An accelerating Universe without Λ in concordance with the last H0 measured value Jorge Alfaro∗, Marco San Mart´ın†,Joaqu´ın Sureda‡
arXiv:1811.05828v1 [astro-ph.CO] 14 Nov 2018
November 15, 2018
Abstract We present a Markov Chain Monte Carlo (MCMC) analysis with the most updated catalog of SN-Ia using an alternative cosmological model named Delta Gravity. This model is based on a new Einstein-Hilbert action obtained by the extension of ˜ This theory predicts an accelerating Universe a new symmetry symbolized as δ. without the need to introduce a cosmological constant Λ by hand in the equations. We obtained a very good fit to the SN-Ia Data, and with this, we found the two free parameters of the theory called C and L2 . With these values, we can predict different cosmological parameters. One of them is the Hubble constant. The last H0 local value measurement is in tension with the CMB Data from Planck.. Therefore is very interesting to analyze it in this theory. DG predicts H0 to be 74.47 ± 1.63 km/(s Mpc). This value is in concordance with the last measurement of the H0 local value, 73.83 ± 1.48 km/(s Mpc) .
This paper provides an analysis using supernova type-Ia Data (SN-Ia Data) updated to 2018  to fit cosmological parameters in an alternative cosmological model known as Delta Gravity (DG)  . The standard knowledge about Cosmology is mainly based on the Standard Cosmological Model called ΛCDM. In this framework, there is a constant Λ known as “Dark Energy” (associated to density of Dark Energy) . This constant is strictly necessary to fit SN Data because it is impossible to obtain a good behavior of the fit if ΩΛ = 0 . ΩΛ contributes to the acceleration of the Universe and it is the only component of the Universe which can produce this effect. Any other component only creates deceleration on the Standard Cosmological Model, where is assumed that General Relativity (GR) works. ∗
Instituto de F´ısica, Pontificia Universidad Cat´olica de Chile. Casilla 306, Santiago, Chile. [email protected]
† Instituto de Astronom´ıa, Pontificia Universidad Cat´olica de Chile. Casilla 306, Santiago, Chile. [email protected]
‡ Instituto de Astronom´ıa, Pontificia Universidad Cat´olica de Chile. Casilla 306, Santiago, Chile. [email protected]
Although the SN Data fit looks good , there is no fundamental physical reason to add the Λ constant in the Einstein Field Equations or add the Λ at the level of the Einstein-Hilbert action . Not only SN Data have been used to fit the ΛCDM model. The Cosmic Microwave Background Radiation (CMB Data) and its Power Spectrum can be used to fit even more cosmological parameters . From here it is possible (assuming GR and ΛCDM work well) to obtain, with a very good constraint, the value ΩΛ = 0.6911 ± 0.0062. In this paper we are testing an alternative cosmological model known as Delta Gravity , that comes from a GR modification. There are two interesting subjects that we want to study from this alternative point of view. The first, The Hubble Constant (H0 ); the second, the “existence” of Dark Energy (associated to Λ 6= 0). A very controversial paper was published on 2016  about a H0 calculation using new parallaxes from Cepheids. The observed value, that is independent from cosmological model, is 73.24 ± 1.74 km Mpc−1 s−1 . This value is 3.4 σ higher than 66.93 ± 0.62 km Mpc−1 s−1 predicted by ΛCDM with Planck. But the discrepancy reduces to 2.1 σ relative to the prediction of 69.3±0.7 km Mpc−1 s−1 based on the comparably precise combination of WMAP+ACT+SPT+BAO observations. This value has been updated  using more precise parallaxes for Cepheids. The H0 updated value at 2018 is 73.52 ± 1.62 km Mpc−1 s−1 . We are interested in analyzing the Delta Gravity model using SN Data, calculating H0 to compare this with Standard Cosmological Model, and trying to give an explanation for the discrepancy between the two values of H0 .
Delta Gravity Foundations
The theory is based on two main ideas. The first is that GR is finite at one loop in vacuum, so it doesn’t need renormalization. The second idea is that DG is based on Delta Gauge Theories ( DGT)   which the main properties are a) New kind of ˜ I , from the originals ΦI , b) The classical equations of motion of ΦI fields are created, Φ are satisfied in the full quantum theory, c) The model lives at one loop, d) The action is obtained through the extension of the original gauge symmetry of the model, introducing an extra symmetry that we call δ˜ symmetry, since it is formally obtained as the variation of the original symmetry. When we apply this prescription to GR we obtain DG. 
