You are an expert at summarizing long articles. Proceed to summarize the following text: 
the merger of two neutron stars , two black holes or a black hole and a neutron star are among the most important sources of gravitational waves ( gw ) , due to the huge energy released in the process . in particular , the coalescence of double neutron stars ( dns ) may radiate about 10@xmath6 erg in the last seconds of their inspiral trajectory , at frequencies up to 1.4 - 1.6 khz , range covered by most of the ground - based laser interferometers like virgo @xcite , ligo @xcite , geo @xcite or tama @xcite . besides the amount of energy involved in these events , the rate at which they occur in the local universe is another parameter characterizing if these mergings are or not potential interesting sources of gw . in spite of the large amount of work performed in the past years , uncertainties persist in estimates of the dns coalescence rate . in a previous investigation , we have revisited this question @xcite , taking into account the galactic star formation history derived directly from observations and including the contribution of elliptical galaxies when estimating the mean merging rate in the local universe . based on these results , we have predicted a detection rate of one event every 125 and 148 years by initial ligo and virgo respectively and up to 6 detections per year in their advanced configurations .    besides the emission produced by the coalescence of the nearest dns , the superposition of a large number of unresolved sources at high redshifts will produce a stochastic background of gw . in the past years , different astrophysical processes susceptible to generate a stochastic background have been investigated . on the one hand , distorted black holes @xcite , bar mode emission from young neutron stars @xcite are examples of sources able to generate a shot noise ( time interval between events large in comparison with duration of a single event ) , while supernovas or hypernovas @xcite are expected to produce an intermediate  popcorn " noise . on the other hand , the contribution of tri - axial rotating neutron stars @xcite , including magnetars @xcite , constitutes a truly continuous background . populations of compact binaries as , for instance , the cataclismic variables are responsible for the existence of a galactic background of gw in the mhz domain , which could represent an important source of confusion noise for space detectors as lisa @xcite . these investigations have been extended recently to the extra - galactic contribution . @xcite , @xcite and @xcite considered cosmological populations of double and mixed systems involving black holes , neutron stars and white dwarfs , while close binaries originated from low and intermediate mass stars were discussed by @xcite .    in this work , using the dns merging rate estimated in our precedent study , we have estimated the gravitational wave background spectrum produced by these coalescences . numerical simulations based on monte carlo methods were performed in order to determine the critical redshift @xmath7 beyond which the duty cycle condition required to have a continuous background ( @xmath8 ) is satisfied . unlike previous studies which focus their attention on the early low frequency inspiral phase covered by lisa @xcite , here we are mainly interested in the few thousand seconds before the last stable orbit is reached , when more than 96% of the gravitational wave energy is released . the signal frequency is in the range 10 - 1500 hz , covered by ground based interferometers . the paper is organized as follows . in  2 , the simulations are described ; in  3 the contribution of dns coalescences to the stochastic background is calculated ; in  4 the detection possibility with laser beam interferometers is discussed and , finally , in  5 the main conclusions are summarized . in order to simulate by monte carlo methods the occurrence of merging events , we have adopted the following procedure . the first step was to estimate the probability for a given pair of massive stars , supposed to be the progenitors of dns , be formed at a given redshift . this probability distribution is essentially given by the cosmic star formation rate @xcite , normalized in the redshift interval @xmath9 , e.g. , @xmath10 the normalization factor in the denominator is essentially the rate at which massive binaries are formed in the considered redshift interval , e.g. , @xmath11 which depends on the adopted cosmic star formation rate , as we shall see later . the formation rate of massive binaries per redshift interval is @xmath12 in the equation above , @xmath13 is the cosmic star formation rate ( sfr ) expressed in m@xmath14 mpc@xmath15yr@xmath16 and @xmath17 is the mass fraction converted into dns progenitors . hereafter , rates per comoving volume will always be indicated by the superscript * , while rates with indexes  z@xmath18 " or z@xmath19 " refer to differential rates per redshift interval , including all cosmological factors . the ( 1+z ) term in the denominator of eq . [ eq - frate ] corrects