Waveforms#
WaveformModes
class#
Bases: WaveformMixin
, TimeSeries
Array-like object representing time-series data of SWSH modes
We generally assume that the data represent mode weights in expansions of functions in terms of spin-weighted spherical harmonics (where the standard spherical harmonics just happen to have spin weight 0).
This object is based on the TimeSeries object, but has many additional methods for manipulating modes.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
input_array |
(..., N, ..., M, ...) array_like
|
Input data representing the dependent variable, in any form that can be
converted to a numpy array. This includes scalars, lists, lists of tuples,
tuples, tuples of tuples, tuples of lists, and numpy ndarrays. It can have
an arbitrary number of dimensions, but the length |
required |
time |
(N,) array_like
|
1-D array containing values of the independent variable. Values must be real, finite, and in strictly increasing order. |
required |
time_axis |
int
|
Axis along which |
required |
modes_axis |
int
|
Axis of the array along which the modes are stored. See Notes below. |
required |
ell_min |
int
|
Smallest value of ℓ stored in the data |
required |
ell_max |
int
|
Largest value of ℓ stored in the data |
required |
Notes
We assume that the modes at a given time, say, are stored in a flat array, in
order of increasing m
and ell
, with m
varying most rapidly — something
like the following:
[f(ell, m) for ell in range(ell_min, ell_max+1) for m in range(-ell,ell+1)]
The total size is implicitly ell_max * (ell_max + 2) - ell_min ** 2 + 1
.
The recommended way of retrieving individual modes is
h_22 = waveform[:, waveform.index(2,2)]
or equivalently,
h_22 = waveform.data[:, waveform.index(2,2)]
For backwards compatibility, it is also possible to retrieve individual modes in the same way as the old NRAR-format HDF5 files would be read, as in
h_22 = waveform["Y_l2_m2.dat"]
Note that "History.txt" may not contain anything but an empty string, because history is not retained in more recent data formats. Also note that — while not strictly a part of this class — the loaders that open waveform files will return a dict-like object when the extrapolation order is not specified. That object can also be accessed in a backwards-compatible way much like the root directory of the NRAR-format HDF5 files. For example:
with sxs.loadcontext("rhOverM_Asymptotic_GeometricUnits_CoM.h5") as f:
h_22 = f["Extrapolated_N2.dir/Y_l2_m2.dat"]
This code is identical to the equivalent code using h5py
except that the call
to h5py.File
is replaced with the call to sxs.loadcontext
. The .dat
datasets are reconstructed on the fly, but should be bitwise-identical to the
output from the HDF5 file whenever the underlying format is NRAR.
Source code in sxs/waveforms/waveform_modes.py
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|
LM
property
#
Array of (ell, m) values in the data
This array is just a flat array of [ell, m]
pairs. It is automatically
recomputed each time ell_min
or ell_max
is changed. Specifically, it is
np.array([
[ell, m]
for ell in range(self.ell_min, self.ell_max+1)
for m in range(-ell, ell+1)
])
abs
property
#
Absolute value of the data
Returns:
Name | Type | Description |
---|---|---|
absolute |
TimeSeries
|
Because the absolute values make no sense as mode weights, this is just a plain TimeSeries object. |
See Also
arg
angular_velocity
property
#
Angular velocity of waveform
This function calculates the angular velocity of a WaveformModes object from its modes — essentially, the angular velocity of the rotating frame in which the time dependence of the modes is minimized. This was introduced in Sec. II of "Angular velocity of gravitational radiation and the corotating frame" http://arxiv.org/abs/1302.2919.
It can be calculated in terms of the expectation values ⟨w|Lᵃ∂ₜ|w⟩ and ⟨w|LᵃLᵇ|w⟩ according to the relation
⟨w|LᵇLᵃ|w⟩ ωₐ = -⟨w|Lᵇ∂ₜ|w⟩
For each set of modes (e.g., at each instant of time), this is a simple linear equation in 3 dimensions to be solved for ω.
See Also
expectation_value_LL expectation_value_Ldt
arg
property
#
Complex phase angle of the data
Note that the result is not "unwrapped", meaning that there may be discontinuities as the phase approaches ±π.
Returns:
Name | Type | Description |
---|---|---|
phase |
TimeSeries
|
Values are in the interval (-π, π]. |
See Also
numpy.angle arg_unwrapped
arg_unwrapped
property
#
Complex phase angle of the data, unwrapped along the time axis
The result is "unwrapped", meaning that discontinuities as the phase approaches ±π are removed by adding an appropriate amount to all following data points.
