interferometry

interferometry.py computes the response and sensitivity functions of space-based interferometer GW detectors (e.g., LISA, Taiji) for the detection of stochastic gravitational wave backgrounds (SGWB).

Adapted from the original interferometry in GW_turbulence (https://github.com/AlbertoRoper/GW_turbulence), created in May 2022

Currently part of the cosmoGW code:

https://github.com/cosmoGW/cosmoGW/ https://github.com/cosmoGW/cosmoGW/blob/main/src/cosmoGW/interferometry.py

Note

For full documentation, visit Read the Docs

Note

See tutorials/interferometry/interferometry.ipynb for usage examples.

To use it, first install cosmoGW:

pip install cosmoGW

Author

• Alberto Roper Pol (alberto.roperpol@unige.ch)

Dates

• Created: 01/05/2022

• Updated: 21/08/2025 (release cosmoGW 1.0: https://pypi.org/project/cosmoGW)

References

• [Caprini:2019pxz]: C. Caprini, D. Figueroa, R. Flauger, G. Nardini, M. Peloso, M. Pieroni, A. Ricciardone, G. Tassinato, “Reconstructing the spectral shape of a stochastic gravitational wave background with LISA,” JCAP 11 (2019), 017, arXiv:1906.09244.

• [Schmitz:2020syl]: K. Schmitz “New Sensitivity Curves for Gravitational-Wave Signals from Cosmological Phase Transitions,” JHEP 01, 097 (2021), arXiv:2002.04615.

• [Orlando:2020oko]: G. Orlando, M. Pieroni, A. Ricciardone, “Measuring Parity Violation in the Stochastic Gravitational Wave Background with the LISA-Taiji network,” JCAP 03, 069 (2021), arXiv:2011.07059.

• [RoperPol:2021xnd]: A. Roper Pol, S. Mandal, A. Brandenburg, T. Kahniashvili, “Polarization of gravitational waves from helical MHD turbulent sources,” JCAP 04 (2022), 019, arXiv:2107.05356.

• [RoperPol:2022iel]: A. Roper Pol, C. Caprini, A. Neronov, D. Semikoz, “The gravitational wave signal from primordial magnetic fields in the Pulsar Timing Array frequency band,” Phys. Rev. D 105, 123502 (2022), arXiv:2201.05630.

cosmoGW.interferometry.OmPLS(Oms, f=<Quantity [1.00000000e-04, 1.00184413e-04, 1.00369167e-04, ...,            9.96321908e-01, 9.98159260e-01, 1.00000000e+00] Hz>, beta=array([-20.        , -19.98666222, -19.97332444, ...,  19.97332444,         19.98666222,  20.        ], shape=(3000,)), SNR=1, T=1, Xi=0)

Compute the power law sensitivity (PLS).

Reference

RoperPol:2021xnd, appendix B (equation B.31).

Parameters

Omsnp.ndarray

GW energy density sensitivity.

fnp.ndarray or astropy.units.Quantity, optional

Frequency array (default: f_ref).

betanp.ndarray, optional

Array of slopes (default: beta_ref from -20 to 20).

SNRfloat, optional

Signal-to-noise ratio threshold (default: 1).

Tfloat, optional

Duration of the observation in years (default: 1).

Xifloat, optional

For polarization signals using the dipole response (default: 0).

Returns

Omeganp.ndarray

GW energy density power law sensitivity (PLS).

cosmoGW.interferometry.Oms(f, S, h0=1.0, comb=False, S2=None, S3=None, S4=None, Xi=False)

Return the sensitivity Sh(f) in terms of the GW energy density Om(f).

Reference for Omega is RoperPol:2021xnd, equation B.18 (seems like it might have a typo, need to investigate this!). Final PLS sensitivites are again correct for a single channel, since the factor of 2 in the SNR compensates for the 1/2 factor here. Reference for combined sensitivities is RoperPol:2021xnd, equations B.37 (GW energy density, combining LISA and Taiji TDI channels A and C), and B.41 (polarization, combining 4 cross-correlation between LISA and Taiji TDI channels AE, AD, CE, CD).