the star formation rate by time dilatation due to the cosmic expansion . in the present work we assume that the parameter @xmath20 does not change significantly with the redshift and thus it will be considered as a constant . in fact , this term is the product of three other parameters , namely , @xmath21 where @xmath22 is the fraction of binaries which remains bounded after the second supernova event , @xmath23 is the fraction of massive binaries formed among all stars and @xmath24 is the mass fraction of neutron star progenitors . according to the results by @xcite,@xcite , @xmath22 = 0.024 and @xmath23 = 0.136 , values which will be adopted in our calculations . assuming that progenitors with initial masses above 40 m@xmath14 will produce black holes and considering an initial mass function ( imf ) of the form @xmath25 , with @xmath26 = 2.35 ( salpeter s law ) , normalized within the mass interval 0.1 - 80 m@xmath14 such as @xmath27 = 1 , it results finally @xmath28 and @xmath29  m@xmath30 . the evaluation of the parameters @xmath22 and @xmath23 depends on different assumptions , which explain why estimates of the coalescence rate of dns found in the literature may vary by one or even two orders of magnitude . the evolutionary scenario of massive binaries considered in our calculations ( see @xcite for details ) is similar to that developed by @xcite , in which none of the stars ever had the chance of being recycled by accretion . besides the evolutionary path , the resulting fraction @xmath22 of bound ns - ns binaries depends on the adopted velocity distribution of the natal kick . the imparted kick may unbind binaries which otherwise might have remained bound or , less probably , conserve bound systems which without the kick would have been disrupted . the adopted value for @xmath22 corresponds to a 1-d velocity dispersion of about 80 km / s . this value is smaller than those usually assumed for single pulsars , but consistent with recent analyses of the spin period - eccentricity relation for ns - ns binaries @xcite . had we adopted a higher velocity dispersion in our simulations ( 230 km / s instead of 80 km / s ) , the resulting fraction of bound systems is reduced by one order of magnitude , e.g. , @xmath22 = 0.0029 . if , on the one hand the fraction of bound ns - ns systems after the second supernova event depends on the previous evolutionary history of the progenitors and on the kick velocity distribution , on the other hand estimates of the fraction @xmath23 of massive binaries formed among all stars depends on the ratio between single and double ns systems in the galaxy and on the value of @xmath22 itself @xcite . we have estimated relative uncertainties of about @xmath31 and @xmath32 , leading to a relative uncertainty in the parameter @xmath20 of about @xmath33 . however , we emphasize that these are only formal uncertainties resulting from our simulations , which depend on the adopted evolutionary scenario for the progenitors . a comparison with other estimates can be found in @xcite . the element of comoving volume is given by @xmath34 with @xmath35^{1/2}\ ] ] where @xmath36 and @xmath37 are respectively the present values of the density parameters due to matter ( baryonic and non - baryonic ) and vacuum , corresponding to a non - zero cosmological constant . a  flat " cosmological model ( @xmath38 ) was assumed . in our calculations , we have taken @xmath36 = 0.30 and @xmath37 = 0.70 , corresponding to the so - called  concordance " model derived from observations of distant type ia supernovae @xcite and the power spectra of the cosmic microwave background fluctuations @xcite . the hubble parameter h@xmath39 was taken to be @xmath40 . @xcite provide three models for the cosmic sfr history up to redshifts @xmath41 . differences among these models are mainly due to various corrections applied , in particular those due to extinction by the cosmic dust . in our computations , we have considered the second model , labelled sfr2 @xcite but numerical results using sfr1 @xcite will be also given for comparison . both rates increase rapidly between @xmath42 and peak at @xmath43 , but sfr1 decreases gently after @xmath44 while sfr2 remains more or less constant ( figure  [ fig - sfr ] ) . the next step consists to estimate the redshift @xmath45 at which the progenitors have already evolved and the system is now constituted by two neutron stars . this moment fixes also the beginning of the inspiral phase . if @xmath46 ( @xmath47 yr ) is the mean lifetime of the progenitors ( average weighted by the imf in the interval 9 - 40 m@xmath14 ) then    @xmath48    once the beginning of the inspiral phase is established , the redshift at which the coalescence occurs is estimated by the following procedure . the duration of the inspiral phase depends on the orbital parameters just after the second supernova and on the neutron star masses . the probability for a given dns system to coalesce in a timescale @xmath49 was initially derived by @xcite , confirmed by subsequent simulations @xcite and is given by    @xmath50    simulations indicate a