pttools.ssmtools.spectrum

Functions for computing GW power spectra

Classes

class pttools.ssmtools.spectrum.NucType(value, names=_not_given, *values, module=None, qualname=None, type=None, start=1, boundary=None)

Bases: str, Enum

Nucleation type

EXPONENTIAL = 'exponential'

SIMULTANEOUS = 'simultaneous'

class pttools.ssmtools.spectrum.SSMSpectrum(bubble, y=None, z_st_thresh=const.Z_ST_THRESH, nuc_type=DEFAULT_NUC_TYPE, nt=const.NTDEFAULT, n_z_lookup=const.N_Z_LOOKUP_DEFAULT, r_star=None, lifetime_multiplier=1, compute=True, label_latex=None, label_unicode=None)

Bases: object

Gravitational wave simulation object

Parameters:

bubble (Bubble)
y (ndarray)
z_st_thresh (float)
nuc_type (NucType)
nt (int)
n_z_lookup (int)
r_star (float)
lifetime_multiplier (float)
compute (bool)
label_latex (str)
label_unicode (str)

compute()

cs: float | None: $c_s({T}_\text{gw})$

label_latex()

Return type:: str

label_unicode()

Return type:: str

plot(fig=None, ax=None, path=None, **kwargs)

Parameters:

fig (Figure)
ax (Axes)
path (str)

Return type:

FigAndAxes

plot_multi(fig=None, path=None, **kwargs)

Parameters:

fig (Figure)
path (str)

Return type:

Figure

plot_spec_den_gw(fig=None, ax=None, path=None, **kwargs)

Parameters:

fig (Figure)
ax (Axes)
path (str)

Return type:

FigAndAxes

plot_spec_den_v(fig=None, ax=None, path=None, **kwargs)

Parameters:

fig (Figure)
ax (Axes)
path (str)

Return type:

FigAndAxes

plot_v(fig=None, ax=None, path=None, **kwargs)

Parameters:

fig (Figure)
ax (Axes)
path (str)

Return type:

FigAndAxes

pow_gw: ndarray | None: $\mathcal{P}_{\text{gw}}(k)$

pow_v: ndarray | None: $\mathcal{P}_{\tilde{v}}(q)$

property source_lifetime_factor: float

Source lifetime correction factor

\[\frac{1}{1 + 2\nu} \left(1 - \left(1 + \frac{\Delta \eta}{\eta_*} \right) \right)^{-1-2\nu}\]

where

\[\frac{\Delta \eta}{\eta_*} = \lambda \frac{2 r_*}{\sqrt{K}}\]

Giombi et al., 2024, eq. 3.13

spec_den_gw: ndarray | None: ???

spec_den_v: ndarray | None: $P_v(q)$

Functions

pttools.ssmtools.spectrum.convert_params(params)

Convert the physical parameters from a list to a tuple if necessary.

Parameters:: params (Tuple[float, float] | Tuple[float, float, str, Tuple[float, ...]])
Return type:: Tuple[float, float] | Tuple[float, float, str, Tuple[float, …]]

pttools.ssmtools.spectrum.gen_lookup(y, cs, n_z_lookup=const.N_Z_LOOKUP_DEFAULT, eps=0.)

Parameters:

y (ndarray) – Input array
cs (float) – Speed of sound $c_s$
n_z_lookup (int) – Number of points for the generated lookup table
eps (float) – Seems to be needed for max(z) <= 100. E.g. 1e-8. Why?

Returns:

Generated lookup array for z

Return type:

ndarray

pttools.ssmtools.spectrum.lookup_limits(y, cs, eps=0.)

Defined on p. 12 between eq. 3.44 and 3.45

Parameters:

y (ndarray)
cs (float)
eps (float)

Return type:

Tuple[float, float]

pttools.ssmtools.spectrum.nu(T, nuc_type=NucType.SIMULTANEOUS, a=1.)

Bubble lifetime distribution function

Hindmarsh et al., 2019 eq. 4.27 & 4.32

Parameters:

T (float | float64 | ndarray) – dimensionless time
nuc_type (NucType) – nucleation type, simultaneous or exponential
a (float)

Returns:

bubble lifetime distribution $\nu$

Return type:

float | float64 | ndarray

pttools.ssmtools.spectrum.parse_params(params)

Parse physical parameters from the tuple.

Parameters:: params (Tuple[float, float] | Tuple[float, float, str, Tuple[float, ...]]) – tuple of physical parameters
Returns:: $v_\text{wall}, \alpha$, nucleation type, nucleation arguments
Return type:: Tuple[float, float, NucType, Tuple[float, …]]

pttools.ssmtools.spectrum.pow_spec(z, spec_den)

Power spectrum from spectral density at dimensionless wavenumber z.

Hindmarsh et al., 2019 eq. 4.18, but without the factor of 2.

Parameters:

z (float | float64 | ndarray) – dimensionless wavenumber $z$
spec_den (float | float64 | ndarray) – spectral density

Returns:

power spectrum

Return type:

float | float64 | ndarray

pttools.ssmtools.spectrum.power_gw_scaled_bag(z, params, npt=const.NPTDEFAULT, filename=None, skip=1, method=ssm.Method.E_CONSERVING, de_method=ssm.DE_Method.STANDARD, z_st_thresh=const.Z_ST_THRESH)

Scaled GW power spectrum at array of z = kR* values, where R* is mean bubble centre separation and k is comoving wavenumber. To convert to predicted spectrum, multiply by $(H_n R_*)(H_n au_v)$, where $H_n$ is the Hubble rate at the nucleation time, and $ au_v$ is the lifetime of the shear stress source.