Strain sensitivities, Omega sensitivities, and OmGW PLS agree with those of reference Caprini:2019pxz, see equation 2.14.

References

RoperPol:2021xnd, equations B.18, B.21, B.37, B.41. Caprini:2019pxz, equation 2.14.

Parameters

fnp.ndarray or astropy.units.Quantity

Frequency array (should be in units of Hz).

Snp.ndarray

Strain sensitivity function.

h0float, optional

Hubble rate uncertainty parameter (default: 1).

combbool, optional

If True, combine two sensitivities (default: False).

S2, S3, S4np.ndarray, optional

Additional sensitivities for combination (default: None).

Xibool, optional

If True, compute sensitivity to polarized GW backgrounds (default: False).

Returns

Omeganp.ndarray

GW energy density sensitivity.

cosmoGW.interferometry.Pacc_f(f=<Quantity [1.00000000e-04, 1.00184413e-04, 1.00369167e-04, ..., 9.96321908e-01, 9.98159260e-01, 1.00000000e+00] Hz>, A=3, L=<Quantity 2500000. km>)

Compute the power spectral density (PSD) of the mass acceleration noise for a space-based interferometer channel.

This function implements the analytical formula for the acceleration noise PSD as described in RoperPol:2021xnd, equation B.25.

Parameters

fastropy.units.Quantity, optional

Frequency array (default: f_ref, units of Hz).

Afloat, optional

Acceleration noise parameter (default: 3 for LISA; Taiji also uses 3).

Lastropy.units.Quantity, optional

Length of the interferometer arm (default: 2.5e6 km for LISA).

Returns

Paccastropy.units.Quantity

Mass acceleration PSD noise (units of 1/Hz).

References

RoperPol2021xnd

cosmoGW.interferometry.Pn_f(f=<Quantity [1.00000000e-04, 1.00184413e-04, 1.00369167e-04, ..., 9.96321908e-01, 9.98159260e-01, 1.00000000e+00] Hz>, P=15, A=3, L=<Quantity 2500000. km>, TDI=True)

Compute the noise power spectral density (PSD) for a space-based interferometer channel and its cross-correlation.

This function calculates the PSD for a single channel (X) and the cross-correlation (XY), or the TDI channels (A and T) if TDI is True.

Implements the analytical formulas from RoperPol:2021xnd, equations B.23 and B.26.

Parameters

fastropy.units.Quantity, optional

Frequency array (default: f_ref, units of Hz).

Pfloat, optional

Optical metrology system noise parameter (default: 15 for LISA).

Afloat, optional

Acceleration noise parameter (default: 3 for LISA).

Lastropy.units.Quantity, optional

Length of the interferometer arm (default: 2.5e6 km for LISA).

TDIbool, optional

If True, compute TDI channel PSDs (A and T). If False, compute single channel and cross-correlation PSDs.

Returns

If TDI is True:

PnAastropy.units.Quantity

Noise PSD of the TDI channel A.

PnTastropy.units.Quantity

Noise PSD of the TDI channel T.

If TDI is False:

Pnastropy.units.Quantity

Noise PSD of the single channel.

Pn_crossastropy.units.Quantity

Noise PSD of the cross-correlation channel.

References

Roper Pol et al., JCAP 04 (2022) 019, https://arxiv.org/abs/2107.05356

cosmoGW.interferometry.Poms_f(f=<Quantity [1.00000000e-04, 1.00184413e-04, 1.00369167e-04, ..., 9.96321908e-01, 9.98159260e-01, 1.00000000e+00] Hz>, P=15, L=<Quantity 2500000. km>)

Compute the power spectral density (PSD) of the optical metrology system (OMS) noise for a space-based interferometer.

Reference

RoperPol:2021xnd, equation B.24.

Parameters

fastropy.units.Quantity, optional

Frequency array (default: f_ref).

Pfloat, optional

OMS noise parameter (default: 15 for LISA, 8 for Taiji).