Delta Gravity Action
Here we present a summary about the fundamental equations which DG is based on. The full deduction and formalism can be found in . From the Einstein-Hilbert action S0 : Z √ R 4 S0 = d x −g + LM 2κ
(where LM = LM (φI , ∂µ φI ) is the Lagrangian associated to the matter field φI ). We obtain the new Action for Delta Gauge Theories (by extension of the new symmetry): Z √ 1 R αβ αβ 4 ˜ + LM − G − κT g˜αβ + LM (2) S = d x −g 2κ 2κ where κ=
˜ µν g˜µν = δg δ √ 2 [ −gLM ] T µν = √ −g δgµν ˜ M = φ˜I δLM + (∂µ φ˜I ) δLM L δφI δ(∂µ φI ) ˜ I are the Matter Field y Delta Matter Field, respectively. where φI and φ˜I = δφ
Equations of Motion
From the Delta Gravity Action, we can obtain the equations of motion associated to gµν and g˜µν . These equations are: Gµν = κT µν 1 1 F (µν)(αβ)ρλ Dρ Dλ g˜αβ + g µν Rαβ g˜αβ − g˜µν R = κT˜µν 2 2
and the two conservation rules are:
Dν T µν = 0 1 1 = T αβ Dµ g˜αβ − T µβ Dβ g˜αα + Dβ g˜αβ T αµ 2 2
where: F (µν)(αβ)ρλ = P ((ρµ)(αβ)) g νλ + P ((ρν)(αβ)) g µλ − P ((µν)(αβ)) g ρλ − P (ρλ)(αβ) g µν 1 αµ βν (g g + g αν g βµ − g αβ g µν ) 4 Dν represents the Covariant Derivative respect to xν , built using gµν . P ((αβ)(µν)) =
T µν and T˜µν for a Perfect Fluid
The Energy-Stress Tensors for a Perfect Fluid in Delta Gravity are  (assuming c is the speed of light equal to 1): Tµν = p(ρ)gµν + (ρ + p(ρ)) Uµ Uν
∂p ∂p ρgµν + ρ˜ + (ρ)˜ ρ Uµ Uν + T˜µν = p(ρ)˜ gµν + (ρ)˜ ∂ρ ∂ρ 1 α α T T (Uν U g˜µα + Uµ U g˜να ) + Uµ Uν + Uµ Uν (ρ + p(ρ)) 2
where U α UαT = 0. Take into account that p is the pressure, ρ is the density and U µ is the four-velocity. For more details you can see .
Cosmology in Delta Gravity
Effective Metric to describe the Universe in a Cosmological Frame
The Effective Metric for the Universe is given by the photon trajectories, because these particles are what we observe and provide us the information from the observables (like the SN Data), showing us the expansion of the Universe. In DG the Effective Metric in FLRW coordinates for the Universe is given by a linear combination of gµν and g˜µν : 
gµν = gµν + g˜µν = −c dt +
1 + Fa (t) 2 R (t) 1 + 3 Fa (t)
dx2 + dy 2 + dz 2
Delta Gravity Friedmann Equations
We need the equations of state for matter and radiation. These equations are: pm (R) = 0 1 pr (R) = ρr (R) 3 Then, from equation 3 we obtain:
√ 2 C √ t(Y ) = 3H0 Ωr0
ρ(R) = ρm (R) + ρr (R) 1 pr (R) = ρr (R) 3 √ 3/2 Y + C(Y − 2C) + 2C R(t) R0 R0 ≡ R(t = t0 ) ≡ 1 ρr Ωr ≡ ρc ρm Ωm ≡ ρc 3H 2 ρc ≡ 8πG Ωr0 + Ωm0 ≡ 1 Y (t) =
(10) (11) (12) (13) (14) (15) (16) (17) (18)
Where t0 is the Age of the universe (at the current time). It is important to highlight r0 that t is the Cosmic Time, R0 is the Standard Scale Factor at the current time, C ≡ ΩΩm0 , where Ωr0 and Ωm0 are the density energies normalized by the Critical Density at the current time, defined as the same as the Standard Cosmology. Furthermore, we have imposed that Universe must be flat (k = 0), so we require that Ωr + Ωm ≡ 1. Using the Second Continuity Equation (6),where T˜µν is a new Energy-Momentum Tensor, two new densities called ρ˜M (Delta Matter Density) and ρ˜R (Delta Radiation Density) associated to this new tensor are defined. When we solve this equation, we find 3ρm0 Fa (Y ) 2 Y3 Fa (Y ) ρ˜R (Y ) = −2ρr0 Y4
ρ˜M (Y ) = −
Using the Second Fields Equations (4) with the solutions (19) and (20) we found (and redefining with respect to Y ): Fa (Y ) = −
L2 √ Y Y +C 3
Relation between the Effective Scale Factor YDG and the Scale Factor Y
Furthermore, we define the Effective Scale Factor (normalized): YDG (t) ≡
RDG (t) RDG (t0 )
Then, we obtain: YDG (L2 , C, Y ) =
Y RDG (t0 ) 5
1 − L2 Y3
Y +C √ 1 − L2 Y Y + C
With the new definition of L2 , the Delta Densities are given by: √ L2 Y +C ρ˜m (Y ) = ρm0 2 Y2 √ Y +C 2L2 ρr0 ρ˜r (Y ) = 3 Y3
Thus if we know C and L2 , it is possible to know the delta densities ρ˜m and ρ˜r . √ Note that the denominator in equation 23 is equal to zero when 1 = √ L2 Y Y + C. Remember that C = Ωr0 /Ωm0 0 and ρ˜r > 0, then L2 must be greater than 0. Then the valid range for L2 is approximately 0 ≤ L2 ≤ 1. C must be positive, and (hopefully) that is a really little value because the radiation is clearly not dominant in comparison with matter. Then, we can analyze cases close to the standard accepted value for Ωr0 /Ωm0 ∼ 10−4 .
