Dark Energy from Virtual Gravitons (GCDM Model vs. ΛCDM Model)

Abstract

The dark energy from virtual gravitons is consistent with observational data on supernovas with the same accuracy as the ΛCDM model. The fact that virtual gravitons are capable of producing a de Sitter accelerated expansion of the FLRW universe was established in 2008 (see references). The combination of conformal non-invariance with zero rest mass of gravitons (unique properties of the gravitational field) leads to the appearance of graviton dark energy in a mater-dominated era; this fact explains the relatively recent appearance of the dark energy and answers the question “Why now?”. The transition redshifts (where deceleration is replaced by acceleration) that follow from the graviton theory are consistent with model-independent transition redshifts derived from observational data. Prospects for testing the GCDM model (the graviton model of dark energy where G stands for gravitons) and comparison with the ΛCDM model are discussed.

1. Introduction

As is known, more than 20 years ago, two research groups independently discovered the accelerated expansion of the universe [1,2]. This effect was called dark energy (DE). The nature of DE is still unknown. A big number of approaches were considered, including quite unusual ones [3,4]. The most popular, apparently, is the hypothesis about scalar fields of different kinds, filling the universe. A fairly complete bibliography (comprising reviews) can be found, e.g., in [5,6,7,8,9,10,11]. One of the most popular theories based on the idea of a scalar field is so-called quintessence [8,12,13,14]. The quintessence is dark energy in the form of a time-varying scalar field which is slowly rolling down toward the minimum of its potential. Another direction of the search attempted to generalize Einstein’s theory in the classical and quantum way (see e.g., [11], and indeed an interesting attempt by [15] to construct an emergent gravity in which “the space-time and gravity emergent together from entanglement structure of an underlining microscopic theory”. Another interesting direction of modern research is associated with attempts to construct a cosmology with a variable cosmological constant [16].

For now, despite the large number of DE models, the community has settled on the simplest choice, which is the hypothesis that the cause of acceleration is the cosmological constant, which was introduced by Einstein himself into his theory more than 100 years ago. Later, he gave up on it; however, after in [17] it was shown that the cosmological constant is the energy of a non-gravitational vacuum, it again acquired the “rights of citizenship”. In the last hundred years, the cosmological constant has appeared every time astrophysics and cosmology have been faced with problems that, it seemed, could not be solved otherwise. For example, at the end of 60-th, when quasars were discovered, and at one time it seemed that they were concentrated near z = 2 [18,19,20], this effect was attempted to be explained by the fact that there is a positive cosmological constant ([21,22,23,24,25]). Later, quasars with large redshifts were discovered, and this hypothesis was dropped. As V. Petrosian remarked [26]: “In the absence of strong evidence in favor of Lemaître models, we must again send back the Lemaître models and along with them the cosmological constant until their next reappearance”. (page 28) 1 The next reappearance took place in 1998–1999 in connection with the discovery of dark energy. The next reappearance took place in 1998–1999 in connection with the discovery of dark energy [1,2] 2. The community has now opted for the ΛCDM model despite known problems with it. As is known, two main problems with it are the 122 order of magnitude difference between the theoretical and observational value of the cosmological constant and the coincidence problem (why dark energy has appeared recently, why now?). A very significant argument in favor of ΛCDM is the fact that it describes the observational data very well, despite these theoretical contradictions. On the other hand, all observational data (with no exceptions) based on the attempts to find the equation of state parameter $w=\mathrm{p}/\rho$, where $\mathrm{p}$ and $\rho$ are the pressure and density of matter filling the universe. In the case of cosmological constant $w_{\Lambda }=-1$, all observational data show that $w=-1$, and this fact is considered by the community as confirmation of the validity of the ΛCDM model; however, the virtual gravitons also produce the de Sitter expansion with $w_G=-1$ (Appendix B). To find the difference between the models, it is necessary to measure w(z) (Section 5).

