Evolution of the early Universe in the scale invariant theory
Andre Maeder





























Andre Maeder
andre.maeder@unige.ch


0000-0001-8744-0444]Andre Maeder 
Geneva Observatory 

chemin des Maillettes, 51 

CH-1290 Sauverny, Switzerland



Analytical solutions are obtained for the early cosmological phases in the scale invariant models with curvature k=0.
They complete the analytical solutions already found in the matter–dominated era by <cit.>. The physical properties 
in the radiative era are derived from the conservation laws and compared to those of current standard models. 
The critical runs of the temperature  T(MeV) 
and of the expansion rate H
of the scale invariant models   with low densities, e.g. Ω_m=0.04, 
 are quite  similar at the time of nucleosynthesis to those of standard models with  Ω_m=0.30,
leading to the same freezing number ratio of neutrons to protons.
 These results are consistent with the fact that the scale invariant models
appear to not require the presence of dark matter.














§ INTRODUCTION
 


It is often ignored that  when the cosmological constant Λ is assumed to be equal to zero, the equations of General Relativity 
are scale invariant, which is also a property present in Maxwell equations of electrodynamics. However,
 for a non-zero Λ,  the field equations of the gravitation no longer show the property of scale invariance.
 A fact, which as discussed by <cit.>,  was one of the reasons of Einstein's disenchantment for the cosmological constant.
 It is thus of interest to examine at what conditions the scale invariant properties of General Relativity may be restored,
 since current cosmological observations appear to support a  positive cosmological constant.
 
A theoretical framework has been developed by <cit.> and <cit.>, in
the so-called co-tensorial calculus based on the Integrable Weyl's  geometry. It offers a consistent basis to account for 
the properties of scale invariance of gravitation to a scale factor λ, 
as also illustrated by several properties studied by <cit.>.
Scale invariant derivatives, modified affine connexions, modified Ricci tensor and curvatures can be obtained  leading to a general scale
invariant field equation. Dirac and  Canuto et al. have  expressed
 an action principle in the scale invariant framework, with   a matter Lagrangian, as a coscalar  of power -4 (varying like λ^-4).
 By considering the variations of this  action, they also obtain the generalization of the Einstein field equation. 
 This is  Equation (7) in <cit.>  from  which the scale invariant cosmological equations  derived. 
 
 In the integrable  Weyl’s space,  the scale factor remains undetermined without any other constraints
 as shown by Dirac and Canuto et al. Thus, these authors were fixing the scale factor by some external 
 considerations based on the so-called  Large Number Hypothesis, an hypothesis often disputed <cit.>. Thus, it seems 
 appropriate  to explore other conditions for setting the gauge.
 The proposition was thus made to fix the gauge factor by simply assuming  the  scale invariance of the empty space <cit.>. 
 This means that the properties of the empty space should not change under a contraction or dilatation of the space-time.
 Indeed, we note, as shown by  <cit.>, that the current
 equation of the vacuum P_vac= - ϱ_vac  c^2  already implies that    
 ϱ_vac should remain constant ”if a volume of
 vacuum is adiabatically compressed or expanded”. On this basis, the cosmological equations derived by <cit.> were greatly
 simplified <cit.>.
 A number of cosmological tests were performed, with positive results. Interestingly enough, these equations were
 then shown to have rather simple analytical solutions  <cit.> for models of a matter dominated Universe with a zero curvature.
 
 In order to express the  motions of free particles, a geodesic equation was obtained  <cit.>
  from a minimum action in the integrable Weyl’s space. In the weak field approximation,
  the geodesic equation leads to a  modification of the Newton equation <cit.>: it then contains
  a (currently very small) additional acceleration term proportional to the velocity of the particles. This equation was applied
  to study the internal motions of clusters of galaxies, the flat rotation curves of spiral galaxies and  the age increase
  of the ”vertical” velocity dispersion of stars in galaxies <cit.>. 
  The interesting result was that the  observational properties of these various systems
  could be accounted without requiring to  the current hypothesis of dark matter, and the same for the radial acceleration relation (RAR)
  of galaxies <cit.>.
  The growth of the density fluctuations in the early Universe was also studied by
  <cit.>. The usual theory is that at recombination the baryons settle in the potential wells formed by dark matter 
  previously assembly during the radiative era. In the scale invariant framework, the growth of the density fluctuation, predicted from
  the corresponding Euler, Poisson and continuity equations, is fast enough
  so that dark matter is not needed to account for the formation of galaxies.
  