Returns:
Name | Type | Description |
---|---|---|
phase |
TimeSeries
|
Values at the initial time are in the interval (-π, π], but may evolve to arbitrary real values. |
See Also
numpy.angle numpy.unwrap arg
bar
property
#
Return waveform modes of function representing conjugate of this function
N.B.: This property is different from the .conjugate
method; see below.
See Also
re : Return modes of function representing the real part of this function im : Return modes of function representing the imaginary part of this function
Notes
This property is different from the .conjugate
(or .conj
) method, in that
.conjugate
returns the conjugate of the mode weights of the function, whereas
this property returns the mode weights of the conjugate of the function. That
is, .conjugate
treats the data as a generic numpy array, and simply returns
the complex conjugate of the raw data without considering what the data
actually represents. This property treats the data as a function represented
by its mode weights.
The resulting function has the negative spin weight of the input function.
We have
conjugate(f){s, l, m} = (-1)**(s+m) * conjugate(f{-s, l, -m})
ell_max
property
#
Largest value of ℓ stored in the data
ell_min
property
#
Smallest value of ℓ stored in the data
eth_GHP
property
#
Spin-raising derivative operator defined by Geroch-Held-Penrose
The operator ð is defined in https://dx.doi.org/10.1063/1.1666410
See Also
eth : Related operator in the Newman-Penrose convention ethbar : Similar operator in the Newman-Penrose convention ethbar_GHP : Conjugate of this operator
Notes
We assume that the Ricci rotation coefficients satisfy β=β'=0, meaning that this operator equals the Newman-Penrose operator ð multiplied by 1/√2.
ethbar_GHP
property
#
Spin-lowering derivative operator defined by Geroch-Held-Penrose
The operator ð̄ is defined in https://dx.doi.org/10.1063/1.1666410
See Also
eth : Related operator in the Newman-Penrose convention ethbar : Similar operator in the Newman-Penrose convention eth_GHP : Conjugate of this operator
Notes
We assume that the Ricci rotation coefficients satisfy β=β'=0, meaning that this operator equals the Newman-Penrose operator ð̄ multiplied by 1/√2.
expectation_value_LL
property
#
Compute the matrix expectation value ⟨w|LᵃLᵇ|w⟩
Here, Lᵃ is the usual angular-momentum operator familiar from quantum physics, and
⟨w|LᵃLᵇ|w⟩ = ℜ{Σₗₘₙ w̄ˡᵐ ⟨l,m|LᵃLᵇ|l,n⟩ wˡⁿ}
This quantity is important for computing the angular velocity of a waveform. Its dominant eigenvector can also be used as a good choice for the axis of a decomposition into modes.
See Also
dominant_eigenvector_LL expectation_value_Ldt angular_velocity
expectation_value_Ldt
property
#
Compute the matrix expectation value ⟨w|Lᵃ∂ₜ|w⟩
Here, Lᵃ is the usual angular-momentum operator familiar from quantum physics, ∂ₜ is the partial derivative with respect to time, and
⟨w|Lᵃ∂ₜ|w⟩ = ℑ{Σₗₘₙ w̄ˡᵐ ⟨l,m|Lᵃ|l,n⟩ ∂ₜwˡⁿ}
This quantity is important for computing the angular velocity of a waveform.
See Also
expectation_value_LL angular_velocity
im
property
#
Return waveform modes of function representing imaginary part of this function
N.B.: This property is different from the .imag
method; see below.
See Also
re : Equivalent method for the real part bar : Return modes of function representing the conjugate of this function
Notes
This property is different from the .imag
method, in that .imag
returns the
imaginary part of the mode weights of the function, whereas this property
returns the mode weights of the imaginary part of the function. That is,
.imag
treats the data as a generic numpy array, and simply returns the
imaginary part of the raw data without considering what the data actually
represents. This property treats the data as a function represented by its
mode weights.
Note that this only makes sense for functions of spin weight zero; taking the imaginary part of functions with nonzero spin weight will depend too sensitively on the orientation of the coordinate system to make sense. Therefore, this property raises a ValueError for other spins.
The condition that a function f
be imaginary is that its modes satisfy
f{l, m} = -conjugate(f){l, m} = (-1)**(m+1) * conjugate(f{l, -m})
[Note that conjugate(f){l, m} != conjugate(f{l, m}).] As usual, we enforce that condition by essentially averaging the two modes:
f{l, m} = (f{l, m} + (-1)**(m+1) * conjugate(f{l, -m})) / 2
modes_axis
property
#
Axis of the array storing the various modes
See the documentation of this class for an explanation of what this means.
