ensures that X  has the same sign as that of X  .So, and indicate respectively the growth and decay of 0 X   0 X   X with advancing time.0 X   marks X as a universal constant, a constant over the eons.
The method is now described briefly.Using rules, given in Section 1.3, of operations involving indices, an equation of physics or cosmology can be transformed into a linear equation in the indices of scalars, which constitute the former equation.Any carefully chosen set of such linear equations, with at least one among them involving the Hubble's constant, is solvable for the unknown indices.Indices, thus obtained, reflect the effect only of the Hubble's expansion.Each of these indices promotes an equation, which links the corresponding scalar to time.The promoted equation will enable scientists to evaluate the scalar for any chosen time, provided its value for any other time is known.
Next, the known constants of physics and cosmology are constants for the present epoch, but may or may not be so over the eons.So, a prerequisite for the application of the method of indices is set up as follows.
"Each scalar except the velocity of light   c , be it a constant or not, is to be regarded from the outset of the analysis as varying with time till the analysis marks it as a constant for all eons.For a scalar X to be so, X  must be zero." Though the Hubble's law plays a pivotal role in this method, it itself is subjected to analysis by this method.And for this, 0 H , the Hubble's Constant of this law, is replaced by H , which is considered as varying with time, 0 H being the value of H at the present epoch.

Definitions and Notations
1) A spatial sphere will be referred to as a "3-sphere".

2)
will denote the radius of the three sphere of the universe.
S 3) M and will denote the mass of matter and antimatter respectively.m 4) The generalised scalar, denoted by X , will represent all scalars individually, when such situation arises.
5) The constants of integration in all equations will be denoted by A without suffices and must not be considered as equal dimensionally or otherwise.
6) Scalars at and 0 0 t  t t  (present epoch) will have suffices "i" and "0" respectively.7) 8) The Hubble's Law states that galaxies move away from an observer with speeds , directly proportional to their respective distances from the observer i.e.

Rules of Operations for the Indices
Rules of operations for indices of two positive scalars, A and , with respect to positive time, are provided below.Whenever used, will denote a positive real number.The rules can be extended to any number of positive scalars.Higher order indices also exist, an example will be provided in Subsection 4.2.

B a
1) If A and are of the same type, )  

 
2. Analysis 0 , the value of q at the present epoch, is the 'deceleration parameter'.As , , and so by Equation (20), Next, as the universe is expanding, , and so, 0

Friedmann's Equations
The Friedmann's equations for complete with the "Cosmological Constant" where 0  is the rest mass density of the universe st .Let where  is a parameter, that sets the level of  .Then, So, the Friedmann's equations are rewritten as:

The Nature of q
Differentiation of Equation ( 28) with respect to time yields: Then, by Equation (21) this becomes: q is non-zero except for 1 2 q  and .So, varies with time; in fact, it decreases from

More about the Friedmann's Equations
Let two quantities,  and , be introduced below [2]: It will be proved in this very subsection that they vary with time.Their values, 0  and 0 , at 0 are the well known "Cosmological Constant Parameter" and "Density Parameter"   0  respectively.


In fact, at , when . means the universe is empty.So, the condition that the universe is empty is , or In de Sitter's universe,   .So, de Sitter's universe is empty.

as a Function of Time q
Equation (31) is written as: By Equation (28) this becomes: Integration of this yields: where A is the constant of integration.Now, Therefore,  and q 0 2.6.Evaluation of At 0 t t  , Equation (45) becomes: By Equation (43), this further changes to: This equation is satisfied by: 2 By Equations ( 5) and ( 22), , and by Equations ( 5) and (17), . So, 2) Index of force: The centripetal force, acting on the revolving body, referred to in above, is Using Equations ( 3), (8), and ( 14) to this : 3) Index of gravitational constant  : The gravi- tational force of attraction between two bodies of mass Then, ; or, , whence 2.9.A Few More Indices Then, by Equation (54), .Now,  :  :  

Why Does the Universe Expand?
That the mass of matter in the universe is decreasing with advancing time is evident from the negative value of M  , obtained in Equation (51).In this decreasing process, M  is the index of the residual mass, and is that of the withdrawn mass (mass that ceases to be so).Equation (51) implies: This proves that the withdrawn mass creates vacuum energy that promotes the accelerative expansion   S     of the universe.Next, Equation (34) can be written as: where M is the residual mass of the universe.The denominator has the dimension of mass.So,  is the , which represents the total rest mass (in nature).Then, the withdrawn rest mass is 2 . So, the proportion of the withdrawn rest mass is: This shows that another measure of the withdrawn mass is  , which is regarded as the driving factor behind the accelerative expansion of the universe.Thus, the accelerative expansion of the universe is due to the vacuum energy created by the withdrawing mass, indicated by   M   and  .

Why Antimatter?
Antimatter is no less a reality than matter [3,4].This fact alone is reason enough to undertake an investigation into the possibility of antimatter, playing a role, similar to that played by matter, in the workings of the universe.It's extreme rarity has so long relegated it to insignificance.
It will be proved in Subsection 3.3 that antimatter is hugely abundant.It will also be explained, in the two subsequent subsections, why, in spite of its abundance, it is rare.