  The object of the present work is to search for the solutions  in the radiative era of the universe in the scale invariant context.
    In Section 2, we derive the analytical solutions for cosmological
  models dominated by relativistic particles, and examine the values at the equilibrium of matter and radiation. In Section 3, 
  the physical properties in the radiative era are studied. The conditions at the time of the cosmological nucleosynthesis 
  are compared to the results of standard models.
    




§ THE COSMOLOGICAL SOLUTIONS IN THE RADIATIVE ERA


We study   the  physical conditions in the radiative era, in particular when the cosmological nucleosynthesis occurs.
At this stage,  the age of the Universe, of 1 second and more, is already 10^43 times larger than the Planck time
and  thus the Universe dynamics is no longer   dominated by quantum effects. Thus, if it occurs that scale invariance 
applies in the present day Universe, it should likely also be the case at the time of nucleosynthesis.


For the matter era, analytical 
solutions were found by <cit.> in the case of a flat Universe model with k=0. Now, we also need the
solutions of the cosmological models in the radiative era.


 

 §.§ The equations and their solutions
 
 
 The hypothesis of scale invariance assumes that the equations should be invariant to a transformation of the form 
  ds'= λ(x^μ) ds <cit.>. The term λ is the scale factor expressing a relation of the form ds'= λ(x^μ) ds, 
where ds' is the line element in general relativity and ds the line element in the supposed more general scale invariant space.
For reasons of homogeneity and isotropy the scale factor depends only on time t,
and the condition of the invariance of the empty space
imposes  that λ∼ 1/t <cit.>. Interestingly enough, this form is also one of those considered by <cit.>.
 The  equations of the scale invariant cosmology  under the above mentioned hypothesis are 
 
    8  π G ϱ/3 = k/R^2+Ṙ^2/R^2+ 2  Ṙλ̇/R λ  ,
 

    -8  π G p  = k/R^2+ 2 R̈/R+Ṙ^̇2̇/R^2
    + 4 Ṙλ̇/R λ  .

The combination of these two equations  leads to

    -4  π G/3  (3p +ϱ)  =  R̈/R + Ṙλ̇/R λ  .

The various terms have their usual significations, with dots representing  time derivatives. These equations only differ from the 
usual ones by the presence in each case  of an additional term containing the product Ṙλ̇/R λ.
We see that if λ(t) is a constant, 
the usual equations of cosmologies are recovered. 
This means that, at any fixed time, the effects that do not depend on time evolution are just those 
predicted by GR.



For the flat space with k=0 in the matter dominated era,
 this equation has an analytical solution of the following form  <cit.>, 

    R(t)   =  (t^3- Ω_m/1- Ω_m)^2/3 ,

where Ω_m = ϱ_m,0/ϱ_c,0 
with  ϱ_c,0= 3   H^2_0/(8  π G), the standard expression of  the critical density at present.
In the above equation, it is assumed that at the
present  time t_0=1, one has R_0 =1.
Interestingly enough, it is also possible  in this context  to have  for a flat space different values of Ω_m.
The expression  of the Hubble expansion  term H(t) resulting from (<ref>) is 

    H(t)   =  2   t^2/t^3- Ω_m ,   so   that   H_0  =  2/1- Ω_m ,

expressed here in the above scales of time and space.
  In Sect. <ref>, these quantities will be used with their appropriate physical units. 
 We see from these two expressions that there is no solutions for Ω_m≥ 1, 
 since the presence of  a significant matter content in the Universe is killing scale invariance <cit.>.




The radiative era of the Universe is dominated by relativisitic particles. The density  is ϱ_rel,

    ϱ_rel  =   K_0  ϱ_γ  ,   with   
     K_0 = 1+ 7/8(4/11)^4/3 N_ν  .

