The stochastic background from cosmic (super)strings: popcorn and (Gaussian) continuous regimes
Tania Regimbau, Stefanos Giampanis, Xavier Siemens, Vuk Mandic
TThe stochastic background from cosmic (super)strings:popcorn and (Gaussian) continuous regimes
Tania Regimbau, ∗ Stefanos Giampanis, † Xavier Siemens, and Vuk Mandic ARTEMIS, Observatoire de la Cˆote d’Azur,Universit´e de Nice Sophia-Antipolis, CNRS, 06304 Nice, France Center for Gravitation and Cosmology,Department of Physics, University of Wisconsin-Milwaukee,P.O. Box 413, Wisconsin, 53201, USA School of Physics and Astronomy, University of Minnesota-Twin Cities, USA (Dated: November 2, 2018)In the era of the next generation of gravitational wave experiments a stochasticbackground from cusps of cosmic (super)strings is expected to be probed and, ifnot detected, to be significantly constrained. A popcorn-like background can be,for part of the parameter space, as pronounced as the (Gaussian) continuous con-tribution from unresolved sources that overlap in frequency and time. We studyboth contributions from unresolved cosmic string cusps over a range of frequenciesrelevant to ground based interferometers, such as LIGO/Virgo second generation(AdLV) and Einstein Telescope (ET) third generation detectors, the space antennaLISA and Pulsar Timing Arrays (PTA). We compute the sensitivity (at 2 σ level)in the parameter space for AdLV, ET, LISA and PTA. We conclude that the pop-corn regime is complementary to the continuous background. Its detection couldtherefore enhance confidence in a stochastic background detection and possibly helpdetermine fundamental string parameters such as the string tension and the recon-nection probability. PACS numbers: 11.27.+d, 98.80.Cq, 11.25.-w, 04.80.Nn, 04.30.Db, 95.55.Ym, 07.05.Kf ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ a s t r o - ph . C O ] J un I. INTRODUCTION
Cosmic (super)strings, formed as linear topological defects during symmetry breakingphase transitions [1, 2], or in string theory inspired inflation scenarios [3–7], may producestrong bursts of gravitational waves (GW) [8–12]. In particular, the emission from cusps,where, for a short period of time, the string reaches a speed very close to the speed of light,in oscillating cosmic (super)string loops, may be strong enough to be detected by the nextgeneration of ground based detectors such as advanced LIGO/Virgo (AdLV) [13–15] andthe planned Einstein Telescope (ET) [16], the space antenna LISA [17] or Pulsar TimingArrays (PTA) such as PPTA in Parkes/Australia, NANOGrav in North America, or EPTAin Europe [18].If strings can inter-commute and form loops that decay gravitationally or through someother channel (for example, Abelian strings), the network evolves toward a scaling regime[19, 20], in which the statistical quantities that describe the network, such as the typicaldistance between strings and the average size of loops produced by the network, scale withthe cosmic time, and the string energy density is a small constant fraction of the radiationor matter density. This regime is possible because the network produces loops which decayby radiating gravitationally, and take energy out of the network. There remains someuncertainty in the size of loops produced by a cosmic string network. One possibility is thatthe size of loops is set by the gravitational back-reaction scale, the scale of perturbations onlong cosmic strings [21, 22]. In this case the loops produced are sufficiently small that theyradiate away radiate away the energy associated with their length in less than, or of order,a Hubble time. Another possibility, suggested by recent numerical simulations [23], is thatloops form at a much larger size comparable to the Hubble length at the time of formation.In this case loops live for a long time decaying gravitationally in many Hubble times.In this work we treat the strings as one-dimensional objects (zero width, or Nambu-Gotoapproximation) and we will consider the case of small loops, leaving the large loop case for afuture publication. The rate and the amplitude of the bursts depend on three parameters, thestring tension µ (we consider Nambo-Goto strings), the reconnection probability p and thetypical size of the closed loops produced in the string network ε . The closest sources can bedetected individually, while unresolved sources at higher redshift contribute to a stochasticbackground, with a popcorn-like noise on top of a (Gaussian) continuous background [8–12].In this paper, we investigate the GW signal for a grid of values in the parameter space.We compare the popcorn and the continuous contributions over a range of frequencies from10 − − Hz and discuss the constraints that could be placed on the parameters by AdLV,ET, LISA and PTA, using the standard cross correlation statistics [24].For cosmic (super)strings we find that the popcorn regime is complementary to the (Gaus-sian) continuous background. The popcorn signature detection would enhance confidence inthe Gaussian stochastic background detection and could also help determine fundamentalstring parameters such as the string tension and the reconnection probability.In section II we compute the rate of expected burst signals from cosmic string cusps. Insection III we compute the stochastic background in the popcorn and continuous regimes.In section IV we discuss the detection of the two signatures and we compute the expectedconstraints in the signal parameters space placed by future AdLV, ET, LISA and PTAexperiments. Finally, we conclude with a summary of the paper and future research thatcan be motivated by this work.