Dynamic Relation between and M m
If M is the mass of a number of particles, then the mass of the same number of antiparticles is given by [5][6][7]: where is a constant.Then, This is the dynamic relation between particles and antiparticles.This is called so, because the ratio varies with time, as shown below.

Abundance of Antimatter
Friedmann's Equation (34) has been rewritten in Equa-tion (58), which is now changed below to it's equivalent equation in indices.
As M is the rest mass of the universe, it is given by: where M is the mass of the universe, and R R c being the average speed of mass.Now, For such , and Equation ( 61) can be written as: By Equations ( 17), ( 21), (53), and (56), the above equation becomes: This does not agree with Equation (51), which gives . So, M of Equation (64) must have a component other than the mass of matter.There can be no such component other than the mass of antimatter.
Now, by Equation (15), So, by Equations ( 65) and (51), we have This, with Equation (48) gives Now, at any time , let a sphere concentric with the 3-sphere of the universe is of radius and have the density Let another concentric sphere of radius R dR  be considered.The density of this sphere can, for all practical purposes, be taken as

dR
, the mass of this sphere is: R , neglecting higher powers of than one.So, .
Then, the average speed of the entire mass of the 3-sphere of the universe is: where is the speed of the galaxies at a distance from the observer, or for all practical purposes, the speed of matter within the spherical shell, having the outer and inner radii and respectively.so, Now, Equation (58) implies: , by Equation (68), the above becomes: At 0 t t  , this becomes: . This relation is obtained as follows: This is the static relation.Then, . Thus the criterion of static relation is fulfilled by Equation (73).
Next, Equation (60 Thus Equations ( 74) and (75) do not agree, seemingly posing a problem.It will be seen in Subsection 3.7 that the dynamism of relation (75) originating from the static relation (74) is the solution to this problem.

Two Time Zones
The concept of two time zones is necessary to show that the dynamism of relation (75) originates from the static relation (74).This concept is there in the theory of Quantum Electrodynamics also.This theory will come up again in Subsection 3.7.
This concept necessitates introducing a new convention of denoting the time derivative and the indices of scalars as is given below.Suffices (or, " " " t "  ) and " " t  (or, " "  ) will be used under this convention.
It must be mentioned here that all indices, occurring upto Subsection 3.4, are of the type .t X  Now, the static relation (73 This is the fundamental relation between matter and antimatter.Also, as the natural time zone for matter is the Positive Time Zone (PTZ), Equation (78) shows that the same for antimatter is the Negative Time Zone (NTZ).Time is progressive in the PTZ and regressive in the NTZ.

The Time Barrier
The PTZ is defined as and the NTZ is defined as So, matter and antimatter, when in the PTZ, moves forward in time and when they are in the NTZ, they move backward in time.The only instant of time common to both PTZ and NTZ is: Since matter and antimatter are trapped in the PTZ and the NTZ respectively when Equation (78) rules, matter cannot cross over to the NTZ as matter, and antimatter cannot cross over to the PTZ antimatter, and so there exists a "time barrier" (except ) to prevent matter and antimatter from meeting.So, antimatter cannot be sighted in the PTZ, when Equation (78) rules.
When Equation (79) rules, antimatter and matter appear in the PTZ and NTZ, not in the NTZ and PTZ respectively.Since they appear in their unusual time zones, their appearances must be extremely rare.So, extremely rare are the appearances of antimatter in the PTZ.Such rare appearances of antimatter result from chance phenomena like cosmic rays striking interstellar matter.

Conversion
Equation ( 78) is the result of the following transformation: This transformation means that matter in the PTZ gets converted to antimatter in the NTZ.So, antimatter increases at the cost of matter, that decreases [vide Equation (51)].This is exactly what the dynamic relation (60) conveys.As transformation (82) originates from the static relation, the problem, that was faced in Subsection 3.4, vanishes.
Next, Equation ( 79) is the result of the following transformation This transformation shows that antimatter, which is in the PTZ and so moves forward in time, gets converted to matter, which is in NTZ and so moves backward in time [8].In the theory of Quantum Electrodynamics also, antimatter moving forward in time is interpreted as matter moving backward in time.

Rarity of Antimatter
When Equation (78) rules, antimatter is hugely abundant in the NTZ at present, and because of the time barrier there is absolutely no chance of antimatter appearing in the PTZ and being detected.But when Equation (79) rules, antimatter appears in the PTZ.It has been explained in Subsection 3.6 that such appearances are extremely rare.Thus it is explained why antimatter is extremely rare in spite of its huge abundance.