See Also
time_axis : Axis of the array along which time varies
n_modes
property
#
Total number of mode weights stored in the data
norm
property
#
Compute the L² norm of the waveform
Returns:
Name | Type | Description |
---|---|---|
n |
TimeSeries
|
|
See Also
numpy.linalg.norm numpy.take norm2 : squared version of this
Notes
The integral of the (squared) magnitude of the data equals the sum of the (squared) magnitude of the modes for orthonormal basis functions, meaning that the L² norm of the function equals the basic Euclidean norm of the modes. We assume that these modes are expanded in a band-limited but otherwise complete orthonormal basis.
re
property
#
Return waveform modes of function representing real part of this function
N.B.: This property is different from the .real
method; see below.
See Also
im : Equivalent method for the imaginary part bar : Return modes of function representing the conjugate of this function
Notes
This property is different from the .real
method, in that .real
returns the
real part of the mode weights of the function, whereas this property returns
the mode weights of the real part of the function. That is, .real
treats the
data as a generic numpy array, and simply returns the real part of the raw data
without considering what the data actually represents. This property treats
the data as a function represented by its mode weights.
Note that this only makes sense for functions of spin weight zero; taking the real part of functions with nonzero spin weight will depend too sensitively on the orientation of the coordinate system to make sense. Therefore, this property raises a ValueError for other spins.
The condition that a function f
be real is that its modes satisfy
f{l, m} = conjugate(f){l, m} = (-1)**(m) * conjugate(f{l, -m})
[Note that conjugate(f){l, m} != conjugate(f{l, m}).] As usual, we enforce that condition by essentially averaging the two modes:
f{l, m} = (f{l, m} + (-1)**m * conjugate(f{l, -m})) / 2
boost(v⃗, ell_max)
#
Find modes of waveform boosted by velocity v⃗
Implements Equation (21) of arxiv.org/abs/1509.00862
Parameters:
Name | Type | Description | Default |
---|---|---|---|
v |
Three-vector representing the velocity of the boosted frame relative to the inertial frame, in units where the speed of light is 1 |
required | |
ell_max |
int
|
Maximum value of |
required |
Returns:
Name | Type | Description |
---|---|---|
wprime |
WaveformModes
|
Modes of waveform measured in boosted frame or of modes from boosted source
measured in original frame. This should have the same properties as the
input waveform, except with (1) different time data [see Notes, below], (2)
a minimum |
Notes
Due to the nature of the transformation, some of the information in the input waveform must be discarded, because it corresponds to slices of the output waveform that are not completely represented in the input. Thus, the times of the output waveform will not just be the Lorentz-transformed times of the input waveform.
Depending on the magnitude β=|v⃗|, a very large value of ell_max
may be
needed. The dominant factor is the translation that builds up over time:
β*T
, where T
is the largest time found in the waveform. For example, if
β*T ≈ 1000M, we might need ell_max=64
to maintain a comparable accuracy as in
the input data.
Because of the β*T
effects, it is usually best to set t=0 at the merger time
— best approximated as self.max_norm_time()
. The largest translation is then
found early in the waveform, when the waveform is changing slowly.
Source code in sxs/waveforms/waveform_modes.py
convert_from_conjugate_pairs()
#
Convert modes from conjugate-pair format in place
This function reverses the effects of convert_to_conjugate_pairs
. See that
function's docstring for details.
Source code in sxs/waveforms/waveform_modes.py
convert_to_conjugate_pairs()
#
Convert modes to conjugate-pair format in place
This function alters this object's modes to store the sum and difference of
pairs with opposite m
values. If we denote the modes f[l, m]
, then we
define
s[l, m] = (f[l, m] + f̄[l, -m]) / √2
d[l, m] = (f[l, m] - f̄[l, -m]) / √2
For m<0 we replace the mode data with d[l, -m]
, for m=0 we do nothing, and
for m>0 we replace the mode data with s[l, m]
. That is, the mode data on
output look like this:
[d[2, 2], d[2, 1], f[2, 0], s[2, 1], s[2, 2], d[3, 3], d[3, 2], ...]
The factor of √2 is chosen so that the norm (sum of the magnitudes squared) at each time for this data is the same as it is for the original data.