II. RATE OF COSMIC STRING BURSTS AND DETECTION REGIMES
Cusps tend to be formed a few times during each oscillation period [25]. The rate ofbursts at the observed frequency f from the redshift interval dz , from loops of length l isgiven by [11], dRdz ( f, z ) = H − ϕ V ( z )(1 + z ) − ν ( l, z )∆( f, l, z ) (2.1)where H is the Hubble constant, ϕ V ( z ) is the dimensionless co-moving volume element(Appendix A), the factor (1+ z ) − corrects for the cosmic expansion, and ν ( z, l ) is the numberof cusps per unit space-time volume from loops with lengths l at redshift z . Because theGW emission is beamed in the direction of the cusp, only a fraction ∆( f, l, z ) (the beamingfraction) can be observed at frequency f . When loops formed are small, so that the lengthis gravitationally radiated away in a Hubble time, l is given by its redshift as: l ( z ) = αt ( z ) (2.2)where t ( z ) = H − ϕ t ( z ) is the Hubble time (Appendix A), α = ε Γ Gµ , Gµ is the tension inPlanck units ( G being the Newton constant), and Γ ∼
50 is a constant related to the poweremitted by loops into GWs [11].In this case: ν ( z ) = 2 n c l ( z ) n ( l ( z ) , z ) (2.3)where n c is the number of cusps per loop oscillation (we will assume n c = 1 in average), and n ( z, l ) is the loop size distribution: n ( l, z ) = ( p Γ Gµ ) − t ( z ) − δ ( l − αt ( z ))) . (2.4)The beaming fraction is given by:∆( f, z ) ≈ θ m ( f, z ) / z < z m (2.5)where θ m ( f, z ) = [ g (1 + z ) f l ( z )] − / (2.6)is the maximum angle that the line of sight and the direction of a cusp can subtend and stillbe observed at a frequency f . The ignorance constant g absorbs factors of O(1), as wellas the fraction of the loop length l that contributes to the cusp [11]. We expect g to be ofO(1) if loops are smooth.The cutoff redshift is solution of the equation θ m ( f, z m ) = 1 and is shown in Fig. 1.Interestingly, the dependence on the string parameters Gµ , ε , p and on the frequency f isgoing to enter through a pre-factor, and we can then re-write the expression of the rate as: dRdz ( f, z ) = B ( f ) F ( z ) with z < z m ( f ) (2.7)with F ( z ) = ϕ − / t ( z ) ϕ V ( z )(1 + z ) − / (2.8)and B ( f ) = g − / n c H / f − / / ( Gµ ) / ε / p . (2.9)The rate as a function of the redshift z for different sets of parameters and for a frequency f = 1 Hz is plotted in Figure 2. The rate increases with z until it reaches the cutoff z m ,with a slope which is larger in the radiation era after z eq ∼ p , the tension Gµ or thetypical length ε . (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) f (Hz) z m ( f ) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) f (Hz) z * ( f ) FIG. 1: Left: Maximal contributing redshift as a function the frequency f . In the radiation era( z > z eq ∼ z m ( f ) = 1 . ∗ Gµεf . Right: Transition redshift between popcorn and(Gaussian) continuous regimes (Λ = 10), as a function of the frequency. The grey continuous linecorresponds to p = 1, ε = 1 and Gµ = 10 − , the black continuous line to p = 1, ε = 1 and Gµ = 10 − , the black dotted line to p = 1, ε = 1 and Gµ = 10 − , the black dashed line to p = 10 − , ε = 1 and Gµ = 10 − , and the black dot-dashed line to p = 1, ε = 10 − and Gµ = 10 − .The vertical dashed line indicates the transition redshift between matter and radiation eras. The GW signal from the population of cosmic strings falls into three different statisticalregimes, characterized by the value of the quantity (see Figure 2):Λ( f, z ) = τ (cid:90) z dRdz ( f, z ) dz (2.10)where τ is the duration of the signal (typically 1 /f at the frequency f [10, 26] ), and theintegral is the inverse of the time interval between successive events arriving from redshift < z . This quantity is often called the duty cycle [26] and is simply the average number ofsources overlapping in a typical frequency band ∼ f , around the frequency f . It can becompared to the overlap function of [27], which is the number of sources present, in average,in a frequency bin ∆ f around the frequency f . Notice that taking ∆ f ∼ /T , where T is the observation time, determines