Lastropy.units.Quantity, optional

Length of the interferometer arm (default: 2.5e6 km for LISA).

Returns

astropy.units.Quantity

OMS PSD noise array (units: 1/Hz).

cosmoGW.interferometry.R_f(f=<Quantity [1.00000000e-04, 1.00184413e-04, 1.00369167e-04, ..., 9.96321908e-01, 9.98159260e-01, 1.00000000e+00] Hz>, L=<Quantity 2500000. km>)

Compute the analytical fit of the response function for a space-based interferometer channel.

Implements the analytical formula from RoperPol:2021xnd, equation B.15.

Parameters

fastropy.units.Quantity, optional

Frequency array (default: f_ref, units of Hz).

Lastropy.units.Quantity, optional

Length of the interferometer arm (default: 2.5e6 km for LISA).

Returns

Rfastropy.units.Quantity

Analytical fit of the response function.

References

RoperPol2021xnd

cosmoGW.interferometry.SNR(f, OmGW, fs, Oms, T=1.0)

Compute the signal-to-noise ratio (SNR) of a GW signal.

\[{\rm SNR} = \sqrt{T \int \left( \frac{\Omega_{\rm GW} (f)} {\Omega_{\rm sens} (f)} \right)^2 df}\]

Reference

RoperPol:2021xnd, appendix B (equation B.30).

Parameters

fnp.ndarray or astropy.units.Quantity

Frequency array of the GW signal.

OmGWnp.ndarray

GW energy density spectrum of the GW signal.

fsnp.ndarray or astropy.units.Quantity

Frequency array of the GW detector sensitivity.

Omsnp.ndarray

GW energy density sensitivity of the GW detector.

Tfloat, optional

Duration of observations in years (default: 1.0).

Returns

float

SNR of the GW signal.

cosmoGW.interferometry.Sn_f(fs=<Quantity [1.00000000e-04, 1.00184413e-04, 1.00369167e-04, ..., 9.96321908e-01, 9.98159260e-01, 1.00000000e+00] Hz>, v=0.00123, interf='LISA', TDI=True, M='MED', Xi=False)

Compute the strain sensitivity using the analytical fit for an interferometer channel.

Parameters

fsnp.ndarray or astropy.units.Quantity, optional

Frequency array (default: f_ref).

vfloat, optional

Dipole velocity of the solar system (default: 1.23e-3).

interfstr, optional

Interferometer (‘LISA’, ‘Taiji’, ‘comb’; default: ‘LISA’).

TDIbool, optional

If True, use TDI channels (default: True).

Mstr, optional

Cross-correlated channel (default: ‘MED’).

Xibool, optional

If True, compute sensitivity to polarized GW backgrounds (default: False).

Returns

tuple

Frequency array and strain sensitivities (see function body).

cosmoGW.interferometry.Sn_f_analytical(f=<Quantity [1.00000000e-04, 1.00184413e-04, 1.00369167e-04, ..., 9.96321908e-01, 9.98159260e-01, 1.00000000e+00] Hz>, P=15, A=3, L=<Quantity 2500000. km>)

Compute the strain sensitivity using the analytical fit for a space-based interferometer channel.

This function uses the analytical noise PSD and response function to calculate the strain sensitivity.

Parameters

fastropy.units.Quantity, optional

Frequency array (default: f_ref, units of Hz).

Pfloat, optional

Optical metrology system noise parameter (default: 15 for LISA).

Afloat, optional

Acceleration noise parameter (default: 3 for LISA).

Lastropy.units.Quantity, optional

Length of the interferometer arm (default: 2.5e6 km for LISA).

Returns

Snastropy.units.Quantity

Strain sensitivity array (units: 1/Hz).

References

RoperPol2021xnd

cosmoGW.interferometry.compute_Oms_LISA_Taiji(interf='LISA', TDI=True, h0=1.0)

Read response functions for LISA and/or Taiji, compute strain sensitivities, and from those, the sensitivity to the GW energy density spectrum Omega_s.