Source code in sxs/waveforms/waveform_modes.py
corotating_frame(R0=quaternionic.one, tolerance=1e-12, z_alignment_region=None, return_omega=False)
#
Return rotor taking current mode frame into corotating frame
Parameters:
Name | Type | Description | Default |
---|---|---|---|
R0 |
quaternionic
|
Value of the output rotation at the first output instant; defaults to 1 |
one
|
tolerance |
float
|
Absolute tolerance used in integration; defaults to 1e-12 |
1e-12
|
z_alignment_region |
None or 2-tuple of floats
|
If not None, the dominant eigenvector of the |
None
|
return_omega |
If True, return a 2-tuple consisting of the frame (the usual returned
object) and the angular-velocity data. That is frequently also needed, so
this is just a more efficient way of getting the data. Default is |
False
|
Notes
Essentially, this function evaluates the angular velocity of the waveform, and then integrates it to find the corotating frame itself. This frame is defined to be the frame in which the time-dependence of the waveform is minimized — at least, to the extent possible with a time-dependent rotation.
That frame is only unique up to a single overall rotation, which can be
specified as the optional R0
argument to this function. If it is not
specified, the z axis of the rotating frame is aligned with the
dominant_eigenvector_LL
, and chosen to be more parallel than anti-parallel to
the angular velocity.
Source code in sxs/waveforms/waveform_modes.py
dominant_eigenvector_LL(rough_direction=None, rough_direction_index=0)
#
Calculate the principal axis of the matrix expectation value ⟨w|LᵃLᵇ|w⟩
Parameters:
Name | Type | Description | Default |
---|---|---|---|
rough_direction |
array_like
|
Vague guess about the preferred direction as a 3-vector. Default is the
|
None
|
rough_direction_index |
int
|
Index at which the |
0
|
See also
WaveformModes.to_corotating_frame quaternionic.array.to_minimal_rotation
Notes
The principal axis is the eigenvector corresponding to the largest-magnitude (dominant) eigenvalue. This direction can be used as a good choice for the axis of a waveform-mode decomposition at any instant https://arxiv.org/abs/1205.2287. Essentially, it maximizes the power in the large-|m| modes. For example, this can help to ensure that the (ℓ,m) = (2,±2) modes are the largest ℓ=2 modes.
Note, however, that this only specifies an axis at each instant of time. This choice can be supplemented with the "minimal-rotation condition" https://arxiv.org/abs/1110.2965 to fully specify a frame, resulting in the "co-precessing frame". Or it can be used to determine the constant of integration in finding the "co-rotating frame" https://arxiv.org/abs/1302.2919.
The resulting vector is given in the (possibly rotating) mode frame (X,Y,Z), rather than the inertial frame (x,y,z).
Source code in sxs/waveforms/waveform_modes.py
evaluate(*directions)
#
Evaluate waveform in a particular direction or set of directions
Parameters:
Name | Type | Description | Default |
---|---|---|---|
directions |
array_like
|
Directions of the observer relative to the source may be specified using
the usual spherical coordinates, and an optional polarization angle (see
Notes below). These can be expressed as 2 or 3 floats (where the third is
the polarization angle), or as an array with final dimension of size 2 or
3. Alternatively, the input may be a |
()
|
Returns:
Name | Type | Description |
---|---|---|
signal |
array_like
|
Note that this is complex-valued, meaning that it represents both polarizations. To get the signal measured by a single detector, just take the real part. |
Notes
To evaluate mode weights and obtain values, we need to evaluate spin-weighted spherical harmonics (SWSHs). Though usually treated as functions of just the angles (θ, ϕ), a mathematically correct treatment (arxiv.org/abs/1604.08140) defines SWSHs as functions of the rotation needed to rotate the basis (x̂, ŷ, ẑ) onto the usual spherical-coordinate basis (θ̂, ϕ̂, n̂). This function can take quaternionic arrays representing such a rotation directly, or the (θ, ϕ) coordinates themselves, and optionally the polarization angle ψ, which are automatically converted to quaternionic arrays.
We define the spherical coordinates (θ, ϕ) such that θ is the polar angle (angle between the z axis and the point) and ϕ is the azimuthal angle (angle between x axis and orthogonal projection of the point into the x-y plane). This gives rise to the standard unit tangent vectors (θ̂, ϕ̂).
We also define the polarization angle ψ as the angle through which we must rotate the vector θ̂ in a positive sense about n̂ to line up with the vector defining the legs of the detector. If not given, this angle is treated as 0.
Examples:
We can evaluate the signal in a single direction:
>>> θ, ϕ, ψ = 0.1, 0.2, 0.3
>>> w.evaluate(θ, ϕ) # Default polarization angle
>>> w.evaluate(θ, ϕ, ψ) # Specified polarization angle
Or we can evaluate in a set of directions:
We can also evaluate on a more extensive set of directions. Here, we construct an equi-angular grid to evaluate the waveform on (though irregular grids are also acceptable as long as you can pack them into a numpy array).
