-- MuziLi -21 Aug 2014

sensitivity depth study

Main work concent of the two-month intership:

1.Read articles about the search for continuous gravitational waves on data from LIGO to study thebackground knowledge and understand general ideas.

2.Calculate the sensitivity depth with the results of upper limits given by the papers.

Background knowledge study:

1.About the datas and data source :

The papers that I have read are all searches the CW on the LIGO data,and most of them use the data from LIGO's fifth science run(S5).

LIGO( Laser Interferometer Gravitational-Wave Observatory )aims to directly detect gravitational waves.The LIGO consists of three detectors:a 4-km interferometer in Livingston,Louisiana(L1)and two in Hanford,Washington, one a 4-km(H1)and 2-km(H2). The fifth science run spanned a nearly two-year period(2005-2007) ,but the datas is not continupus due to environmental or instrumental disturbances or for scheduled maintenance period.By the end of the S5 run, the cunmulative duty factor of L1 is 66%, while the H1 is 78% and H2 is 79% During LIGO's fifth Science Run, all three detectors had displacement spectral amplitudes very near their design goals of 1.1×10^-19m•Hz^-0.5 in their most sensitive frequency band near 150Hz for the 4-km detectors and H2 interferometer(2-km) was roughly a factor of two less sensitive than the other two over most of the relevant band.The sensitivity is not stable, has a roughly 10‰ daily variation from anthropogenic activity as well as gradual improvement toward the end of the run.

After the completion of S5, initial LIGO was upgraded with certain Advanced LIGO technologies that resulted in an improved-performance configuration dubbed Enhanced LIGO.Its aim was to achieve twice the sensitivity than initial LIGO by the end of the run.Science Run 6 (S6) began in July 2009 with the enhanced configurations on the 4 km detectors.It concluded in October 2010, and the disassembling of the original detectors began. An estimated four-year long effort to install and commission the Advanced LIGO detectors is currently underway.

Detector noise curves for Initial and Advanced LIGO as a function of frequency.The characteristic strain of potential astrophysical sources are also shown.

To be detectable the characteristic strain of a signal must be above the noise curve.(This graph is from wilipedia)

eLISA:evolved Laser Interferometer Space Antenna EPTA:European Pulsar Timing Array

2.The waveform model for CW signals:

The articles I have read are all searches for the continuous gravitational waves that might be radiated by nearly,rapidly spinning neutron stars, the signal is in the LIGO frequency band.The spinning neutron stars may generate continuous gravitational waves through non-axisymmetric distortions of the neutron star, unstable oscillation modes in the fluid part of the star and free precession. The signal is a qusi-periodic wave but the frequency changes slowly due to the energy loss through gravitational wave and other possible mechanisms.As the detector on the Earth moves relative to the sources,the signal exhibits amplitude and phase modulations.

The two polarizations h×,+(t) have the form:

h0 is the wave amplitude and is model dependent. For the non-axisymmetric neutron stars, it is given by: The ellipticity ε is defined as:, it's uncertain and model dependent.The f is the GW signal's frequency,which is also twice the rotational frequency of the star. As the energy loss due to the GW emission and other form, the f changes and it can be expessed as . Finally, the received signal need to be modified due to the detectors moves relative to the wave source, this is given by the non-relativistic Doppler expression : .[t:GPS time, τ:SSB time, v:the detectors velocity with respect to the SSB frame,n:the unit vector pointing to the neutron star,given by(cosαcosδ,sinαcosδ,sinδ)]

The received signal at the detector is:. The F+,×are the detector beam pattern functions which depend on the sky position and the relative polarizaiton angle Ψ of the wave-frame.

The eight parameters : phase evolution parameters

the other parameters:

For the all-sky searches for unknown neutron stars must cope with the a large parameter space volum, we have to focus on four parameters******at least. But for some neutron stars whose position have been known like CasA, and some searches that just analyze particular area, for example,the search from the Galactic Center.

3.Sensitivity depth:

The sensitivity depth is defined as:

The Sh is the power spectra density(PSD)，which stands for the detector noise.

As we known, no gravitational wave signals have been found for present searches, so they report upper limits on the intrinsic gravitational wave strain amplitude h0, as a function of frequency.The results are usually at the 90%confidence level, denoted h090%. It means that 90% of the signal injections at this amplitude would be recovered and are more significant than the most significant candidate from the actual search in that frequency band.

The signals that received by detectors might consist of the noise and the gravitational waves component, but the GW signal is too weak to be found. Hence, in the case of noise is the same， the larger sensitivity depth values are, the more the data analysis method is. For example, the sensitivity values is 30, it means that the noise is 30 times stronger than the weakest signal we can found. If the signal exceed the upper limit that we given in results, it would be likely to be found. So the larger values means we can dig more deeply through the noise to find the CW signals.

