Cosmological solution of the relativistic theory of gravitation 
and supernova observational data

© A.V. Genk1,2,3, A.A.Tron 2

1 University “INGECON”, St.-Petersburg, Russia 
2 SEU “Earth and Universe” St.-Petersburg, Russia  
3 Email:  genk-av@yandex.ru 

Abstract:  We derived the expression for the observable quantity - the luminosity distance - within the relativistic theory of  gravitation (RTG) in the case of zero pressure (p = 0 ) ("Dust Universe"). It contains both visible (m) and absolute (M) magnitudes as functions of redshift  z  and mass of graviton mg}.  The comparison of theoretical curve with non-linear regression m(z) obtained from the Ia supernova observations, reveals that  for  mg < 1.8∙10-66 g (including the case  mg→ 0, corresponding to the GR solution with Щ = 1), theory is in good agreement with all Ia supernova observations at the 95% confidence level, and, thus these data do not require accelerated expansion for their interpretation.

1. Introduction.

The effect of the accelerated cosmological expansion was first discovered in supernova SN Ia [1-3]
and now is believed to be commonly accepted among the professional community. To explain this acceleration two most popular concepts are usually being attracted: one deals with the suggestion of the existence of non zero cosmological constant, another is the hypothesis about very special substance in the Universe - the so called "quintessence" with negative pressure.
Nevertheless, the comparatively limited number of SN observations (about 100 flashes) together with
significant scattering in data, leaves room for other approaches. The point is as follows: does there exist any cosmological model which is consistent with all  SN Ia observations made so far and which has no accelerated expansion? The positive answer, if exists, would allow to reject $\Lambda$ term connected with non-zero vacuum energy е0 with negative pressure p0 = – е0 [4-8] or with exotic substance - "quintessence", which possess the unusual equation of state: pq = – (1– н) еq, (0 < н <2/3) [9-12]. Are there any reasons to attract exotic "essences" with hardly comprehensible physical nature if we have good agreement between simple theory and observations? One of the alternatives to both "dark energy" and acceleration is considered in the work [13].
In the present work we consider the cosmological solution of the relativistic theory of gravitation (RTG) [14,15] with non-zero graviton mass mg, at the stage of dust-like matter (н  = 0) and compare it with
experimental supernova data Ia. At this point we are not dealing with experiments studying the CMB anisotropies data [16-19] which are believed to confirm the accelerated expansion. We think that interpretation of these data requires more detailed analysis (at least within the RTG), because it, in
considerable degree, is model dependent in contrast with independent data from supernova measurements.
As will be shown below, all the data can be, under the certain constrains, well adjusted with RTG theoretical curves for dust matter without "quintessence".Thus, the idea of accelerated expansion cannot be regarded  as firmly established under the present state of experimental data.
In Part 2 we discuss the RTG equations (with mg ≠ 0) and their general solution for homogeneous and isotropic Universe. In Part 3 we derive the RTG expression for luminosity distance in the case of dust-like 
(н  = 0) Universe as a function of redshift z. In Part 4 we compare these functions with the results of statistical processing of supernova SN Ia experimental data - the nonlinear regression curve based on the
model under discussion and taking account for actual scattering of photometric data.
In Part 5 we summarize and discuss the main results.

2. Cosmological solution in RTG.

The equations of the relativistic theory of gravitation (RTG) are written as follows [14,15], (in units
G = ħ = c = 1):

								(1)
											(2)

In (1),(2)  mg –  graviton rest mass;  gмн  and  гмн –  metric tensors of correspondingly Riemann effective and basic flat Minkovski space-time; –  energy-momentum tensor (EMT) density of a source in metric gмн;
Dм – covariant derivative in respect to flat metric  гмн . The equation (2) is a general covariant field equation (not a coordinate condition!) which follows from conservation of the full EMT.    In the case of homogeneous and isotropic Universe it follows from equation (2) [14,15], that Universe is flat (parameter k in Robertson-Walker metric is equal to zero, which implies that space geometry of Universe is flat). Thus, the effective Riemann metric takes the form

.						(3)

If we introduce the proper time   and denote R2(ф) = U1/3, then the metric (3)is written as

							(4)

