1

12 marzo 2014 – ore 17.30

THE GAP ( EGUT – ESU ) AND THE EVOLUTION

OF THE UNIVERSE

Antonino Zichichi INFN and University of Bologna, Italy

CERN, Geneva, Switzerland Enrico Fermi Centre, Rome, Italy

World Federation of Scientists, Beijing, Geneva, Moscow, New York

Purpose of this note is to call attention on the possible connection with the evolution of our Universe of the energy GAP which exist between the energy level EGUT [where the three gauge couplings α! (QED), α! (QFD) and α! (QCD) converge towards a common origin, α!"#] and the energy level ESU [where the Relativistic Quantum String Theory (RQST) has established the origin of the Gravitational Forces (i.e. the String Unification Scale ESU)].

The existence of the GAP illustrated in Figure 1 between EGUT and ESU, is based on the most exact study of the evolution with Energy of the three gauge couplings α!  α!  α!, reported in [1]. The conclusion of this paper is the consequences of the existence of this GAP should be extensively studied.

2

Figure 1

3

The Fundamental Forces of Nature should all start at ESU. The consequences of the GAP in understanding the evolution of our Universe is one of the most interesting problems in front of us.

The starting point is at the time of the 1979 EPS Geneva Conference, when «the three gauge couplings, 𝛼!  𝛼!  𝛼!, were not converging in a point but in a sort of triangle». These words are of Rudolf Mössbauer [2]. The three gauge couplings α!  α!  α! do converge towards a unique value α!"#  ≃  

!  !"

at the energy  α!"# ≃  5  ×  10!"  GeV if the existence of supersymmetry is introduced in the evolution equations of α!  α!  α! [3]. The energy evolution of the three couplings

α! (q2) α! (q2) (1)

α! (q2)

had never before been performed with the new condition [4] based on the energy dependence, not only of the gauge couplings themselves as indicated in (1), but also with the energy dependence of the masses: i.e. the EGM (Evolution of Gaugino Mass) effect [4].

This inclusion produced a factor (700)−1, nearly three orders of magnitudes, for the threshold of supersymmetry breaking.

4

Suppose the convergence of the three couplings (α!α!α!) is computed taking into account the evolution of each couplings with q2, neglecting the variation of the masses associated with the physics of the given gauge–group, i.e. U(1) for α! , SU(2) for α!  and SU(3) for α! . Suppose the “prediction” is E!"!#

(!)  = 700 TeV. Using the same model the prediction becomes 1 TeV, if the EGM effect [4] is included. The search for the lightest supersymmetric particle would become possible for LHC.

The first reason for the study of the evolution with q2 of α!  α!  α! was in order to see what was needed [3] for the three gauge couplings to converge towards the same point at the energy EGUT.

A detailed study of this convergence, when all experimental uncertainties are taken into account [5], gave as result that, if we take all “best” values, the convergence level where they become equal

α!   =  α! = α!   ≃  !  !"

is at  E!"#   ≃  5  ×  10!"  GeV

which is more than three orders of magnitude below the Planck Energy level.

5

Let us take for granted that this is indeed the case; i.e. the three couplings converge at  E!"#  .

For the Planck Energy level, taking into account the RQST (Relativistic Quantum String Theory) approach to include Gravitational Forces in the game, it is necessary to multiply  E!"#$%&   by 𝛼! and the result goes down to

 E!"   ≃  10!"  GeV.

This means that the Gravitational Forces start to operate at

 E!" ≃  10!". Suppose that this is correct, and suppose that there are

no other forces in this energy range, from ≃ 10!" Gev to ≃ 10!" Gev.

Problem. What happens in the energy range of the GAP

 E!"        and      E!"#    

where the only forces are the Gravitational. In this Energy interval the Universe consists only of what

the Gravitational Forces can do. They can only produce masses without any charge and without other properties, which are generated by the other Fundamental Forces with their elementary particles. All that the primordial Universe can consist of are masses as Black–Holes. Let us call them primordial Black–Holes.

6

At present Galaxies and Stars are supposed to have their origin in Space-Time quantum–fluctuations.