Useful equations for Cosmology
Here we present the equations that are useful to fit the SN Data and obtain cosmological parameters that are presented in the Results Section.
The relation between the cosmological redshift and the scale factor is preserved in DG: 1 (26) 1+z It is important to take into account that the Current Time is given by t0 → Y (t0 ) → YDG (Y = 1) = 1, where YDG is normalized. YDG =
The proof is the same as GR, because the main idea is based on the light traveling through a null geodesic described by the Effective Metric given by (9) in DG . Taking into account that idea, we can obtain the following expression: √ Z (1 + z) C 1 Y dY √ √ dL (z, L2 , C) = c (27) 2 100 h Ωr0 Y (t1 ) Y + C YDG (t) Notice that Y = 1 today. To solve Y (t1 ) at a given redshift z, we need to solve (22) and (26) numerically. Furthermore, the integrand contains YDG (t) that can be expressed in function of Y in (23). Do not confuse c (speed of light) with C, a free parameter to be fitted by SN Data. The parameter h2 Ωr0 can be obtained from the CMB. The CMB Spectrum can be described by a Black Body Spectrum, where the energy density of photons is given by ργ0 = aT 4 6
From statistical mechanics, we know the neutrinos are related by : ρν0
7 =3 8
Then, Ωr0 h2 = Ωγ h2 + Ων h2
(28) is a value that only depends on the temperature of the Black Body Spectrum of CMB. So we can add this value like a known Cosmological Parameter. We only need to know the values C and L2 . Take into account that it is impossible to know the value of Ωr0 without any SN Data because to calculate this, we need to know the H, but this is not a known parameter for us. We want to predict this value using SN Data and don’t want to introduce any value from CMB-ΛCDM analysis, like Planck 2015 .
The distance modulus is the difference between the apparent magnitud m and the absolute magnitud M of an astronomical object. Knowing this we can estimate the distance d to the object, provided that we know the value of the Absolute Magnitude MV . This will be discussed in the next pages. d (29) µ = m − M = 5 log10 10 pc
Effective Scale Factor
In DG, the “size” of the Universe is given by YDG (t), so every cosmological parameter that in the GR theory was built up from the standard scale factor a(t), in DG will be built from YDG (t). This value is equal to 1 at the current time, because it is the RDG normalized by RDG (Y = 1).
In DG we will define the Hubble Parameter as follows: H DG (t) ≡
R˙ DG (t) RDG (t)
So, the Hubble Parameter is given by: H DG (t) =
RDG Notice that all the DG parameters are written as function of Y .
In DG we will define the Deceleration Parameter as follows: q DG (t) = −
¨ DG RDG R 2 R˙ DG
Then, q DG (t) = −
2 dt −1
Densities of Common Matter and Radiation: Ωm0 and Ωr0
We have Ωm0 + Ωr0 = 1 and C = Ωr0 =
Ωr0 , Ωm0
then: 1 C
Ωm0 = 1 − Ωr0
It is important to realize that this equation only expresses a relation, or a proportion, between the energy density for Common Matter and Common Radiation Densities, and does not express a real percentage of composition of the Universe because in DG we also have Delta Matter and Delta Radiation.
Fitting the SN Data
We are interested in the viability of DG as a real Alternative Cosmology Theory that could explain the Accelerating Universe without Λ.
