Asteroseismic sensitivity to internal rotation along the red-giant branch

Transport of angular momentum in stellar interiors is currently not well understood. Asteroseismology can provide us with estimates of internal rotation of stars and thereby advances our understanding of angular momentum transport. We can measure core-rotation rates in red-giant stars and we can place upper bounds on surface-rotation rates using measurements of dipole ($l=1$) modes. Here, we aim to determine the theoretical sensitivity of modes of different spherical degree towards the surface rotation. Additionally, we aim to identify modes that can potentially add sensitivity at intermediate radii. We used asteroseismic rotational inversions to probe the internal stellar rotation profiles in red-giant models from the base of the red-giant branch up to the luminosity bump. We used the inversion method of multiplicative optimally localised averages (MOLA) to assess how well internal and surface rotation rates can be recovered from different mode sets and different synthetic rotation profiles. We confirm that dipole mixed modes are sufficient to set constraints on the average core-rotation rates in red giants. However, surface-rotation rates estimated with only dipole mixed modes are contaminated by the core rotation. We show that the sensitivity to the surface rotation decreases from the base of the red-giant branch until it reaches a minimum at 0.6-0.8$L_\text{bump}$ due to a glitch in the buoyancy frequency. Thereafter a narrow range of increased surface sensitivity just below the bump luminosity exists. Quadrupole and octopole modes have more sensitivity in the outer parts of the star. If observed, quadrupole and octopole modes enable us to distinguish between differential and solid body rotation in the convection zone. To obtain accurate estimates of rotation rates at intermediate radii, acoustic oscillation modes with a spherical degree of $l\approx10$ are needed.