Recovery of general relativity in massive gravity via the Vainshtein mechanism

We study in detail static spherically symmetric solutions of nonlinear Pauli-Fierz theory. We obtain a numerical solution with a constant density source. This solution shows a recovery of the corresponding solution of general relativity via the Vainshtein mechanism. This result has already been presented by us in a recent paper, and we give here more detailed information on it as well as on the procedure used to obtain it. We give new analytic insights into this problem, in particular, for what concerns the question of the number of solutions at infinity. We also present a weak-field limit which allows one to capture all the salient features of the numerical solution, including the Vainshtein crossover and the Yukawa decay.

Figure 1

Plot of the metric functions $-\nu$ and $\lambda$ vs $R/R_V$, in the full nonlinear system and the decoupling limit (DL), with a star of radius $R_{\odot }={10}^{-2}{R}_V$ and $m\times R_V={10}^{-2}$. For $R\ll R_V$, the numerical solution is close to the GR solution (where in particular $\nu \sim -\lambda \sim -R_S/R$ for $R>R_{\odot }$). For $R\gg R_V$, the solution enters a linear regime. Between $R_V$ and $m^{-1}$, where the DL is still a good approximation, one has $\nu \sim -2\lambda \sim -4/3\times R_S/R$. At distances larger than $m^{-1}$ the metric functions decay à la Yukawa as appearing more clearly in the inset. The latter shows the same solution but for larger values of $R/R_V$, and in the range of distance plotted there, the numerical solutions are indistinguishable from the analytic solutions of the linearized field equations (24). This plot was already presented as Fig. 1 of Ref. 26.

Figure 2

Plot of the difference between the numerical solution for $\lambda$, ${\lambda }_{\mathrm{NUM}}$ and the one of GR, ${\lambda }_{\mathrm{GR}}$. This is compared to the computation of the same quantity using the first term in the expansion in $m^2$ proposed by Vainshtein, $|\Delta {\lambda }_V|=(\sqrt{2}/3)(mR{)}^2\sqrt{R_S/R}$. Both functions are plotted vs $R/R_V$, for a source of radius $R_{\odot }={10}^{-3}{R}_V$ and a choice of parameters such that $a\equiv m\times R_V={10}^{-2}$. It can be seen that the numerical solution agrees very well with Vainshtein’s expression given by Eq. (46) for $R\ll R_V$. For $R\sim R_V$, the expansion in $m^2$ is, as expected, no longer a good approximation of the solution.

Figure 3

Plot of $-\nu \times a^{-4}$ vs $R/R_V$ (the $a^{-4}$ factor is included for convenience such that, in the DL, all plotted theories would exactly coincide) inside the source of radius $R_{\odot }={10}^{-3}{R}_V$, for three different values of $a\equiv m\times R_V$. Along with the numerical solution, the GR analytical solution for $\nu$ is plotted for each value of $a$. It can be seen that the numerical solutions perfectly agree with the GR behavior, while slightly differing from the DL expression for large $a$. This illustrates the fact that inside the source, the nonlinearities coming from the GR terms of Eq. (8), and which are not taken into account in the DL, are important. For small $a$, both the GR and the numerical solutions agree with the DL solution, which encodes for most of the physics, in agreement with the fact that the DL corresponds to the limit $a\to 0$.

Figure 4

Plot of the functions $P/\rho$ and $P_{\mathrm{GR}}/\rho$ vs $R/R_V$, for a source of radius $R_{\odot }={10}^{-2}{R}_V$, of pressure $P$, and of constant density $\rho$, and for a parameter choice such that $a\equiv m\times R_V={10}^{-2}$. The function $P/\rho$ is found numerically, while $P_{\mathrm{GR}}/\rho$ is the pressure computed in GR and is given by the analytical expression (60). The two curves are indistinguishable from each other, illustrating the fact that Vainshtein’s conjecture is valid inside the star.