whether sources create a confusion background in theframework of single source detection [27], which is not the purpose of this work, where weare interested in the detection of the background itself.Based on the value of Λ we distinguish the following three regimes:1. shot noise : at low redshift, the number of sources is small enough for the time interval (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) z d R / d z ( f = H z , z ) (cid:239) (cid:239) z max (cid:82) ( f = H z ) FIG. 2: Left: Rate of cosmic strings per interval of redshift at the frequency f = 1 Hz. Right:Average number of bursts overlapping at the frequency f = 1 Hz, as a function of the maximalredshift z max . For other frequencies, the rate can be deduced from these plots by multiplying dR/dz by f / . The grey continuous line corresponds to p = 1, ε = 1 and Gµ = 10 − , the black continuousline to p = 1, ε = 1 and Gµ = 10 − , the black dotted line to p = 1, ε = 1 and Gµ = 10 − , theblack dashed line to p = 10 − , ε = 1 and Gµ = 10 − , and the black dot-dashed line to p = 1, ε = 10 − and Gµ = 10 − . between events to be long compared to the duration of a single event. Sources areseparated by long stretches of silence and the closest ones may be detected individually.2. popcorn noise : when the redshift and the number of sources increases, the timeinterval between events becomes comparable to the duration of a single event. Thenumber of sources present at the frequency f is a Poisson process, sometimes there isno GW signal, sometimes sources overlap, and the sum of the amplitudes at a giventime is still unpredictable. These signals, which sound like crackling popcorn, areknown as popcorn noise .3. continuous : the number of sources is large enough for the time interval betweenevents to be small compared to the duration of a single event. Sources overlap at thefrequency f to create a continuous background (there is always a GW signal present)that is Gaussian in nature (due to the central limit theorem, if the number of sources islarge, the sum of their amplitudes has a Gaussian distribution) and can be confoundedwith the detector noise.[8–10] first discussed the presence of a popcorn-like noise on top of a continuous stochasticbackground for cosmic strings . They adopted the value of Λ = 1 as the limit between thecontinuous and popcorn regime. In this paper we follow [28] and consider a more conservativevalue of Λ = 10, in order to ensure also Gaussianity. It is worth pointing out that the resultsare not too sensitive to this choice. III. THE STOCHASTIC BACKGROUND
The spectrum of the gravitational stochastic background is usually characterized by thedimensionless parameter [24]: Ω gw ( f ) = 1 ρ c dρ gw d ln f (3.1)and for the case of cosmic strings is given by[12]:Ω gw ( f ) = 4 π H f (cid:90) z m ( f ) z ˜ h ( f, z ) dRdz ( f, z ) dz (3.2)where we have added up individual contributions at all redshifts, excluding the very nearby( z ≈ − or 5 kpc), and where the gravitational strain produced by a cosmic string cuspis [8–12], ˜ h ( f ) = Af − / Θ( f h − f )Θ( f − f l ) . (3.3)The low frequency cutoff of the gravitational wave signal, f l , is determined by the size ofthe cusp – a scale which is typically cosmological. As a result the low frequency cutoffof detectable radiation is determined by the low frequency behavior of the instrument (forterrestrial detectors, for instance, by seismic noise). The high frequency cutoff depends onthe angle between the line of sight and the direction of the cusp θ , as f h = [ l ( z )(1 + z ) θ ] − . (3.4)The amplitude A of a cusp from a loop of length l at a redshift z is given by [11], A = g Gµl / H (1 + z ) / ϕ r ( z ) . (3.5)= g Γ / H / ( Gµ ) / ε / ϕ t ( z ) / (1 + z ) / ϕ r ( z ) . (3.6)Here g is an ignorance constant that absorbs the uncertainty on exactly how much of thelength l is involved in the production of the cusp and ϕ r ( z ) is the dimensionless properdistance (Appendix A).We define the stochastic background as the sum of the continuous and popcorn contri-butions: Ω gw ( f ) = Ω popgw ( f ) + Ω contgw ( f ) (3.7)where Ω popgw ( f ) = 4 π H f (cid:90) z ∗ ( f ) z ˜ h ( f, z ) dRdz ( f, z ) dz (3.8)and Ω contgw ( f ) = 4 π H f (cid:90) z m ( f ) z ∗ ( f ) ˜ h ( f, z ) dRdz ( f, z ) dz (3.9)where z ∗ ( f ) is the redshift at which Λ( z ∗ ( f )) = 10 (see Fig. 1)The shape of the continuous background is determined by a very sharp rise at f ≈ . × − ε − ( Gµ ) − where sources at low redshift start to become observable ( θ m ( f , z ) = 1),followed by a decrease ( ∼ f − / ), and a flat region where most of the sources belong tothe radiation era and where the increase of the rate with frequency cancels the decrease oftheir contribution to Ω gw . The popcorn background has a similar behavior, except for theflat part of the spectrum. It raises more smoothly at lowest frequencies because the upperlimit of the integral of Eq. 3.8 (the transition redshift z ∗ ) increases slower with frequencythan z m . Near f there is a very small region where the popcorn background dominates as z m is still in the popcorn regime or very close to the transition redshift so that there is aninsignificant continuous contribution. Sometimes this region is not be visible in the plots dueto our numerical precision. Figures 3, 4 and 5 compare the two contributions for differentsets of parameters. Depending on the parameters, the popcorn regime may overwhelm thecontinuous background in the frequency range of AdLV, ET, LISA and PTA. The parameters Gµ , ε and p affect the amplitude of the power spectra through an overall scaling factor F s ∝ ε − / ( Gµ ) / p − but also through the bounds z m ∝ εGµ and z ∗ of the integrals overredshift. In the case of ε the two effects compensate so that finally, the amplitude doesn’tdepend on this parameter. Another interesting feature is that the parameter p doesn’t affectthe two backgrounds the same way. In fact, unlike z m ( f ), the popcorn maximal redshiftdepends on the parameter p . Decreasing the reconnection probability p increases the overallfactor F s but also reduces z ∗ , canceling part of the gain.Figure 6 shows the corresponding regions in the plane ε - Gµ with constant ratio R =Ω popgw / Ω contgw , for frequencies f = 100 , − and 10 − Hz, typical for terrestrial detectors (AdLV,ET), LISA and PTA, and for two different values of the parameter p ( p = 1 and p = 10 − ). (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) f (Hz) (cid:49) g w ( f ) G µ =10 (cid:239) p=1 FIG. 3: Popcorn (grey) and continuous (black) contributions to Ω gw ( f ) for p = 1, Gµ = 10 − and ε = 1 (continuous line), 10 − (dashed line) and 10 − (dot-dashed line) . Increasing ε shifts thespectra toward lower frequencies but doesn’t affect the amplitude. For p = 1, in the denser regions from dark black to heavy grey, the popcorn backgroundcontribution dominates and is expected to significantly contribute to the signal-to-noise ratio(see section IV). For p = 10 − , the background is always dominated by the continuous con-tribution. In the complementary light grey regions the continuous contribution dominates.Decreasing the frequency, the popcorn dominated area is shifted toward larger values of ε .For the LISA typical frequency f = 10 − , the popcorn dominated area is slightly reduced,while for the PTA frequency f = 10 − , most of this region disappears outside the range ofvalues for ε , so that the continuous background dominates in almost all of the parameterspace, except for the largest values of both ε and Gµ . The effect of decreasing p is similarto decreasing f and the region where the popcorn contribution dominates is even smaller at0 (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) f (Hz) (cid:49) g w ( f ) (cid:161) =10 (cid:239) p=1 FIG. 4: Popcorn (grey) and continuous (black) contributions to Ω gw ( f ) for p = 1, ε = 10 − and Gµ = 10 − (continuous line), 10 − (dashed line) and 10 − (dot-dashed line). Increasing Gµ shiftsthe spectra toward lower frequencies, increases the amplitude and the relative importance of thecontinuous contribution compared to the popcorn contribution. LISA and PTA frequencies.
