The Simulation of Lisa and Data Analysis E.Plagnol for the LISA_APC group
Outline The simulation of LISA : LISACode Motivations LISACode The sensitivity curve in different situations The strategy The EMRIs and Time-Frequency analysis Data Analysis and the Lisa Mock Data Challenge The Analysis of Training and Challenge 111a
LISACode : the motivation Simulation and detector development European Effort (ESA/DAST) Comparison between different codes Data Analysis
LISACode : Basic Principles and structure of the code Inputs : Gravitational Waves (and noise!). Outputs : Time sequences : phasemeters and TDI Filter 4
The LISA sensitivity curves : X1 Lisa is fixed : no flexing or Sagnac TDI first generation of course. Isotropic distribution of sources J.Y.Vinet : Analytic + numerical integration 5
The LISA sensitivity curves : X1 Standard noises (Pre-Phase A report) : inertial mass, optics and laser. Isotropic distribution of sources 100 Power (Hz -1 ) 10-36 10-38 10-40 Inertial mass noise Noise Power TDI X1s1 combination LISA fixed Optical-path noise RMS GW response / h 1 0,01 0,0001 RMS GW response Averaged over the sky TDI X1s1 combination Lisa fixed Long Wavelengths 10-42 LISACode 10-6 Analytic calculation (J.Y.Vinet) Analytic calculation (J.Y.Vinet) LISACode 10-44 0,0001 0,001 0,01 0,1 f (Hz) 10-8 0,0001 0,001 0,01 0,1 f (Hz) 6
The LISA sensitivity curves : X1 h = 5 Sensitivity Noise Yr * Resp GW Sensitivity 10-19 10-20 10-21 10-22 Sensitivity of LISA Averaged over the sky TDI X1s1 combination Lisa fixed Validation of LISACode 10-23 10-24 Analytic calculation (J.Y.Vinet) LISACode 0,0001 0,001 0,01 0,1 f (Hz) 7
The LISA sensitivity curves : X2 Lisa on realistic orbits : Sagnac + Flexing TDI 2nd generation 10-38 Noise Power for X2s1 TDI combination LISA on orbit 10-19 10-20 Sensitivity of LISA Averaged over the sky TDI X2s1 combination LISA on orbit Power (Hz -1 ) 10-40 10-42 Sensitivity 10-21 10-22 10-44 10-46 MeanPSDNoise LocalMeanPSDNoise 0,0001 0,001 0,01 0,1 f (Hz) 10-23 10-24 Analytic Calculation (J.Y.Vinet) : 1st generation LISACode 0,0001 0,001 0,01 0,1 f (Hz) 8
Modifying the Armlengths L Analysis of table 4.1 of Pre-Phase A Report Only the shot noise varies with L 10-36 10-37 10-38 Noise Power : X2s1 L = 5 Mkm L = 2 Mkm 10-19 10-20 Lisa Sensitivity X2s1 L = 5 Mkm L = 2 Mkm PSD (Hz-1) 10-39 10-40 10-41 Sensitivity 10-21 10-22 10-42 10-23 10-43 10-44 0,0001 0,001 0,01 0,1 f (Hz) 10-24 0,0001 0,001 0,01 0,1 f (Hz) 9
Status and Evolution of the code LISACode is finalised : present version 1.2 GW : monochromatic, binaries, input files, Realistic orbits, Noise : Laser, inertial mass, shot noise, Phasemeter : filtering and sampling, TDI : 1st and 2nd generation. Non standard combinations are possible, Inputs by ASCI files for configuration files and GW, output by ASCI files. Executes on most platforms : Mac, Unix, Windows The future... XML inputs/outputs Galactic confusion noise (finalised) more inbedded GW types : MBHB, EMRIs, more complex noise functions, phasemeters, A user friendly interface The Developers A.Petiteau (APC) G.Auger (APC) H.Halloin (APC) S.Pireaux (Artemis) E.Plagnol (APC) T.Regimbeau (Artemis) J.Y.Vinet (Artemis)
A User Friendly Interface
Data Analysis and the Lisa Mock Data Challenge at APC (Paris) One of the aims of LISACode is to analyse data and extract the physical parameters of the GW emitter. In order to support the LISA project, a Mock Data Challenge has been established mid 2006. A number of Training (known parameters) and challenges (unknown parameters) of increasing complexity have been issued. We have started with the simplest: Training and Challenge 111a Monochromatic GW : 1 year samples of TDI Xf, Yf and Zf 7 Parameters : frequency, amplitude and β, λ, ι, ψ and φ
Monochromatic GW : The parameters 7 Parameters frequency, Amplitude and β, λ, ι, ψ and φ A+ = A (1+cos 2 (ι)) cos(2π f t+φ) Ax = 2 A cos(ι) cos(2π f t+φ) ψ is the polarisation angle β and λ define the directions of the source in the Barycentric Ecliptic Plane Reference System.