Calculate sensitivity depth:

1.《Einstein@Home all-sky search for periodic gravitational wave in LIGO S5 data》

the frequency band:[50,1190]Hz the frequency derivative range:[-20,1.1]×10¯10 Hz/s

data: the fifth LIGO science run(S5),collected from the GPS times of 815155213s(Fri NOV 04 16:00:00 UTC 2005)to 875145614s(SUN Sep 30 00:00:00 UTC 2007), and this search only used the data from H1 and L1.

data analyse: uses a non-coherent Hough-transform method :sums weighted binary counts.depending upon whether the normalized power in an SFT bin exceeds a certain threshold.

And before the Hough transform, the F-statisic was performed with each segment, and the threshold is 5.2. The volunteer computing project Einstein@Home has been created to address this need.

the upper limits given by the search :(given in the paper ,Fig 9)

the sensitivity depth I computed with the upper limits h0 given by the papers: (fig1)

I download the upper limits eh0 values of S5R5 file online , which was the supplemental material of the paper, the first graph also was ploted according to this data.

_ upper-limits: http://journals.aps.org/prd/abstract/10.1103/PhysRevD.87.042001#supplemental__

About the Sh values, I got them from this paper's author, Doctor Paola.Leaci.She has computed a while ago.

psd: https://www.lsc-group.phys.uwm.edu/cgi-bin/pcvs/viewcvs.cgi/einsteinathome/einsteinathome1/HierarchicalSearch/PostProcessingS5R5/HarmonicMeanPSD_NonNarrB_S5R5H1L1.dat?rev=HEAD&cvsroot=einsteinathome&content-type=text/vnd.viewcvs-markup

the command I used in Matlab:

a.the function that can be used to calculated the psd average of the different parts:

the upper limits was given in each frequency band, like this paper, the upper limits was computed in every 0.5 Hz, so the more proper correspondence was the mean psd of the band rather than just the the psd value that was in the same frequency. For this paper, the author decided to write frequencies indicate the left border, I call it in main program.

%calculate the averge of different parts of PSD

%the inpur parameters are: f: the frequency of the upperlimit; fn: the frequency of psd; psd: psd values; position: the part you want to calculate the average, it must be the character variable,'l'.'s' or 'c'.

function averagepsd=part_average(f,fn,psd,position)
n=length(f);

averagepsd=zeros(n,1);

if position=='r'

for i=1:n

p1=find(fn==(f(i)-0.25))+1;

p2=find(fn==f(i));

meannr=mean(psd(p1:p2));

averagepsd(i)=meannr;

end

elseif position=='l'

for j=1:n

p1=find(fn==f(j));

p2=find(fn==(f(j)+0.25))-1;

meannl=mean(psd(p1:p2));

averagepsd(j)=meannl;

end

elseif position=='c'

for k=1:n

p1=find(fn==f(k));

p2=find(fn==(f(k)+0.25))-1;

meannc=mean(psd(p1-(p2-p1+1)/2:p2+(p2-p1+1)/2));

averagepsd(k)=meannc;

end

end

b, the script that calculates the sensitivity depth, plots the graph and records the result in a dat file:

clear

load('Noiselevel.dat')

load('upperlimits.dat')

f=upperlimits(:,1);

h0=upperlimits(:,2);

frequency=Noiselevel(:,1);

psd=Noiselevel(:,2);

position=input('which part related to h0:','s')

averagepsd=part_average(f,frequency,psd,position);

sdepth=sqrt(averagepsd)./h0;

%record the results in a dat file

result=[f,h0,averagepsd,sdepth]';

fid1=fopen('results.dat','w');

fprintf(fid1,'frquency ul noise sdpeth\n');

fprintf(fid1,'%10.6f %e %e %f\n',result);

fclose(fid1);

% plot the f—sensitivity

semilogy(f,sdepth,'.')

title('f-sensitivity depth','fontsize',20)

xlabel('frequency','fontsize',16)

ylabel('sensitivity depth','fontsize',16)

2.《All-sky search for periodic gravitational waves in the full S5 LIGO Data》

the frequency band:[50,800]Hz the frequency derivative range :[-6,0]×10¯9 Hz/s

data: the fifth LIGO science run(S5),collected from the GPS times of 816070843s( NOV 15 06:20:30 UTC 2005)to878044141s(Nov 02 13:08:47 UTC 2007), and this search only used the data from H1 and L1.

data analyse: the main PowerFlux code was run to establish upper limits and produce lists of outliers with signal-to-noise ratio(SNR) great than 5. Then, the Loosely Coherent pipeline was used to reject or confirm collected outlier.(PowerFlux shows butter results in frequency bands lacking severe spectral artifacts)

the upper limits given by the search :(given in the paper ,Fig1):

worst-case: indicates to linear polarizations with Τ=Π/2; the best-case: for the most sensitivity circular polarization(Τ=0 or Τ=Π).

the sensitivity depth I computed with the upper limits h0 given by the papers: (fig2)

1. I download the upper limits eh0 values file online , which was the supplemental material of the paper.

upper limits:http://journals.aps.org/prd/abstract/10.1103/PhysRevD.85.022001#supplemental

2.About the Sh values( S5spectradata.mat , I got them from this paper's author, professor Keith Rilesi..

psd:https://login.ligo.org/idp/Authn/RemoteUser

3.The H1 and L1 psd values were given separately, so I combined them with the harmonic average. ,In factstrict harmonic mean wouldn't be quite right because one has to weight for the number of SFTs used from each detector in a given band, and in general, one won't get the same number for H1 and L1.If you want better than O(5-10%), then it will be necessary (I think) to deal with those details.

4.The results that I computed is recorded in a dat file(result2.dat).and was supplied in the attachment.

the command I used in Matlab

clear

load('S5spectradata.mat')

load('upperlimits.dat')

f=upperlimits(:,1);

h0linear=upperlimits(:,2);

h0circ=upperlimits(:,3);

h=S5H1amppsdwt;

l=S5L1amppsdwt;

psd=(2*l.*h)./(l+h);

position=input('which part related to h0:','s')

averagepsd=part_average(f,frequency,psd,position);

sdepthlinear=averagepsd./h0linear;

sdepthcirc=averagepsd./h0circ;

result=[f,h0linear,h0circ,averagepsd,sdepthlinear,sdepthcirc]';

fid1=fopen('results.dat','w');

fprintf(fid1,'frquency ul-l ul-cir noise sdpeth-l sdepth-cir\n');

fprintf(fid1,'%10.6f %e %e %e %f %f\n',result);

fclose(fid1);

semilogy(f,sdepthlinear,f,sdepthcirc)

title('f-sensitivity depth','fontsize',20)

xlabel('frequency','fontsize',16)

ylabel('sensitivity depth','fontsize',16)

legend('worst case(linear)','best case(circular)','fontsize','14')

3.《A directed search for continuous gravitational waves from the Galactic Center》

the frequency band:[78,496)Hz the frequency derivative range :-7.86×10^-8Hz/s at the highest frequency

data: the data used for the search comes from two of the three initial LIGO detector, H1 and L1, collected during the S5. Their data comprises 630 segments, each spanning 11.5 hours of coincident data with the best sensitivity to a CGW signal from the GC.

data analyse:

1. They use the hierarchical approch (known as the global correlation transform)and divide the data into single segments which are coherently analyzed and afterwards incoherently combined.

2.Use the search algorithm HierarchSearchGCT that is part of the LAL/LALapps Software Suite.

3.The four phase evolution parameters only leave two of them to be search, because the search only a single sky position, the Galactic Center,α=4.650 rad and δ= -0.506 rad. The initial search is sensitive to sources within a distance R less than 8 pc around the Sgr A*.

the upper limits given by the search :(given in the paper ,Fig1):

the sensitivity depth I computed with the upper limits h0 given by the papers: (fig3)

1.the data I used to calculate the sensitivity depth was given by Map. The data can be found in the ATLAS cluster and their positions are:

psd:/home/bbehnke/gcsearch/data/harmonic_psd_of_used_data/psd_used_sfts

upper limits:/home/bbehnke/gcsearch/postprocessing/sensitivity_studies/analytic_upper_limits/upper_limit_results_shortened

2.I used the interpolation method with the nearest points to get the psd values that respond to the upper-limit h0.

the command I used in Matlab

clear

load('upperlimits.txt')

load('Noiselevel.txt')

frequency=upperlimits(:,2);

h0=upperlimits(:,3);

f=Noiselevel(:,1);

noise=Noiselevel(:,2);

psd=interp1(f,noise,frequency,'linear','extrap');

sdepth=sqrt(psd)./h0;

semilogy(frequency,sdepth,'.');

results=[frequency,h0,psd,sdepth]';

fid=fopen('results.dat','w');

fprintf(fid,'frquency h0 noise sdpeth\n');

fprintf(fid,'%f %e %e %f\n',results);

title('f-sensitivity depth','fontsize',20)

xlabel('frequency','fontsize',16)

ylabel('sensitivity depth','fontsize',16)

fclose(fid);