minimum coalescence timescale @xmath51 yr but a considerable number of systems have a coalescence timescale higher than the hubble time . the normalized probability in the range @xmath52 yr up to 20 gyr implies @xmath53 . therefore , the redshift @xmath54 at which the coalescence occurs after a timescale @xmath49 is derived from the equation @xmath55 which was solved in our code by an iterative method . the resulting distribution of the number of coalescences as a function of @xmath54 is shown in figure  [ fig - dist_coalescence ] for both star formation rates sfr1 and sfr2 , while the corresponding coalescence rate per redshift interval , @xmath56 , is shown in figure  [ fig - sfr ] . in the same figure , for comparison , we have plotted the formation rate @xmath57 ( eq . [ eq - frate ] ) . notice that the maximum of @xmath56 is shifted towards lower redshifts with respect to the maximum of @xmath57 , reflecting the time delay between the formation of the progenitors and the coalescence event . the coalescence rate @xmath56 , does not fall to zero at @xmath58 because a non negligible fraction of coalescences ( @xmath59 for sfr2 and @xmath60 for sfr1 ) occurs later than @xmath58 . the nature of the background is determined by the duty cycle defined as the ratio of the typical duration of a single burst @xmath61 to the average time interval between successive events , e.g. , @xmath62 the critical redshift @xmath7 at which the background becomes continuous is fixed by the condition @xmath63 . since we are interested in the last instants of the inspiral , when the signal is within the frequency band of ground based interferometers , we took @xmath64 s , duration which includes about @xmath65 of the total energy released ( see table  [ tbl - duration ] ) . from our numerical experiments and imposing d = 1 , one obtains @xmath66 when the sfr2 is used and @xmath67 for the sfr1 . about 96% ( 94% in the case of sfr1 ) of coalescences occur above such a redshift , contributing to the production of a continuous background . sources in the redshift interval @xmath68 ( sfr2 ) or @xmath69 ( sfr1 ) correspond to a duty cycle d = 0.1 and they are responsible for a cosmic  popcorn " noise . the gravitational fluence ( given here in erg @xmath70hz@xmath16 ) in the observer frame produced by a given dns coalescence is :    @xmath71    where @xmath72 is the distance luminosity , @xmath73 is the proper distance , which depends on the adopted cosmology , @xmath74 is the gravitational spectral energy and @xmath75 the frequency in the source frame . in the quadrupolar approximation and for a binary system with masses @xmath76 and @xmath77 in a circular orbit : @xmath78 where the fact that the gravitational wave frequency is twice the orbital frequency was taken into account . then @xmath79 assuming @xmath80 one obtains @xmath81 erg hz@xmath82 . the spectral properties of the stochastic background are characterized by the dimensionless parameter @xcite :    @xmath83    where @xmath84 is the wave frequency in the observer frame , @xmath85 is the critical mass density needed to close the universe , related to the hubble parameter @xmath86 by ,    @xmath87    @xmath88 is the gravitational wave flux ( given here in erg @xmath70hz@xmath16s@xmath16 ) at the observer frequency @xmath89 , integrated over all sources at redshifts @xmath90 , namely @xmath91    instead of solving analytically the equation above by introducing , for instance , an adequate fit of the cosmic coalescence rate , we have calculated the integrated gravitational flux by summing individual fluences ( eq .  [ eq - fluence ] ) , scaled by the ratio between the expected number of events per unit of time and the number of simulated coalescences or , in other words , the ratio between the total formation rate of progenitors ( eq .  [ eq - nevents ] ) and the number of simulated massive binaries , e.g. ,    @xmath92    the number of runs ( or @xmath93 ) in our simulations was equal to @xmath94 , representing an uncertainty of @xmath95 0.1% in the density parameter @xmath4 . using the sfr2 , the derived formation rate of progenitors is @xmath96 , whereas for the sfr1 one obtains @xmath97 . for each run , the probability distribution @xmath98 defines , via monte carlo , the redshift at which the massive binary is formed . the beginning of the inspiral phase at @xmath45 is fixed by eq . [ eq - zb ] . then , in next step , the probability distribution of the coalescence timescale and eq .  [ eq - h0_tau ] define the redshift @xmath54 at which the merging occurs . the fluence produced by this event is calculated by eq . [ eq - fluence ] , stored in different frequency bins in the observer frame and added according to the equation above .    