[Paper Review] Weak gravitational lensing in brane-worlds

L. Gergely, Barbara Darázs|arXiv (Cornell University)|Feb 20, 2006

Galaxies: Formation, Evolution, Phenomena3 citations

TL;DR

This paper derives the second-order weak gravitational lensing deflection angle for brane-world black holes with mass $m$ and tidal charge $q$, using perturbation theory in $\varepsilon = m/b$ and $\eta = q/b^2$. The key result is a deflection formula that includes second-order corrections in both mass and tidal charge, revealing that negative tidal charges significantly enhance lensing, potentially explaining dark matter-like effects.

ABSTRACT

We derive the deflection angle of light rays caused by a brane black hole with mass m and tidal charge q in the weak lensing approach, up to the second order in perturbation theory. We point out when the newly derived second order contributions become important.

Motivation & Objective

• To extend weak gravitational lensing theory in brane-world scenarios beyond first-order approximations.
• To investigate the influence of tidal charge $q$—a non-electromagnetic, bulk-induced parameter—on light deflection, independent of mass.
• To determine when second-order contributions in $m$ and $q$ become significant in lensing observables.
• To express the deflection angle in terms of the minimal approach distance $r_{\text{min}}$, enabling observational comparison.
• To explore the astrophysical implications of negative tidal charges, including potential demagnification and enhanced lensing effects.

Proposed method

• Formulate the null geodesic equations for the brane-world metric $ds^2 = -f(r)dt^2 + f^{-1}(r)dr^2 + r^2(d\theta^2 + \sin^2\theta d\varphi^2)$ with $f(r) = 1 - 2m/r + q/r^2$.
• Introduce the radial variable $u = 1/r$ and derive the differential equation $u'' + u = 3mu^2 - 2qu^3$ for the photon trajectory.
• Perform a perturbative expansion in small parameters $\varepsilon = m/b$ and $\eta = q/b^2$, solving up to second order.
• Solve the resulting linearized equations for $u_1, v_1, u_2, v_2, w_2$ using trigonometric and polynomial terms in $\varphi$.
• Extract the deflection angle $\delta\varphi$ from the asymptotic behavior of the solution, including terms up to $\mathcal{O}(\varepsilon^2, \eta^2, \varepsilon\eta)$.
• Re-express the deflection angle in terms of the minimal approach distance $r_{\text{min}}$ via inversion of the relation $r_{\text{min}} = b(1 - \varepsilon + \frac{1}{2}\eta + \cdots)$.

Experimental results

Research questions

• RQ1 How do second-order corrections in mass and tidal charge affect the weak lensing deflection angle in brane-world models?
• RQ2 Under what conditions do second-order terms dominate over first-order terms in the deflection angle?
• RQ3 Can negative tidal charges lead to observable lensing effects that differ qualitatively from standard general relativity?
• RQ4 What is the deflection angle expressed in terms of the physical minimal approach distance $r_{\text{min}}$ rather than the impact parameter $b$?
• RQ5 Could tidal charges with $q > 0$ or $q < 0$ explain dark matter-like lensing without invoking unseen mass?

Key findings

• The second-order deflection angle is given by $\delta\varphi = \frac{4m}{r_{\text{min}}} - \frac{3\pi q}{4r_{\text{min}}^2} + \frac{(15\pi - 16)m^2}{4r_{\text{min}}^2} + \frac{57\pi q^2}{64r_{\text{min}}^4} + \frac{(3\pi - 28)mq}{2r_{\text{min}}^3}$, valid to second order in $m/r_{\text{min}}$ and $q/r_{\text{min}}^2$.
• When $16m r_{\text{min}} = 3\pi q$, the first-order deflection terms cancel, making second-order terms the leading contribution to lensing.
• For $16m r_{\text{min}} < 3\pi q$, the first-order deflection becomes negative, implying demagnification of distant sources—unlike in standard GR.
• Negative tidal charges ($q < 0$) significantly enhance the deflection angle, potentially explaining strong lensing effects attributed to dark matter.
• The deflection angle formula includes all second-order contributions: $\varepsilon^2$, $\eta^2$, and $\varepsilon\eta$, which were previously neglected in first-order treatments.
• The result generalizes previous Reissner-Nordström lensing formulas by allowing $q$ to be independent of $m^2$, as in brane-world models.