Parameters

interfstr, optional

Interferometer (‘LISA’, ‘Taiji’, ‘comb’; default: ‘LISA’).

TDIbool, optional

If True, use TDI channels (default: True).

h0float, optional

Hubble rate uncertainty parameter (default: 1.0).

Returns

tuple

fs : frequency array OmSA : GW energy density sensitivity for channel A OmST : GW energy density sensitivity for channel T

cosmoGW.interferometry.compute_interferometry(f=<Quantity [1.00000000e-04, 1.00184413e-04, 1.00369167e-04, ..., 9.96321908e-01, 9.98159260e-01, 1.00000000e+00] Hz>, L=<Quantity 2500000. km>, TDI=True, order=1, comp_all=False, comp_all_rel=True)

Numerically compute the monopole (order=1) or dipole (order=2) response functions of a space-based GW interferometer channel.

This function integrates over sky directions and computes the response functions for interferometer channels (or TDI channels if TDI is True). It can compute all relevant response functions, including those for geometric symmetries.

Implements the formalism from RoperPol:2021xnd, appendix B (eqs. B.13, B.16).

Parameters

fastropy.units.Quantity, optional

Frequency array (default: f_ref, units of Hz).

Lastropy.units.Quantity, optional

Length of the interferometer arm (default: 2.5e6 km for LISA).

TDIbool, optional

If True, compute TDI channel response functions (A, E, T). If False, compute XYZ channel response functions.

orderint, optional

Moment of the response function (1 for monopole, 2 for dipole).

comp_allbool, optional

If True, compute all response functions (monopole and dipole, X and A channels).

comp_all_relbool, optional

If True, compute only relevant response functions (not identically zero or equal).

Returns

tuple

Monopole or dipole response functions, depending on order and options. See function body for details.

References

RoperPol2021xnd

cosmoGW.interferometry.compute_response_LISA_Taiji(f=<Quantity [1.00000000e-04, 1.00184413e-04, 1.00369167e-04, ..., 9.96321908e-01, 9.98159260e-01, 1.00000000e+00] Hz>, dir0='resources/detectors_sensitivity/', dir_HOME=None, save=True, ret=False)

Compute LISA and Taiji’s monopole and dipole response functions.

Uses the compute_interferometry() routine and refines the response functions at low frequencies.

Parameters

fnp.ndarray or astropy.units.Quantity, optional

Frequency array (default: f_ref).

dir0str, optional

Directory to save results (default: dir_sens).

savebool, optional

If True, saves results as output files (default: True).

retbool, optional

If True, returns the results from the function (default: False).

Returns

MAAnp.ndarray

Monopole response function of the TDI channel A.

MTTnp.ndarray

Monopole response function of the TDI channel T (Sagnac channel).

DAEnp.ndarray

Dipole response function of the TDI correlation of the channels A and E.

MXXnp.ndarray

Monopole response function of the interferometer channel X.

MXYnp.ndarray

Monopole response function of the correlation of interferometer channels X and Y.

DXYnp.ndarray

Dipole response functions of the correlation of the interferometer channels X and Y.

cosmoGW.interferometry.read_MAC(dirr0='/LISA_Taiji/', dir_HOME=None, M='MAC', V='V')

Read the V response functions of the cross-correlated channels of the LISA-Taiji network.

Reference

RoperPol:2021xnd, see figure 18. Data from Orlando:2020oko, see figure 2.

Parameters

dirr0str, optional

Directory where to save the results (default: ‘detector_sensitivity/LISA_Taiji/’).

Mstr, optional

Channel to be read (‘MAC’, ‘MAD’, ‘MEC’, ‘MED’; default: ‘MAC’).

Vstr, optional

Stokes parameter. Use ‘I’ to read the intensity response (default: ‘V’).

Returns

tuple

fastropy.units.Quantity

Frequency array in Hz.

MACnp.ndarray

Response function array.

cosmoGW.interferometry.read_all_MAC(V='V')

Read all relevant TDI cross-correlated response functions between LISA and Taiji (AC, AD, EC, ED channels) using read_MAC.