The dark energy at the epoch of last scattering can contribute to the observed CMB anisotropies. We discuss it in Appendix E.

As we already mentioned, the goal of this paper is to compare the ΛCDM model with the GCDM model where G stands for gravitons. To do so, we are going to calculate distance modules for both models and compare them (Section 3 and Section 4). We will see that the GCDM model is consistent with SNe Ia observations with the same accuracy as the ΛCDM model. At the same time, the GCDM model is free from the contradictions that the ΛCDM model encounters.

2. Dark Energy from Virtual Gravitons

A graviton theory of dark energy was presented in [27]. A rigorous mathematical foundation of the theory was given in [28,29]. The present paper is dedicated to the comparison of graviton theory with supernova observations only, so we do not touch upon any theoretical problems. Nevertheless, we believe that it is necessary to remind the reader of at least basic information of graviton theory. It was shown in [28] that quantum fluctuations of the metric (gravitons) and their back reaction on the isotropic and homogeneous (on average) background provide the mechanism for cosmological acceleration (Appendix A). The dark energy effect is a consequence of the vacuum polarization and graviton creation by the non-stationary gravitational field of the Universe. The energy density of gravitons is a functional of the background geometry. In the non-empty Universe, the background geometry is defined by all contributing cosmological subsystems—by gravitons, matter, and radiation. The combination of conformal non-invariance with zero rest mass of gravitons (unique properties of the gravitational field) leads to a macroscopic quantum effect: condensation of gravitons in a quantum state with the wavelength of the order of the distance to the horizon. The same unique properties of the gravitational field are the causes for the appearance of dark energy in our era (Appendix D), and it answers the old unanswered question “Why now?”.

In the process of the evolution of the Universe, the density of gravitons, because of their condensation, starts dominating over the sum of the energy density of other subsystems of the cosmological media. The self-consistent state of background and gravitons, which evolves asymptotically, represents a self-polarized vacuum in the de Sitter space. The regime of the de Sitter-like expansion is beginning to form in the current Universe. Three new exact solutions for the one-loop quantum gravity were presented in [28]. The de Sitter solution is one of these. All exact solutions can be found when the theory is presented as Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy equations for moments of the graviton spectral function (Appendix B).

In this paper, we show that dark energy of graviton origin, with the case of a zero cosmological constant, is consistent with SNe Ia observational data with the same accuracy as the ΛCDM model. In terms of observational data, the difference between these models lies beyond the accuracy of observations (Section 4). We show also that the appearance of graviton DE during the modern epoch of the Universe evolution is a direct consequence of the combination of conformal non-invariance with zero rest mass of gravitons, which are unique features of the gravitational field (Appendix C). In Section 5 and Section 6, we discuss observational tests that might distinguish GCDM and ΛCDM models.

3. Distance Modules

To fit supernova data, we use distance modulus as follows

$$\mu (z,h,{\Omega }_0)=25+5\log (\frac{3000}{h}(1+z)R(z))$$

(1)

$$R(z)={\displaystyle \underset{0}{\overset{z}{\int }}\frac{d{z}^{\prime }}{\stackrel{⌢}{H}(z^{\prime })}}$$

(2)

$$\stackrel{⌢}{H}(z)={[{\Omega }_{\mathrm{m}}{(1+z)}^3+\Omega (\mathrm{z})]}^{1/2}$$

(3)