The Strategy A direct X2 search is NOT practical 1. Determine (approximately) the frequency f (FFT) 2. Divide the total time sample into N subsets 3. Determine β, λ and ψ (Χ 2 ) 4. Re-determine the 7 parameters by minimisation (Χ 2 ) with respect to the Fourrier components The present problem Defining the errors on the data and on the parameters
Training 111a The frequency Challenge 111a 10-22 Challenge 111a 1 year set 15 sec sampling time FFT GW amplitude_fft 10-23 10-24 10-25 10-26 10-6 10-5 0,0001 0,001 0,01 0,1 Frequence (Hz) 5 10-24 4 10-24 Training 111a 1 year, 15sec sampling FFT 8 10-24 7 10-24 GW Challenge 111a 1 year set 15 sec sampling time FFT 6 10-24 amplitude_fft 3 10-24 2 10-24 amplitude_fft 5 10-24 4 10-24 3 10-24 1 10-24 GW < f > = 9.93604679e-03 Hz f = 0.0003179 Hz 0 0,000992 0,0009925 0,000993 0,0009935 0,000994 Frequence (Hz) 2 10-24 1 10-24 0,001062 0,001062 0,001063 0,001063 Frequence (Hz)
The spread of the frequency The spread of the frequency is due to: The modulation of the amplitude, The Doppler effect due to the motion of Lisa. 9,934 10-1 LMDC Training 111a F = 0.993035 mhz Xf Frequency (mhz) 9,932 10-1 9,930 10-1 9,928 10-1 Yf Zf 9,926 10-1 0,0 0,20 0,40 0,60 0,80 1,0 Time (year)
β, λ :The modulation formula 3 assumptions: low frequencies 2πfL<<1 The variations of the envelopes are << f hx(t) = ρ h+(t-τ) or ρx hx(t) = ρ+ h+(t-τ) TDI Michelson : 17
Training 111a N subsets and determination of β, λ and ψ with the 3 TDI observables The minima are related mostly to the source direction (β, λ) 3 2.5 Xf > Yf > Zf pour des fenetres en temps de 11 jours (donnees [MLDC col2] x 2.5) Resultats : beta 26.8793, lambda 61.6391, rhop 33.2275, rhoc 33.6701, tau 243.2828 soit : beta a 0.079289%, lambda a 0.13003%, rhop a 5.644%, rhoc a 4.3734%, tau a 3.3901% Xf Yf Zf Data Real param Fit 2 1.5 1 Error on β = 0.29 Error on λ = 0.47 0.5 0 0 20 40 60 80 100 120 140 160 180 200 The optimum determination seems to be obtained for 64 (overlapping) samples of 11 days
2.5 2 The blind Challenge 111a Xf > Yf > Zf pour des fenetres en temps de 11 jours (donnees [MLDC111ab col2] x 2.5) Resultats : beta 54.6321, lambda 64.9757, rhop 30.9551, rhoc 30.1232, tau 230.3561 β = 54.6 λ = -65 Data Fit 1.5 Xf Yf Zf 1 0.5 0 0 20 40 60 80 100 120 140 160 180 200
The general fit on the 7 parameters From the FFT, 20 frequencies are considered, centred on the mean frequency. This gives 20 amplitudes and phases or, equivalently, 20 vectors. The X2 is calculated using the vector difference between the fit and the data. The error on the amplitude is extracted from the noise to the left and right of signal. 5,000 10-2 Training 111a 6,000 10-2 Challenge 111a Xf Yf Zf Xf Yf Zf Amplitudes of the Fourrier components 4,000 10-2 3,000 10-2 2,000 10-2 1,000 10-2 Amplitudes of the Fourrier components 5,000 10-2 4,000 10-2 3,000 10-2 2,000 10-2 1,000 10-2 0,000 10 0 0,00 10,00 20,00 30,00 40,00 50,00 60,00 0,000 10 0 0,00 10,00 20,00 30,00 40,00 50,00 60,00 2,000 10 2 ii Training 111a 2,000 10 2 ii Challenge 111a 1,500 10 2 Xf Yf Zf 1,500 10 2 Xf Yf Zf Phases of the Fourrier components 1,000 10 2 5,000 10 1 0,000 10 0-5,000 10 1-1,000 10 2 Phases of the Fourrier components 1,000 10 2 5,000 10 1 0,000 10 0-5,000 10 1-1,000 10 2-1,500 10 2-1,500 10 2-2,000 10 2 0,00 10,00 20,00 30,00 40,00 50,00 60,00 ii -2,000 10 2 0,00 10,00 20,00 30,00 40,00 50,00 60,00 ii
The final parameters f =1.06273044e-3 mhz A =0.63 10-22 β = 54.6 λ = 291. ι = 55.0 ψ = -113.0 φ = -61.0 Maximum of TDI 1,500 10-1 Challenge 111a 1,000 10-1 5,000 10-2 0,000 10 0 Xf Yf Zf 0 1/3 2/3 0 1/3 2/3 0 1/3 2/3 Time (year) Open problems: definition of the errors of the data and of the X2 determination of the error on the parameters.
The difficulties are ahead! A monochromatic GW, over 1 year with a high S/N is the simplest problem... and it can be optimised. More complicated scenari are included in the LMDC multiple overlapping GW smaller time samples with and without chirp EMRIs
EMRIs produce a wide variety of waveforms 14-17 parameters Circular Lens-Thiring (spin-orbit) excentric general This translates into multiple frequencies and complex time-frequency patterns
Time-Frequency Analysis
EMRIs The simultaneous study of multiple frequencies The possible connection of different time-frequency lines We are looking into wavelets type analysis and image processing methods.
Summary LISACode LISACode is a sophisticated software simulator of LISA which impacts both the technical development of LISA and the data analysis. It is readily available to the public and is permanently upgraded, both in efficiency and versatility. Data Analysis Our data analysis effort is starting. We believe we are on the right track but many new tools have still to be developed and understood. In many instances, the correct estimation of the errors (data and parameters) is an issue.