IV. PARAMETER SPACE CONSTRAINTS
In this section we concentrate on the detection of the stochastic background withplanned/proposed future detectors and the bounds they can place in the parameter space.For a given detector we assume that the gravitational wave emission from a single cosmicstring cusp is present for approximately τ = 1 /f L , where f L is the lowest observable detectorfrequency. We then compute the popcorn and continuous contributions to the stochastic1 (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) f (Hz) (cid:49) g w ( f ) G µ =10 (cid:239) (cid:161) =10 (cid:239) FIG. 5: Popcorn (grey) and continuous (black) contributions to Ω gw ( f ) for Gµ = 10 − , ε = 10 − and p = 1 (continuous line), 0 . − (dot-dashed line). Increasing p shiftsthe popcorn spectrum toward lower frequencies, decreases the amplitude of both the continuousand popcorn spectra and the relative importance of the continuous contribution compared to thepopcorn contribution. background using Eq. 3.8 and 3.9, requiring that Λ( f ) is smaller or larger than 10.For terrestrial interferometers (AdLV, ET) we use the frequency domain method of cross-correlation between pairs of detectors [24]. This technique has been shown to be optimalfor continuous stochastic backgrounds and to perform nearly optimally for the popcorncontribution down to very small values of Λ [29] (much smaller than our threshold value ofΛ = 10). The signal-to-noise ratio ( SN R ), for an integration time T , obtained by cross-2 log(G µ ) l og ( (cid:161) ) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239)
20 0.511.5 log(G µ ) l og ( (cid:161) ) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239)
20 0.10.20.30.40.50.60.7 log(G µ ) l og ( (cid:161) ) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239)
20 0.511.5 log(G µ ) l og ( (cid:161) ) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239)
20 0.10.20.30.40.50.60.7 log(G µ ) l og ( (cid:161) ) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239)
20 0.511.5 log(G µ ) l og ( (cid:161) ) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239)
20 0.10.20.30.40.50.60.7
FIG. 6: ratio R = Ω popgw / Ω contgw for p = 1 (left) and for p = 10 − (right), and from top to bottom at f = 100, 10 − and 10 − Hz. For p = 1, in the denser regions from dark black to grey, the popcornbackground contribution dominates and is expected to significantly contribute to the signal-to-noise ratio (see section IV). For p = 10 − , on the other hand the background is always dominatedby the continuous contribution. Notice that the greyscale color bar has different scales in the twocases. The small sharp area at the right bottom of the two last plots of the first column is not anartefact, but corresponds to the lower frequency part of the spectrum where the number of sourcesis larger in the popcorn regime than in the continuous regime (see discussion in section III). In thewhite region in the bottom left corner, both the popcorn and the continuous backgrounds are null(Ω gw = 0). SN R = 3 H F π √ T (cid:20)(cid:90) ∞ df γ ( f )Ω ( f ) f P ( f ) P ( f ) (cid:21) / (4.1)where γ is the normalized overlap reduction function characterizing the loss in sensitivitydue to the separation and the relative orientation of the detectors, F = < F + F × > =2 sin ( α ) / α is the opening angle betweenthe interferometer’s arms ( π/ π/ P and P are the strain noisepower spectral densities of the two detectors.For LISA and PTA we simply compare the GW signal to the expected sensitivity (usingthe LISA strain noise power spectral density of the standard Michelson configuration [17]and the Parkes PTA’s projected sensitivity [30]). It is worthwhile to notice that for LISAit may be possible to combine the symmetrized Sagnac with the Michelson configurationand nearly achieve the sensitivity of cross correlating two LISA detectors [31, 32]. Howevergiving the uncertainties on the planned configuration, we preferred not to consider it in thispaper.Figures 7, 8, 9 and 10 show regions in the ε - Gµ plane where both the popcorn and thecontinuous contributions can be potentially constrained. We compute the sensitivity (at2 σ level), assuming one year integration time, and following [33]. For comparison purpose,we also show the bounds from Big Bang Nucleosynthesis (BBN) and CMB observations,including the future Planck experiment [12, 33] on the GW stochastic background formedbefore BBN ( z ∼ × ) and before the CMB photons decoupled ( z ∼ Gµ (left side in the ε - Gµ plane) the individual SNR contributions (popcornand continuous) are small and are not expected to be detected. For large Gµ and small ε values (bottom right region) both popcorn and continuous contributions are likely to beprobed.Figure 11 shows contours in the plane ε - Gµ with constant ratio between the popcornand continuous contributions to the SNR, for ground based detectors and for LISA. Forhigh values of p , the popcorn contribution can overwhelm the continuous background insome region of the parameter space. Even though there is a non negligible region where thepopcorn background gives the largest SNR (see Fig 11), the area of the parameter space4probed by the popcorn contribution is always included in the area probed by the continuousbackground. This may change in the future with the development of specific data analysistechniques that could perform up to a few times better than the standard cross correlationstatistic used in this paper [29, 34, 35]. Also, for cosmologies other than the standard oneused in this paper, for example (see Appendix A), if the redshift at the transition betweenthe matter and radiation era z eq were larger than ∼ V. CONCLUSION