In (4) R(ф) – scale factor, b – integration constant. It can be shown [14,15] that from causality conditions, always present implicitly in RTG,  there follows the requirement for constant  b in (4): R2(R4 – b) ≤ 0. So, for  b in (4) one can choose   where is a maximum scale factor value. According to RTG [15], the unlimited growth of the scale factor R(ф), i.e. the unlimited expansion of Universe  is not possible.      Substituting the expression for metric (4) to the RTG equations (1),(2), and taking into account the expression for EMT in this case (,   where с – density, P – pressure,    – 4-velocity), we arrive to the equation for homogeneous and isotropic Universe in the form  [15]:

							(5)

,							(6)
  where , and  mg – graviton rest mass, which value in all cases does not exceed 
10 -65 g   (see. [15] and below, part.4). 
Equation (5) immediately implies that evolution of Universe in RTG is cyclic.  It means that scale factor varies within Rmin ≤ R(ф) ≤  Rmax (where Rmin << 1, Rmax >> 1 –  stationary points where shrinking (expansion) stops and where dR/dф = 0). Hence the matter  density varies within сmin ≤ с(ф) ≤  сmax,  and minimal density which corresponds to Rmax,  equals   .  Thus   RTG with the nonzero graviton rest mass, equations  (5),(6),   avoids both the cosmological singularity (Rmin ≠ 0, сmax is finite) and infinite Universe expansion (Rmax is also finite).
One can get consequence from equations (5),(6):
  									(7)
From the general equation of state in the form P = нc2с (0 ≤ н ≤ 1/3),   where  н = 1/3  corresponds to the ultra relativistic “radiation dominant”   state and н = 0  –   to the “dust like”  non relativistic matter state (including the present epoch), it follows that the solutions of (7) are: 
  a) for  н = 1/3:  
  b) н = 0:   , where Rmax is connected with сmax by relation 
.
 The solution of (5) for the scale factor R(ф) is written for these cases as a quadrature for   inverse function ф(R).
       1) For the radiation dominant epoch (0≤ ф ≤ ф0, н = 1/3,  Rmin ≤ R ≤  R0) we get:

  .								(8)
 
a) It follows from (8), that at the early stages of evolution (ф → 0, R > ≈ Rmin) we arrive to the   asymptotic ,  i.e. the uniformly accelerated expansion (). 
 b) Under R >> Rmin  we  get   and   .
       2) For the "dust like"  case  (ф0 ≤ ф ≤ фmax , н = 0,  R0 ≤ R ≤  Rmax) we get ():

 .						(9)

The approximate analytic expression for quadrature (9) was first obtained in [15] (Logunov A.A. et al, 2001,) in the form:
 ,									(10)
where .  Under  ф >> ф0  we get from (10): .
In order to evaluate the validity and accuracy of the approximations adopted in [15] (Logunov A.A. et
al, 2001,) to derive (10), we performed the numerical integration of (9) using the Maple-8 Package. The calculations were made within the following ranges of mg and сmax: mg = 10-67ч10-65 g,  сmax = 106ч1048 g/cm3. It turned out that approximation (10) remains a very good one within these intervals, with relative error not exceeding 10-6.
Thus the RTG cosmological solution R(ф), derived from (8) - (10), yields the accelerated expansion
 only at the most earlier stages  (ф << ф0)  with deceleration   at all other stages of evolution up to фmax.
It follows then, from the above discussion, that under  ф >> ф0, including the present moment, that one can get with very good accuracy:
 ,										(11)
where  .  From (11) we get for Hubble function:
 . 									(12)
At the present moment ф =  фc,  H(ф) =  Hc, where we accept, according to most of astronomical dates,  Hc = 
= (65± 15) (km/sec)/Mpc. Introducing the notion (where we using (12)): 
,									(13)
we get from (11),(13) for the present value of the scale factor:
.										(14)
Exactly this expression will be used below for the analysis of luminosity distance and comparison with observations.

