ON THE DIMENSION OF THE SPACE IN THE EARLY UNIVERSE

V.V.Skobelev

                                     Moscow State Industrial University; E-mail: v.skobelev@inbox.ru

 Abstract. It is shown that if we consider fermions (quarks, electrons) generated by superstrings in the early Universe a degenerate ideal quantum gas, the entropy of the system has a local maxima in spaces with a number of dimensions equal to  Thus, there is reason to believe that the Universe was formed in the space with just such a number of dimensions. An argument in favor of the dominance of space   is brought. With further evolution, if we continue to adhere to the fundamental principle of the second law of thermodynamics, due to changes in temperature and composition, the space apparently started to match the maximum entropy.

I. INTRODUCTION

The modern theories do not exclude the possibility of the existence of spaces with the number of dimensions more than, some of which might be hidden from us. For example, in the popular theory of superstrings the dimension of the space  [1,2]is possible, while the number of dimensions of our space is three. In the present article we attempt to clarify the issue in relation to the first instants of the Universe (about 0.01s from the "start") when, apparently, the question of the dimension of the space in which it could continue to evolve, was “being solved”. We assume that in these first moments of the Universe it was governed by the laws of quantum mechanics, while the majority of other laws of physics had not yet "joined", partially because it was not an ordinary matter, but there were only superstrings primarily in their excited states, quarks and electrons. The second law of thermodynamics, according to which the most probable states get implemented, has to deal not only with physical but with mathematical laws, and we further assume that in this era the Universe has evolved in accordance with the fundamental law of nature. In this regard, the fermion excitations of superstrings form in the space with the highest entropy level. That is why the problem of computing of the entropy of any of the fermion components as a function of space dimension rises. For simplicity, we assume that the system of fermions is a degenerate Fermi gas. There are good reasons to do that: because of the high concentration of fermions in an initially small volume of the Universe of their kinetic energy at the Fermi level could considerably exceed the temperature value. Note that the hypothesis of an extremely high temperature of the early Universe at the stage of the Big Bang [2] relates, as we believe, to a later point when the number of particles reaching the point at which we have the usual statistical description of a system with temperature as a parameter of its condition, and the second law of thermodynamics was an absolute exact law of nature. If the number of fermions is limited and does not reach the point at which the probability of states with maximum entropy is practically equal to unity, the second law of thermodynamics indicates only a preferred direction of evolution of the system with the realization of such states, and the parameter  in the Fermi-Dirac distribution is not the temperature in the conventional sense, and approval of an extremely large value of the deficit of the particles immediately after the start of their generation by superstrings does not make sense. However, we further refer to this parameter as to the temperature for major simplicity.       

II. ENTROPY OF THE FERMION GAS IN - DIMENSION SPACE

The total number of fermions in space of  dimensions is determined by the basic formula of Fermi-Dirac statistics: 

                                                                                  ,                                                    (1)             

Where is the statistical weight of state with a given value of the relativistic kinetic energy, which we further consider independent from,   - the chemical potential, - normalization lengths of the axes,  - total angle in the space of dimensions which can be expressed in terms of - function [3]:

                                                                       .                                                                                        (2)

Determining the concentration in the space of  dimensions by equation

                                                                         ,                                                                              (3)

we get:

                                                                  .                                                                        (4)

 

      In the case of a completely degenerate Fermi gas  we find:

                                                                              ,                                                                     (5)

Where  is the Fermi momentum, for which we therefore have:

                                                                           .                                                                          (6)     

In accordance with the ordinary differential equation for - potential [4]

                                                                                                                                        (7)

 the entropy can be found from its known value:

                                                                            .                                                                               (8)

In our case, the -potential per unit  "volume" is  equal to

                                                                    .                                          (9)

Integration by parts gives:

                                                                 ,,

 

and with the change of variable  we obtain: 

                                                                        .                                                 (10)    

Furthermore, we use the asymptotic expansion of the integral form (10) with respect to temperature [4] in the case of a degenerate Fermi gas, which we put down in a generalized form as follows: 

                                                                            .                       (11)

Note that the contribution to (8) gives only the second term in (11) with explicit dependence on temperature, since the derivative (8) is taken at a constant, and the dependence upon differentiation should not be considered. The index of summation in (11) takes odd values ​​(see also [4]) with maximum, and the way of determining of it at concrete will be indicated lower, and the coefficient  equal to:

                                                                         ,                                                                     (12)

where  is the Riemann function in area is defined by equation [5]: 

                                                                        .

