Constraints on Galileons from the positions of supermassive black holes

Galileons are scalar field theories which obey the Galileon symmetry $\phi \to \phi +b+c_{\mu }{x}^{\mu }$ and are capable of self-acceleration if they have an inverted sign for the kinetic term. These theories violate the strong equivalence principle, such that black holes (BHs) do not couple to the Galileon field, whereas nonrelativistic objects experience a fifth force with strength $\mathrm{\Delta }G/G_{\mathrm{N}}$ relative to gravity. For galaxies falling down a gradient in the Galileon field, this results in an offset between the center of the galaxy and its host supermassive BH. We reconstruct the local gravitational and Galileon fields through a suite of constrained N-body simulations (which we dub CSiBORG) and develop a Monte Carlo-based forward model for these offsets on a galaxy-by-galaxy basis. Using the measured offset between the optical center and active galactic nucleus of 1916 galaxies from the literature, propagating uncertainties in the input quantities and marginalizing over an empirical noise model describing astrophysical and observational noise, we constrain the Galileon coupling to be $\mathrm{\Delta }G/G_{\mathrm{N}}<0.16$ at $1\sigma$ confidence for Galileons with crossover scale $r_{\mathrm{C}}\gtrsim H_0^{-1}$.

Figure 1

Observed BH–galaxy offset as a function of redshift for the galaxies used in this work. The upper panel gives the physical offsets and the low panel gives the angular offsets. For redshift we use zdist from the NSA.

Figure 2

Corner plot of the constraints on the strength of the coupling to the Galileon field, $\mathrm{\Delta }G/G_{\mathrm{N}}$, and the noise parameters at $r_{\mathrm{V}}=100\mathrm{Mpc}$. The contours show the 1 and $2\sigma$ confidence intervals. Each dataset is consistent with $\mathrm{\Delta }G/G_{\mathrm{N}}=0$, with $\mathrm{\Delta }G/G_{\mathrm{N}}>0.16$ ruled out at $1\sigma$ confidence when we combine the datasets. Note that while $\mathrm{\Delta }G/G_{\mathrm{N}}$ is assumed universal, ${\sigma }_{\mathrm{obs}}$, $\nu$ and $f$ are sample-specific.

Figure 3

$1\sigma$ constraint on $\mathrm{\Delta }G/G_{\mathrm{N}}$ as a function of average Vainshtein radius, $r_{\mathrm{V}}$. We expect $r_{\mathrm{V}}\sim 10\mathrm{Mpc}$ for crossover scales $\sim H_0^{-1}$. $r_{\mathrm{V}}>L_{\mathrm{eq}}$ corresponds to crossover scales much larger than the observable universe (see Fig. 5).

Figure 4

Constraints on $\mathrm{\Delta }G/G_{\mathrm{N}}$ at $r_{\mathrm{V}}=100\mathrm{Mpc}$ at $1\sigma$ confidence using the OF13 data. Left: a power law halo density profile of index $n$ is assumed. To determine the density profile, we use $\mathtt{N}\_\mathtt{AM}=200$ mock catalogues obtained through abundance matching. We re-run the analysis at $n=0.5$ with $\mathtt{N}\_\mathtt{AM}=100$ and see that the constraint is unchanged. Right: using our fiducial density profile, we vary the box size, $L^{\prime }$, used to add in long wavelength modes when reconstructing the Galileon field. $L^{\prime }\sim 1\mathrm{Gpc}$ corresponds to no extra long wavelength modes. We also show the constraints from different noise models at $L^{\prime }\sim 6\mathrm{Gpc}$, and if we reduce the number of constrained simulations to 50.

Figure 5

Constraints on the coupling of a cubic Galileon to matter, $\alpha$, as a function of the crossover scale, $r_{\mathrm{C}}$, from lunar laser ranging (LLR) [108], the BH at the center of M87 [29, 30] and this work. The shaded regions are excluded at $1\sigma$ confidence. We also plot the $\alpha -r_{\mathrm{C}}$ curves for the normal and self-accelerating branches of the DGP model [Eq. (7)].

Figure 6

Constraints on the Horndeski parameters $c_2$ and $c_3$ for the cubic Galileon tracker solution $\dot{\overline{\phi }}H=\xi H_0^2$ at $1\sigma$ confidence. The blue region corresponds to $\xi >0$ and the orange region is for $\xi <0$. Lines of constant $r_{\mathrm{V}}$ are given by the curves ${\xi }^3{c}_3=\text{constant}$, where $\xi$ is proportional to $\alpha$ along these curves. Probing smaller $r_{\mathrm{V}}$ in future work will push the constraint to the upper right, as shown by the arrows.