Physical parameters

vw = params[0] scalar (required) [0 < vw < 1]
alpha = params[1] scalar (required) [0 < alpha_n < alpha_n_max(v_w)]
nuc_type = params[2] string (optional) [exponential* | simultaneous]
nuc_args = params[3] tuple (optional) default (1,)

Steps:

Getting velocity field spectral density
Geeting gw spectral density
turning SD into power

Parameters:

z (ndarray) – array $z = qR_*$
params (Tuple[float, float] | Tuple[float, float, str, Tuple[float, ...]]) – physical parameters, see the description above
npt (ndarray | Tuple[int, int, int]) – number of points
filename (str) – path to load A2 values from
skip (int)
method (Method)
de_method (DE_Method)
z_st_thresh (float)

Returns:

scaled GW power spectrum

Return type:

ndarray

pttools.ssmtools.spectrum.power_v_bag(z, params, npt=const.NPTDEFAULT, filename=None, skip=1, method=ssm.Method.E_CONSERVING, de_method=ssm.DE_Method.STANDARD, z_st_thresh=const.Z_ST_THRESH)

Power spectrum of the velocity field in the Sound Shell Model.

vw = params[0] scalar
alpha = params[1] scalar
nuc_type = params[2] string [exponential* | simultaneous]
nuc_args = params[3] tuple default (1,)

Parameters:

z (ndarray) – array $z = qR_*$
params (Tuple[float, float] | Tuple[float, float, str, Tuple[float, ...]]) – physical parameters, see the description above
npt (ndarray | Tuple[int, int, int]) – number of points
filename (str) – path to load A2 values from
z_st_thresh (float) – not used
skip (int)
method (Method)
de_method (DE_Method)

Returns:

power spectrum of the velocity field

Return type:

ndarray

pttools.ssmtools.spectrum.spec_den_gw_scaled(z_lookup, P_v_lookup, y=None, cs=const.CS0, Gamma=const.GAMMA, source_lifetime_factor=1., nz_int=None)

Spectral density of scaled gravitational wave power

Parameters:

z_lookup (ndarray) – Lookup table for the $z = qL_f$ values corresponding to P_v_lookup
P_v_lookup (ndarray) – $\bar{U}_f^2 \tilde{P}_v (z)$, a lookup table for the spectral density of the Fourier transform of the velocity field, not the spectral density of plane wave coefficients, which is lower by a factor of 2.
y (ndarray) – $y = kL_f = kR*$ corresponding to z_lookup. If not given, will be created from z_lookup.
cs (float) – Speed of sound (in the broken phase after the phase transition)
Gamma (float) – Mean adiabatic index $\Gamma = \frac{\bar{w}}{\bar{e}}$
source_lifetime_factor (float)
nz_int (int)

Returns:

$\hat{\mathcal{P}}$ Eq. 3.33 of Chloe’s thesis, which should be ($3\Gamma \bar{U}_f$) Eq. 3.47 Eq. 3.46 converted to the spectral density and divided by (H L_f)

Return type:

Tuple[ndarray, ndarray] | callable | CPUDispatcher

The factor of 3 comes from the Friedmann equation 3H^2/(8pi G)

pttools.ssmtools.spectrum.spec_den_v(bub, z, a, nuc_type, nt=const.NPTDEFAULT[1], z_st_thresh=const.Z_ST_THRESH, cs=None, return_a2=False)

The full spectral density of the velocity field

This is twice the spectral density of the plane wave components of the velocity field

Returns:

$P_{\tilde{v}} = 2 * P_v(q)$ of Hindmarsh et al., 2019 eq. 4.17

Parameters:

bub (Bubble)
z (ndarray)
a (float)
nuc_type (NucType)
nt (int)
z_st_thresh (float)
cs (float)
return_a2 (bool)

pttools.ssmtools.spectrum.spec_den_v_bag(z, params, npt=const.NPTDEFAULT, filename=None, skip=1, method=ssm.Method.E_CONSERVING, de_method=ssm.DE_Method.STANDARD, z_st_thresh=const.Z_ST_THRESH)

Get dimensionless velocity spectral density $\bar{P}_v$.

Gets fluid velocity profile from bubble toolbox or from file if specified. Convolves 1-bubble Fourier transform $|A(q T)|^2$ with bubble wall lifetime distribution $\nu(T \beta)$ specified by “nuc_type” and “nuc_args”.

Parameters:

z (ndarray) – array $z = qR_*$
params (Tuple[float, float] | Tuple[float, float, str, Tuple[float, ...]]) – tuple of vw (scalar), alpha (scalar), nuc_type (string [exponential* | simultaneous]), nuc_args (tuple, default (1,))
npt (ndarray | Tuple[int, int, int]) – number of points
filename (str)
skip (int)
method (Method)
de_method (DE_Method)

Returns:

dimensionless velocity spectral density $\bar{P}_v$

Attributes

pttools.ssmtools.spectrum.DEFAULT_NUC_TYPE = NucType.EXPONENTIAL: Default nucleation type