figure  [ fig - omega ] shows the density parameter @xmath4 as a function of the observed frequency derived from our simulations . the density parameter @xmath4 increases as @xmath99 at low frequencies , reaches a maximum amplitude of about @xmath100 around 670 hz in the case of sfr2 and a maximum of @xmath101 ) around 630 hz , in the case of sfr1 . a high frequency cut - off at @xmath102 hz ( @xmath103 hz for sfr1 ) is observed , corresponding approximately to the frequency of the last stable orbit at the critical redshift @xmath104 ( @xmath105 for sfr1 ) . calculations performed by @xcite , in spite of the similar local merging rates , indicate that the maximum occurs at lower frequencies ( @xmath106 100 hz ) with an amplitude ( scaled to the hubble parameter adopted in this work ) lower by a factor of seven . however , as those authors have stressed , their calculations are expected to be accurate in the frequency range 10 @xmath107hz up to 1 hz , since they have set the value of the maximum frequency @xmath108 to about that expected at a separation of three times the  last stable orbit " ( lso ) , e.g. , @xmath109 . thus , a direct comparison with our results is probably meaningless .    a more conservative estimate can be obtained if one adopts a higher duty cycle value , namely , @xmath110 , corresponding to sources located beyond @xmath111 ( @xmath112 for sfr1 ) . our results are plotted in fig . [ fig - omega ] , which includes , for comparison , the  popcorn " noise contribution arising from sources between @xmath113 ( @xmath69 for sfr1 ) , corresponding to the intermediate zone between a shot noise ( @xmath114 ) and a continuous background ( @xmath115 ) . when increasing the critical redshift ( or removing the nearest sources ) , the amplitude of @xmath4 decreases and the spectrum is shifted toward lower frequencies . the amplitude of the  popcorn " background is about one order of magnitude higher than the continuous background , with a maximum of about @xmath116 ( @xmath117 for sfr1 ) around a frequency of @xmath118 khz .    since some authors use , instead of @xmath4 , the gravitational strain @xmath119 defined by @xcite as @xmath120 we show this quantity in figure  [ fig - sh ] . because the background obeys a gaussian statistic and can be confounded with the instrumental noise background of a single detector , the optimal detection strategy is to cross - correlate the output of two ( or more ) detectors , assumed to have independent spectral noises . the cross correlation product is given by @xcite :    @xmath121    where @xmath122 is a filter that maximizes the signal to noise ratio ( @xmath123 ) . in the above equations , @xmath124 and @xmath125 are the power spectral noise densities of the two detectors and @xmath126 is the non - normalized overlap reduction function , characterizing the loss of sensitivity due to the separation and the relative orientation of the detectors . the optimized @xmath127 ratio for an integration time @xmath128 is given by @xcite : @xmath129    in the literature , the sensitivity of detector pairs is usually given in terms of the minimum detectable amplitude for a flat spectrum ( @xmath4 equal to constant ) @xcite , e.g. ,    @xmath130    the expected minimum detectable amplitude for the main pair of detectors in the world , after one year integration , are given in table  [ tbl - sensitivity ] , for a detection rate @xmath131 and a false alarm rate @xmath132 . the power spectral densities expressions used for the present calculation can be found in @xcite . @xmath133 is of the order of @xmath134 for the first generation of interferometers combined as ligo / ligo and ligo / virgo . their advanced counterparts will permit an increase of two or even three orders of magnitude in sensitivity ( @xmath135 ) . the pair formed by the co - located and co - aligned ligo hanford detectors , for which the overlap reduction function is equal to one , is potentially one order of magnitude more sensitive than the hanford / livingston pair , provided that instrumental and environmental noises could be removed . however , because the spectrum of dns coalescences _ is not flat _ and the maximum occurs out of the optimal frequency band of ground based interferometers , which is typically around @xmath136 hz , as shown in figure  [ fig - snr ] , the s / n ratio is slightly reduced . considering the co - located and co - aligned ligo interferometer pair , we find a signal - to - noise ratio of @xmath137 ( @xmath138 ) for the initial ( advanced ) configuration . unless the coalescence rate be substantially higher than the present expectations , our results indicate that their contribution to the gravitational background is out of reach of the first and the second generation of interferometers . on the other hand , the sensitivity of the future third generation of detectors , presently in discussion , could be high enough to gain one order of magnitude in the expected s / n ratio . examples are the large scale cryogenic gravitational wave telescope ( lcgt ) , sponsored by the university of tokyo and the european antenna ego ( sathyaprakash , private communication ) . ego will incorporate signal recycling , diffractive optics on silicon mirrors , cryo - techniques and kw - class lasers , among other technological improvements . a possible sensitivity for this detector is shown in figure  [ fig - sh ] , compared to the expected sensitivity of advanced ligo . around 650 hz , the planned strain noise @xmath139 is about @xmath140 hz@xmath141 for the advanced ligo configuration while at this frequency , the planned strain noise for ego is @xmath142 hz@xmath141 , which represents a gain by a factor of @xmath143 . considering two interferometers located at the same place , we find a signal - to - noise ratio @xmath144 . on the other hand , the popcorn noise contribution could be detected by new data analysis techniques currently under investigation , such as the search for anisotropies @xcite that can be used to create a map of the gw background @xcite , the maximum likelihood statistic @xcite , or methods based on the probability  event horizon " concept @xcite , which describes the evolution , as a function of the observation time , of the cumulated signal throughout the universe . the peh of the gw signal evolves fastly from contributions of high redshift populations , forming a real continuous stochastic background , to low redshift and less probable sources that can be resolved individually , while the peh of the instrumental noise is expected to evolve much slower . consequently , the gw signature could be distinguished from the instrumental noise background . in this work , we have performed numerical simulations using monte carlo techniques to estimate the occurrence of double neutron star coalescences and the gravitational stochastic background produced these events . since the coalescence timescale obeys a well defined probability distribution ( @xmath145 ) , derived from simulations of the evolution of massive binaries @xcite , the cosmic coalescence rate does not follow the cosmic star formation rate and presents necessarily a time - lag . in the case where the sources are supernovae or black holes , the gravitational burst is produced in a quite short timescale after the the formation of the progenitors . therefore , the time - lag is negligible and the comoving volume where the progenitors are formed is practically the same as that where the gravitational wave emission occurs , introducing a considerable simplification in the calculations . this is not the case when ns - ns coalescences are considered , since timescales comparable or even higher than the hubble timescale have non negligible probabilities . the maximum probability to form a massive binary occurs at @xmath146 , depending slightly on the adopted cosmic star formation rate , whereas the maximum probability to occur a coalescence is around @xmath147 . we have found that a truly continuous background is formed only when sources located beyond @xmath148 ( @xmath149 for the sfr1 case ) , including 96% ( 94% for sfr1 ) of all events and the critical redshift corresponds to the condition @xmath8 . sources in the redshift interval @xmath113 ( @xmath69 for sfr1 ) produce a  popcorn " noise . our computations indicate that the density parameter @xmath4 has a maximum around 670 hz ( 630 hz for sfr1 ) , attaining an amplitude of about of @xmath100 ( @xmath150 for sfr1 ) . the low frequency cutoff around 1.2 khz corresponds essentially to the gravitational redshifted wave frequency associated to last stable orbit of sources located near the maximum of the coalescence rate . the computed signal is below the sensitivity of the first and the second generation of detectors . however , using the planned sensitivity of third generation interferometers , we found that after one year of integration , the cross - correlation of two ego like coincident antennas , gives an the optimized signal - to - noise of @xmath151 . the  popcorn " contribution is one order of magnitude higher with a maximum of @xmath152 ( @xmath117 for sfr1 ) at @xmath153 khz . this signal , which is characterized by the spatial and temporal evolution of the events as well as by its signature , can be distinguished from the instrumental noise background and adequate data analysis strategies for its detection are currently under investigation @xcite .
in this work , numerical simulations were used to investigate the gravitational stochastic background produced by coalescences occurring up to @xmath0 of double neutron star systems . 
 the cosmic coalescence rate was derived from monte carlo methods using the probability distributions for forming a massive binary and to occur a coalescence in a given redshift . 
 a truly continuous background is produced by events located only beyond the critical redshift @xmath1 . 
 events occurring in the redshift interval @xmath2 give origin to a  popcorn " noise , while those arising closer than @xmath3 produce a shot noise . 
 the gravitational density parameter @xmath4 for the continuous background reaches a maximum around 670 hz with an amplitude of @xmath5 , while the  popcorn " noise has an amplitude about one order of magnitude higher and the maximum occurs around a frequency of 1.2 khz . 
 the signal is below the sensitivity of the first generation of detectors but could be detectable by the future generation of ground based interferometers . correlating two coincident advanced - ligo detectors or two ego interferometers , the expected s / n ratio are respectively 0.5 and 10 .