To analyze this we used the most updated Type Ia Supernovae Catalog. We obtained the Data from Scolnic . We only needed the distance modulus µ and the redshift z to the SN-Ia to fit the model using the Luminosity Distance dL predicted from the theory. The SN Ia are very useful in cosmology  because they can be used as standard candles and allow to fit the ΛCDM model finding out free parameters like ΩΛ . We are interested in doing this in DG. The main characteristic of the SN-Ia that makes them so useful is that they have a very standardized absolute magnitude close to −19.  From the observations we only know the apparent magnitude and the redshift for each SN-Ia. We have two options: try to fit the absolute magnitude MV or use a standardized Absolute Magnitude obtained by an independent method that does not involve ΛCDM model, or any other assumptions. In consequence, we define two kinds of fits, MV Fixed Fit and MV Free Fit. Both methods try to find the values of the two free parameters C and L2 for DG, but the latter tries to find MV too. In the MV Fixed Fit we will use the MV = −19.23 ± 0.05 . The 8
value was calculated using 210 SN-Ia Data from . This value is very important for us, because it is independent from the Model, because Riess et. al. are trying to calculate the H0 local value, and they did not use any particular Cosmological Model. So, the value used here is independent from the model, and it was found by fitting Data using calibration by Cepheids to calculate the H0 local value. Anyway, we did the independent MV fit to clear up any doubt about the DG model behavior. We used 1048 SN-Ia Data in  1 . All the SN-Ia are spectroscopically confirmed. In this paper, we have used the full set of SN-Ia presented in . They present a set of spectroscopically confirmed PS1 SN-Ia and combine this sample with spectroscopicallyconfirmed SN-Ia from CfA1-4, CSP, PS1, SDSS, SNLS and HST SN surveys. At  they used the SN Data to try to obtain a better estimation of the Dark Energy state equation. They define the distance modulus as follows: µ ≡ mB − M + αx1 − βc + ∆M + ∆B
where µ is the distance modulus, ∆M is a distance correction based on the host-galaxy mass of the SN and ∆B is a distance correction based on predicted biases from simulations. Furthermore, α is the coefficient of the relation between luminosity and stretch, β is the coefficient of the relation between luminosity and color and MV is the absolute B-band magnitude of a fiducial SN-Ia with x1 = 0 and c = 0. In this paper we are not interested in the specific corrections to observational magnitudes of SN-Ia. We only take the values extracted from  to analyze the DG model. The SN Data are the redshift zi and (µ + M )i with the respective errors.
We need to establish a relation between redshift and the apparent magnitude for the SN-Ia: dL (z, C, L2 ) (36) [µ + M ] − M = 5 log10 10 pc where dL (z, L2 , C) is given by (27) and [µ + M ] are the SN-Ia Data given at  defined as (35). It is important to take into account the units of dL . In this expression we have as free parameters: MV (the dependence on MV is Fixed or Free), C and L2 to be found by fitting the model to the points (zi , [µ + M ]i ). We have two different fits, DG MV Fixed Fit, and DG MV Free Fit. For the MV fixed fit we used MV = −19.23. It is important to remember that Ωr0 h2 is independent from H0 = 100h, and this value only depends on TCM B = 2.725 K. Taking into account this temperature, we obtain Ωr0 h2 = 0.0000418 (28). 1
Scolnic’s Data are available at https://archive.stsci.edu/hlsps/ps1cosmo/scolnic/
For GR we use the following expression [µ + M ] − M = 5 log10
dL (z, H0 , Ωm0 ) 10 pc
where dL (z, H0 , Ωm0 ) is given by: c(1 + z) dL (z, H0 , Ωm0 ) = H0
du p (1 − Ωm0 )u4 + Ωm0 u
and [µ + M ] are the SN Data given at  defined as (35). It is important to take into account the units of dL . Remember that we are always working on a flat Universe, and in GR standard model the Ωr0 is negligible. We have the same degrees of freedom as DG. Note that we are including Dark Energy as ΩΛ0 ≡ ΩΛ ≡ 1 − Ωm0 in GR.
To fit the SN-Ia Data to GR and DG models, we used Markov Chain Monte Carlo (MCMC). This routine was implemented in python 3.6 using PyMC2. 2
Basically, MCMC consists on Fitting a model, characterizing its posterior distribution. It is based on Bayesian Statistics. We used the Metropolis-Hastings algorithm. Initially we propose initial distributions for the parameters that we want to fix, and then PyMC2 will give us the posterior probability distribution for these parameters. We want to find the best fitted parameters for DG and GR models. These parameters will be C, L2 (and MV ) for DG and H0 , ΩM (and MV ) for GR. The initial proposed probability distribution for the parameters are Normal Distributions and we show them with the posterior probabilities in the Figure 3 for DG and in Figure 4 for GR, in the Results and Analysis Section. For the MCMC method assuming MV free, we only show the Posterior Probability contours maps, because they will be useful for the final analysis (Figures 19 and 20).