There, ϱ_γ is the  density of photons. The term K_0 accounts for the neutrino contribution to the density
of relativistic particles <cit.>, where N_ν is the number of neutrino types. For a number N_ν=3, the value is
K_0=1.6813.  The appropriate conservation law 
for relativistic particles has to be accounted for. According to <cit.>, 
it is ϱ_rel  R^4 λ^2= const., thus 
Equation (<ref>) writes, 

    λ^2 Ṙ^2 R^2 + 2 Ṙ R^3 λ̇ λ = -  C_rel  ,

with

    C_rel  =  8  π G ϱ_rel R^4 λ^2/3  .

The differential equation to be solved for R(t) is thus,

    Ṙ^2 R^2   t  -2 ṘR^3 - C_rel  t^3=   0  .

We  now examine whether 
 the above differential equation also possesses analytical solutions. We try to develop R(t) as a product,
 
    R(t) = t   v(t)  ,   thus  Ṙ= v + t  v̇ .

 Thus, Equation (<ref>) leads to
 
    v̇  =  ±√(%s/%s)v^4+C_relv^2 t^2  ,   and  v dv/√(v^4+C_rel)  = ±dt/t  .

The equation is separable and from  <cit.> we obtain as an integral
 
    ln (√(v^4+C_rel) + v^2 )   = ±  2 ln t + const.

 Considering solutions of R(t) growing with time, we take  the sign ”+” in the above equation
 and obtain
 
    √(v^4+C_rel) + v^2   = c_2   t^2   ,

 where c_2 is a constant, which will be fixed below from the relevant boundary conditions.
 The present solution does not apply at time t_0, but
 only in the radiative era, thus  it is not possible to directly express the constants C_rel and c_2
 as functions of  cosmological parameters (like H_0) at the present time. We get after some simple manipulations
 
    v   =  (c_2^2t^4 - C_rel/2    c_2 t^2)^1/2

 and for the expansion factor
 
    R(t)   =  t  (c_2^2t^4 - C_rel/2    c_2 t^2)^1/2  =
       1/√(2)(c_2   t^4 -C_rel/c_2)^1/2 .

 The derivative of the expansion factor is
 
    Ṙ  =  (c_2^2   t^4 - C_rel/2    c_2 t^2)^1/2+t/2 c^2_2  t^4+C_rel/c_2   t^3 (2  c_2 t^2/c^2_2 t^4- C_rel)^1/2  ,

 so that the Hubble expansion rate at a given time in the radiative era is,
 
    H   =  Ṙ/R  =  1/t+1/2t^2/R^2 c^2_2  t^4+C_rel/c_2  t^3  .

 There is thus an analytical solution of the scale invariant equation in the radiative era, which may greatly simplify all
 studies. The solution depends  on the constants C_rel and c_2, 
 which we shall relate below to the  densities of matter and of relativistic particles, respectively ϱ_m  
 and ϱ_rel at the time when these two densities become equal.



 §.§ Values of the constants and properties at the density  equilibrium
   

In the radiation era, according to Equation (<ref>) which expresses  the appropriate conservation law, 
the density of relativistic particles at a given time t is,

    ϱ_rel  =  3   C_rel/8  π G R^4 λ^2  .

In the matter era, the corresponding expression obtained from the matter conservation law is <cit.>,

    ϱ_m  =  3   C_m/8  π G R^3 λ  .

At the time t_eq when both densities become equal, we have the following relation between the constants,

    C_rel =  C_m R_eq/t_eq =
      4  Ω_m/(1-Ω_m)^2 R_eq/t_eq  ,

where R_eq and t_eq are the values at equilibrium and the constant 
C_m=  4  Ω_m/(1-Ω_m)^2
 in the matter dominated era,  as given by Equation (66) in <cit.>.
 The term Ω_m is the 
 usual parameter of  matter density as discussed in Section (<ref>).  
 

The constant c_2 is fixed by expression (<ref>) applied  to the time of equilibrium,

    c_2= v^2_eq+√(v^4_eq+C_rel)/t^2_eq  ,
      with    v_eq  =  R_eq/t_eq .

The expression  of v_eq in terms of basic quantities is given below in Equation (<ref>).
The values of R_eq and t_eq are fixed thanks to the conservation laws. In the matter and radiative eras,
we respectively have at a time t,

    ϱ_m  R^3 λ  =  Ω_m ϱ_c,0  ,    and  ϱ_rel  R^4 λ^2    =   ϱ_rel,0  ,

where the index _0 indicates values at the present time t_0.
The equality of the two densities defines the equilibrium point, we get

    R_eq λ_eq  = R_eq/t_eq (=  v_eq)   = 
     K_0  ϱ_γ,0/Ω_m ϱ_c,0  .