where $\mu (z,h)$ is distance modules, the Hubble constant $\mathrm{H}=\mathrm{h}\cdot 100 \mathrm{km}/\mathrm{s}\cdot \mathrm{Mpc}$. The first term in Formula (3) represents the energy density of non-relativistic matter, and the second one represents the energy density of DE as a function of z. In the case of cosmological constant, $\Omega \left(\mathrm{z}\right)={\Omega }_0={\Omega }_{\Lambda }=\mathrm{const}$. In the case of DE, $\Omega \left(\mathrm{z}\right)$ is the fraction of DE in the total energy balance. Let us denote distance modules for the cosmological constant case as ${\mu }_{\Lambda }(\mathrm{z}, \mathrm{h}, {\Omega }_0)$ Our program is to compare runs for ${\mu }_{\mathrm{G}}(z,h,\Omega (\mathrm{z})) \mathrm{and} {\mu }_{\Lambda }(\mathrm{z}, \mathrm{h}, {\Omega }_0)$ to show that the difference between them is smaller than observational errors. We do not include CMB input into Formula (3). The input of radiation into the current energy balance is very small, it is of the order of ${\Omega }_{\gamma }=5.38\cdot {10}^{-5}$ [30]. For example, for the Planck case with ${\Omega }_{DE}=0.685$ the input of radiation to the region z = 1000 is about 8%; it is reasonable to assume that the effect of radiation on the dynamics of expansion can be neglected when its contribution to the dynamics becomes no more than 8%.

The usual practice is to use the parameter of the equation of state $\mathrm{w}$ and determine it from the observational data (see e.g., [1]). The equation of the state of dark energy is unknown, therefore for lack of a better option, the dark energy equation of state can be taken in the $\mathrm{p}=\mathrm{w}\epsilon$ form (sometimes in more general forms such as $\mathrm{p}/\rho {=\mathrm{w}}_0{+\mathrm{w}}_1\cdot \mathrm{f}\left(\mathrm{a}\right)$). After that, one can look for the w parameter which provides the best fit. Usually, the supernova data are used in combination with other data. All of them show that $\mathrm{w}\approx -1$, and this fact, as it is generally accepted, leads to the idea that the DE effect is due to the cosmological constant. In the case of dynamical DE (and in the case of gravitons, particularly), the second term of Formula (3) must follow from the solution of Einstein’s equations for the model. In the case of cosmological constant, the Friedmannian equations for the flat Universe have a well-known exact analytical solution (see e.g., [31])

$$\mathrm{a}\left(\mathrm{t}\right)={\left({\Omega }_{\mathrm{m}}/{\Omega }_{\Lambda }\right)}^{1/3}{\left(\sin \mathrm{h}\left[3\sqrt{{\Omega }_{\Lambda }}Ht/2\right]\right)}^{2/3}$$

This allows us to calculate all values of interest analytically. Unlike the cosmological constant case, there is no exact analytical solution to the system consisting of gravitons and non-relativistic matter; thus, we must go for the numerical integration of Einstein’s equations (in our case they are the equations of the BBGKY chain) for the graviton model.

4. GCDM vs. ΛCDM

In this section, we will show that the difference between ${\mu }_{\Lambda }(\mathrm{z}, \mathrm{h}, {\Omega }_0)$ and ${\mu }_{\mathrm{G}}(\mathrm{z}, \mathrm{h}, \Omega (\mathrm{z}\left)\right)$ for all cases of interest is less than the observational errors in SNe Ia database. In other words, we will show that both ΛCDM and GCDM are statistically indistinguishable. As it was mentioned by the Planck team [32], there is a “tension” between WMAP and Planck data. In accordance with WMAP, today’s value of DE is ${\Omega }_0=0.721$ [33]; meanwhile, Plank’s data give ${\Omega }_0\approx 0.685$ [32]. Recent data obtained from the analysis of the large-scale structure give ${\Omega }_{\mathrm{m}}=0.339\pm 0.032/0.031$ for the ΛCDM model [34]. The combination of these data with available baryon acoustic oscillation, redshift space distortion, SNe Ia data and with Planck CMB lensing leads to the following content of the nonrelativistic matter and Hubble constant in the universe for ΛCDM model [34] ${\Omega }_{\mathrm{m}}=0.306\pm 0.004/0.005; \mathrm{h}=0.680\pm 0.004/0.003$; this leads to the following DE contents, which we need for our calculations: ${\Omega }_0=0.694$. We consider three cases ${\Omega }_0=0.721$ (Bennett et al., 2012), ${\Omega }_0=0.685$ [32] and ${\Omega }_0=0.694$ [34]. In this work, we use Union 2.1 compilation from the Supernova Cosmology project [35,36]. A direct link to the observed supernova data used in calculations is shown in the reference [35].