If at this time only Gravitational Forces were present, they could generate only primordial Black–Holes. The origin of the Galaxies should be these primordial Black–Holes, now at the centre of each Galaxy. Their masses are in the range of 10!  ÷  10! solar masses; the total mass of each Galaxy being of the order of 10!!  solar masses:

 M!"#"$%. ≃  2  ×  10!!  𝓂⊙,

the “primordial” Black–Hole acted as seeds for each Galaxy formation. Imagine we could see what the matter of a primordial Black–Hole is made of. We would find that this matter has zero protons, zero neutrons and zero electrons.

At the time of the GAP only the Gravitational Forces were present. The Universe at that time knew nothing of SU(3) ×  SU(2) × U(1).

In this primordial Universe the Einstein equation was operative and the solution of the Einstein equation found by Schwarzschild was at work.

The Schwarzschild solution of the Einstein equation corresponds to such a high density of matter that – apart from

7

radiative effects – nothing can escape the gravitational attraction. John Wheeler called this solution: “Black–Hole”.

The smallest Black–Hole is given by the Planck scale where the unity of length is

ℓ𝓁!"#$%& =  10!!!  𝑐𝑚.

There is no limit for the largest length. If we take the present radius of our Universe

ℓ𝓁!"#$%&'% ≅ 10!"  𝑐𝑚

the ratio of the two lengths is a very large number

ℓ𝓁!"#$%&'%ℓ𝓁!"#$%&

 ≃ 1062 .

If we take the Schwarzschild formula in order to know what is the mass which correspond to this Radius, the answer is the mass of our Universe

𝑚!"#$%&'% ≅    8   ∙  10!!  𝑔𝑟   ≃  10!" gr ;

since 𝑚  (𝑝,𝑛, 𝑒)  ≃  10!!"  𝑔𝑟, the total number of nucleons in the Universe is

𝑁!"#$%&!'  ≃  10!"  .

According to Peter Bergmann [6] the mass-energy of the Universe should be zero. The remarkable fact is that if we take as mass of the Universe the value coming from the number of

8

Galaxies, the mass of the Universe turns out to be [7] the one predicted by the Schwarzschild condition.

The conclusion is that the evolution of the smallest Black–Hole has obeyed to the Schwarzschild equation producing our Universe, which looks like a very big Black–Hole.

And now the problem arises: are Black–Holes phenomena scale-invariant? If Black–Hole properties are scale invariant Small and Big Black–Holes should be such that in their inner structure the law of physics known to us should no longer be valid.

But, our Black–Hole allows the study all laws of physics which go from the Standard Model to the extrapolation called Beyond Standard Model (BSM). One of the possibilities for BSM being the Superworld [5].

It is therefore established that the laws of Physics we know remain valid well inside a Black–Hole, provided that this Black–Hole is as large as our Universe.

This finding could be related with the properties theoretically mentioned by Gerardus 't Hooft [8]. He says on page 77: «If the original amount of material was big enough, the contraction will proceed, and, in the limit of zero pressure and purely radial, spherically symmetric motion, the

9

equations can easily be solved exactly. We obtain flat Space-Time inside, and a pure Schwarzschild metric outside. As the ball contracts, a moment will arrive when the Schwarzschild horizon appears. From that moment on, an outside observer will no-longer detect any radiation from the shell, but a Black–Hole instead». All we need is to apply T-reversal to the 't Hooft Black–Hole and have it stay fixed when the Schwarzschild horizon appears.

Our Universe is a static Black–Hole. If somebody from outside our Universe would like to see

what there is in a sphere whose radius is the radius of our Universe (𝑅!  ≃  10!"  𝑐𝑚 ) he will find our Black–Hole, which is the Universe were we have life and knowledge.

Planck discovered [9] what we now call “the Planck Universe”. In his universal outlook of the world – independent of our restricted environment – Planck wanted that the fundamental units of Mass, Length and Time must depend only on the values of the Fundamental Constants of Nature: the speed of light c, the constant of action h, and the gravitational constant G.

In this Universe the units of Length, Mass, Time and Temperature must be independent of special bodies or substances such as it is the case for our units of Length

10

(centimetre), Mass (gram), Time (second) and Temperature (degree Celsius or Kelvin).