The numerical value of v_eq is given by the above expression, then   t_eq  is obtained 
by using the expression for R(t) in the matter dominated era, as given
by Equation (<ref>). We first write, 

    v_eq  =   R_eq/t_eq =  1/t_eq (t_eq^3- Ω_m/1- Ω_m)^2/3  ,

and then search the solution of this equation, which  is

    t_eq  =  (v_eq^3/2  (1-Ω_m)/2+ 
     √(v_eq^3   (1-Ω_m)^2
    +4  Ω_m)/2)^2/3  .

Since, v_eq is known and Ω_m is chosen,
 the above equation gives the value of t_eq and 
then of R_eq by (<ref>).
The redshift at the equality of 
the two densities is  obtained from expression (<ref>) by,

    (1+z_eq)    t_eq  =  Ω_m ϱ_c,0/K_0  ϱ_γ,0  =   1/v_eq  .




Equations (<ref>) and (<ref>) provide the constants C_rel and c_2.
The numerical values of the various input parameters seen above are the following ones,

    H_0    = 3.2408 · 10^-18 h     [s^-1] 
    ϱ_rel,0 =  K_0  a  T^4/c^2 =  4.6485 · 10^-34  K_0      [g/cm^3]  
    ϱ_m,0 =  Ω_m ϱ_c,0  = 
    1.8788 · 10^-29 h^2  Ω_m    [g/cm^3]  
    
    v_eq =  R_eq/t_eq  =   2.4741 · 10^-5K_0/Ω_m  h^2  .

Table 1 gives for different values of Ω_m  the values of the useful parameters  for the model description in 
the early phases according to the above equations  and constants.










§ PHYSICAL PROPERTIES IN THE RADIATIVE ERA
   

We now examine the evolution of the physical properties during the radiative era. The conservation laws of matter 
and radiation densities  (<ref>), together with the  constants given by (<ref>) lead to

    ϱ_m (t)  =   1.8788 · 10^-29  h^2  Ω_mt/R^3(t)    [g/cm^3]   , 
    ϱ_rel (t) = K_0  ϱ_γ  =
       4.6485  · 10^-34   K_0  t^2/R^4(t)    [g/cm^3]   , 
    
    T(t)  =   2.726  t^1/2/R(t)    [°K]  .

There, R(t) is given by Equation (<ref>).
The initial  t_in   of the Universe for a given model is obtained by setting R(t)=0 in this equation,

    t_in  =  C^1/4_rel/c^1/2_2   ,

which applies in the scales where 
t_0=1 and  R_0=1. We notice that the values obtained by this last expression 
are very close to those obtained by imposing R(t)=0 in Equation (<ref>) for the matter-dominated era,
which gives simply t_in= Ω_m^1/3. The more appropriate values (<ref>) for the initial time
in the radiative era
 are generally  slightly larger than Ω_m^1/3,  with a minor difference concerning only the sixth decimal.
  (It may also be recalled  <cit.> that t_in=0 
in the above mentioned scale only when Ω_m=0.)

If one considers very early epochs of the Universe, we may write  the time as t = t_in+ Δ t.
 This quantity Δ t  represents  the age counted from the origin of the Universe, 
in a scale where the interval 1-t_in is the present age of the Universe. 
Thus, the age  τ, the cosmic time expressed in seconds, for a given
event at the time t may be written,

    τ  =   4.355 · 10^17 Δ t/1- t_in  [s]  ,

where a present age of 13.8 Gyr has been adopted.
 This relates the ages τ and Δ t for a given cosmological model (since t_in depends
 on the model).  In the very early stages, the quantity Δ t is very small.
For example, for the cosmological nucleosynthesis which occurs during, say, the first three minutes Δ t < 4.1 · 10^-16,
 whatever the cosmological model.
Before the  equilibrium time of matter and radiation, say of the order of 10^5 yr, one 
has Δ t < 7 · 10^-6.
Thus, we may consider that during the whole radiative phase  the 
value of Δ t is  ≪ 1 for any flat scale invariant model.  This leads to an interesting  simplification of the expression of R(t) 
given by Equation (<ref>). We may write,

    t^4  =  (t_in+Δ t)^4  ≅  t^4_in+ 4   t^3_in Δ t   ,   and   t^2   ≅   t^2_in+ 2   t_in Δ t   .
 