We calculate distance modules $\mu (\mathrm{z}, \mathrm{h}, {\Omega }_0)$ using Formula (3). As we already mentioned, in the distinction of the generally accepted method of calculation of $\mu (\mathrm{z}, \mathrm{h}, {\Omega }_0)$ which uses the hypothesis on the equation of state parameter of DE $\mathrm{p}=\mathrm{w}\epsilon$, with the following determination of $\mathrm{w}$, we calculate $\Omega \left(\mathrm{z}\right)$ directly by numerical integration of Einstein equations presented in the form of BBGKY chain (Equations (A7) and (A8)). In such an approach, the cosmological constant case corresponds to $\Omega \left(\mathrm{z}\right)=\Omega \left(0\right)={\Omega }_0=\mathrm{const}$ in Formula (3); thus, we must compare ${\mu }_{\Lambda }(\mathrm{z}, \mathrm{h}, {\Omega }_0)$ calculated for $\Omega \left(\mathrm{z}\right)={\Omega }_0=\mathrm{const}$ (cosmological constant) with ${\mu }_{\mathrm{G}}(\mathrm{z}, \mathrm{h}, \Omega (\mathrm{z}\left)\right)$ calculated for $\Omega \left(\mathrm{z}\right)$, i.e., graviton dark energy.

Our computational model has only one free parameter ${\Omega }^0{}_{\mathrm{DE}}$ which is a fraction of the graviton energy in the total energy balance at the start of the calculation run, (Appendix C). Each run started from some value of ${\Omega }^0{}_{\mathrm{DE}}$ and finished when Ω(z) becomes equal to Ω₀. Attention. In our calculations, the scale factor is increasing from the past to the future, i.e., from a = 1 to $a=a_{\mathrm{today}}$, where $a_{\mathrm{today}}$ corresponds to the observed today DE fraction of the full energy, which we denote as ${\Omega }_0\equiv {\Omega }_{DE}(z=0)$. First, we conducted a series of runs to localize the interval of ${\Omega }^0{}_{\mathrm{DE}}$ where the statistical deviation between calculated distance modulus and their values from the database for SNe Ia (Formulas (1)–(3) above) is minimal. We found that the interval 0.000125 < ${\Omega }^0{}_{\mathrm{DE}}$ < 0.000162 provide the statistical deviation is about 1 sigma. Second, we found the best fitting ${\Omega }^0{}_{\mathrm{DE}}$ separately for the three cases Ω₀ = (0.721, 0.694, 0.685).

We ran our numerical simulations for the following initial conditions, ${\Omega }^0{}_{\mathrm{DE}}=0.000162; 0.000154; 0.000142; 0.000125$; these runs automatically lead to the following initial ${\mathrm{z}}_0$ for all ${\Omega }_0$ chosen (see

Table 1. Calculated ${\Omega }_0$ and ${\mathrm{z}}_0$ for different initial ${\Omega }^0{}_{\mathrm{DE}}$.

$${\mathbf{\Omega }}^0{}_{\mathbf{DE}}$$	$${\mathbf{\Omega }}_0$$	$${\mathbf{z}}_0$$
0.000162	0.721	1058
	0.694	1000
	0.685	982.6
0.000154	0.721	1106–1108
	0.694	1047–1048
	0.685	1028–1029
0.000142	0.721	1211–1212
	0.694	1145–1146
	0.685	1124–1125
0.000125	0.721	>1318
	0.694	1297–1298
	0.685	1272–1275

).

The SLS is situated at $\mathrm{z}\approx 1100$ with the width of this region $\Delta \mathrm{z}=90$ [37]; thus, we get the initial conditions for such runs in the following regions of interest.

