Combined cosmological and solar system constraints on chameleon mechanism

The chameleon mechanism appearing in the massive tensor-scalar theory of gravity can effectively reduce locally the nonminimal coupling between the scalar field and matter. This mechanism is invoked to reconcile large-scale departures from general relativity, supposedly accounting for cosmic acceleration, to small-scale stringent constraints on general relativity. In this paper, we carefully investigate this framework on cosmological and solar system scales to derive combined constraints on model parameters, notably by performing a nonambiguous derivation of observables like luminosity distance and local post-Newtonian parameters. The likelihood analysis of type Ia supernovae data and of an admissible domain for the parametrized-post-Newtonian parameters clearly demonstrates that the chameleon mechanism cannot occur in the same region of parameter space as the one necessary to account for cosmic acceleration with the assumed Ratra-Peebles potential and exponential coupling function.

Figure 1

Representation of the conditional 68% (in blue—darker area) and 95% (in green—lighter area) confidence regions in the $h^2{\tilde{\Omega }}_{m0}–\alpha$ plane for different assumed values of coupling constant $k$ from the analysis of the UNION SN Ia Hubble diagram. The star represents the best fit.

Figure 2

Top: Representation of the age of the Universe with respect to the observed matter density ${\tilde{\Omega }}_{m0}$ for models characterized by $\alpha =0.5$. Bottom: Representation of the chi squared per degrees of freedom ${\chi }^2/n_{\mathrm{dof}}$ (number of degree of freedom) with respect to the observed matter density ${\tilde{\Omega }}_{m0}$ for models characterized by $\alpha =0.5$.

Figure 3

Dark (blue) line: data representing $\Lambda$ with respect to $\alpha$ for different values of $k$ and ${\tilde{\Omega }}_{m0}$. It can be seen that the influence of $k$ and ${\tilde{\Omega }}_{m0}$ is very weak compared to $\alpha$. Light (red) line: curve resulting from a fit on these data [see (33)].

Figure 4

Evolution of the scalar field as a function of the Jordan frame scale factor $\tilde{a}$ (the observable one) for different initial conditions ${\varphi }_i$ and for the model characterized by ($k=1$, ${\tilde{\Omega }}_{m0}=0.197$, and $\alpha =0.5$). The minimum of the effective potential [see relation (37)] is represented by the darkest (blue) lines. Top: curves on a log-log scale, we can see that the scalar field reaches the minimum of the effective potential for all the initial conditions. Bottom: linear $x$ axis, we see that the scalar field does not follow the minimum of the effective potential from $\tilde{a}\sim 0.1$ and the evolution of the scalar field is independent of the initial condition ${\varphi }_i$ (all curves are superimposed).

Figure 5

Evolution of the three time scales characterizing the field response time (40), the Hubble damping (38), and the evolution of the effective potential (41) as a function of the Jordan frame cosmic scale factor $\tilde{a}$ for the same model as in Fig. 4. We can see the intersection between $t_{\varphi }$ and $t_{\mathrm{damp}}$ occurred at the transition scale factor ${\tilde{a}}_t$ that corresponds to the scale factor where the field leaves the minimum of the potential (see Fig. 4).

Figure 6

(a) Evolution of the scalar field (continuous lines) and of the minimum of the effective potential ${\varphi }_{\min }$ (37) (dash-dotted lines) for different values of the coupling constant $k$. For small values of the coupling constant, the scalar field leaves ${\varphi }_{\min }$ quite early. The transition scale factor ${\tilde{a}}_t$ where the field leaves the potential increases with the coupling constant $k$. For $k=15$, the scalar field is still in the minimum of the effective potential at the present epoch. (b) Evolution of the scalar field (continuous lines) and of the minimum of the effective potential ${\varphi }_{\min }$ (37) (dash-dotted lines) for two models with $k=10$ located in the local minima of the ${\chi }^2$ (see Fig. 2). The evolution of the model located in the second minimum of the ${\chi }^2$ located in the low ${\tilde{\Omega }}_{m0}$ minimum is very fast. (c) Evolution of the transition scale factor ${\tilde{a}}_t$ with the coupling constant $k$.

Figure 7

Representation of the different terms of the acceleration factor for different coupling constants $k$. The different terms ${\tilde{q}}_M$ for the matter contribution, ${\tilde{q}}_Q$ for the quintessence contribution, and ${\tilde{q}}_{\mathrm{NMC}}$ for the nonminimal coupling contribution are given by Eq. (45). The first three curves $k=1$, 10, 20 represent a model in the global minimum of the ${\chi }^2$ while the last curve $k=10$-left represents a model in the second minimum (left) of the ${\chi }^2$ (see Fig. 2). For a model in the global minimum of the ${\chi }^2$, the GR contribution is of the same order of magnitude for different values of $k$; the quintessence potential contribution decreases when the coupling constant increases to leave room for the contribution coming from the nonminimal coupling. The model in the low ${\tilde{\Omega }}_{m0}$ minimum of ${\chi }^2$ (left) characterized by a lower value of the ${\tilde{\Omega }}_{m0}$ has a much faster evolution.