Results and Analysis
We present the results for DG and GR fitted Data, and with these values we obtain different cosmological parameters. We divide the results into two fits: DG Fit and GR Fit. Each section has two kind of fitting method, one taking MV like a fixed value, and another taking MV like a free value to be fixed. 2
The DG and GR models describe very well the mB vs z SN-Ia Data. It’s important to note that Λ 6= 0 is strictly needed in GR frame, because in other cases the well-behaved curved doesn’t appear. Also, we can see that DG model describes the SN-Ia very well without Λ. So, essentially, we have the same behavior in DG, but the acceleration Universe appears explained without needing Λ, or anything like “Dark Energy”. For both fitted curves, the residual plots are normally distributed and don’t exhibit any bad signal.3 In Table 1, we present the coefficients of determination (r2 ) and residual sum of squares (RSS) for both fitted models: MV fixed model r2 RSS DG 0.99709 21.39 GR 0.99708 21.44 Table 1: MV fixed statistical parameters. Both coefficients of determination are very good and the RSS are similar for both cases. The fitted parameters for GR and DG models are: MV fixed DG model Value L2 0.455 C 0.000169
Error 0.008 0.000003
Table 2: MV fitted parameters using MCMC for DG.
MV fixed GR model Ωm0 h2
Value 0.28 0.549
Error 0.01 0.004
Table 3: MV fitted parameters using MCMC for GR. Also, we show the priors for DG and GR, with the Posterior Probabilities obtained from PyMC2 in Figures 3 and 4 assuming MV fixed. 3
A problem could be that we have more Data at low redshift than high. But this is a typical problem about SN-Ia Data. Furthermore, this is the most extensive sample for Cosmology use at this date. 
Figure 1: Fitted curve for DG model assuming MV = −19.23. On the right corner, the residual plot for the fitted Data.
Figure 2: Fitted curve for GR standard model assuming MV = −19.23. On the right corner, the residual plot for the fitted Data.
(a) Prior and Posterior Probabilities for L2 in(b) Prior and Posterior Probabilities for C in DG, assuming MV fixed. DG, assuming MV fixed.
Figure 3: Priors and Posterior probabilities for DG assuming MV fixed. The red lines indicate the best fitted normal distribution for the Posterior Probability Distribution for the respective parameters.
(a) Prior and Posterior Probabilities for Ωm0 in(b) Prior and Posterior Probabilities for h2 in GR, assuming MV fixed. GR, assuming MV fixed.
Figure 4: Prior and Posterior probabilities for GR assuming MV fixed. The red lines indicate the best fitted normal distribution for the Posterior Probability Distribution for the respective parameters. For both models, we obtained a “narrow” Normal Posterior Distribution. Furthermore, we present the posterior probability density maps for GR and DG in Figure 5.
(a) Posterior probability density maps with for(b) Posterior probability density maps for DG. GR. Combination for Ωm0 and h2 . Combination for L2 and C.
Figure 5: Posterior probability density plots obtained from MCMC assuming MV fixed for GR and DG models. Note that for both plots in Figure 5 the distributions are well defined, and for each parameter we obtain a Gaussian-like distribution. For both models, the combination of parameters constrained a region in the 2D-density plot. The fitted values for both models converged very well. We can see the convergence in Figure 6.
(a) Evolution of values with steps in GR.
(b) Evolution of values with steps in DG.
Figure 6: Convergence of values for GR and DG assuming MV fixed. In Figure 6 we can see that for both models the evolution of the parameters is convergent. This is in concordance with the posterior probabilities shown in Figure 5.
We applied two convergence tests for MCMC analysis. The first is known as Geweke . This is a time-series approach that compares the mean and variance of segments from the beginning and end of a single chain. This method calculates values named z-scores (theoretically distributed as standard normal variates). If the chain has converged, the majority of points should fall within 2 standard deviations of zero. 4 . The plots are shown in Figure 7. 4
(a) Evolution of z-scores with steps in GR.
(b) Evolution of z-scores with steps in DG.