The right column in the Table 1 shows what the initial values are for the observational values of dark energy (second column). The first column shows four randomly selected EDEs, each markedly less than 0.009. We present here the results of numerical simulation for the ${\Omega }^0{}_{\mathrm{DE}}=0.000154$ for all three cases ${\Omega }_0=0.685, {\Omega }_0=0.721 \mathrm{and} {\Omega }_0=0.694$. The results for all three other cases are similar.

To start with, we choose the initial value of the energy density of DE as ${\Omega }^0{}_{\mathrm{DE}}=0.000154$ at the initial point ${\mathrm{z}}_0\approx 1028$ for ${\Omega }_0=0.685$. Such a run produces $\Omega \left(\mathrm{z}\right)$ shown on the Figure 1. For all figures below $\Omega \left(\mathrm{z}\right)$ is the fraction of DE in the total energy balance.

Graviton DE as a function of z is shown in Figure 2 for the region of interest, i.e., from z = 0 to z = 2, where supernovas observed are situated:

To compare $\Omega \left(\mathrm{z}\right)$, obtained by numerical integration of BBGKY chain Equations (A7) and (A8) with the observational data, a high accuracy analytical approximation of it in the whole interval $0\le \mathrm{z}\le 1028$ was obtained in the form of

$$\Omega \left(z\right)=\frac{\alpha +\gamma z+\epsilon z^2+\eta z^3+\iota z^4}{1+\beta z+\delta z^2+\varsigma z^3+\theta z^4},$$

where

$\alpha$ = 0.685, $\beta$ = 0.62974, $\gamma$ = −0.046398, $\delta$ = 0.10038, $\epsilon$ = 0.109381, $\varsigma$ = 0.115417, $\eta$ = 0.0058062, $\theta$ = 0.0057869, $\iota$ = −4.135292 × 10⁻⁶

Such $\Omega \left(\mathrm{z}\right)$ generates distance modules which are showed together with the observational errors in Figure 3.

The Figure 4 below visually demonstrates the fact that the difference $\Delta$ between ΛCDM and GCDM models is less than observational errors. The same fact follows from the consideration of statistical sums (chi-square criterions). The fit proceeds by minimizing the χ² statistical criterion

$${\chi }_{\mu }{}^2={\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}\frac{{({\mu }_{\mathrm{i}}-{\mu }_{\mathrm{i}}{}^{\mathrm{obs}})}^2}{{\sigma }_{\mu ,\mathrm{i}}{}^2}}$$

where the sum is over all supernova defined in the Union 2.1 compilation [35,36]; ${\mu }_{\mathrm{i}}{}^{\mathrm{obs}}$ is the ‘observed’ distance modulus to the i-th supernova in the Union 2.1 of supernovas; ${\sigma }_{\mu ,\mathrm{i}}$ is the observational error (standard deviation) in the value estimation ${\mu }_{\mathrm{i}}{}^{\mathrm{obs}}$; ${\mu }_{\mathrm{i}}$ is a theoretical estimation for the same distance as ${\mu }_{\mathrm{i}}{}^{\mathrm{obs}}$. The statistical sum S is defined as

$$\mathrm{S}={\chi }_{\mu }{}^2/2$$

If the theoretical estimation is correct, and experimental errors of observed values ${\mu }_{\mathrm{i}}{}^{\mathrm{obs}}$ are also correct (being independent of each other), and normally distributed, then ${\chi }_{\mu }{}^2$ follows the Chi-squared distribution [38]. When the number of members in the sum is equal N, the average (mean) value of ${\chi }_{\mu }{}^2$ is equal to N, and the variance is equal to 2 N [39]. Practically, if N ≥ 50, at its extremum, the Chi-squared distribution becomes very close to the normal distribution. In other words, the probability that the estimated ${\chi }_{\mu }{}^2$ will deviate from the average value, N, on more than three-sigma, ±3·sqrt (2 N), is about 100% − 99.7% = 0.3%. Mostly, with the probability 68%, the deviation should be observed within one-sigma interval about the mean value. To be exact, when one is fitting experimental data with a model having m free parameters, the value of χ² in average reduces by the factor (N − m)/N [40]. In our case we used models with only one fitting parameter (m = 1) and many observations (N = 580), so we can safely neglect this factor in the chi-squared statistics. For the ${\Omega }_0=0.685$ case of cosmological constant we get the minimal statistical sum ${\mathrm{S}}_{\Lambda }^{0.685}=284.52$ for h = 0.7.

