qnm.schwarzschild.approx¶
Analytic approximations for Schwarzschild QNMs.
The approximations implemented in this module can be used as initial guesses when numerically searching for QNM frequencies.
Functions
Schw_QNM_estimate (s, l, n) |
Give either large_overtone_expansion() or dolan_ottewill_expansion() . |
dolan_ottewill_expansion (s, l, n) |
High l asymptotic expansion of Schwarzschild QNM frequency. |
large_overtone_expansion (s, l, n) |
The eikonal approximation for QNMs, valid for l >> n >> 1 . |
-
qnm.schwarzschild.approx.
Schw_QNM_estimate
(s, l, n)[source]¶ Give either
large_overtone_expansion()
ordolan_ottewill_expansion()
.The Dolan-Ottewill expansion includes terms with higher powers of the overtone number n, so it breaks down faster at high n.
Parameters: - s: int
Spin weight of the field of interest.
- l: int
Multipole number of interest.
- [The m parameter is omitted because this is just for Schwarzschild.]
- n: int
Overtone number of interest.
Returns: - complex
Analytic approximation of QNM of interest.
-
qnm.schwarzschild.approx.
dolan_ottewill_expansion
(s, l, n)[source]¶ High l asymptotic expansion of Schwarzschild QNM frequency.
The result of [1] is an expansion in inverse powers of L = (l+1/2). Their paper stated this series out to L^{-4}, which is how many terms are implemented here. The coefficients in this series are themselves positive powers of N = (n+1/2). This means the expansion breaks down for large N.
Parameters: - s: int
Spin weight of the field of interest.
- l: int
Multipole number of interest.
- [The m parameter is omitted because this is just for Schwarzschild.]
- n: int
Overtone number of interest.
Returns: - complex
Analytic approximation of QNM of interest.
References
[1] SR Dolan, AC Ottewill, “On an expansion method for black hole quasinormal modes and Regge poles,” CQG 26 225003 (2009), https://arxiv.org/abs/0908.0329 .
-
qnm.schwarzschild.approx.
large_overtone_expansion
(s, l, n)[source]¶ The eikonal approximation for QNMs, valid for l >> n >> 1 .
This is just the first two terms of the series in
dolan_ottewill_expansion()
. The earliest work I know deriving this result is [1] but there may be others. In the eikonal approximation, valid when \(l \gg n \gg 1\), the QNM frequency is\[\sqrt{27} M \omega \approx (l+\frac{1}{2}) - i (n+\frac{1}{2}) .\]Parameters: - s: int
Spin weight of the field of interest.
- l: int
Multipole number of interest.
- [The m parameter is omitted because this is just for Schwarzschild.]
- n: int
Overtone number of interest.
Returns: - complex
Analytic approximation of QNM of interest.
References
[1] V Ferrari, B Mashhoon, “New approach to the quasinormal modes of a black hole,” Phys. Rev. D 30, 295 (1984)