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, , appearing in an equation of physics or cosmology, is denoted by and given by

where, t denoting time. in the denominator ensures that has the same sign as that of.

So, and indicate respectively the growth and decay of with advancing time. marks 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, 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 to be so, 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, , the Hubble’s Constant of this law, is replaced by, which is considered as varying with time, being the value of at the present epoch.

1) A spatial sphere will be referred to as a “3-sphere”.

2) will denote the radius of the three sphere of the universe.

3) and will denote the mass of matter and antimatter respectively.

4) The generalised scalar, denoted by, will represent all scalars individually, when such situation arises.

5) The constants of integration in all equations will be denoted by without suffices and must not be considered as equal dimensionally or otherwise.

6) Scalars at and (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.

where i.e. the Hubble’s Constant. Now, as and is to be replaced by (as proposed in Section 1.0.1), then is the Hubble’s Law which can now be written in the following new form:

9) 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 used

Rules of operations for indices of two positive scalars, 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.

1) If and are of the same type,

But,

2)

3)

4)

5)

6)

7)

8)

9)

10)

11)

12) For,

Since and are of the same type, both being lengths, by rule 1 of Subsection 1.3, Equation (4) is written as [1]

As,; and so, , whence Equation (17) becomes

So,

If

then

, the value of at the present epoch, is the ‘deceleration parameter’. As, , and so by Equation (20), , whence

Next, as the universe is expanding, , and so,

. Then, , whence

Also, ,

The Friedmann’s equations for complete with the “Cosmological Constant”, are [2]

where 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:

Differentiation of Equation (28) with respect to time yields:

Then, by Equation (21) this becomes:

So,

is non-zero except for and. So, varies with time; in fact, it decreases from to.

Let two quantities, and, be introduced below [2]:

It will be proved in this very subsection that they vary with time. Their values, and, at are the well known “Cosmological Constant Parameter” and “Density Parameter” respectively.

Equation (28) implies:, whence

Next, Equation (29) implies:whence

Equations (35) and (36) together imply:

Also, Equations (35) and (37) together imply:

Now, by Equation (2).

Since by Equation (23), , then

Next, by Equation (39),. Then,

Then,

Equations (33) and (41) together imply:

By Equation (35) this becomes:

Equations (35) and (37) respectively show that and vary with, which has been shown in Section 2.3 to be varying with time. So, and also vary with time.

In fact, at, when, and.

And, when, and. means the universe is empty. So, the condition that the universe is empty is, or, or.

In de Sitter’s universe,. So, by Equation (33),. So, de Sitter’s universe is empty.

Equation (31) is written as:

By Equation (28) this becomes:

Integration of this yields:

where is the constant of integration.

Now,

So,; then,

So,

At; so,. Then,

or,

whence

Therefore,

At, Equation (45) becomes:

By Equation (43), this further changes to:

This equation is satisfied by:

Then, by Equation (43),

1) Index of mass of matter: Angular momentum of a body of mass, revolving in a circular orbit of radius with linear velocity in a direction tangential to the orbit, is constant; i.e., where is the constant angular momentum. Using Equation (13) this is transformed to:

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 :

whence

3) Index of gravitational constant: The gravitational force of attraction between two bodies of mass and at a distance is:

, whence.

Then,; or, , whence

1) Index of for: For any

Then,

2) Index of: By Equation (16), as,

Then, by Equation (54),. Now, whence

3) Index of: (by Equation (6))

As and so,. Then,

So, by Equation (31),

4) Index of: By Equation (6),. Then, , as . Then, by Equation (31),

That the mass of matter in the universe is decreasing with advancing time is evident from the negative value of, obtained in Equation (51). In this decreasing process, 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 of the universe. Next, Equation (34) can be written as:

where is the residual mass of the universe. The denominator has the dimension of mass. So, is the proportion of to, which represents the total rest mass (in nature). Then, the withdrawn rest mass is. 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 and.

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.