Planck included also the Boltzmann’s constant k which converts the units of Energy into units of Temperature. This allowed Planck to have a fundamental value also for the Temperature. Here are the orders of magnitudes of Planck’s units:

Length = (G ⋅ h /c3)1/2 ≃   10−33 cm Time = (G ⋅ h /c5)1/2 ≃   10−44 sec Mass = (h ⋅ c /G)1/2 ≃   10−5 gr Temperature = K−1 ⋅ (hc5/G)1/2 ≃   1032 Kelvin.

It is remarkable the way Planck considered these quantities: «In the new system of measurement each of the four preceding constants of Nature (G, h, c, k) has the value one». This is the meaning of measuring Lengths, Times, Masses and Temperatures in Planck’s units.

These quantities had a special meaning for Planck [9]: «These quantities retain their natural significance as long as the Law of Gravitation and that of the propagation of light in a vacuum and the two principles of thermodynamics remain valid; they therefore must be found always to be the same, when measured by the most widely differing intelligence according to the most widely differing methods».

11

When Planck was expressing his ideas on the meaning of his fundamental natural units there was neither the Big-Bang nor the cosmic evolution. The very instant of the cosmic expansion is the Planck-Time and the corresponding density of the Universe is the Planck density.

In our world the density is dictated by the fact that we are made by atoms and therefore the basic quantities are the mass of the nucleon (mN ≅ 10−24 gr) and the radius of the atom (10−8 cm) which is dictated by the electric charge and the mass of the electron.

Let us call the value of this “density” the “atomic density”:

ρatomic ≅ Mnucleon

Ratom3 ≅ 10−24gr

10−8cm( )3 = 10−24gr

10−24cm3 = 1 grcm3 .

The density of water is typical of the atomic density. In the above formula we neglect details like (4/3 π) in front of R!"#$!  to estimate the “atomic volume”.

When we go from water to lead the “atomic density” increases by an order of magnitude.

This is due to the increase in the mass of the nucleus by two orders of magnitude

€

mPbnucleus ≅ 102 Mnucleon

and a correspondent increase by an order of magnitude in the

12

atomic volume. The next possible density – many orders of magnitudes

higher – is the “nuclear” density: 1015 times greater than the “atomic” density.

The reason being the value of the nuclear radius, which is of the order of one Fermi-unit (10−13 cm), i.e. five orders of magnitude smaller than the atomic radius.

Atomic and nuclear bindings are specific for a given Element of the Mendeleev Table and do not change when the amount of the given Element changes.

For both forms of matter, atomic and nuclear, the density does not change when the amount of matter increases: one ton of lead has the same density as one kilogram of lead.

For matter where the binding force is gravitational (without any other forces being involved), the density decreases when the amount of mass increases.

More precisely the density decreases with the square of the mass. This is the great discovery of Schwarzschild [10] and is coming from the relation which exists between the Radius of a Black–Hole,

RBH ,

and the mass of the same Black–Hole,

MBH .

13

Here is the Schwarzschild formula:

€

R BH = 2G MBH

c2 = K MBH ≅ 1.5 × 10−28 ⋅ cm ⋅ gr −1 ⋅ MBH (2)

with G being the Gravitational Constant, c the speed of light and

K =

€

2Gc2 ≅ 1.5 × 10−28 ⋅ cm ⋅ gr −1 .

In the Table below we show the values of the matter density in our world today and in the world we come from, i.e. the Planck Universe.

Every The World day’s we World come from

Human Body Density Planck Density

1gr/cm3 1093 ⋅ gr/cm3

Our World The Planck Universe

cm 10−33 cm

gr 10−5 gr

sec 10−44 sec

14

When the mass is the value of the Planck unit given in the previous Table. i.e.

MPlanck = 2.2 × 10−5 gr,

the correspondent Radius of the Black–Hole is

€

R BHPlanck ≅ 1.5 × 10−28 × 2.2 × 10−5 × cm ≅ 3.3 × 10−33 cm.

The Black–Hole Radius increases linearly with its mass

value, as shown in Figure 2.