The GW emission from the population of cusps of small loops of cosmic strings fall intotwo regimes with very different statistical properties: sources at large redshift overlap tocreate a Gaussian and continuous stochastic background, while close sources create a non-Gaussian and non-continuous popcorn-like signal. The Gaussian continuous background iscompletely characterized by its spectral properties and can be detected by the standard crosscorrelation methods in the frequency domain. The popcorn background is less predictable,as it may show important variations in the time domain.In this paper, we investigated the popcorn and continuous (Gaussian) contribution tothe background mapping it onto the cosmic string parameter space. The transition betweenthe two regimes depends on the frequency (the larger the frequency, the smaller the tran-sition redshift z ∗ ) and on the cosmic string parameters, in particular on the reconnectionprobability p (the larger is this parameter, the larger is the rate and the smaller is z ∗ ).We found that the popcorn contribution may dominate in different regions of the param-eter space and over different frequency ranges. It is therefore worthwhile to develop dataanalyses methods that can better capture the popcorn signature as well as the continuous.Future gravitational wave experiments, such as Advanced LIGO/Virgo, Einstein Telescope,LISA or PTA, may be able to observe both gravitational wave signatures.We computed the sensitivity (at 2 σ level) in the cosmic strings parameter space for AdLV,ET, LISA and PTA. The deduced regions cover a large area of the parameter space but thepopcorn contribution to the stochastic background is expected to be more pronounced athigher frequencies where ground based detectors operate. The case of large loop cosmicstrings, of relevance to LISA and PTA, will be the subject of a following study.5 FIG. 7: AdLV sensitivity (at 2 σ level) in the plane (( Gµ )) − ε for string parameters p = 0.001,0.01, 0.1, 1. The continuous and popcorn contributions, in light and darker grey colors respectively,represent the part of the parameter space that is expected to be probed in a search for a stochasticbackground using a pair of coincident and co-located AdLV detectors and integrating over a year.The solid dark line indicates the sensitivity region (to the right of the curve) arising from thesum of the popcorn and continuous contribution. For comparison we also show the Big BangNucleosynthesis (BBN) and CMB observation limits on Ω, including the future Planck experiment[12, 33] on the GW stochastic background formed before BBN ( z ∼ . × ) and before the CMBphotons decoupled ( z ∼ In this paper we used the same method for the popcorn and for the continuous back-grounds, but specific data analysis techniques that could perform up to a few times betterthan the cross-correlation statistics when the background is not Gaussian have been pro-posed [29, 34, 35] and are currently investigated in the LIGO/Virgo collaboration.6
FIG. 8: ET-B (ET in broadband sensitivity configuration) sensitivity (at 2 σ level) in the plane(( Gµ )) − ε for string parameters p = 0.001, 0.01, 0.1, 1. An important aspect of detecting the popcorn and continuous background, which furthermotivates the development of specific data analysis techniques in this case, is the possibilityof pinning down the parameter space as additional statistical information is carried by thepopcorn sector of the spectrum. As shown in this work, in part of the parameter spacethe relative contribution of the popcorn and continuous backgrounds to the measured signalto noise ratio crucially depends on both the string tension and the reconnection probabil-ity. Breaking the degeneracy in these parameters requires, as expected, two measurements.Further work on parameter estimation is currently pursued by some of the authors.7
FIG. 9: LISA sensitivity (at 2 σ level) in the plane (( Gµ )) − ε for string parameters p = 0.001, 0.01,0.1, 1. Acknowledgments
The authors thank J.Romano and G. Cella for careful reading and valuable comments.X.S. acknowledges the support from NSF grant PHY-0970074, PHY-0955929 and PHY-0758155. S. G. acknowledges the support from NSF grant PHY-0970074 and UWM’s Re-search Growth Initiative.
