The optimal approach of detecting stochastic gravitational wave from string cosmology using multiple detectors

String cosmology models predict a relic background of gravitational wave produced during the dilaton-driven inflation. It's spectrum is most likely to be detected by ground gravitational wave laser interferometers (IFOs), like LIGO, Virgo, GEO, as the energy density grows rapidly with frequency. We show the certain ranges of the parameters that underlying string cosmology model using two approaches, associated with 5% false alarm and 95% detection rate. The result presents that the approach of combining multiple pairs of IFOs is better than the approach of directly combining the outputs of multiple IFOs for LIGOH, LIGOL, Virgo and GEO.


I. INTRODUCTION
Stochastic gravitational wave background, which has two origins, is one target of gravitational wave interferometers (IFOs). It might result from an extremely large number of weak astrophysical gravitational wave sources, like compact stars in binary system (see e.g. [1] for more details). It also might result from some processes of very early universe, like phase transitions or amplification of vacuum fluctuations in inflationary (see e.g. [2,3,4] for reviews). In the latter case of origin, the gravitational waves carry the earlier information of the universe than that shown by electromagnetic waves. One of the most interesting processes in the early universe is from the string cosmology [5,6], which predicts a quite different gravitational wave background spectrum from that predicted by other cosmological models for early universe. That the energy density grows rapidly with frequency [7] means that the ground IFOs may be the best detectors. Several large scale ground IFOs are in operation: Laser Interferometric Gravitational Wave Observatory (LIGO) [8] in Livingston (LIGOL) and in Hanford(LIGOH), Virgo [9] near Pisa and GEO [10] in Hanover.
Two approaches of combining 2N detectors to improve the detection ability to the stochastic gravitational wave back ground are proposed in [11]: (i)correlating the outputs of a pair of detectors, then combing the multiple pairs, and (ii) directly combing the outputs of 2N detectors. As shown in [12], for 2N detectors with equal noise level, the data observation time and the overlap functions, the optimal approach is to combine multiple pairs of two detectors comparing to directly combing 2M (M ≤ N ) detectors. But the real detectors should not be of identical noise levels and overlap functions. We plot the proposed noise curves of detectors in Fig.  1 and the overlap reduction functions in Fig. 2. A * zhuzh@bnu.edu.cn number of authors [13,14,15,16] have used the approach of combining LIGOH and LIGOL to detect the string cosmology gravitational wave background. A recent work [17] has shown that the approach of combining multiple pairs of IFOs using Virgo and LIGO and GEO can improve the detection ability to the stochastic gravitational wave background illustrated by a simulated isotropic gravitational wave background generated with an astrophysically-motivated spectral shape.
In this paper we compare the two approaches by constraining the parameter space of gravitational wave background predicted from string cosmology using LIGOH, LIGOL, Virgo, and GEO. Our result shows that the approach of combining multiple pairs of IFOs is better than the approach of directly combining the outputs of multiple IFOs for those real IFOs at their designed noise levels. Our paper is organized as follows. In Sec. II we review the two approaches of detecting a stochastic background using multiple detectors. In Sec. III after a brief review of the gravitational wave background produced by string cosmology, we implement the two approaches using four IFOs: LIGOH, LIGOL, Virgo and GEO at their designed noise levels. Our conclusion will be provided in Sec. IV shows.

II. TWO APPROACHES OF DETECTING A STOCHASTIC BACKGROUND USING MULTIPLE DETECTORS
It has been shown [11] that after correlating signals of two detectors for time T (we take T = 10 7 sec = 3months) the squared ratio of "Signal" (S) to "Noise" (N ) is given by an integral over frequency f : where P i (f ) is the one-side noise power spectral density which describes the instrument noise n i (f ) in frequency domain: Eq. 1 is under the assumption that the noise of the detectors are (i) stationary, (ii) Gaussian, (iii) statistically independent of one another and of the stochastic gravitational wave background, and (iv) much larger in magnitude than the stochastic gravitational wave background.
γ(f ) is the so-called overlap reduction function first calculated by Flanagan [18], which shows the co-response of two detectors. This is a dimensionless function of frequency f , which is determined by the relative positions and orientations of two detectors. Explicitly, whereΩ is a unit vector specifying a direction on the twosphere, ∆ x := x 1 − x 2 is the separation vector between the central stations of the two detector sites, and is the i th detector's response to a zero frequency, unit amplitude, A = +, × polarized gravitational wave, wherê X a i andŶ a i are unit vectors pointing in the direction of the detector arms. The overlap reduction function γ(f ) in Eq. 3 is normalized for coincident and coaligned detectors: γ(0) = 1. we refer the reader to [2,11] for more details about the overlap reduction function γ(f ). Two approaches were shown in [11] for multiple IFOs, the optimal approach of combing multiple detector pairs: and the optimal approach of directly combing 2N detectors: In order to detect a stochastic background with 5% false alarm and 95% detection rate, the total optimal signal to noise ratio SN R opt threshold should be 3.29.