In this case the standard deviation of the statistical sum for ΛCDM model is equal to

$${\sigma }_{0.685}^{\Lambda }=\sqrt{S_{\Lambda }^{0.685}}=16.87$$

The calculated statistical sum for GCDM model and its standard deviation here are

$$S_{\mathrm{DE}}^{0.685}=283.68 \mathrm{and}{\sigma }_{\mathrm{DE}}^{0.685}=16.84$$

The difference $S_{\mathrm{DE}}^{\Lambda }-S_{\mathrm{DE}}^{0.685}=0.84$ << ${\sigma }_{\mathrm{DE}}^{0.685}$, ${\sigma }_{\Lambda }^{0.685}$. In other words, these two models (ΛCDM and GCDM) are indistinguishable, so that the visual result shown on the Figure 5 is confirmed by direct calculation of statistical sums.

The best fit to the cosmological constant model ${\Omega }_0=0.721$ and $\mathrm{h}=070$ 4 leads to statistical sum and the standard deviation as follows:

$${\mathrm{S}}_{\Lambda }^{0.721}=281.125$$

(4)

$${\sigma }_{\Lambda }^{0.721}=\sqrt{S_{\Lambda }^{0.721}}=16.77$$

(5)

For the ${\Omega }_0=0.721$ case, the calculations started at the point ${\mathrm{z}}_0=1106$. Numerical integration leads to the dark energy distribution showed on the Figure 5.

Graviton DE if Figure 6 shown for the same region of interest 0 < z < 2 as in Figure 4 but for ${\Omega }_0=0.721$; this dependence obtained by numerical integration of the BBGKY chain may be described by the following simple approximation

$$\begin{gathered} \Omega \left(z\right)=\frac{1}{\alpha +\beta z+c{z}^2} \\ \alpha =1.389213, \beta =0.9601587, c=0.00370186 \end{gathered}$$

Figure 7 (similar to Figure 3) shows the calculated fitting curve for the graviton DE model but with h = 0.7 and ${\Omega }_0=0.721$.

The statistical sum for the Ω₀ = 0.721 is calculated the same way as for Ω₀ = 0.685 case. The dependence of the statistical sum on the Hubble constant is illustrated in Figure 8 for the ${\Omega }_0=0.721$ case.

In this case, we get

$${\sigma }_{\mathrm{DE}}^{0.721}=17.03$$

(6)

Comparing Formulas (4) and (5) with Formula (6), we get an expected result of

$$S_{\mathrm{DE}}^{0.721}-S_{\Lambda }^{0.721}=8.825 < {\sigma }_{\mathrm{DE}}^{0.721}, {\sigma }_{\Lambda }^{0.721}$$

Again, the difference between statistical sums less than standard deviations which means that both cases statistically indistinguishable. Visually, this fact can be seen in Figure 9.

Similar to the Figure 9, in the Figure 10 below shown the case ${\Omega }_0=0.694$ from the Table 1, corresponding to ${\mathrm{z}}_0=1047–1048$, Again the difference between ΛCDM and GCDM models is less than the observational errors shown as circles.

Below, in Figure 11 and Figure 12 we show the graviton DE obtained for the case ${\Omega }_0=0.694$.

The behavior of $\Omega \left(\mathrm{z}\right)$ on the interval $0\le \mathrm{z}\le 3$ is shown in Figure 12.