Appendix A: Cosmological functions
The dimensionless cosmological functions ϕ t ( z ), ϕ r ( z ) and ϕ V ( z ) were calculated usinga vanilla Λ-CDM model. We adopted the cosmological parameters derived from 7 years ofWMAP observations [36]: H = 72 km s − for the Hubble parameter, Ω m = 0 .
279 for the8
FIG. 10: PTA sensitivity in the plane (( Gµ )) − ε for string parameters p = 0.001, 0.01, 0.1, 1.PTA’s sensitivity in Ω gw corresponds to a 0 .
1% false alarm rate and a 95% detection rate upperbound (using [30] table 3 for α = − σ level. density of matter, Ω r = 8 . × − for the energy density of radiation, and assuming a flatuniverse Ω Λ = 1 − Ω m − Ω r .The dimensionless cosmological time at redshift z is given by: ϕ t ( z ) = (cid:90) z dz (cid:48) (1 + z (cid:48) ) E (Ω , z (cid:48) ) , (A1)where E (Ω , z ) = (cid:112) Ω Λ + Ω m (1 + z ) + Ω r (1 + z ) (A2)The dimensionless proper distance at redshift z by: ϕ r ( z ) = (cid:90) z dz (cid:48) E (Ω , z (cid:48) ) (A3)9 log (G µ ) l og ( (cid:161) ) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239)
20 0.20.40.60.811.21.41.6 log (G µ ) l og ( (cid:161) ) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239)
20 0.20.40.60.811.21.41.6 l og ( (cid:161) ) log (G µ ) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239)
20 0.20.40.60.811.21.41.6 log (G µ ) l og ( (cid:161) ) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239) (cid:239)
20 0.511.5
FIG. 11: ratio R = SN R pop /SN R cont for p = 1, for a pair of separated AdLV (Livingston andHanford LIGO detectors), a pair of coincident and co-located (CC) AdLV (the 2 LIGO Hanforddetectors) (top right), two V-shaped co-located ET detectors (bottom left) and LISA (bottomright). In the denser black regions the popcorn background has a larger contribution in the signal-to-noise ratio. In the lighter grey regions, on the other hand, the continuous contribution dominates.In the white region in the bottom left corner, there is negligible GW signal from cosmic strings inthe considered frequency range. And the dimensionless volume at redshift z by: ϕ V ( z ) = 4 π ϕ r ( z ) (1 + z ) E (Ω , z ) (A4)The redshift at the transition between the matter and the radiation dominated eras canbe deduced form Eq. A2: z eq = Ω m Ω r − ∼ [1] A. Vilenkin and E. Shellard, Cosmic strings and other Topological Defects (Cambridge Uni-versity Press, 2000).[2] M. B. Hindmarsh and T. W. B. Kibble, Reports on Progress in Physics , 477 (1995), URL http://stacks.iop.org/0034-4885/58/i=5/a=001 .[3] N. Jones, H. Stoica, and S.-H. H. Tye, Journal of High Energy Physics , 51 (2002), arXiv:hep-th/0203163.[4] N. T. Jones, H. Stoica, and S.-H. H. Tye, Physics Letters B , 6 (2003), arXiv:hep-th/0303269.[5] S. Sarangi and S.-H. H. Tye, Physics Letters B , 185 (2002), arXiv:hep-th/0204074.[6] G. Dvali and A. Vilenkin, Journal of Cosmology and Astroparticle Physics , 10 (2004),arXiv:hep-th/0312007.[7] E. J. Copeland, R. C. Myers, and J. Polchinski, Journal of High Energy Physics , 13 (2004),arXiv:hep-th/0312067.[8] T. Damour and A. Vilenkin, Phys. Rev. Lett. , 3761 (2000), URL http://link.aps.org/doi/10.1103/PhysRevLett.85.3761 .