 3. Luminosity distance in RTG

Assuming that the source of light radiating at the redshift z = (щ–щc)/щc  and at coordinate distance r at the moment   ф <  фc  is seen now at ф =  фc  (with coordinate r = 0), we get:
												(15)
From (11), (14), (15) we get:
 .										(16)
Coordinate distance is given by the relation:
 . 											(17)
The  luminosity distance is defined (see, for example, Weinberg, 1972 [20]) as:
 ,												(18)
where  L - is the intrinsic luminosity of the source in  erg/sec  and  l  is the observable flux in erg/cm2sec.
Using the standard procedure (see Weinberg, 1972 [20]), the expression for metric (4) and (15),(17), we arrive at:
 .								(19)
Applying here (11) and (14) we get
 .								(20)
Let us express the integral in (20) in the form:
 , 							(21)
where there introduced variable p (using (11),(15))

										(22)
and  y  introduced earlier in (13). Since always y < 1, then  pc <    (with maximum at ф = фmax). So,  at present epoch we have pc << 1.
Integral in (21) is expressed through the hypergeometric function (as a result of the power expansion): .  Keeping two first terms in this expansion, we get:
,										(23)
with relative error з = дI/I  falling fast with diminishing  p  (for example, з < 10-5  for p < 0.1). From (20), using approximation (23) we get the RTG luminosity distance as a function of z and mg in form

 .			(24)

In General Relativity we have the well known relation for the luminosity distance (Zeldovich,  Novikov, 1975 [21]),   which in the case of  q = Ѕ  has the form:
 .									(25)
It easy to see, as one can expect, that under pc → 0, i.e. vanishing mass of  graviton mg, expression (24) transforms to (25).
For nearby sources z << 1 and   pc ≠ 0 (i.e. mg ≠ 0)  from (24)  there it follows:
.				(26)
It is interesting to note, that from (24) there it follows for two sources with  redshifts z1 and z2:
.								(27)    This relation, in principle, allows to determine the mass of graviton mg ,  provided we know the fraction of luminosity distances  df1/df2  with sufficient  accuracy. Unfortunately, the distances are determined with big errors and this method  cannot be applied directly and statistical approach is needed. (See p.4).

Now, let us rewrite the relation for luminosity distance (24) in terms of stellar  magnitudes, as commonly accepted in observational astrophysics. Using the known relations  between magnitudes and fluxes [Weinberg,1972]:   erg/cm2sec; erg/sec, where m and M are correspondingly the visible and absolute bolometric magnitudes, definition (18) is written as:
,								(28)
where (m – M) is the so called  distance module. Using the relation (28), we arrive to the equation for luminosity distance (24) in  terms of distance module:

,								(29)

where  F(z,pc)  has been defined in (24). In (29) c=3·105 km/sec , so  (25 + 5 lg c) ≈ 52.38; Hc = (65±15) (km/sec)/Mpc,   so 5 lg Hc ≈ (9.0±0.5).  Usually, the value M for the Ia supernova is taken to be -19.5m  [1-3]. Total uncertainty in M – 5 lg Hc is at least  ±0.5m.
In the case of small  z << 1 and pc = 0 (i.e. mg = 0) (29) turns to the known expression in GR:

.							(30)
 If  pc ≠ 0  (i.e. mg ≠ 0)  and z << 1, taking into account (26), one gets:

 .						(31)

These relations (29)-(31) will be used for comparison with observational data below (See p.4).
4. The SN Ia  observational data and RTG model

The supernova observational data available up to now [1-3] can be summarized in the Fig 1, where the visible bolometric magnitudes of supernova are plotted against redshifts of their host galaxies.

Figure 1. SN Ia effective bolometric magnitudes vs redshift data.  From the work [2] (Perlmutter S. et al. A.J. 517, 565 (1999))

The diagram at Fig.1 consists of two different groups of data: one (Hammy et al. A.J., 1996) is the observations of the low redshift host galaxies with errors of order of 0.25m and more later data with higher redshifts and with larger error bars up to 0.5m – 0.7m.
Two groups of theoretical curves in standard cosmological model for Л = 0 and for Л ≠ 0  and for different values of  Щ are also plotted on the Fig.1. It should be noted from the very beginning that the vertical shift (along the mB axis) of the theoretical curve mB(z) directly depends on absolute magnitude M and Hubble constant H (see, for example, the equ.(30)). The uncertainties in this quantities can lead to arbitrary vertical shift up to  ±0.5m and the only way to fix it is to built the regression line for all set of available data. Let us note, that there are no regression lines and/or confidence belts on Fig.1.
To provide statistically correct basis for supernova data analysis we collect all the available data for both small and large redshifts and construct overall regression between mB and z using the general nonlinear regression. This regression together with 95% confidence belt, obtained with the TableCurveFitting and Maple-8 statistical packages, are plotted in Fig.2. Initial model for building the regression is the Equ.(29) which yields the overall regression in the form:

,								(32)

where K is fixing the regression curve along the mB axis and b is describing the  nonlinearity with larger values of z. 
The regression model gives the following values of K and b: K = 24.11 ± 0.05, b = 0.30 ± 0.11 with overall r.m.s. about 0.25m.