The minimal statistical sums for the case ${\Omega }_0=0.694$ are ${\mathrm{S}}_{\mathrm{DE}}^{0.694}=284.9$ and ${\mathrm{S}}_{\Lambda }^{0.694}=283.0$, respectively. Obviously, ${\mathrm{S}}_{\Lambda }^{0.694}-{\mathrm{S}}_{\mathrm{DE}}^{0.694}=1.9\ll 1\sigma$, where 1${\sigma }_{\Lambda }$ = 16.8 and $1{\sigma }_{\mathrm{DE}}=16.9$. Thus, in this case we also have observational errors greater than the difference between models.

Thus, the results of this section confirm the fact that ΛCDM and GCDM models are statistically indistinguishable; moreover, they show that the Hubble constants h providing minimums for statistical sums are 0.709, 0.711, 0.714 for the cases considered which are close to the [43] result Ho = 69.8 ± 0.6 (stat) ± 1.6 (sys) km/s Mpc.

5. What Are the Prospects for the GCDM Testing and Comparison with ΛCDM Model?

We see at least two possible ways to test the difference between GCDM and ΛCDM models in the frame of supernova data. One of them is comparison of equations of state for GCDM and ΛCDM models. And the second one is comparison of transition redshifts at which the deceleration of expansion changes to acceleration.

Our program computes both energy density and pressure of GCDM from the start ${\mathrm{z}}_0$ to the end when $\Omega \left(\mathrm{z}\right)={\Omega }_0$. So, we can get the dependence $\mathrm{p}(\rho )$ for GCDM and compare it with $\mathrm{p}=-\rho$ for ΛCDM model. Figure 13 shows this comparison for the Planck case ${\Omega }_0=0.685$.

Below, in the Figure 14 shown the differences in the equation of state between GCDM and ΛCDM models.

Obviously, to compare these two models we must have w(z) determined observationally. Usual practice is to suggest that w = const and to look for this constant from observational data.

In attempt to consider $\mathrm{w}\ne \mathrm{const}$, Planck Collaboration [32] tested the model ${\mathrm{w}=\mathrm{w}}_0{+\mathrm{w}}_{\mathrm{a}}(1-\mathrm{a})$ where a is a scale factor [44]. In z terms it reads

$${\mathrm{w}=\mathrm{w}}_0{+\mathrm{w}}_{\mathrm{a}}(1-{\mathrm{a})=\mathrm{w}}_0{+\mathrm{w}}_{\mathrm{a}}\cdot \frac{\mathrm{z}}{1+\mathrm{z}}$$

(7)

It was found in [32] that

$${\mathrm{w}}_0={{-1.04}_{{-}_{0.69}}}^{+0.72};\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathrm{w}}_{\mathrm{a}}<1.32 (95\%; \mathrm{Planck}+\mathrm{WP}+\mathrm{BAO})$$

(8)

As we see from Equation (8), w(0) can significantly differ from −1. Thus, even light modification of the model Equation (7) can be used for the successful determination w(z) and comparison GCDM and ΛCDM models.

6. Transition Redshifts

Recall that the Friedmannian equation for deceleration/acceleration in our case reads

$$\mathrm{q}=-\frac{ä}{{aH}^2}=\frac{\kappa }{{6H}^2}[{\rho }_{\mathrm{de}}{+3\mathrm{p}}_{\mathrm{de}}+{\rho }_{\mathrm{m}}{\left(\frac{a_{\mathrm{today}}}{a}\right)}^3]$$

(9)

In the case of cosmological constant, $\kappa {\rho }_{\mathrm{de}}=\Lambda ,\hspace{0.17em}\kappa {\mathrm{p}}_{\mathrm{de}}=-\Lambda$, Equation (9) leads to the well-known result that the redshift ${\mathrm{z}}_{\mathrm{q}=0}$ where deceleration is changed for the acceleration, i.e., where $\mathrm{q}=0$ reads

$${1+\mathrm{z}}_{\mathrm{q}=0}=\left(\frac{2{\Omega }_{\Lambda }}{1-{\Omega }_{\Lambda }}\right){}^{1/3}$$

