Finite size effects in DBI and Born-Infeld for screened spherically symmetric objects

TL;DR

This work analyzes finite-size effects on the linear response of spherically symmetric objects in Born-Infeld electromagnetism and Dirac-Born-Infeld scalar theories. Building on point-like results where odd multipoles above the dipole have vanishing response, the authors compute the background screening profile for a finite sphere and solve polar and axial perturbations in exterior and interior regions, using hypergeometric solutions and surface matching. They show that finite size induces nonzero corrections to previously vanishing multipoles, yielding a hierarchy between even and odd multipoles that depends on the dimensionless radius $x_0=r_0/r_s$, with polar corrections scaling as $\mathcal{O}(x_0)$ and axial corrections as $\mathcal{O}(x_0^3)$, and they connect these effects through ladder operator structures that relate multipoles. The results suggest that measurements of far-field potentials can simultaneously diagnose the screening scale and the object’s size, with potential implications for dark sector models employing BI/DBI-type interactions.

Abstract

We study finite size effects on the linear response of spherically symmetric objects in Born-Infeld (BI) electromagnetism and Dirac-Born-Infeld (DBI) scalar field theories. Previous works show that the linear response coefficients for a point-like source vanish for odd multipoles above the dipole, a feature that resembles the vanishing of Love numbers for black holes. This work goes beyond the point-like idealisation and considers a sphere of finite radius. We find that the vanishing of the linear response coefficients ceases as they acquire a correction due to the finite size of the object. This introduces a hierarchy between the even and odd multipoles of the response coefficients determined by the separation of scales between the radius of the sphere and the screening scale of non-linearities. From a phenomenological viewpoint, the hierarchy between the odd and even multipoles would give access to the screening scale and the object's radius by measuring the behaviour of the potentials at infinity.

Figures (3)

• Figure 1: Electric field profile for a sphere in BI and DBI (solid-orange). The case of standard linear theories, i.e., Maxwell electromagnetism and a canonical massless scalar field, is also shown in dashed line for comparison. We have indicated the radius of the sphere (that we choose as $r_0=10^{-1}r_{\text{s}}$) with a vertical line, and the screened shell is shown in light green shading. Far from the sphere we observe the standard $1/r^2$ decay. Inside the screened shell, the field saturates to $\Lambda^2$ and is suppressed with respect to the linear theories. In this screened shell, the derivative of the non-linear function $\mathcal{K}_Y$ (yellow line) coincides with the field of the linear theory, as expected because $1/\mathcal{K}_Y$ is precisely the screening factor and the background field is constant in that region. Finally, the screening ceases in the inner unscreened core and we recover again the behaviour of the linear theories.
• Figure 2: In the left panel we show the solutions for a few multipoles normalised to the amplitude of the external perturbation for a sphere of radius $x_0=10^{-1}$. We show the exact numerical solution (solid) together with the analytical solution (dashed) obtained by matching at the surface of the sphere taking into account the discontinuity in the first derivative. We can clearly observe how the perturbation grows as it transits through the screened shell (shaded region) and it has the usual behaviour $\sim x^{\ell+1}$ in the asymptotic region and the unscreened core. In the right panel we plot the linear response coefficients where we can see the hierarchy between odd and even multipoles explained in the main text.
• Figure 3: Same as in Fig. \ref{['Fig:Axial']} for the axial perturbations. In the right panel we can see again a hierarchy between even and odd multipoles. As explained in the main text, this hierarchy is more pronounced for the axial perturbations since the correction due to finite size effects scales as $x_0^3$ instead as the correction $x_0$ obtained for the polar case.