Figure 2. Regression line, 95% confidence belt, original data from [1-3]  and theoretical curves mB(z).

Fig.2 contains the following curves: 1 - regression line (32); 2 - asymptotic of (32)   under the small values of z; 3 - RTG curve with mg → 0, i.e. the curve  corresponding to GR model with Щ = 1, which turns to curve 2 under z <<1$;  4 - curves 4a and 4b are the upper and lower borders of the 95% confidence belt
correspondingly; 5 - RTG curve with  mg → 0, but with asymptotic 4a. It should be stressed that statistically speaking the real m(z) curve can occupy any position inside the 95% confidence belt and, thus, one can put theoretical asymptotic for  z << 1 at any place within the belt, including its upper border --- the curve 4a. In later case the curve 5 practically wholly occurs inside the confidence belt, including its part near z = 1.
As seen from Fig.2, all curves only slightly differ from strait line (curve 2). For the reasons of obviousness, on the Fig.3 they are presented together with original data after the subtraction of asymptotic of regression line (32), i.e. line 2, Fig.2.   So, on Fig.3 the vertical axis correspond to the quantity 
Дm = mB – mass-regr, where  mass-regr is the  asymptotic of regression value m. This procedure looks to be correct, because we subtract the statistically grounded regression, instead of any sort of theoretical curves containing at least two not so good known parameters M and Hc, as sometimes is being done.
The curve on Fig.3 correspond to that on the Fig.2 with addition of two items. First, here are plotted two RTG theoretical curves 6I and 6II  (equ. (29)) corresponding to two values of graviton mass: pc = 0.2 and pc = 0.4. Second, here we approximately took into account the fact that the uncertainties in observed magnitude values tend to increase with z, and we added curves  4a2  and 4b2 which are the 95% confidence borders corrected for the data error bars. From Fig.3 it is clearly seen that curve 5 (with pc = 0) and curve 6I with pc = 0.2) with asymptotic 4a1 (z  → 0) practically wholly lie inside the confidence belt, and if we take 4a2 - 4b2 border lines,  then this will be true for the curve 6II too. The curves 6I and 6II have the same
asymptotic 4a1 which is possible due to above mentioned uncertainties in M and Hc. 
Thus, we can state that RTG theoretical curves with graviton mass  mg < (1.8 ± 0.5)· 10-66 g are in satisfactory agreement with the existing Ia supernova observations. It means that one cannot state the acceleration of Universe as a firmly established point. This situation can be changed only when the massive of observational date will be significantly increased (at least to one  hundred data with  z ≈ 1).

  
Figure 3. Curves from Fig.2 with regression asymptotic subtracted.

5. Conclusions

The analysis made above allows to formulate the following conclusions (we are formulated theirs at first in [22]):

1. The explicit expressions for the observable quantity - the luminosity distance   - were derived within the RTG. They contain both visible  m  and absolute  M  magnitudes,  redshift  z  and graviton mass mg (the later is included in parameter  ).
2. The SN Ia data were statistically processed and non-linear approximation   function obtained together with regression curves m(z) and confidence bars.      These results are compared with theoretical curves.
3. It is shown that RTG curves with mg < (1.8 ± 0.5)· 10-66 g, including the extreme one with 
mg → 0  (this one corresponds to the General Relativity solution with  q = Ѕ  and Щ = 1 ), are in good - on the 95% level - agreement with all observational   data available up to the present moment. It means that the model under discussion   principally doesn't require any acceleration of the Universe expansion, which thus
cannot be regarded as an established fact.
4. The significant increase in SN Ia observational data is required as the only way to  put more accurate limits of mg and to support or rule out the model  with н= 0  and without acceleration.

Acknowledgements.  Authors express their deep appreciations to Academicians  A.A. Logunov    and S.S. Gershtein  for fruitful discussions and encouragement, to Professor V.L. Gorohov for consultations in statistical data processing and to Dr. N.P.Tkachenko for cooperative remarks.

  

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