Figure 7: Convergence of values for GR and DG assuming MV fixed. In both plots it is possible to observe that the most part of the z-scores fall within 2σ, so the method is convergent for both models based on the Geweke criterion. Another convergence test is the Gelman-Rubin statistic. The Gelman-Rubin diagnostic uses an analysis of variance approach to assessing convergence. This diagnostic uses multiple chains to check for lack of convergence, and is based on the notion that if multiple chains have converged, by definition they should appear very similar to one another; if not, one or more of the chains has failed to converge (see PyMC 2 documentation). ˆ close to one because this is the indicator that In practice, we look for values of R shows convergence. We ran 16 chains for DG model. We show the results in Figures 8, 9 and 10. The first plot shows the L2 and C predicted values for every chain of the MC. The second and third Figure show the convergence of L2 and C. All the chains converge to a similar value assuming different priors. These final value predicted for every chain can be visualized in 8. From all these chains, is clear that the DG MCMC analysis is convergent for the two free parameters.
Figure 8: Gelman Rubin test for DG model assuming MV = −19.23. The Gelman Rubin ˆ coefficient test was run with 16 different chains, all with different L2 and C priors. The R (Gelman Rubin coeficient) was calculated for each parameter.
Figure 9: Gelman Rubin test for DG model assuming MV = −19.23. There are 16 chains with different priors. All the chains converge to a L2 ≈ 0.455.
Figure 10: Gelman Rubin test for DG model assuming MV = −19.23. There are 16 chains with different priors. All the chains converge to a C ≈ 0.000169 . Looking the 16 chains, is clear that the MCMC analysis for DG model is convergent.
Cosmic Time and redshift
To calculate the Cosmic Time in DG, we used the equation 12. The redshift is obtained by numerical solution from equation 26. Meanwhile for GR model, we obtained the Cosmic Time from the integration of the First Friedmann Equation and solving t(Ωm0 , H0 ). Here we have included ΩΛ = 1 − Ωm0 and we did Ωk (k = 0) and Ωr0 = 0. The integral for the First Friedmann Equation can be analytically solved (from equation 41): Z t= 0
+ (1 − Ωm0
2 ln dx = √ 3 1 − Ωm0 )x2
√ −Ωm0 a3 + Ωm0 + a3 + 1 − Ωm0 a3/2 √ Ωm0 (39)
where t in 39 is the Cosmic Time for GR. We plot the results in Figure 11:
Figure 11: Cosmic Time for GR and DG assuming MV = −19.23. The behavior of Cosmic Time dependence with redshift for both models is very similar.
Hubble parameter and H0
With the fitted parameters found by MCMC for GR and DG, we can find H(t) and H0 . Note the superscript for GR as GR and DG as DG . For GR H0 is easily obtained from the h2 fitted (H0 = 100h). H GR (t) can be obtained using the first Friedmann equation 2 8πG ρm0 ρr0 a˙ 2 = + + ρ (40) H = Λ0 a 3 a3 a4 Taking into account that Ωm0 + Ωr0 + ΩΛ0 = 1, Ωr0 ≈ 0, and ρc0 = ρ ΩXi 0 = ρXc0i 0 for every Xi component in the Universe, we obtain Ωm0 2 2 + (1 − Ωm0 ) H = H0 a3
3H02 , 8πG
With (41), we obtain H GR (t) and using (31) we obtain H DG (t). For the actual time we evaluate H RG at a = 1 and for DG we evaluate H DG at YDG = 1 obtaining the Hubble constant H0GR and H0DG . We present the results from both models and we compare these values with measurements: Model Planck 2015  Riess 20185  5
H0 ( km/(s Mpc) ) Error 67.74 0.46 73.52 1.62
The calibration was made including the new MW parallaxes from HST and Gaia.
Riess 20186  GR DG
73.83 74.08 74.47
1.48 0.24 1.63
Table 4: H0 values founded by MCMC with SN-Ia Data. Furthermore, we tabulate Planck  and Riess  H0 values.
Figure 12: Hubble Parameter for DG and GR fitted models assuming MV = −19.23 6
The calibration was made considering the external constrains on the parallax offset based on Red Giants.
Figure 13: Comparative for Hubble Constant assuming MV fixed for DG and GR models
Age of the Universe
The age of the Universe in DG is calculated using (12). t(Y ) only depends on C and not on L2 . In GR we calculate the age of the Universe using 39. With these expressions, we can compare the behavior between cosmic time and the scale factor in GR (or the effective scale factor in DG).
Figure 14: The size of the Universe vs age of the Universe. In th DG model, the size of the Universe YDG depends on cosmic time t and on C. The blue line indicates the Effective Scale Factor in DG. The gray zone shows the error associated to YDG . For GR, the Scale Factor a depends on cosmic time t and on Ωm0 . The red line indicates the Scale Factor evolution in GR. The gray zone shows the error associated to a (these are tiny). In Figure 14, it is possible to see the evolution for YDG (t) in time. At t = 28.75 Gyr, YDG goes to infinity, and the Universe ends with a Big Rip, so, in this model the Universe has an end (in time). Also, we see the dependence between the scale factor a and cosmic time t. The Universe has no end (in time) in GR.