 The expansion factor becomes,
 
    R(t)   = 
      1/√(2)(c_2   t^4 -C_rel/c_2)^1/2 ≅ 1/√(2)(c_2    t^4_in+4  c_2    t^3_inΔ t -C_rel/ c_2 )^1/2  ,

 which may then be written  with  expression (<ref>), 
 
    R(t)  ≅ √(2)  c^1/2_2 t_in^3/2 Δ t^1/2  ,

 and
 
    R(t)  ≅  B  Δ t^1/2  ,

 
    with  B = √(2)  c^1/2_2 t_in^3/2  =  √(2)  C^3/8_rel/c^1/4_2  .

 Thus, the time dependence of the expansion factor R(t) for  the early Universe in the scale invariant context  is the same
 as in standard models. The Hubble term H becomes
 
    H(t)   =  1/2 1/Δ t  ,

 which applies to the scale where t_0=1. 
 
 
 
 

     
 In the early phases, we can approximate expressions (<ref>) by,
 
    ϱ_m (t)  ≅   1.8788 · 10^-29  h^2 
     Ω_mt_in/R^3(t) ≅ 1.8788 · 10^-29/2^3/2  h^2 
     Ω_mc^1/4_2/C^7/8_rel 1/Δ t ^3/2   [g/cm^3]   , 
    ϱ_rel (t)   ≅  ≅   K_0  ϱ_γ =
      4.6485  · 10^-34   K_0  t^2_in/R^4( t) ≅   4.6485  · 10^-34/4   K_0   1/C_rel 1/Δ t ^2   [g/cm^3]   , 
      
    T(t)   ≅   2.726  t^1/2_in/R( t) ≅ 2.726/√(2)1/C^1/4_rel 1/Δ t^1/2    [°K]  .

We have used the fact that t= t_in + Δ  t   ≅   t_in.


The expansion factor
is given by Equation (<ref>), with Equations (<ref>) and (<ref>) for 
t_in and B. The values of C_rel and c_2 for different models
are given in Table 1. In the above expressions, the time Δ t is in the scale where t_0 =1.  We need these equations 
in the current units. With Equation (<ref>), we have with  the time τ in seconds,

    ϱ_m (τ)   ≅        1.9091· 10^-3   h^2 
     Ω_mc^1/4_2/C^7/8_rel (1-t_in)^3/2 1/τ^3/2   [g/cm^3]   , 
    ϱ_rel (τ)   ≅    22.0409   K_0   1/C_rel(1-t_in)^2 1/τ ^2   [g/cm^3]   .

 The temperature expressed in  °K  or in MeV as a function of the time in sec is
 
    T(τ)   ≅ 1.272 · 10^9/C^1/4_rel (1-t_in)^1/2 1/τ^1/2    [°K]  , 
    
     T(τ)    ≅ 0.1096/C^1/4_rel (1-t_in)^1/2 1/τ^1/2    [MeV]  .

 The Hubble rate expressed in seconds is, consistently with Equations (<ref>) and (<ref>),
 
    H(τ)  ≅ 1/2 1/τ    [s^-1]  ,

 which is the same as in standard models.
 The above relations allow us to describe the physical conditions in the scale invariant  models with k=0 of the early Universe.
 Different values of the density parameter Ω _m lead to different models.
 
 
 

§ THE PHYSICAL CONDITIONS AT THE TIME OF THE COSMOLOGICAL NUCLEOSYNTHESIS


      
     
 
 
 

It is interesting to compare  the physical conditions predicted by the scale invariant models 
at the time of the nucleosynthesis with the corresponding values in standard models. The key parameters are the 
  temperature T  and the expansion rate H.
 The runs  T(τ), with the time τ is  in seconds, are illustrated for some standard and scale invariant
 models in Fig. <ref>. The  results of the standard models by  <cit.> and <cit.> are
 very similar (black line), while the results by <cit.> are slightly different. We see that the scale invariant models
with Ω_m=0.30 show  a significantly higher temperature at a given time than the standard models.
For  Ω_m=0.04, there is a nice agreement. This is interesting since in the scale invariant 
context, there is no need to call for an unknown dark matter, thus the matter content  Ω_m
 is only that resulting from the baryons, currently of the order of 0.04 <cit.>.
 