If is the mass of a number of particles, then the mass of the same number of antiparticles is given by [5-7]:

This implies

where is a constant. Then, , whence

This is the dynamic relation between particles and antiparticles. This is called so, because the ratio varies with time, as shown below.

Then,

Friedmann’s Equation (34) has been rewritten in Equation (58), which is now changed below to it’s equivalent equation in indices.

As is the rest mass of the universe, it is given by:

where is the mass of the universe, and, being the average speed of mass. Now, , and; so,. For such, and consequently for, has been evaluated in Equation (55) as. Then

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, 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. This prompts replacement of by, i.e., by in Equation (64) as given below.

Now, by Equation (15), .

Equation (60) So,

Since during the expansion of the universe, ,

. Then,

So, by Equations (65) and (51), we have

Then, at,

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. Then the mass of this sphere is

, where.

Let another concentric sphere of radius be considered. The density of this sphere can, for all practical purposes, be taken as, since is infinitesimally small. Then, , the mass of this sphere is:, 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,

So, by Equation (40), , whence

Therefore,

Now, Equation (58) implies:

As, by Equation (68), the above becomes:

Replacement of by changes this to:

At, this becomes:

Equations (37) and (48) together give . Values of and are taken from Subsection 1.2 and from Equation (50). With these values, Equation (70) gives

From this with the help of Equation (66), and are evaluated as given below:

The static relation between and is characterised by. This relation is obtained as follows:

Equation (59), whence

This is the static relation. Then,

This implies. Thus the criterion of static relation is fulfilled by Equation (73). Next, Equation (60), whence

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.

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,) and (or,) will be used under this convention.

1) and will respectively be denoted by and.

2) and will respectively be denoted by and.

Also, two relations between and are deduced below.

a)

whence

b) Equation (76) implies

It must be mentioned here that all indices, occurring upto Subsection 3.4, are of the type

Now, the static relation (73), whence by Equation (76),

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.

Next, relation (73) whence by Equation (76)

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.

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.

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 (time, radius of the 3- sphere 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.

Equation (39); so, by Equation (23),

where

Equation (39) also implies:

Now, Equation (12)

So, by Equation (51),

whence

Next, Equations (87) and (88) together imply:

whence

By Equations (90) and (91)

Equation (76) whence

Equation (92). So, by Equation (93),.

Then, by Equation (77), this becomes:

or, , whence.

Now, because the LHS and the RHS, , where is a constant. Then, , whence

Let the four corners and of a rectangular strip of paper represent respectively the paired scalars and, as shown in the figure. Then, transformations (82) and (83) can be written as below.

Now, when this strip is given a single twist and the edge is joined with the edge in such a way that and coincide with and respectively, a Mobius strip is formed.

Then, transformations (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, 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.

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, is symptomatic of the expansion of the universe. The universe will be on the threshold of inflecting into the contraction phase when

Let the moment of inflexion be, and all the scalars at this moment have the suffix “T”. Then, by Equation (97),

. By Equation (60),. So,

Then, by Equations (71) and (72),

Next, by Equation (51), , or,

.

or,

Integration of this gives: , where is the constant of integration.

Now, since, and . Also,.

Then,; or,

Then,. By Equations (48) and (71),. So,

Next, let

Then, by Equation (46), , and by Equation (45),

Then,. So, .

By Equation (99), this becomes:

By Equation (47),. Also, .

So, by Equation (102),. For this value of, Equation (103) gives:

So, inflexion occurs after years from now.

Let in the post-inflexion period, time be denoted by and other scalars have the suffix “c”. Also, let and be the radii of the 3-sphere, and and be the Hubble’s Constant therein at times and respectively. will indicate the reversal of the “Arrow of Time”.

moves from to, while moves from to. So, each value of must correspond to a unique value of. So,

Also, as the universe contracts,

Next, and, and so, . So,

Also,

Now,

where is the constant of integration. Then, by Equation (108),; or,

Then,

So, by Equation (107),

Hence,

Differentiation of this with respect to gives:

Next, moves from to, while moves from to. So, each value of must correspond to a unique value of. So,

Then, , or,. Inequality (109), so, indicates

So, the “Arrow of Time” reverses at the moment of inflexion, i.e. at.

