If the Redshift Depends on the Pressure then the Acceleration of the Universe Can Be Explained ()
1. Introduction
The equation of continuity with the classical linear equation of state 
[1-6] where 
is the pressure, 
is the density of matter and 
is the coefficient, leads to negative pressure values for 
. The 
CDM model predicts effective value for 
which is negative [7,8]. Then for a small scale factor 
in the beginning of the Universe the total amount of matter in the Universe would be negligible. When the scale factor increased the matter appeared literally from nothing.
We propose new correction to redshift which can be useful in cosmology. This explains the appeared acceleration of the Universe. High values of the mass density are consistent with the experimental data on Supernova Ia within this model without the cosmological constant
. We compared this model with the experimental data of the Supernova Cosmology Project Supernova Ia compilation. We assumed that 
and
, hence 
(yet 
and curvature is positive). We assumed also that
, where 
is the parameter of this model and 
is the redshift. We found the optimal values
and
.
The quality of this regression was as high as it was in the 
CDM model. Yet the value of 
was much less in the absolute value than in the 
CDM model. The acceleration of the Universe appeared to be a pure observational effect due to the negative pressure.
2. The Equation of Continuity
The Einstein equations with the cosmological constant 
are
(1)
where
• 
• the Einstein gravitational constant [9, p.~347]• 
• the Newton gravitational constant.
To solve these equations we should know the energymomentum tensor
. For synchronous coordinates the energy-momentum tensor is
(2)
where 
is the density of matter and 
is the pressure of matter. The covariant divergence of the energymomentum tensor is zero:
,
.
If
,
.
Considering (22) we obtain
for the 
component. Assume that the pressure is proportional to the density:
.
for the dust matter this value is 
and the value for radiation is 
[9, p.~123]. Therefore
and
(3)
Considering physical dimensions of values, the Equation (3) is better as
.
Hence
.
Let us compare this equation with the change of volume of the Universe for closed models. The FriedmannRobertson-Walker metric in the cosmic time and in the polar coordinates is
(4)
where
, 
, 
and 
is the curvature parameter
. The 3-volume element at 
is
.
The volume of the Universe at 
is
(5)
Since our Universe is homogeneous, the total amount of matter in the Universe at some 
is
(6)
where
. Hence for 
and 
sufficiently small the total amount of matter in the Universe will be negligible: 
when 
and
. Cosmology with 
violates the law of conservation of matter (conservation of leptonic and baryonic numbers [10]). The idea that matter originated from radiation is not a good idea because 
is too large. The matter - antimatter asymmetry also cannot be explained in this way. Cosmology with negative pressure 
contained a “smooth bounce” from the collapsing to expanding stage and this was also due to the same fact. Since the matter disappeared, the gravitation force was negligible and we obtained the smooth bounce with the horizontal tangent [11-13]. The only equation which is fully consistent with (5) and conservation of leptonic and baryonic numbers is
(7)
This is due to the equation
. The Equation (3) could work for the appropriate epoch only.
Let us try to interpret the covariant divergence of the 
4-vector. If currents are negligible (as in the comoving frame), then
will get an additional nonzero component and 
will increase or decrease additionally. It seems reasonable because 
is the energy and its change is the redshift. Hence this equation should give a contribution to the redshift (and not bound the matter components). Since
and
due to (5), this covariant divergence is
where 
is the pressure. Assume that
(8)
Here
is the energy density per unit volume. Integrating
with
, we obtain
(9)
Since the Hubble law is
and both contributions seem to be multiplicative, we get
(10)
Hence the redshift depends on the matter equation of state. This correction to redshift 
may be useful in cosmology.
3. Correction to the Distance Due to Pressure
3.1. The Hubble Diagram
Recall the distance formula. Define 
-coordinate by
,
.
Then 
if 
and 
if
. Hence 
where
(11)
The Friedmann-Robertson-Walker metric in the cosmic time becomes
The space part of the metric is
.
If 
and 
then 
and
. For a photon
, therefore 
and
(12)
where
.
Now we use our main hypothesis:
.
Then
.