After the transition to dimensionless variables we write the contribution we’re interested in   in the form

                                                                      .             (13)

Here,   ,.  For the further, we also give the expression of the dimensionless momentum at the Fermi level, which, according to (6), is equal to

                                                                      = ,                                         (14)

where we used the formula (2), and the expression is written for the quarks (). For simplicity we also believe that our system is in the -dimensional cube with side. Since the cube in a  -dimensional space is obtained by designing a cube in -dimensional space on its face, then at the dimension change of the space, as also   should be regarded as constant; in this connection in (14) the notation    is introduced (in the same time, ). Obviously,  should be perceived as effective size of the early Universe. As will be seen, a non-relativistic case is realized in it:

                                                                             ,                                            (15)

                                                                          .                                                                           (15a)      

Inequality (15) can always be observed for an appropriate choice of "free" parameter . Performing the (15), only the last term in parentheses (13) makes the contribution. Next, we consider the values ​​of the derivative in various cases:  

                        ,                                                                                                      (16)

1)even,                  ,                                                                                                             (16a)

2)even,                        ,                                                                                                                    (16b)

3) odd,             ,                                                                                                             (16c)

4)odd, , ,                                                                          (16d)                          

 

 

The expression (16d) is the key for the purposes of this article. Combining expressions (8), (12) - (16), we find the total entropy of the degenerate non-relativistic quarks system, which in our units is a dimensionless quantity:  

                                                ,                                                        (17)

                                                .                                                       (17a)

 Note that these expressions are valid when the non-relativistic condition (15) and the corresponding limits on the temperature,  which in view of (15a)) can be written in a more convenient form as a condition on possible values of parameters  :  

                                                                                                                                                        (17b)

                                                             .                                                                            (17c)

It is possible to check that the expression (17)  at    with the "replacement" of the quarks for electrons and values ​​coincide with the corresponding formula of the book [4], in which, as in most other studies, only the linear term of the expansion of entropy with respect to temperature was considered.

 

In the ultrarelativistic case

                                                             ,                                                                   (18)

 when the first summand in parentheses in (13) dominates, we obtain:

                                                        ,                                                                (19)            

                                                       ,                                               (19à)                                                                                                                   

with the temperature limitation, having form

                                                                  ,                                                                               (19b)

                                                              .                                                                           (19c)

 

 

In the formula (19) , if  is even, and, if  is odd. As seen from (19), there is no fundamental difference between "odd" and "even" spaces in the sense of presence of maxima and minima of the entropy in the ultrarelativistic case (the entropy increases monotonically with increasing of), which can be determined also by a specific calculation, taking for example, neglecting the temperature dependence of the chemical potential. Likewise, a) the calculation of the degeneracy temperature and entropy of the degenerate boson gas in -dimensional space, and b) the entropy of photon gas, lead to the same conclusion about the monotonic variation of entropy. The results obtained by standard methods [4] are as follows:

a)                                                                ,                                                             (20a)  

                                                                 ,                                                           (20b) 

b)                                                              ,                                                                           (21)

  

 

where the value is given by formula (19c).