Figure 8

(a–b) Evolution of the density parameter $\tilde{\Omega }$ corresponding to the quintessence contribution [${\tilde{\Omega }}_Q$ given by (50a) and represented by the dark (blue) lines] and to the nonminimal coupling contribution [${\tilde{\Omega }}_{\mathrm{NMC}}$ given by (50b) and represented by the light (red) lines] different coupling constants: (a) $k=1$ and $k=20$; (b) two models characterized by $k=10$ and in the different local minima of the ${\chi }^2$. (c) Evolution of the equation of state parameter ${\tilde{\omega }}_{\mathrm{DE}}$ as a function of the cosmic scale factor for a best-fit model with different coupling constants and for a model in the second minimum of the ${\chi }^2$ with $k=10$.

Figure 9

Ratio ${\varphi }_{\min }({\tilde{\rho }}_{\infty })/{\varphi }_0$ between the value of $\varphi$ minimizing the cosmological effective potential $V_{\mathrm{eff}}$ and ${\varphi }_0$ the value of the field given by the cosmological evolution. The different lines correspond to different values of the coupling constant $k$ and the filled areas represent different values of ${\tilde{\Omega }}_{m0}$.

Figure 10

Representation of the effective potential for a large and small value of the matter density $\tilde{\rho }$. This illustrates that, as $\tilde{\rho }$ decreases, the effective potential becomes wider and the minimum shifts to a higher value. The effective potential inside a body (star or planet) is represented by the light (red) line and outside the body by the dark (blue) line.

Figure 11

Representation of the evolution of the scalar field as a function of the radius coordinate in the two regimes. Light (red) line: in the thin-shell regime, we see that the field is frozen in the minimum of the potential inside the star. Dark (blue) line: in the thick-shell regime, the field does not reach the minimum of the effective potential inside the star.

Figure 12

Representation of the evolution of the scalar field around the Sun for a model characterized by the parameters: $\Lambda ={10}^{-9}\mathrm{GeV}$, $\alpha =0.5$, $k=1$, and ${\tilde{\rho }}_{\infty }={10}^{-27}\mathrm{kg}/{\mathrm{m}}^3$. With these parameters, we obtain a thick shell around the Sun. Top: comparison of the field profile around the neighborhood of the Sun obtained numerically by solving the full BVP problem (55) and by using the analytical solution. Bottom: relative difference between the numerical solution and the analytical approximation. One can see that the analytical solution is very good far from the Sun but is less efficient in the Sun.

Figure 13

Representation of the evolution of the scalar field around the Sun for a model characterized by the parameters $\Lambda ={10}^{-2}\mathrm{GeV}$, $\alpha =3.3$, $k=1$, and ${\tilde{\rho }}_{\infty }={10}^{-27}\mathrm{kg}/{\mathrm{m}}^3$. With these parameters, we obtain a thin shell around the Sun. Top: comparison of the field profile around the neighborhood of the Sun obtained numerically by solving the BVP problem related to the scalar field and by using the analytical solution. Bottom: relative difference between the numerical solution and the analytical approximation. One can see that the analytical solution is very good far from the Sun but is less efficient in the Sun.

Figure 14

Representation of the thin-shell parameters $\epsilon$. The different lines represent different values of the coupling constant and the filled areas represent different values of $\alpha$ between 0.5 and 3.

Figure 15

Representation of couples $(\alpha ,\Lambda )$ admissible with the Cassini constraint (53) for different values of the coupling constant $k$ and representation of the couples $(\alpha ,\Lambda )$ needed to explain cosmological observations. The density at infinity of the solar system $\tilde{\rho }(r=\infty )={10}^{-27}\mathrm{kg}/{\mathrm{m}}^3$ which corresponds to the cosmological matter density.

Figure 16

Representation of couples $(\alpha ,\Lambda )$ admissible with the Cassini constraint (53) for different values of the coupling constant $k$ and representation of the couples $(\alpha ,\Lambda )$ needed to explain cosmological observations. The density at infinity of the solar system $\tilde{\rho }(r=\infty )={10}^{-21}\mathrm{kg}/{\mathrm{m}}^3$ which roughly corresponds to the galactic matter density.