The Three Fundamental Scalars
The three fundamental scalars (time, radius of the 3sphere and mass), associated with antimatter, are related to the corresponding scalars, associated with matter, as given below.where the suffix "a" indicates association with antimatter. 
Now, when this strip is given a single twist and the edge AB is joined with the edge in such a way that CD A and coincide with and respectively, a Mobius strip is formed.Let the moment of inflexion be , and all the scalars at this moment have the suffix "T".Then, by Equation Then, by Equations ( 71) and (72), 52 3.738643084 10 Kg Next, by Equation (51), or, Integration of this gives: Then, Then, . By Equations ( 48) and ( 71 12.42726525 10 Kg 1 Then, by Equation (46),     .Also, Antimatter decreases in the contracting universe with regressive time   t , as given in relation (113), while matter increas hown in rela appearing antimatter is converted transformation (114), which is part built "Mobius transformation", discussed in Subsection 4. ve expansion of the universe.With the mass of matter preponderating atter, contraction rules over expansion.es, as s tion (112).The disto matter by the of the reversed in-4 and now reversed.As antimatter in the universe is plentiful, the con rsion is "continual", and matter increases at the cost of decreasing antimatter.

Back to t  0
Relation (112) implies that with regressive time increasing mass of matter effects contraction of the universe, while relation (113) shows that with regressive time decreasing mass of antimatter effects over that of antim As time is regressive, it is back to 0 t  , when by relation (112), 0 S  and M   , and by relation (113), 0 m  .Thus , , t S M and m are back to their starting values at 0 t  .

Symmetry between Matter and Antimatter
Th start atter and antimatter at a xcep e of e preponderance of matter over antimatter, denied antimatter a role, as important as the in their denotations, with those in the WORLD, enotations.e universe, discussed so far, did not with symmetry between m , nor this symmetry was there ny instant e t at t T  .Absence of this symmetry, becaus th role of matter, in the workings of the universe.
In the next section, the concept of the "Antiworld" is introduced and it is shown that symmetry between matter and antimatter is never disturbed.

What the ANTIWORLD Is
The ANTIWORLD is the mirror image of the WORLD.It is like the WORLD except that the fundamental scalar therein, having the subscript "A" bear the following relations having no subscript in their d where by Equation( 60), M m  

Isolation of Two WORLDS fro
fo ation ( Next, the trans rm 96) implies: This means that growth rates of A M in the ANTIWORLD m in the WORLD are equal.
The initial values and growth rates of A M in the ANTIWORLD being correspondingly equal to those of M in the WORLD implies that the value of A M at any instant quals that of Addition of Equations ( 128) and (130) gives: of th

Equation (123) implies that wit
This shows that symmetry between matter and antimatter is maintained throughout the expansion phase e universe.

Roles P by Antimatter and Matter in the ANTIW LD
, which conveys that in the ANTIWORLD antimatter preponderates over matter.

Then,  
is indicative of the expansion o the ANTIWORLD, which will be on the threshold o cting into the contractio

The Moment of Inflexion
It has been seen in 132 Subsection 4.6 that the WORLD inflects into the contraction phase at t T  , when Equations ( 128) and (130) for t T  are respectively Equations ( 133) and (135) together give: This, with Equation (134), gives:

132), ANTIWORLD inflects into the c
This, read with Equation ( shows that the ontraction phase when

at . The mass of antimatter as well as that of matter
is given by:

where 0 H 9 )
i.e. the Hubble's Constant.Now, as 0 R R    R and 0 H is to be replaced by H (as proposed in Section 1.0.1),then R H R   is the Hubble's Law which can now be written in the following new form: In all numerical calculations, following values of the Gravitational Constant (G), the Hubble's Constant (H) and the velocity (c) of light for the present epoch will be

H
and 0 are taken from Subsection 1.2 and 0 from Equation (50).With these values, Equation (70) gives The static relation between M and m is characterised by 0

Figure 1 .
Figure 1.In the figure all the Particles in the positive time zone represented by gets converted into the corresponding antiparticle in the negative time zone represented by  A t M ,

4 . 5 .
(95)  and (96) together constitute a Mobius transformation operating within the universe[9].So, under this built-in Mobius transformation in the universe, particles (in the PTZ), represented by   , A t M , undergo conversion to the corresponding antiparticles (in the NTZ), represented by .Plenty of matter in the universe ensures "continual" conversion of matter to antimatter.Such a scenario supports Equation (60) as well as the pair of Equations (92) and (94) taken together.Roles Played by Matter and Antimatter Relation (92) shows that with advancing time, decreasing matter effects the expansion of the universe, while increasing antimatter effects its contraction [vide relation(94)].Due to matter preponderating over antimatter during the expansion phase, the contraction remains eclipsed.So, m M  is symptomatic of the expansion of the universe.The universe will be on the threshold of inflecting into the contraction phase when ).

. The ANTIWORLD Is Expanding with the WORLD
With the reversal of the "Arrow of Time", the arrow of the transformation (125) gets reversed ) implies that with regressive time increasing mass of anti a ter effects contraction of the ANTIWORLD, while relation (146) shows that with regressive time decreasing mass of matter effects its expansion.With the mass of antimatter preponderating over that of matter, contraction rules over expansion.As time is regressive, it is back to 0 , when, by re A t  0 A M  .Thus, , ,