The value of 
is the function of
, therefore
where
(13)
Finally
(14)
where
(15)
Suppose a source has an absolute luminosity
, its luminosity (bolorimetric) distance is defined in terms of the measured flux 
(16)
The measured flux is
(17)
one factor of 
arising from the decrease in total energy due to the red shift of the energy of the individual photons, and the other factor of 
arising from the increased time interval between the detection of incoming photons due to the fact that two photons separated by a time 
at emission, will be separated by a time 
at the time of detection, where
[14, p.~41].
hence
and
.
applying our main hypothesis (10) we get
.
recalling the definition of
we get
for
.
therefore the luminosity distance is
(18)
We compared this model with the experimental data of the Supernova Cosmology Project Supernova Ia compilation1. In general no more than two parameters can be determined from these statistical problems [15]. We assumed that 
and
, hence
(yet 
and curvature is positive). We assumed also
(19)
where 
is the parameter of this model. We found the optimal values
and
.
the Hubble diagram with these parameters is presented on the figure 1. The quality of this regression is as high as it is in the ΛCDM model. High values of the mass density are consistent with the experimental data on Supernova Ia within this model. Yet the value of 
is much less in the absolute value than in the ΛCDM model.
The Taylor expansion of the luminosity distance (18) is
(20)
The optimal values for this simplified case for 
are
and
.
3.2. Comparing with the Riess Expansion
In the flat Universe 
we can compare (20) with the Equation (10) in [16, p.~25]:
(21)
therefore the retardation parameter
.
If the parameter introduced in this paper
we obtain acceleration. Hence the acceleration of the Universe appears as a pure observational effect due to the negative pressure.
3.3. Fate of the Universe
Note that we have chosen 
and 
while interpreting the Supernova Ia experimental data. We had to make some assumptions on these parameters because large number of unknown parameters cannot be obtained from this simple regression. Yet since experimental data are consistent with 
and
, then one can say that there is a solution of the Friedman equation wich accelerates for a short time in the beginning but there is a limit for the scale factor
. We can smoothly glue another solution where the Universe shrinks.
The question of the fate of matter at the end of the Universe is the most complicated. We assume that the matter does not disappeared at the end of the Universe. The matter with some unknown critical density should reproduce the hydrogen and provide condition for another Big Bung. The remnants of the Big Bung are black holes. We assume that all their singularities are topologically equivalent and lead to the same space-time moment in the past where the collapsing stage of the previous Universe ends and the new Big Bung begins.
4. The Section Curvatures
For completeness let us review the Christoffel symbols for the Friedmann-Robertson-Walker metric (4) and the section curvatures. The Christoffel symbols for the timelike dimension (i,j = 1,2,3) are
(22)
and
,
. The Christoffel symbols for the spacelike dimensions are
these are all the nonzero components.
Let 
be linear independent vectors of the tangent space for the manifold 
at the point
. The section curvature 
is defined as
(23)
where 
is the Riemannian curvature map [17, p.~94]. Compared with the Riemannian case we have changed sign in this formula because the signature of our space is Lorentzian. With local coordinates
.
Angle brackets mean scalar product. The denominator is the square of the area of the parallelogram based on the vectors 
and
. If we choose another basis in the same two-dimensional plane 
defined by 
and
, then the section curvature does not change. Hence the value 
depends on the plane 
only. Therefore the value 
is assigned 
and is called the section curvature of the (pseudo)Riemannian manifold 
at the point 
in the two-dimensional direction
. In local coordinates the section curvature in the direction 
− 
is
(no summation for
). The section curvature of the FRW space in the directions 0 - 1, 0 - 2, 0 - 3 is
.
In general for any vectors 
such as
, 
, 
the section curvature
. The section curvature of the FRW space in the directions 1 - 2, 1 - 3, 2 - 3 is
.
Also for any vectors 
such as 
and 
(other components are arbitrary),
.
Please note that for 
and 
the section curvatures of the spacetime at present are zero. Indeed for the directions 1 - 2, 1 - 3, 2 - 3 the section curvatures
.
It seems sensible that without matter the spacetime should be flat. The section curvatures for the directions 0 - 1, 0 - 2, 0 - 3 are 
where 
is the retardation parameter. At present time
.
In the absence of matter at present for 
it will be zero also.
5. Conclusion
We propose new correction to the Hubble's law which can be useful in cosmology. This correction and our assumption 
produce a model which is capable to explain the acceleration of our Universe. This model is also consistent with the parameters 
and 
corresponding to nearly flat and topologically closed Universe. The value 
means that the expansion of the Universe will change to shrinkage yet we should observe acceleration at present. The assimmetry between matter and antimatter is also explained within this model. The matter does not appear from nothing at the beginning of the Universe; it is the same matter that worked at the previous cycle. The matter does not dissappear at the end of the Universe; it will just reproduce the hydrogen. The antimatter is not holes in the Dirac sea of states with negative energy; it is the matter with opposite set of quantum numbers. During the final state of the cycle (which is equally the first state of the new cycle) all properties of matter are equalized.
NOTES