 
 
 At energies above 1 MeV, when electrons and positrons are still in equilibrium with radiation,
the equilibrium between neutrons and protons is maintained by the weak interactions,
ν + n  ↔ p+e^-,   n+e^+↔ p+ν and  n → p+e^-+ν.
The ratio of the numbers of neutrons to protons is then defined by a Boltzman distribution of the form 
 n_n/n_p≈  e^Q/(kT), 
 where Q= 1.293  MeV is the energy corresponding to the mass difference of neutrons and protons.
 At some  energy  below 1 MeV,  the  breakdown of the equilibrium  makes
  the first two  exchange rates to become rapidly negligible, only the third one is remaining, slowly reducing the number of free neutrons
  until deuterium is formed.
 
  
  
  Fig. <ref>  shows as a a function of temperature  
  the rates  Γ (n → p)  τ_n  and 
   Γ (p → n)  τ_n <cit.>, 
   the Γ-rates are  the products  σ  v   n, expressed in sec^-1, of the corresponding
    cross-sections, velocities and concentrations.
   At energies above 1 MeV, we see  the  equality of the two rates.  Below it, the rate Γ (p → n)
 rapidly vanishes,  while the neutron  slowly decays to protons at a timescale τ_n = 885.7 sec,
  so that  the product Γ (n → p) τ_n tends towards unity. 
  The expansion rate τ_n  H
  is also illustrated.  When the Γ-rates are higher than the expansion rate H, the nuclear process are faster and thus  tend
  towards equilibrium. At the opposite, when the Γ-rates become lower, 
  the cosmological expansion  and the 
  associated  cooling rapidly dominate, the nuclear interactions cease, except for the neutron disintegration.
  The crossing temperature of the  τ_n  H-  and τ_n Γ-lines defines the critical temperature,
  fixing the fraction of neutrons, which after the disintegration of a small part of them, 
will finally be turned to helium-4,  when deuterium formation starts 
at 0.085 MeV,



Fig. <ref> also shows the expansion  rates    τ_n  H  of the standard models 
together with the values of the scale invariant models already considered above. 
From Equations (<ref>) and (<ref>), we  eliminate the time τ  and obtain the relation
between H   in s^-1 and T in MeV,

    H(τ)   =  41.6245  · C^1/2_rel (1-t_in)   T^2  .

We see as before that
 the scale invariant model with Ω_m=0.30 somehow deviates from the standard models and would
 suggest a lower value of the freezing temperature for the neutron-proton decoupling. The model with a lower matter density
 (baryonic)  of Ω_m=0.04 shows an excellent agreement with the standard models,
the minor  differences being of the same order as the differences between the  various  standard models.  
The estimated  freezing  energy for neutrons  is around 0.6 to 0.7 MeV. The agreement of the
standard models and the low density scale invariant model is encouraging and makes worth  a detailed study 
of the cosmological nucleosynthesis in the scale invariant framework.





§ CONCLUSIONS


The runs of T(MeV) and of the expansion rate H are essential parameters for the cosmological nucleosynthesis, 
which is used  in standard models to fix the   value of the density parameter Ω_m, typically found 
around 0.30. It is thus amazing that the scale invariant models, which usually do not need the presence of dark matter
as shown by a number of tests
<cit.>, find an excellent agreement with the conditions
of the standard models for much lower values of Ω_m of about 0.04. Such a density parameter leads
to the same value of the freezing energy of the neutron to proton ratio as in standard models.
These positive results are stimulating for the further study of the cosmological nucleosynthesis in the scale invariant context,
they come after a series of other positives tests as mentioned in the introduction.






 

 



 
 
Acknowledgments: 
The author expresses his deep gratitude to his wife and to D. Gachet for their encouragements, as well to Dr. Vesselin G. Gueorguiev
for his continuous support and constructive remarks.









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