[9] T. Damour and A. Vilenkin, Phys. Rev. D , 064008 (2001), URL http://link.aps.org/doi/10.1103/PhysRevD.64.064008 .[10] T. Damour and A. Vilenkin, Phys. Rev. D , 063510 (2005), URL http://link.aps.org/doi/10.1103/PhysRevD.71.063510 .[11] X. Siemens, J. Creighton, I. Maor, S. R. Majumder, K. Cannon, and J. Read, Phys. Rev. D , 105001 (2006), URL http://link.aps.org/doi/10.1103/PhysRevD.73.105001 .[12] X. Siemens, V. Mandic, and J. Creighton, Phys. Rev. Lett. , 111101 (2007), URL http://link.aps.org/doi/10.1103/PhysRevLett.98.111101 .[13] G. M. Harry and the LIGO Scientific Collaboration, Classical and Quantum Gravity ,084006 (2010).[14] the Advanced LIGO Team (2007), URL https://dcc.ligo.org/cgi-bin/DocDB/ShowDocument?docid=m060056 .[15] G. Losurdo and the Advanced Virgo Team (2007), URL https://tds.ego-gw.it/ql/?c=1900 . [16] M. Punturo et al., Classical and Quantum Gravity , 194002 (2010), URL http://stacks.iop.org/0264-9381/27/i=19/a=194002 .[17] P. L. Bender and the LISA Study Team (1998), URL http://list.caltech.edu/lib/exe/fetch.php?media=documents:early:prephasea.pdf .[18] R. N. Manchester, in American Institute of Physics Conference Series , edited by M. B. et al.(2011), vol. 1357, pp. 65–72, 1101.5202.[19] P. McGraw, Phys. Rev. D , 3317 (1998), arXiv:astro-ph/9706182.[20] D. Spergel and U.-L. Pen, ApJL , L67 (1997), arXiv:astro-ph/9611198.[21] F. Dubath, J. Polchinski, and J. V. Rocha, Phys.Rev. D77 , 123528 (2008), 0711.0994.[22] X. Siemens, K. D. Olum, and A. Vilenkin, Phys.Rev.
D66 , 043501 (2002), gr-qc/0203006.[23] J. J. Blanco-Pillado, K. D. Olum, and B. Shlaer, Phys.Rev.
D83 , 083514 (2011), 1101.5173.[24] B. Allen and J. D. Romano, Phys. Rev. D , 102001 (1999), URL http://link.aps.org/doi/10.1103/PhysRevD.59.102001 .[25] N. Turok, Nuclear Physics B , 520 (1984).[26] A. Buonanno, G. Sigl, G. G. Raffelt, H.-T. Janka, and E. M¨uller, Phys. Rev. D , 084001(2005), URL http://link.aps.org/doi/10.1103/PhysRevD.72.084001 .[27] P. A. Rosado, Phys. Rev. D , 084004 (2011), 1106.5795.[28] D. Coward and T. Regimbau, New Astronomy Reviews p. 461 (2006).[29] S. Drasco and E. E. Flanagan, Phys. Rev. D , 082003 (2003), URL http://link.aps.org/doi/10.1103/PhysRevD.67.082003 .[30] F. A. Jenet, G. B. Hobbs, W. van Straten, R. N. Manchester, M. Bailes, J. P. W. Verbiest,R. T. Edwards, A. W. Hotan, J. M. Sarkissian, and S. M. Ord, The Astrophysical Journal , 1571 (2006), URL http://stacks.iop.org/0004-637X/653/i=2/a=1571 .[31] C. J. Hogan and P. L. Bender, Phys. Rev. D , 062002 (2001).[32] A. Vecchio, Classical and Quantum Gravity , 1449 (2002), URL http://stacks.iop.org/0264-9381/19/i=7/a=329 .[33] B. P. Abbott et al., Nature , 990 (2009), URL http://dx.doi.org/10.1038/nature08278 .[34] N. Seto, Phys. Rev. D , 043003 (2009), URL http://link.aps.org/doi/10.1103/PhysRevD.80.043003 .[35] D. M. Coward and R. R. Burman, Monthly Notices of the Royal Astronomical Society , 362 (2005), ISSN 1365-2966, URL http://dx.doi.org/10.1111/j.1365-2966.2005.09178.x .[36] D. Larson, J. Dunkley, G. Hinshaw, E. Komatsu, M. R. Nolta, C. L. Bennett, B. Gold,M. Halpern, R. S. Hill, N. Jarosik, et al., Astrophysical Journal Supplement192