Figure 5

Plot of $a^{-2}\mu$ vs $R/R_V$ (the $a^{-2}$ factor is included for convenience such that, in the DL, all plotted theories would exactly coincide) for a source of radius $R_{\odot }={10}^{-3}{R}_V$ for three different values of $a\equiv m\times R_V=0.005$, 0.05, and 0.1 as well as in the DL case (which corresponds to $a\to 0$). The three DL regimes of Table  are clearly distinguishable: for $R<R_{\odot }$, $\mu \sim \mathrm{const}$; for $R_{\odot }<R\ll R_V$, $\mu \sim \sqrt{(8R_V)/(9R)}$; for $R_V\ll R\ll m^{-1}$, $\mu \sim 2R_V^3/(3R^3)$. For $R\gg m^{-1}$, we observe the Yukawa decay of the linear solution (24). At the resolution of the picture, the thee numerical solutions are hard to distinguish for distances $R$ smaller than $m^{-1}$. The next figure shows a zoom of this figure at small distances (namely inside the source) where the differences between the three solutions are easily seen.

Figure 6

Plot of $a^{-2}\mu$ vs $R/R_V$ (the $a^{-2}$ factor is included for convenience such that, in the DL, all plotted theories would exactly coincide) for a source of radius $R_{\odot }={10}^{-3}{R}_V$ for three different values of $a\equiv m\times R_V=0.005$, 0.05, and 0.1 as well as in the DL case. The picture is a zoom of Fig. 5, inside the star. One can check that even if the three curves for $a=0.005$, 0.05, and 0.1 are close to each other and to the DL solution, they differ inside the star due to nonlinearities not captured by the DL. One notes that the full solution goes toward the DL solution when $a\equiv R_V\times m$ tends toward 0.

Figure 7

Plot of the function $\mu$ (${\mu }_N$ as given by the numerical integration) vs $R/R_V$, for a source of radius $R_{\odot }={10}^{-2}{R}_V$ and of constant density $\rho$, and for a choice of parameters such that $a\equiv m\times R_V={10}^{-2}$. The plot has been obtained using an direct integration à la Runge-Kutta. We also added the plot of series ${\mu }_S={\mu }_0^{(x)}+{\mu }_1^{(x)}{x}^2+{\mu }_2^{(x)}{x}^4$, which was introduced in the main text as an expansion inside the star: the two curves are indistinguishable from each other.

Figure 8

Plot of the function $a^{-2}\mu$ (the $a^{-2}$ is added to make the comparison between $\mu$ and its DL behavior easier), vs $R/R_V$, for a source of radius $R_{\odot }={10}^{-3}{R}_V$, of constant density $\rho$ and a choice of parameters such that $a\equiv m\times R_V={10}^{-1}$. The three curves correspond, respectively, to the weak-field solution of Eq. (58), to the solution of the fully nonlinear theory and to the DL solution. One can see that the weak-field model allows one to recover the general behavior of the full solution, except inside the star (as seen in the panel showing a zoom of the plots for small $R$) where it is very close to the DL, and thus misses some nonlinearities.

Figure 9

Plot of the numerical solution for $\delta \mu (\zeta )$ for the choice of parameters $b=0.1$ and $\delta \mu (\zeta =10)=1$. We also plotted the leading behavior $\delta \mu (\zeta )=\exp (-3\sqrt{\frac{\zeta }{b}}\log \zeta )$ of Eq. (37), and the first series expansion correction obtained from Eqs. (c5, c13). In terms of the $\zeta$ variable, this first order series expansion expression reads $\delta \mu (\zeta )=\frac{1}{(\log \zeta {)}^{5/4}(1+\log \zeta {)}^{1/4}}\exp (\frac{\log \zeta }{4}-\frac{1}{\sqrt{b}}(3\log \zeta -\frac{15}{2}+\frac{9}{8\log \zeta })\sqrt{\zeta })$. One can see that while the leading behavior already encodes the main behavior [exponentially decreasing $\delta \mu (\zeta )$], the first correction coming from the series expansion improves greatly the agreement with the numerical solution, confirming our analytical analysis of the system (c2).