For parameters ${\Omega }_{\Lambda }=0.72$ that we use in our simulations one gets ${\mathrm{z}}_{\mathrm{q}=0}\approx 0.73$. For the Planck parameters ${\Omega }_{\Lambda }=0.685$, we obtain ${\mathrm{z}}_{\mathrm{q}=0}\approx 0.63$. So, for the ΛCDM model one gets ${\mathrm{z}}_{\mathrm{q}=0}\approx 0.63–0.73$ considering a tension between WMAP and Planck data. For the GCDM model we have to expect [28]

$${1+\mathrm{z}}_{\mathrm{q}=0}\approx {\left(\frac{{\Omega }_0}{1-{\Omega }_0}\right)}^{1/3}$$

Which leads to ${\mathrm{z}}_{\mathrm{q}=0}=0.37$ for WMAP data and ${\mathrm{z}}_{\mathrm{q}=0}=0.29$ for the Planck data. Now what we get from our numerical simulations. Figure 15 shows the case ${\Omega }_0=0.72$.

Below, in Figure 16 shown the case ${\Omega }_0=0.685$.

Direct measurements of the transition point parameter ${\mathrm{z}}_{\mathrm{q}=0}$ were made by several authors (see below). All these works must be divided into two categories. The first group of authors directly used the ΛCDM model in their studies. The second group of authors considered the model-independent approach. As expected, the first group have obtained ${\mathrm{z}}_{\mathrm{q}=0}$ close to numbers following from [45] ${\mathrm{z}}_{\mathrm{q}=0}<0.7$; [46] found 0.82 $\pm$ 0.08; [47] ${\mathrm{z}}_{\mathrm{q}=0}=0.74\pm 0.05$; [48] ${\mathrm{z}}_{\mathrm{q}=0}=0.7$; [49] ${\mathrm{z}}_{\mathrm{q}=0}=0.7$; [50] ${\mathrm{z}}_{\mathrm{q}=0}=0.65$; [51] ${\mathrm{z}}_{\mathrm{q}=0}\approx {0.78}_{-0.27}^{+0.08}$. The second group of authors (model-independent results) found that ${\mathrm{z}}_{\mathrm{q}=0}$ is much closer to our numbers; they are [52] ${\mathrm{z}}_{\mathrm{q}=0}=0.35\pm 0.07$; [53] ${\mathrm{z}}_{\mathrm{q}=0}\approx {0.29}_{-0.06}^{+0.07}$; [42] ${\mathrm{z}}_{\mathrm{q}=0}\approx 0.43\pm 0.07$; [54] ${\mathrm{z}}_{\mathrm{q}=0}\approx 0.3$; [55] ${\mathrm{z}}_{\mathrm{q}=0}=0.4\pm 0.1$ (99.9% confidence level).

The observational determination of transition redshifts is an ill-defined problem due to the necessity to measure the second derivative of scale factor $\ddot{a}$. Such a procedure produces significant errors that can be seen from the above references. Nevertheless, the increase in accuracy with the model-independent approach to measuring transition redshifts can help to make a choice between these models.

7. Conclusions

The GCDM model of graviton dark energy is consistent with the supernova observational data with the same accuracy as the ΛCDM model. The graviton DE naturally explains the reason why the DE appears during the matter dominated era. The transition redshifts that follow from graviton theory are consistent with the model independent transition redshifts found from observational data. The way to find out which of the models (GCDM or ΛCDM) better fits the observational data lies in the need to abandon the practically standard assumption that $\mathrm{w}=\mathrm{const}$ and consider $\mathrm{w}=\mathrm{w}\left(\mathrm{z}\right)$ when analyzing observational data. Another prospect for evaluating the GCDM model and comparison with ΛCDM lies in increasing the accuracy of transition redshifts measurements.