>>> n_theta = n_phi = 2 * w.ell_max + 1
>>> equiangular = np.array([
[
[theta, phi]
for phi in np.linspace(0.0, 2*np.pi, num=n_phi, endpoint=False)
]
for theta in np.linspace(0.0, np.pi, num=n_theta, endpoint=True)
])
>>> w.evaluate(equiangular)
Source code in sxs/waveforms/waveform_modes.py
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|
index(ell, m)
#
Mode index of given (ell,m) mode in the data
Parameters:
Name | Type | Description | Default |
---|---|---|---|
ell |
int
|
|
required |
m |
int
|
|
required |
Returns:
Name | Type | Description |
---|---|---|
idx |
int
|
Index such that self.LM[idx] == [ell, m] |
Source code in sxs/waveforms/waveform_modes.py
interpolate(new_time, derivative_order=0, out=None)
#
Interpolate this object to a new set of times
Note that if this object has "frame" data and the derivative order is nonzero, it is not entirely clear what is desired. In those cases, the frame is just interpolated to the new times, but no derivative or antiderivative is taken.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
new_time |
array_like
|
Points to evaluate the interpolant at |
required |
derivative_order |
int
|
Order of derivative to evaluate. If negative, the antiderivative is returned. Default value of 0 returns the interpolated data without derivatives or antiderivatives. Must be between -3 and 3, inclusive. |
0
|
See Also
scipy.interpolate.CubicSpline :
The function that this function is based on.
antiderivative :
Calls this funtion with new_time=self.time
and
derivative_order=-antiderivative_order
(defaulting to a single
antiderivative).
derivative :
Calls this function new_time=self.time
and
derivative_order=derivative_order
(defaulting to a single derivative).
dot :
Property calling self.derivative(1)
.
ddot :
Property calling self.derivative(2)
.
int :
Property calling self.antiderivative(1)
.
iint :
Property calling self.antiderivative(2)
.
Source code in sxs/waveforms/waveform_modes.py
max_norm_index(skip_fraction_of_data=4)
#
Index of time step with largest norm
The optional argument skips a fraction of the data. The default is 4, which means that it skips the first 1/4 of the data, and only searches the last 3/4 of the data for the max. This must be strictly greater than 1, or the entire data is searched for the maximum of the norm.
Source code in sxs/waveforms/waveform_modes.py
max_norm_time(skip_fraction_of_data=4)
#
Return time at which largest norm occurs in data
See help(max_norm_index)
for explanation of the optional argument.
Source code in sxs/waveforms/waveform_modes.py
rotate(quat)
#
Rotate decomposition basis of modes represented by this waveform
This returns a new waveform object, with missing "frame" data.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
quat |
array
|
This must have one quaternion or the same number of quaternions as the number of times in the waveform. |
required |
Source code in sxs/waveforms/waveform_modes.py
to_corotating_frame(R0=None, tolerance=1e-12, z_alignment_region=None, return_omega=False, truncate_log_frame=False)
#
Return a copy of this waveform in the corotating frame
The corotating frame is defined to be a rotating frame for which the (L² norm of the) time-dependence of the modes expressed in that frame is minimized. This leaves the frame determined only up to an overall rotation. In this
Parameters:
Name | Type | Description | Default |
---|---|---|---|
R0 |
quaternionic
|
Initial value of frame when integrating angular velocity. Defaults to the identity. |
None
|
tolerance |
float
|
Absolute tolerance used in integration of angular velocity |
1e-12
|
z_alignment_region |
None, 2-tuple of floats
|
If not None, the dominant eigenvector of the |
None
|
return_omega |
bool
|
If True, return a 2-tuple consisting of the waveform in the corotating frame (the usual returned object) and the angular-velocity data. That is frequently also needed, so this is just a more efficient way of getting the data. |
False
|
truncate_log_frame |
bool
|
If True, set bits of log(frame) with lower significance than |
False
|
Source code in sxs/waveforms/waveform_modes.py
to_inertial_frame()
#
Return a copy of this waveform in the inertial frame
Source code in sxs/waveforms/waveform_modes.py
truncate(tol=1e-10)
#
Truncate the precision of this object's data in place
This function sets bits in the data to 0 when they have lower significance than
will alter the norm of the Waveform by a fraction tol
at that instant in
time.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
tol |
float
|
Fractional tolerance to which the norm of this waveform will be preserved |
1e-10
|
Returns:
Type | Description |
---|---|
None
|
This value is returned to serve as a reminder that this function operates in place. |
See also
TimeSeries.truncate
Source code in sxs/waveforms/waveform_modes.py
WaveformMixin
base class#
Bases: ABC