The density is given by the mass over the volume

€

ρBH

= MBHVBH

= MBH

K ⋅ MBH( )3

The result is that the Black–Hole density decreases with the square of the Black–Hole mass

€

ρBH

= K −3 ⋅ MBH−2

The remarkable fact is however that the extensions of

the world we leave in is now (1029 cm) and – as shown in Figure 3 – its density satisfies the same relation of the world we come from, whose radius was 10−33 cm and its density

ρ!"#$%&  ≅  54  ×  1093  gr  ×  cm−3.

15

It turns out that the Planck density and radius satisfy the Black–Hole conditions exactly as the present day density of the Universe and its radius satisfy the Black–Hole conditions: our Universe has the characteristic as if we come from a Black–Hole and we are still in a Black–Hole [7]. The Universe where we are seems to be the proof that a Black–Hole can expand its radius by something like 62 orders of magnitudes going from 10−33 cm up to 1029 cm. The basic quantity in this expansion being the density, which seems to follow the conditions dictated by the Schwarzschild solution [10] of the Einstein equation.

What is needed is an equation which describes the evolution of the Radius and the evolution of mass of the Universe as a function of Time.

Suppose we were able to find – from basic principles – the function which describes the Universe, 𝜓!"#$%&'% . This function must evolve in Time

𝜕𝜓!  (𝑅,𝑀)𝜕𝑡  

in such a way that the correlation between Radius and mass of the Universe obeys the Schwarzschild equation (2) already quoted:

𝑅! 𝑡 ≅ 1.5  ×  10!!"   ∙ 𝑐𝑚   ∙  𝑔𝑟!!   ∙𝑀!(𝑡) (2).

16

The evolution of 𝜓!"#$%&'% must describe the change of density of the Universe, from

𝜌!"#$%&'%   𝑡 = 0  ≅    10!"   ∙ 𝑔𝑟   ∙  𝑐𝑚!!

down to

𝜌!"#$%&'%   𝑛𝑜𝑤  ≅    5  ×  10!!"   ∙ 𝑔𝑟   ∙  𝑐𝑚!!

i.e. along a change of density by 123 powers of ten as illustrated in Figure 3.

It should be pointed out that the Einstein equation establishes a correlation between mass-energy and the curvature of Space-Time: high curvature corresponds to high values for the mass-energy. Since high curvature corresponds to small Radius, the mass-energy goes like 1/R. On the other hand in the Schwarzschild equation (2) the mass MBH increases with the Radius RBH.

In fact the Radius of the Horizon produced by the point-like mass, M, goes nearly with M and depends only on the fundamental constants G and c,

𝑅!" =   !!!!  𝑀!"  ≃ 1.5  ×  10!!"   ∙ 𝑐𝑚   ∙  𝑔𝑟!!   ∙𝑀!" (2)

The equation (2) is only one point in the Einstein equation which gives the Radius versus the mass-energy, as illustrated in Figure 4.

17

Figure 2

THE EVOLUTON OF THE UNIVERSE FOLLOWING THE SCHWARZSCHILD EQUATION

18

Figure 3: The Figure shows the relation which exists between the value of the Black–Hole radius (RBH) and the corresponding density (ρBH), from the Planck scale to the Universe scale now.

19

Figure 4

All possible Einstein equations are represented by the functions 𝑅 =   !

! in Figure 4. What is needed is the function

𝜓!"#$%&'%

which evolves with Time along the line (R = M) given by the Schwarzschild condition (Figure 4).

The Einstein equation and the Schwarzschild solution ignore the existence of the SU(3) × SU(2) × U(1). The convergence of the three couplings α!  α!  α! at EGUT is at least

20

two orders of magnitudes below ESU. In this energy GAP the primordial Black-Holes are produced, which are the seeds of all Galaxies. The other Black–Holes produced in the Universe now have in their inner structure protons, neutrons and electrons.

If we could see the inner structure of these Black–Holes we would find that the matter they are made is the one familiar to us, i.e. made with (p, n, e). The primordial Black–Holes, as said before, are made with matter whose charge is only the gravitational charge.

Note 1

It should be pointed out that the Boltzmann’s constant k, represents the “quantum” of Entropy, or the minimum amount of “caos”.

21

REFERENCES [1] A Study of the Various Approaches to MGUT and αGUT F. Anselmo, L. Cifarelli and A. Zichichi, Nuovo Cimento 105A, 1335

(1992).