III. DISCUSSION

Note that the role of "primary" bosons in the early Universe, could be claimed by axions, "responsible" for the CP-invariance of strong interactions [6], and having a small mass   [7,8], and also by Higgs bosons [9], by which the Standard Model particles acquire mass. In this case the bosons can also be viewed as excited states of superstrings [2]. But, as already noted, neither they nor the photons “chose” the space with given values of invariants in which the early Universe existed. The situation is completely different, taking into account the entire temperature range (11) in the case of non-relativistic fermion gas, the entropy value if which is given by (16), (17). That is, at even, according to (16a, b), series (17) is cut off "automatically" on, if  is even, and if  is odd. At the odd  the number (17) by a factor !!  in (16d) diverges generally because, in general, the decomposition (11) used is asymptotic and not convergent [7]. The number must be torn off "by force" from physical considerations. Here are the relevant arguments. From (16d) it is seen that in the spaces    at the values ​​ the contribution to the sum (17) is positive, as at  , thereby increasing its value, and in the spaces  it is negative, reducing the value of the sum at. This means that in the spaces  there will be maximum entropy, and in the spaces  there will be minimum entropy. Further, because of the factor  in (16c, d) the peaks will indefinitely increase with increasing of  , as in the above-mentioned cases of boson and ultrarelativistic fermion gases, which is unacceptable from physical considerations. Thus, there should be such a maximum value  that at  the second law of thermodynamics in the terms we use is not applicable. Further, in the space because of the same factor  the negative entropy will be achieved at the lowest value of the index of summation  rather than in spaces such as  with a lower value. To make entropy remain positive in these spaces, the number (17) in them should be cut at. As far as at the same parameters  the values ​​of the entropy in the spaces  are not actually independent, the temperature series (17) in the spaces  should be truncated at the same value.

In fact, let us put down the equation of the form (17)  for the two types of "odd spaces": 

                                                                      ,                                                            (22a)

                                                                  .                                                           (22b)

 

With the first equation we can find function; substituting it in (22b), and then we have:

                                                                         .                                              (23)

Since the functionis defined up to  inclusively, the expansion  in series by  should be done with the same precision as required. Only the value of local maxima in the spaces  depends on the way of "breaking" the series (17)  while they stay themselves, allocating these spaces among others.

Thus, although we do not have a clear recipe of definition  (perhaps this value can be determined from the condition of small fluctuations of the entropy, we are studying this issue), it is clear from the foregoing that the graph of dependence of the entropy on   for the non-relativistic degenerate fermion gas should be a broken line with local maxima at  and with minima at  which must be constructed also considering the approximate constraints (17b) on the temperature.    

 From the steps of the calculations we have done,  another conclusion can be made. As seen, the inequality (17b) permits a wider range of parameter values ​​in the area ,  at the lowest value, i.e. at  . Since the expansion of the range is actually equivalent to increasing of the statistical weight of the quarks in this space, then above all the spaces of the type  it is considered most likely that the early Universe could get in space  (which is consistent with the concept of five-dimensional cosmological string model [10]), which partially explains its transition due to changes in temperature and composition in the "neighbor" odd space, to which the maximum entropy related in future.

 

 

 

                                                                                                                                      

                                                         REFERENCES:                                                                   

1.      Veneziano G., Nuovo Cim. V.57A, P.190, 1968.                                                         

2.      Greene Brian, The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory, Vintage Series, Random House Inc.,2000.

3.    N. Ya. Vilenkin, Special functions and the theory of groups representations, (in Russian), Ìoscow, Nauka,  1965.

4.    L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics, Vol.5: Statistical Physics: Part 1, Butterworth-Heinemann, Oxford, 2000. 

             5. I.S. Gradshteyn and I.M. Ryzhik,  Tables of Integrals, Sums, Series, and Products, Acad. Press, New York, 1980. 

6.    R.D. Peccei and H.R, Quinn, Phys. Rev. Lett.,V.38, P.1440, 1977; Phys. Rev. V. D16, P. 179,1977.                                   

7.    J.E. Kim, Phys. Rep., V.150, P.1,1987.

8.    H.Y. Cheng, Phys. Rep., V. 158,P.1, 1988.

9.    P.W. Higgs, Phys. Lett., V.12,P.132,1964.

10. G.C. Samanta, S. Debata , J. Mod. Phys., V. 3, P. 180, 2012.