Deceleration Parameter q(t)
For DG, we used the equation (33). To evaluate at current time, we evaluate a = 1 for GR, and RDG = 1 for DG. In Figure 15, we can see the evolution in time for both GR and DG models.
(a) Evolution of Deceleration Parameter in GR.(b) Evolution of Deceleration Parameter in DG.
Figure 15: Deceleration Parameter assuming MV fixed. Actually, we tabulate the deceleration parameter for both models in Table 5 Model DG GR
q0 Error -0.664 0.002 -0.57 0.02
Table 5: q0 values founded by MCMC with SN-Ia Data. In both models q0 < 0, then the Universe is accelerating.
Relation with Delta components
In DG we are interested in determining the Delta composition of the Universe. Using the equations 24 and 25, we can obtain the densities for Delta Matter and Delta Radiation with the C and L2 fitted values. ρ˜m0 = 0.22777ρm0 = 0.22773ρc0 ρ˜r0 = 0.68330ρr0 = 0.000115ρc0
In the expressions 43 and 42, we have obtained the current values for Delta densities.
Figure 16: Temporal evolution of density components for DG assuming MV fixed. The vertical axis is normalized by critical density at current time ρc0 . On the top right corner, there is a zoom in very close to YDG = 0 showing the transition between Delta Matter and Delta Radiation (Delta components), and the transition between Matter and Radiation (common components). In general, the Common Density is higher than the Delta Density. The Common Components are dominant compared with Delta components. Matter is always dominant compared with radiation (in both cases). See Figure 16. Note that the four components diverge (in density) at the beginning of the Universe, and the Delta Components show a “constant-like” behavior for YDG > 0.4. (specially Delta Matter that is clearly dominant compared to the Delta Radiation). In both the Common Components and Delta Components, there is a transition between matter and radiation that is indicated in the zoom included in Figure 16. These transitions occur at very early stage of the Universe. Both transitions are indicated in Figure 16. It’s important to remember that in DG we don’t know the ρc0 , but we know the densities of each component in units of ρc0 , because they are given by C and L2 fitted values from SN Data.
Importance of L2 and C
To understand the role that L2 and C are playing in the DG model, we need to plot some cosmological parameters in function of both coefficients. We are interested in analyzing the accelerating expansion of the Universe in function of these two parameters, so we plotted H0 in Figure 17 and q0 in Figure 18,
(a) C values go from 0 to 6 to explore various (b) The C values are bounded to very little valUniverses, even a Universe wholly dominated by ues, nearly close to the C fitted value obtained radiation. by MCMC.
Figure 17: H0 for a different combination of L2 and C values. The fitted values found by MCMC analysis is indicated in the Figure. In Figure 17, we can see there is a big zone prohibited, because there the parameters become complex numbers. The only allowed values are colored. Note that in the 17a almost all the allowed H0 values are close to 0. Only the contour of the colored area shows H0 6= 0. The 17b is the same as the left one, but zooming the vicinity of the fitted values obtained from MCMC analysis. These range of C and L2 are reasonable to make an analysis. Note that H0 has a strong dependence of C and L2 . Remember that L2 has only sense between values 0 and 1, because we only want to allow positive Delta Densities and, from the equation 23, the denominator could be equal to 0.
(a) C values go from 0 to 6 to explore various (b) The C values are bounded to very little valUniverses, even a Universe wholly dominated by ues, nearly close to the C fitted value obtained radiation. by MCMC.
Figure 18: q0 for different combination of L2 and C values. The fitted values found by MCMC analysis is indicated in the Figure. The Figure 18 is very interesting because it shows the dependence of the current value of acceleration of the Universe expressed by the deceleration parameter q0 . If we examine the parameters zone close to the fitted values in the Figure 18b, we can note that the acceleration of the Universe only depends on the value of L2 . This is a very important result from the DG model. The accelerating Universe is given by the L2 parameter. This 24
parameter appears naturally like a integration constant from the differential equations when we solved the field equations for DG model. Then, in this model, and exploring the closest area to the Universe with a little amount of radiation compared to matter, we found that a higher L2 value, higher the acceleration of the Universe (current age): q0 becomes more negative when L2 → 1 independently of C.
MCMC with MV free
If we leave MV free to be fixed by MCMC analysis, we obtain a non-convergent result. In DG model, the C, L2 and MV parameters are dependent. This can be visualized in Figure 19
(b) Evolution of values with steps. (a) Posterior probabilities densities.
