Long-lived quasinormal modes and echoes in the Einstein–Gauss–Bonnet–Proca theory

Abstract We study quasinormal modes and time-domain profiles of a massive scalar field in the background of black holes arising in Einstein–Gauss–Bonnet–Proca gravity. The black holes in this theory possess primary Proca hair , which modifies the effective potential and gives rise to distinctive dynamical phenomena. Using three complementary numerical techniques – the WKB method with Padé resummation, time-domain integration, and the Frobenius (Leaver) method – we obtain accurate spectra of quasinormal frequencies. Our analysis shows that increasing the scalar-field mass leads to arbitrarily long-lived states, known as quasi-resonances, a behavior shared by both the fundamental mode and the first overtone. The real part of the quasinormal mode decreases for the first and higher overtones and starting from the second overtone it reaches zero at some critical value of mass of the field. When the effective potential develops an additional peak, the late-time signal exhibits a sequence of echoes. Furthermore, the time-domain evolution reveals distinct regimes of intermediate power-law tails $$\sim t^{-4/3-\ell }\sin (A(\mu )t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> <mml:mo>-</mml:mo> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:msup> <mml:mo>sin</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and a universal asymptotic tails $$\sim t^{-5/6}\sin (\mu t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>5</mml:mn> <mml:mo>/</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msup> <mml:mo>sin</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .