Data Analysis for the Sthocastic Background

Transcription

1 Data Analysis for the Sthocastic Background Andrea Viceré May 28, 2009 Istituto di Fisica dell Università di Urbino Via S.Chiara 27, I Urbino, ITALY Abstract The purpose of these lessons is to introduce the basic data analysis techniques for the detection of a stochastic background of GWs. We will discuss the way to detect an isotropic, gaussian background due to cosmological sources, then we will see how possible anisotropies might be detected, and finally we will briefly mention the issue of non-cosmological backgrounds. Typeset by FoilTEX

2 The cosmological background of GWs In analogy with the e.m. cosmological background, the Earth should be bathed by an uncoherent superposition of waves coming from any possible direction ˆΩin the sky. Consequently we expect that a detector l will receive a signal h l (t) = p ( ) + dωf p l ( ˆΩ) d f e i2π f t x l ˆΩ c h p ( f, Ω) ˆ (1) where p = +, represents the wave polarization, x l (t) represents the position of the detector l in a suitable inertial frame, and the components ĥ p ( f, ˆΩ) of the wave are assumed to be random variables, whose statistic determines the kind of signal one is receiving. The quantity F p l ( ˆΩ) has been met already a few times and represents the antenna pattern of the detector l, that is its response to a wave of polarization p coming from a direction ˆΩ in the sky. The signal received by two detectors l, m will be different, in general, for two reasons: a different antenna pattern, f.i. because of a different orientation in space; a different location x, which introduces a direction dependent delay, or equivalently a direction dependent dephasing between signals from different locations ˆΩ in the sky. Typeset by FoilTEX 1

3 Assumptions about the cosmological background What can we assume as for the statistics of the cosmological background? Stationarity we expect that the background does not depend on time, at least on human timescales; in other words, any correlation between variables h l (t), h m (t ) will depend only upon a time difference t t. Homogeneity we assume, at least as a first approximation that the statistical characteristics of the GW background do not depend on the direction. Of course, such an approximation is expected to be very good for cosmological backgrounds, but could be evaded badly by backgrounds of astrophysical origin, like those resulting by galactic sources! Gaussianity the statistical properties of the signal should be accounted for only in terms of second order momenta; any higher order correlation would be predictable in terms of second order. If these assumptions are all correct, it turns out that variables h p ( f, ˆΩ) obey a simple statistical relations: h p ( f, ˆΩ) h q( f, ˆΩ ) = 1 4π δ( f f )δ 2 ( ˆΩ ˆΩ 1 )δ pq H( f ) (2) 2 where H( f ) is a one-sided spectrum (mirrored to negative values of f ) expressing the energy content as a function of frequency of the waves. Typeset by FoilTEX 2

4 Is it possible to detect the background with a single detector? In a single detector, the GW background would be just an extra gaussian noise, whose strength we can easily estimate; we have h l (t)h l (t ) = pq ( ) ( + dωdω F p l ( ˆΩ)F q l ( ˆΩ ) d f d f e i2π f t x l ˆΩ c e i2π f t x l ˆΩ ) c < h p ( f, ˆΩ) h q( f,ω ) > (3) and then exploiting the δs we obtain directly h l (t)h l (t ) = p 1 4π + dω[f p l ( ˆΩ)] 2 d f e i2π f (t t ) 1 H( f ) (4) 2 that is, apart a factor of order unit, the correlation is just the Fourier transform of the spectrum 1 2H( f ). Now, Penzias and Wilson were able to assess the existence of the e.m. background by using a single microwave detector, because they could demonstrate that the extra noise they were observing was not explained by the detector noises. Is this possible for the stochastic background of cosmological origin? To this end, H( f ) should stand above the detector noise spectral density S n ( f ). Typeset by FoilTEX 3

5 Energy density of the background GWs Recall that the actual h tensor is written in terms of the polarizations as[2] h i j (t) = p d f where e p i j ( ˆΩ) are the wave polarization tensors dω h p ( f,ω)e i2π ft e p i j ( ˆΩ) (5) ê + = ˆx ˆx ŷ ŷ, ê = ˆx ŷ + ŷ ˆx (6) It is therefore possible to write down the energy density of the gravitational wave background, that we recall is just ρ gw = c3 ḣi j (t)ḣ i j (t) (7) 32πG with some boring calculation is immediate to find that ρ gw = 4c3 32πG or equivalently, per unit logarithmic frequency d f (2π f ) 2 H( f ) (8) dρ gw d log f = πc3 2G f 3 H( f ) (9) Typeset by FoilTEX 4

6 The dimensionless energy density It is customary to introduce the quantity Ω GW ( f ) 1 ρ c dρ gw d log f (10) in terms of the critical energy density ρ c, in turn defined as ρ c = 3H2 0 c3 8πG (11) where the Hubble constant H 0 = h km/(s Mpc). One obtains h 2 0Ω GW ( f ) = 4π2 h 2 0 3H 2 0 f 3 H( f ) (12) as a dimensionless measure of the spectral strength of the GW background, independent on the experimental uncertainty on h 0. Recall the most recent estimates 0.50 < h 0 < Typeset by FoilTEX 5

7 Can a single detector see the signal, then? The short answer is no. The plausible values of H( f ) are far too small to emerge from the background noise of a single detector, even an advanced one. Taking into account the existing bounds, one should aim at values h 2 0Ω GW ( f ) 10 6 at least. Recalling that 1Mpc = km, one has that H( f ) 3H2 0 4π 2 h 2 f Hz2 0 f 3 This quantity is to be compared with the square of the best sensitivity of an advanced detectors, say ( / Hz) 2, for instance at 100Hz: the signal is about a factor 10 below the detector noise. We will see that it is possible to go much below if one exploits the two detector correlations. Typeset by FoilTEX 6

8 Two detector correlation We have seen that in a single detector it is unlikely to dig the GW background signal out of the noise. However, the fact that the background is present in all detectors, with some degree of correlation, suggest that by cross-correlating the output of different detectors one can assess the presence of the signal[1]. To simplify the calculations, let us forget about antenna patterns and wave polarizations, working like the waves were scalar; we will reintroduce the needed modifications at end. Under this simplification, we need to compute < h l (t)h m (t ) >= 1 4π dω d f e i2π f (t t ) e i2π f x l x m c ˆΩ H( f ) (13) where we have already exploited the δ s. To compute the integral dω = dφd cosθ we just orient the ẑ axis as the difference x l x m. Easily we obtain < h l (t)h m (t ) >= d f e i2π f (t t ) sin(2π f x lm(t,t )/c) H( f ). (14) 2π f x lm (t,t )/c Differently from the case of coincident points, we have an oscillating factor which depends on the difference in the position of the detectors at times t, t ; a dependency due to Earth translation! Typeset by FoilTEX 7

9 The overlap reduction function If the spectrum H( f ) is broad band, without dominating spectral features, then the integral giving the correlation decays rapidly as t t gets large, because of the oscillating factor e i2π f (t t ). Under this assumption, one can limit oneself to time intervals where x lm (t,t ) = r lm is a constant, just the distance of the detectors. One can write therefore < h l (t)h m (t ) >= d f e i2π f (t t ) γ(r lm, f )H( f ) (15) where γ(r, f ) sinc[2π f r] is the so-called overlap reduction function, an oscillating factor depending on the product f r. This expression is easy to interpret: for instance,for a given detectors distance, say 1000km, the response displays maxima and minima as a function of frequency, with the first lobe ending at about 150Hz. Typeset by FoilTEX 8

10 An intuitive explanation Recall we assume the Earth being bathed, from all directions, by random gravitational waves. If two detectors are separated by a distance corresponding to half a wavelength λ, it will happen that depending on the direction of the wave propagation, the two detectors will see a signal perfectly in phase or perfectly in opposition of phase; as a consequence, the signal will average to zero, and give the first zero. In our example, at 150Hz, corresponding to λ/2 = 1000km. For frequencies smaller than 150Hz, one will have a predominance of in phase correlation, for frequencies above a dominance of out of phase correlation. However, the 1/r factor in the sinc will also depress higher order lobes. Typeset by FoilTEX 9

11 Some examples for close detectors This plot is made for real GW s and shows only the correlations among CNRS-INFN detectors, including the resonant bars. Note that because we are not considering scalar waves, the orientation matters, and at small frequencies the overlap does not go to 1. Typeset by FoilTEX 10

12 More examples including LIGO detectors This plot shows LLO against other detectors; ALLEGRO is a resonant bar practically coincident and coaligned with LLO (Livingston, Louisiana), that s why the overlap is essentially 1. Then up to ~100Hz there is a reasonable overlap among the two LIGO sites (LLO-LHO). Typeset by FoilTEX 11

13 The detection algorithm In principle, the correlation < h l (t)h m (t) > of the signal at any detector pair could be considered a measurement of the SB. However, we have the freedom to choose the way we make the correlation so as to maximize the signal wrt the noise, much as in the case of Wiener filtering. The idea is to introduce an observable which is a linear combination of the correlation at different times t, t : x = + dt T /2 T /2 dt h l (t)h m (t )Q(t t ) (16) and then we want to find a function Q which in some sense maximizes the SNR. Note that in building this average of correlations, we have set one of the integration limits to an interval [ T /2, T /2], while leaving the other running in [, + ]. This assumption will come handy later in simplifying the calculations. Computing the expectation value one obtains (see previous formulas) µ = + dt T /2 T /2 + dt d f e i2π f (t t ) H( f )γ(r f )Q(t t ) (17) which can be readily integrated in t, giving the Fourier transform of the function Q, that Typeset by FoilTEX 12

14 we can start calling a filter : µ = T /2 T /2 + dt d f H( f )γ(r f ) Q( f ) = T + d f H( f )γ(r f ) Q( f ). (18) Recall that for writing down an SNR we need not only the expectation value of the observable, but also its variance in absence of signal. The observable, at zero signal, is just x = + dt T /2 T /2 dt n l (t)n j (t )Q(t t ) (19) and has zero mean because the two detector noises are assumed to be uncorrelated (is this reasonable at all times?). The calculation of the variance is direct, albeit a bit tedious (exercise!); to be completed, it requires to remember that, for large T, approximately One obtains T /2 T /2 σ 2 = E[x 2 ] = T where S l ( f ) is the noise spectral density of the l-th detector. dt e i2π( f f )t δ( f f ) (20) d f S l ( f )S m ( f ) Q( f ) 2 (21) Typeset by FoilTEX 13

15 The signal to noise ratio Taking the ratio of µ over σ, one obtains by definition the SNR: SNR = T + d f H( f )γ(r f ) Q( f ) + d f S l( f )S m ( f ) Q( f ) 2 (22) which grows as T, independently on our choice of Q. Now we want to maximize SNR 2 (easier) which is equivalent to maximizing a ratio of the form R = [ A( f ) Q( f )d f ] 2 B( f ) Q( f ) 2 d f ; (23) this is a simple exercise (do it!), requiring to equate to zero the variation δr/δ Q( f ): one easily obtains Q = A/B, that is H( f )γ(r f ) Q( f ) = (24) S l ( f )S m ( f ) hence the SNR results to be SNR = T [ + d f H2 ( f )γ 2 ] 1/2 (r f ) (25) S l ( f )S m ( f ) which is the quantity to measure if we have any hint about H( f ). Typeset by FoilTEX 14

16 Wait; was the calculation for scalar waves? Indeed: we have simplified the treatment, but the final results are not really that different. In fact the only modification comes with the overlap reduction function, which should be defined as follows γ lm ( f ) 5 8π p=+, dωe i2π f ˆΩ x/c F p l ( ˆΩ)F p m( ˆΩ) (26) and therefore takes into account not only the distance of the detectors, but also their relative orientation, as we have already seen when considering a few real cases. Recall also to express H( f ) in terms of what the theorists use, namely Ω gw ( f ) = 1 ρ c dρ gw d log f H( f ) = 3H2 0 4π 2 c f h 3 0 Ω GW( f ) (27) Again, we underline that in order to set up a detection procedure one needs a guidance about H( f ); or one can check for different shapes of this spectral quantity, and look for the most likely one. Typeset by FoilTEX 15

17 Detection algorithm How does one proceed in practice, in order to estimate the observable? Recall its form x = + dt T /2 T /2 dt s l (t)s m (t )Q(t t ) (28) There are a few steps to be walked: 1. measure the spectral densities S l ( f ), S m ( f ), relying on the assumption that these are by far dominated by the detector noise. 2. Define Q( f ) for each spectrum H( f ) one is interested into. 3. For each time interval labeled k, of duration T, take the Fourier transform of the data s l, s m at each detector. 4. Build an estimate of the optimal correlation in Fourier domain: recall in fact that convolutions in time domain become correlations in frequency domain! x k = + d f s l ( f ) s m ( f ) Q( f ) (29) 5. Average the values of the samples x k,building the sample mean and sample variance: ˆµ N = 1 N N x k ; ˆσ N 2 = 1 N k=1 N 1 (x k ˆµ N ) 2 (30) k=1 Typeset by FoilTEX 16

18 note that the sample mean will have, in turn, a variance ˆσ N / N 6. Use a statistical criterion to assess whether the observed value ˆµ N is consistent or not with the hypothesis that the average µ of the distribution is not zero. There is a question though which might come to one s mind. Given a total observation time T tot, how should I divide it up in sub-intervals T, to be averaged over, in order to build the estimate ˆµ? Is there a criterion? Well, as long as T is sufficiently large, so that consecutive estimates x k can be considered statistically uncorrelated, the choice does not matter. In fact, ˆµ N / ˆσ N is an estimate of the SNR in a single interval, which we have seen is T. Taking the ratio of the sample mean and the sample variance of the mean ˆµ N ˆσ N / N = ˆµ N ˆσ N N N T = Ttot ; (31) the proportionality constant depends on spectral quantities, not varying with T. In real life it is necessary to take into account that the level of noise in detectors is not really constant, hence for instance the filters to be applied to the data may vary in time. Another subtle point is that all was derived under the assumption of independency of the noises in the detectors. For interferometers sitting in the same vacuum, like the two LIGO s at the Hanford site, this might not be the case. Typeset by FoilTEX 17

19 Only detector pairs? There are more than two detectors; does this help? For instance, can one build useful higher order correlations? No, the only useful statistic is given by pair correlations. Higher order statistics are just trivially computable from the pair correlations. However, checking all possible pairs can help in rejecting spurious effects, if any. Typeset by FoilTEX 18

20 The big picture: detectability The shaded regions represent COBE, pulsar and binary pulsar, BBN limits The red and purple curve are possible signals due to strings (cosmological or not). The orange curve is a prediction of GWs resulting from amplification of vacuum fluctuations during inflation. Green areas are accessible to detectors. Typeset by FoilTEX 19

21 Non isotropic backgrounds? If there are anisotropies in the background, one expects a different statistic for the correlations: hp ( f, ˆΩ)h q ( f, ˆΩ ) = δ pq δ 2 ( ˆΩ ˆΩ )H( f )P( ˆΩ) (32) where the function P will now depend on the location in the sky[3]. Such an effect would have an obvious signature; a stellar day modulation due to Earth rotation, which would have to be discriminated against instrumental modulations going as the solar day goes. How to detect? By decomposing P in spherical harmonics, then do the same exercise as before in performing the correlation, taking now into account this geometrical effect to set an independent limit for each multipole component. Necessarily, such an effect will require greater SNR than detecting an isotropic background, and the greater the multipole the larger is the signal required for detecting the effect. Typeset by FoilTEX 20

22 Limits on the multipoles contributing to anisotropies[3] Typeset by FoilTEX 21

23 A sky map of the stochastic background? One can deduce the distribution of the signal in the sky by solving the so called inverse problem; the single detector will receive, in a frequency band f, a signal h(t, f ) f = dωf(ω,t)l (Ω, f ) f where L is the luminosity of the sky in that direction and at that frequency. One such expression is valid for each detector in a network; with at least three of them, the relation can be inverted and yield a solution for L. The angular resolution is limited by λ/d, where D is the distance of the detectors. Typeset by FoilTEX 22

24 Non gaussian backgrounds Any source of gravitational waves that we are unable to resolve individually will contribute to a background that will appear as a noise in the instrument. But not necessarily Gaussian: the number of sources may not be large enough to have the central limit theorem valid. In fact, the noise statistics will rather be characterized by isolated events, Poisson distributed: how to detect them is a current research topic[5]. Typeset by FoilTEX 23

25 Conclusions The detection of GW from stochastic background is particularly difficult: the signal is known to be weak, except for somewhat exotic string models, that I personally find somewhat too optimistic. The standard cosmological background, equivalent to the e.m. one, is possibly not accessible to ground based detectors. Despite these somewhat pessimistic statements, I believe there is hope; in particular, the astrophysical backgrounds have not received yet enough attention; those due to binaries might be close to detectability for advanced detectors, provided the analysis strategy is smart enough. References [1] B.Allen and J.Romano, Phys. Rev. D 59 (1999) 102 [2] M.Maggiore, Physics Reports 331 (2000) [3] B.Allen and A.C.Ottewill, Phys. Rev. D 56 (1997) 545 [4] J.Cornish, CQG 18 (2001) 4277 [5] D.Coward, T.Regimbau, New Astronomy Reviews 50 (2006) 461 Typeset by FoilTEX 24