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)

Give either large_overtone_expansion() or dolan_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)

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)

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)