Dynamics and cosmological constraints on Brans-Dicke cosmology

We investigate observational constraints on the Brans-Dicke cosmological model using observational data coming from distant supernovae type Ia, the Hubble function $H(z)$ measurements, information coming from the Alcock-Paczyński test, and baryon acoustic oscillations. Our analysis is based on the modified Friedmann function resulting form dynamical investigations of Brans-Dicke cosmology in the vicinity of a de Sitter state. The qualitative theory of dynamical systems enables us to obtain three different behaviors in the vicinity of this state. We find for a linear approach to the de Sitter state ${\omega }_{\mathrm{BD}}={-0.8606}_{-0.1341}^{+0.8281}$, for an oscillatory approach to the de Sitter state ${\omega }_{\mathrm{BD}}={-1.1103}_{-0.1729}^{+0.1872}$, and for the transient de Sitter state represented by a saddle-type critical point ${\omega }_{\mathrm{BD}}={-2.3837}_{-4.5459}^{+0.4588}$. We obtain the mass of the Brans-Dicke scalar field at the present epoch as $m_{\varphi }\sim H_0$. The Bayesian methods of model comparison are used to discriminate between obtained models. We show that observational data point toward vales of the ${\omega }_{\mathrm{BD}}$ parameter close to the value suggested by the low-energy limit of the bosonic string theory.

Figure 1

Posterior constraints for investigated model 1a (linear approach to the de Sitter state). One-dimensional plots: solid lines denote fully marginalized probabilities, dotted lines show mean likelihood. Two-dimensional plots: solid lines denote 68% and 95% credible intervals of fully marginalized probabilities, the colors illustrate mean likelihood of the sample. Estimations were made using $\mathit{Union}\mathit{2.1}+H(z)+\mathit{Alcock}-\mathit{Paczy}\mathit{ń}\mathit{ski}+\mathit{BAO}$ data set. For the numerical results see Table 1.

Figure 2

Posterior constraints for investigated model 1b (transient de Sitter evolution). One-dimensional plots: solid lines denote fully marginalized probabilities, dotted lines show mean likelihood. Two-dimensional plots: solid lines denote 68% and 95% credible intervals of fully marginalized probabilities, the colors illustrate mean likelihood of the sample. Estimations were made using $\mathit{Union}\mathit{2.1}+H(z)+\mathit{Alcock}-\mathit{Paczy}\mathit{ń}\mathit{ski}+\mathit{BAO}$ data set. For the numerical results see Table 1.

Figure 3

Posterior constraints for investigated model 2 (oscillatory approach to the de Sitter state). One-dimensional plots: solid lines denote fully marginalized probabilities, dotted lines show mean likelihood. Two-dimensional plots: solid lines denote 68% and 95% credible intervals of fully marginalized probabilities, the colors illustrate mean likelihood of the sample. Estimations were made using $\mathit{Union}\mathit{2.1}+H(z)+\mathit{Alcock}-\mathit{Paczy}\mathit{ń}\mathit{ski}+\mathit{BAO}$ data set. For the numerical results see Table 1.

Figure 4

The fully marginalized probabilities (solid lines) and mean likelihood (dotted lines) for the present value of the phase space variable $x(a_0)$ calculated for the models $1a$ (top) and $1b$ (middle) from Eq. (39a) and for the model 2 (bottom) from Eq. (40a). The maximal probability is located at the negative values indicating that at present the effective gravitational coupling constant increases.

Figure 5

The fully marginalized probabilities (solid lines) and mean likelihood (dotted lines) for the parameter of the Brans-Dicke theory ${\omega }_{\mathrm{BD}}$ calculated for the models $1a$ (top) and $1b$ (middle) from Eq. (45) and for the model 2 (bottom) from Eq. (46).