Reference

RoperPol:2021xnd, see figure 18. Data from Orlando:2020oko, see figure 2.

Parameters

Vstr, optional

Stokes parameter. Use ‘I’ to read the intensity response (default: ‘V’).

Returns

tuple

fsastropy.units.Quantity

Frequency array in Hz.

M_ACnp.ndarray

Response function for AC channel.

M_ADnp.ndarray

Response function for AD channel.

M_ECnp.ndarray

Response function for EC channel.

M_EDnp.ndarray

Response function for ED channel.

cosmoGW.interferometry.read_detector_PLIS_Schmitz(dirr0='/power-law-integrated_sensitivities/', dir_HOME=None, det='BBO', SNR=10, T=4)

Read power law integrated sensitivities from Schmitz:2020syl.

Parameters

dirr0str, optional

Directory where the PLS are stored (default: ‘detector_sensitivity/power-law-integrated_sensitivities’).

detstr, optional

GW detector name (check available detectors in default directory).

SNRfloat, optional

Signal-to-noise ratio (SNR) of the resulting PLS (default: 10).

Tfloat, optional

Duration of the mission in years (default: 4).

Returns

tuple

fnp.ndarray

Frequency array.

Omeganp.ndarray

Power law integrated sensitivity curve.

cosmoGW.interferometry.read_response_LISA_Taiji(dir0='resources/detectors_sensitivity/', dir_HOME=None, TDI=True, interf='LISA')

Read response functions for LISA or Taiji interferometers. TDI channels are defined following RoperPol:2021xnd, appendix B.

Parameters

dir0str, optional

Directory containing sensitivity files (default: dir_sens).

TDIbool, optional

If True, read response functions for TDI channels; otherwise XYZ.

interfstr, optional

Interferometer name (‘LISA’ or ‘Taiji’).

Returns

fsastropy.units.Quantity

Frequency array in Hz.

MAsnp.ndarray

Response function MA (or MX for XYZ).

MTsnp.ndarray

Response function MT (or MXY for XYZ).

cosmoGW.interferometry.read_sens(dir0='resources/detectors_sensitivity/', SNR=10, T=4, interf='LISA', Xi=False, TDI=True, chan='A')

Read sensitivity curve for a given interferometer.

For LISA and Taiji the sensitivity can be given for each chanel X, Y, Z or on the TDI chanels A=E, T (A chanel is the default option, and the relevant for Omega sensitivity).

Parameters

dir0str, optional

Directory containing sensitivity files (default: ‘detector_sensitivity’).

SNRfloat, optional

Signal-to-noise ratio threshold (default: 10).

Tfloat, optional

Observation time in years (default: 4).

interfstr, optional

Interferometer name (‘LISA’, ‘Taiji’, ‘comb’, ‘muAres’).

Xibool, optional

If True, include helical sensitivity and PLS.

TDIbool, optional

If True, use TDI channels.

chanstr, optional

Specific channel (‘A’, ‘X’, ‘Y’, ‘Z’, ‘T’).

Returns

fsastropy.units.Quantity

Frequency array in Hz.

Omeganp.ndarray

Sensitivity curve array.

OmegaPLSnp.ndarray

Power law sensitivity curve array.

Xinp.ndarray, optional

Helical sensitivity curve array (if Xi is True).

XiPLSnp.ndarray, optional

Helical power law sensitivity curve array (if Xi is True).

cosmoGW.interferometry.refine_M(f, M, A=0.3, exp=0)

Refine the response function by appending a low-frequency term.

Parameters

fnp.ndarray or astropy.units.Quantity

Frequency array (should be in units of Hz).

Mnp.ndarray

Response function to be refined at low frequencies.

Afloat, optional

Amplitude of the response function at low frequencies (default: 0.3 for LISA monopole).

expint, optional

Exponent of the response function at low frequencies (default: 0).

Returns

fsnp.ndarray

Refined array of frequencies.

Msnp.ndarray

Refined response function.