[2] At the time of the 1979 EPS Geneva Conference the three gauge couplings 𝛼!  𝛼!  𝛼! were not converging in a point, but in a sort of triangle, as emphasized by R.M. Mössbauer, “Searching for the Superworld”, S. Ferrara and R.M. Mössbauer (eds), World Scientific Series in 20th Century Physics Vol. 39 (2007), page 4.

[3] New Developments in Elementary Particle Physics

A. Zichichi, Rivista del Nuovo Cimento 2, n. 14, 1 (1979). The statement on page 2 of this paper, «Unification of all forces needs first a Supersymmetry. This can be broken later, thus generating the sequence of the various forces of nature as we observe them», was based on a work by A. Petermann and A. Zichichi in which the renormalization group running of the couplings using supersymmetry was studied in order to obtain the convergence of the three couplings at the same point. This work was not published, but perhaps known to a few. The statement quoted is the first instance in which it was pointed out that supersymmetry should play an important role in the convergence of the gauge couplings.

[4] “The Evolution of Gaugino Masses and the SUSY Threshold A. Zichichi, A. Petermann, L. Cifarelli and F. Anselmo, Il Nuovo Cimento

106A, 581 (April 1992).

The Simultaneous Evolution of Masses and Couplings: Consequences on Supersymmetry Spectra and Thresholds

F. Anselmo, L. Cifarelli, A. Petermann and A. Zichichi, Il Nuovo Cimento 105A, 1179 (1992).

The Effective Experimental Constraints on MSUSY and MGUT F. Anselmo, L. Cifarelli, A. Petermann and A. Zichichi, Il Nuovo Cimento

104A, 1817 (1991).

[5] Understanding Where The Supersymmetry Threshold Should Be A. Zichichi from Proceedings of the Workshop on "Ten Years on SUSY

Confronting Experiment", CERN, Geneva, Switzerland, 7-9 September 1992, CERN-PPE/92-149, CERN/LAA/MSL/92-017 (7 September 1992)

22

and CERN-TH 6707/92-CERN-PPE/92-180 (CERN, Geneva, 1992), 94. All References to these works are in the Special Volume “Searching for the Superworld”, S. Ferrara and R.M. Mössbauer (eds), World Scientific Series in 20th Century Physics Vol. 39 (2007).

[6] INTERNATIONAL SCHOOL OF COSMOLOGY AND GRAVITATION P.G. Bergmann, 1979–Spin, Torsion, Rotation and Supergravity; 1982–

Unified Field Theories of More Than 4 Dimensions Including Exact Solutions; 1985–Topological Properties and Global Structure of Space-Time; 1987–Gravitation Measurements, Fundamental Metrology and Constants;

1990–Symposium on the Problem of the Cosmological Constant in Honor of Peter Gabriel Bergmann's 75th Birthday;

P.G. Bergmann and Zheniju Zhang, 1991–Black Hole Physics; P.G. Bergmann, V. De Sabbata and T.-H. Ho, 1993–Cosmology and

Particle Physics; P.G. Bergmann, V. De Sabbata and H.-J. Treder, 1995–Quantum Gravity; P.G. Bergmann, G. 't Hooft and G. Veneziano, 1998–From the Planck

Length to the Hubble Radius.

[7] It is as if we come from a Black–Hole and we are in a Black–Hole A. Zichichi, to be published (2014).

[8] The Holographic Principle G. 't Hooft, in Proccedings of the Erice 1999 Subnuclear Physics School

“Basics and Highlights in Fundamental Physics”, page 77, World Scientific (2001).

[9] Über Irreversible Strathlungsvorgänge M. Planck, S.B. Preuss. Akad. Wiss. 5, 440–480 (1899). The problem of the Fundamental Units of Nature was also presented by

M. Planck in a series of lectures he delivered in Berlin (1906) and published as “Theorie der Wärmestrahlung”, Barth, Leipzig, 1906, the English translation is “The Theory of Heat Radiation”, (trans. M. Masius) Dover, New York (1959).

[10] Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie

K. Schwarzschild, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 7: 189–196 (1916).