III. DETECTING A STRING COSMOLOGY BACKGROUND USING TWO APPROACHES BY FOUR IFOS
String cosmology, also denoted in the literature as the "pre-Big-Bang (PBB) models" depicts a different view of PBB era from the "slow-roll" (standard) inflation. The dilaton-driven inflation phase is well understood, followed by a sting phase which is currently not known. The "minimal" PBB model [19] describes the sting phase following the dilaton-driven inflation phase is a constant  [22,23,24,25] curvature phase. At the end of the string phase, the dilaton reaches the present vacuum expectation value and stops. The spectrum of gravitational wave background predicted during the periods of dilaton-driven inflation phase and string phase in string cosmology is discussed in [20]. The simplest model, in which the approximate form of the spectrum is [21] Ω gw (f ) =  FIG. 3: The region of parameter space for which the gravitational wave stochastic produced in the "dilaton + string" case is detectable by the approach of combining multiple pairs of IFOs (labeled "combine pairs") and the approach of directly combining four IFOs (labeled "directly combine"). With 5% false alarm and 95% detection rate, the region above of the curves shows the detectable parameter space of the stochastic gravitational wave background produced in the "dilaton + string" case by networks of detectors. It is clear that the "combine pairs" approach has higher detection ability for the gravitational wave background. But the "combine pairs" approach is still far away from observing the stochastic gravitational wave background spectrum. The region in parameter space for which the gravitational wave stochastic background is observable by two approaches with 5% false alarm and 95% detection rate. The region above of the curves shows the detectable parameter space of the stochastic gravitational wave background produced in the "dilaton only" case by networks of detectors. It is clear that the "combine pairs" approach has more chance to detect the gravitational wave background.
is the logarithmic slope of the spectrum produced in the string phase. The spectrum depends on four parameters: the frequency f s and the fractional energy density Ω s gw produced at the end of the dilaton-driven inflation phase, the maximal frequency f 1 above which gravitational radiation is not produced and the maximum fractional energy density Ω max gw which occurs at frequency f 1 . In this paper we follow [13] setting f 1 = 1.3 × 10 10 Hz H r 5 × 10 17 GeV 1/2 (8) and assuming no late entropy production and making reasonable choices for the number of effective degrees of freedom. H r is the Hubble parameter at the string phase. The "reduced" Hubble parameter h 100 is in the range 0.4 ≤ h 100 ≤ 0.85 by observations. By virtue of Eq. 5 and Eq. 6, using given multiple IFOs, for any given set of parameters we may numerically evaluate the optimal signal to noise ratio SN R opt ; if this value is greater than 3.29 then with 5% false alarm and 95% detection rate, the background can be detected by those IFOs. We compare the two approaches for constraining the string cosmology gravitational wave background space parameter adopting LIGOH, LIGOL, Virgo and GEO at their designed noise levels [22,23,24]. The location information of the different GW observatories were obtained from [25] and references therein. The regions of detection ability to parameters space for the approach of combining multiple pairs of IFOs (labeled "combine pairs") and the approach of directly combining four IFOs (labeled "directly combine") for spectrum in Eq. 7 (labeled "dilaton + string" case) are shown in Fig. 3. We have assumed h 100 = 0.63 [26] and H r = 5 × 10 17 GeV.
The parameter β is determined by the basic physical parameters of string cosmology models which are not well known. Just to show the different detection abilities of two approaches, a phenomenological model is adopted. This model is the "dilaton only case" in [13], assuming NO stochastic background is produced during the string phase of expansion, i.e.
It is phenomenologically interesting as it is a model whose spectrum peaks in the real IFOs band. Fig. 4 shows the regions of detection abilities to parameter space for "combine pairs" approach and "directly combine" approach. We also plot the Big Bang nucleosynthesis (BBN) bounds [13,27] to see the detective chance for the assuming spectrum in Eq. 7 and Eq. 10. Note that in the "dilaton + string" case, the slope β appearing in Eq. 7 must satisfy the constraint β ≥ 0 (see e.g. [28]). As a consequence, Ω max gw = Ω gw (f 1 ) is always larger than Ω s gw = Ω gw (f s ). So, there is another bound for Ω s gw . In this paper, by assuming h 100 = 0.63 and H r = 5 × 10 17 GeV, Ω gw (f 1 ) ≃ 0.25 × 10 −6 sets a tighter bound. We also plot this tighter bound (labeled "β" bound) in Fig. 3.

IV. CONCLUSION
Two optimal approaches of combing multiple detectors to detect a stochastic gravitational wave back ground are proposed in [11]. As shown in Fig. 1 and Fig. 2, the real detectors should not be of identical noise levels and overlap functions. It is necessary to compare two optimal approaches of detecting the stochastic gravitational wave back ground for any given real IFOs. In this short paper, we have compared two approaches of combing four real ground IFOs (LIGOH, LIGOL, Virgo, GEO) to show the detection ability of the stochastic gravitational wave background predicted by string cosmology of the early universe. As shown in Fig. 3 and Fig. 4, it is clear that the approach of combing multiple pairs of IFOs shows more detective chance to string cosmology in both "dilaton + string" case and "dilaton only" case than the approach of directly combing outputs of those four IFOs. In the " dilaton only" case, as shown in Fig. 4, both approaches have the chance to observe the spectral peak between 50 Hz and 100 Hz. The approach of combing multiple pairs of IFOs could detect that background at a little higher frequency up to 160 Hz. In the " dilaton + string" case, both approaches are also far away from observing the stochastic gravitational wave background spectrum. We hope the next generation IFOs will constrain the parameter tighter.