Equation (92) is reproduced below:

Equation (94) is reproduced below:

Reversal of the “Arrow of Time” at reverses the arrows of transformations (95) and (96) as shown below.

Antimatter decreases in the contracting universe with regressive time, as given in relation (113), while matter increases, as shown in relation (112). The disappearing antimatter is converted to matter by the transformation (114), which is part of the reversed inbuilt “Mobius transformation”, discussed in Subsection 4.4 and now reversed. As antimatter in the universe is plentiful, the conversion is “continual”, and matter increases at the cost of decreasing antimatter.

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 expansion of the universe. With the mass of matter preponderating over that of antimatter, contraction rules over expansion.

As time is regressive, it is back to, when by relation (112), and, and by relation (113),. Thus and are back to their starting values at.

The universe, discussed so far, did not start with symmetry between matter and antimatter, nor this symmetry was there at any instant except at. Absence of this symmetry, because of the preponderance of matter over antimatter, denied antimatter a role, as important as the 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.

The ANTIWORLD is the mirror image of the WORLD. It is like the WORLD except that the fundamental scalar therein, having the subscript “A” in their denotations, bear the following relations with those in the WORLD, having no subscript in their denotations.

where by Equation(60),

Time in the ANTIWORLD is, by Equation (116) negative. It advances from to. So, its range is:

On the other hand, time in the WORLD is positive and advances from to, its range being

These two ranges show that neither of the WORLDS has in the other’s range, and so the WORLD has no past beyond, or the ANTIWORLD has no past beyond. This means that except at, the two WORLDS are completely isolated from each other by this time barrier.

The relation between, the Hubble’s Constant in the ANTIWORLD, and, that in the WORLD, is obtained, using Equations (116) and (117), as is given below.

By Equation (77), , and so, , whence

This shows that the ANTIWORLD expands from (at) to (at) along with the WORLD, which expands from at to at.

As at, and growth rates of and are equal (vide Equation (122)), at any instant in the WORLD implies at in the ANTIWORLD. That is, at, the ANTIWORLD is exactly at that stage of expansion, whereat the WORLD is at.

Equation (94) implies. Then, by Equation (77), . Of this, the L.H.S. and the R.H.S., and so, , where is a constant. Then,; or,; or, , whence

Equation (92) implies. Then, by Equation (77), , whence. The L.H.S. of this, and the R.H.S., and so, , where is a constant. Then,; or, ; whence

Let the points, , , and in the figure of Subsection 4.4 be represented in the ANTIWORLD by, , , and respectively. Then, transformation (95) implies:

whence,

Next, the transformation (96) implies:

whence,

Now, let, , , be the four corners of a rectangular strip of paper. When this strip is given a single twist and the edge is joined with the edge in such a way that and coincide with and respectively, a Mobius strip is formed.

Then, transformations (125) and (126) together constitute a Mobius transformation, operating within the ANTIWORLD. So, under this built-in system of Mobius transformation, anti-particles (in the NTZ) represented by undergo conversion to the corresponding particles (in the PTZ) represented by. Plenty of antimatter in the ANTIWORLD ensures “continual” conversion of antimatter to matter. Such a scenario supports Equation (60) as well as Equations (123) and (124).

As matter is very rare in the NTZ, transformation (126) rarely occurs.

Equation (123) shows that in the ANTIWORLD, at. Also, Equation (92) shows that in the WORLD, at So, in the ANTIWORLD and in the WORLD have the same initial values.

Next,. So, by Equation (118),

So, , whence

This means that the growth rates of in the ANTIWORLD and in the WORLD are equal.

The initial value and growth rate of in the ANTIWORLD being correspondingly equal to those of in the WORLD implies that the value of at any instant equals that of at the corresponding instant. That is

Next, Equation (124) shows that in the ANTIWORLD at. Also, Equation (94) shows that in the WORLD, at. That is, in the ANTIWORLD and in the WORLD have the same initial values.