Figure 19: MCMC analysis assuming MV free parameter in DG. The same occurs with GR model,
(b) Evolution of values with steps. (a) Posterior probabilities densities.
Figure 20: MCMC analysis assuming MV free parameter in GR. In both models the process is non-convergent (evolution did not find a “constant value” 25
with the evolution of the steps) and the posterior density plots are degenerate for h2 vs M in GR and for C and M for DG. Note that L2 in DG and Ωm0 reach the convergence. In this case it is needed to find a confidence “range” for the predicted values and constraint the parameters. This process is equivalent to assume MV as a fixed parameter from the beginning. We want to show the degeneration here and explain why we decided to choose the MV fixed MCMC analysis. The MV used in the MCMC analysis is independent from cosmological model. In both models, we obtained a degeneracy between two parameters. For DG, the dependence appears between C and MV . The dependence can be fitted by a second order polynomial. This is shown in Figure 19. The polynomial is given by (44) C = 8.59 × 10−5 M 2 + 3.15 × 10−3 M + 2.9 × 10−2
If we evaluate the equation (44) MV at the MV assumed for M = −19.23 fixed case, we obtain C = 0.000168688 and this value represents a percentage error compared to the C from MV fixed case about 0.03%. Therefore, the MCMC with MV free and the fitted polynomial consider the MV fixed case. For GR, we did the same procedure, but in this model the dependence appears between h2 and MV The polynomial is showed in Figure 20 and is given by (45) h2 = 0.177M 2 + 3.15 × 7.335M + 75.896 × 10−2
If we evaluate the equation (45) MV at the MV assumed for M = −19.23 fixed case, we obtain h2 = 0.550783 and this value represents a percentage error compared to the h2 from MV fixed case about 0.3%. Therefore, the MCMC with MV free and the fitted polynomial consider the MV fixed case.
Here we have studied the cosmological implications for a modified gravity theory, named Delta Gravity. The results from SN-Ia analysis indicate that DG explains the accelerating expansion of the Universe without include Λ or anything like “Dark Energy”. The acceleration is naturally produced by the DG equations. We assumed that MV = −19.23 is a suitable value calculated from . We want to emphasize the very important fact that this value was obtained by local measurements and calibrations of SN-Ia, and then, it is independent from any cosmological model. Assuming this, the procedure presented does not use ΛCDM assumptions. We only assume that the calibrations from Cepheids and SN-Ia are correct; therefore, the absolute magnitude MV = −19.23 for SN-Ia is reasonably correct. Furthermore, we generalize the procedure finding a dependence expressed by equation (44) for C. Then, if it is needed to change the MV value, we can recalculate all the procedure only changing C in function of MV . Note that C is degenerated with MV but L2 is independent from MV . This is 26
very important, because L2 establishes the acceleration of the Universe in DG; so, even in the case MV could be very imprecisely known, L2 does not change very much. In this case, the Universe is accelerating and this result is stable under any change of the priors for the MCMC analysis. The acceleration in DG is given by L2 6= 0. L2 also determines that the Universe is ˜ made of delta matter and delta radiation. This can be associated to the new field: φ. Therefore, this theory can be interpreted as a phantom model (from the Action). DG needs L2 6= 0 to explain Dark Energy, and this implies that there must exist a new kind of energy density that we have called Delta Matter and Delta Radiation. It is not clear if this Delta Composition are real particles, or not. Also, DG can predict a high value for H0 . This aspect is very important because the current H0 value is in tension  between Sn-Ia analysis and CMB Data. In Figure 13 we can visualize that DG predict a H0 in concordance with the last H0 measurement. GR also predicts a high H0 value with the same assumptions, but it needs to include Λ to fit the SN-Ia Data. The most important point about this, is that the local measurement of H0 is independent of the model. 7 , furthermore, the discrepancy about H0 value could be indicating new physics beyond the Standard Cosmology Model Assumptions, and maybe, one possibility could be the modification of GR. Another difference between DG and GR models, is that DG model predicts a Big Rip that is dominated by the L2 value. This is shown in Figure 14. The most important differences between DG and Standard Cosmological Model are the explanation about “Dark Energy” (the relation with L2 ) and that the H0 calculated in DG is in concordance with the latest measurement of H0 local value.
Acknowledgements. The work of M. San Martin has been partially financed by Beca Doctorado Nacional Conicyt; Fondecyt 1150390; CONICYT-PIA-ACT1417.J. Sureda has been partially financed by CONICYT-PIA-ACT1417; Fondecyt 1150390.The work of J. Alfaro is partially supported by Fondecyt 1150390, CONICYT-PIA-ACT1417.
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