Application of Equations (76) and (77) on Equation (127) gives:

; or,; or,; or, whence

This means that growth rates of in the ANTIWORLD in the WORLD are equal.

The initial values and growth rates of in the ANTIWORLD being correspondingly equal to those of in the WORLD implies that the value of at any instant equals that of at the corresponding instant. That is, , whence

Addition of Equations (128) and (130) gives:

This shows that symmetry between matter and antimatter is maintained throughout the expansion phase of the universe.

Equation (123) implies that with progressive time decreasing antimatter effects expansion of the ANTIWORLD, while Equation (124) implies that with progressive time increasing matter effects contraction of the ANTIWORLD.

In the WORLD, matter preponderates over antimatter, i.e.. So, by Equations (128) and (130), this becomes, which conveys that in the ANTIWORLD antimatter preponderates over matter. Then, is indicative of the expansion of the ANTIWORLD, which will be on the threshold of inflecting into the contraction phase when

It has been seen in Subsection 4.6 that the WORLD inflects into the contraction phase at, when

Equations (128) and (130) for are respectively

and

Equations (133) and (135) together give:

This, with Equation (134), gives:

This, read with Equation (132), shows that the ANTIWORLD inflects into the contraction phase when. The mass of antimatter as well as that of matter at is given by:

Let in the post-inflexion period, scalars have the suffix “”. Also, let and be the radii of the 3-sphere, and and be the Hubble’s Constant therein at times and respectively. will indicate the reversal of the “Arrow of Time”.

moves from to, while moves from to. So, each value of must correspond to a unique value of. Then,

Also, as the ANTIWORLD contracts,

As and; , and so,

Then^,

whence

Next,. As ,. So, . On integration, this yields:

where is the constant of integration. Then,

So,

Therefore,

which implies: or, whence,

Differentiation of this with respect to yields:

Now, moves from to, while moves from to. So, each value of must correspond to a unique value of. So,

Then, either; or,.

In the former case, , which contradicts Equation (140). So,

This indicates that there is the reversal of the “Arrow of Time” at.

Equation (123) is reproduced below:

Equation (124) is reproduced below:

With the reversal of the “Arrow of Time”, the arrow of the transformation (125) gets reversed as follows:

Matter decreases in the contracting ANTIWORLD with regressive time, as given in relation (146), while antimatter increases, as shown in relation (145). The disappearing matter is converted to antimatter by the transformation (147), which is a part of reversed in-built “Mobius transformation” discussed in Subsection 5.6 and now reversed.

As matter in the ANTIWORLD is plentiful, the conversion is continual, and antimatter increases at the cost of decreasing matter.

Relation (145) implies that with regressive time increasing mass of antimatter 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, when, by relation (145), and, and by relation (146),. Thus, and are back to their starting values at.

Equation (145) shows that in the ANTIWORLD at,. Also, Equation (112) shows in the WORLD at. So, in the ANTIWORLD and in the WORLD have the same initial values.

Next, Equation (127) shows that, i.e., the growth rates of in the ANTIWORLD and in the WORLD are equal.

The initial value and growth rate of being correspondingly equal to those of implies that the value of at any instant equals that of at the corresponding instant. That is,

Next, Equation (146) shows that at. Also, Equation (113) shows that at. That is, in the ANTIWORLD and in the WORLD have the same initial value.

Application of Equation (77) on Equation (127) gives:

whence

The initial value and growth rate of in the WORLD being correspondingly equal to those of in the ANTIWORLD implies that the value of at any instant equals that of at the corresponding instant. That is, whence

Addition of Equations (148) and (150) gives:

This shows that the symmetry between matter and antimatter is maintained throughout the contraction phase of the universe.

It has been proved in subsection 5.7 that this symmetry is maintained throughout the expansion phase of the universe. So, this symmetry is absolute in the universe.