Constraining the distance to inspiralling NS-NS with Einstein Telescope
aa r X i v : . [ a s t r o - ph . C O ] M a y CONSTRAINING THE DISTANCE TO INSPIRALING NS-NS WITHEINSTEIN TELESCOPE
I. KOWALSKA-LESZCZYNSKA , T. BULIK Astronomical Observatory, University of Warsaw, Al Ujazdowskie 4, 00-478 Warsaw, Poland
Einstein Telescope (ET) is a planned third generation gravitational waves detector located inEurope1. Its design will be different from currently build interferometers: First, ET will belocated underground in order to reduce the seismic noise. The arms length will be 10 km,and the configuration of arms will be different from all interferometers build so far i.e. therewill be three tunnels in a triangular shape. ET will consist of three interferometers rotatedby a 60deg with respect to each other in one plane. One of the biggest challenges for ET willbe to determine sky position and distance to observed sources. If an object is observed in afew interferometers simultaneously one can estimate the position using traingulation from timedelays2, but so far there are no plans for a network of third generation detectors. Anotherpossibility to deal with that problem is by using multimessenger approach, because redshift andsky position could be recovered from electromagnetic observations. However, in most cases ofET detection there will be only gravitational signal. In this paper we present a novel method ofestimating distance and position in the sky of merging binaries. While our procedure is not asaccurate as the multimessenger method, it can be applied to all observations, not just the oneswith electromagnetic counterparts.
For simplicity let us consider the case of observation of a double neutron star. In gravitationalwaves we will be observing directly two quantities: signal to noise ratio ( ρ ), which is a compli-cated function of the source properties, as well as the detector characterization, and redshiftedchirp mass ( M z = (1 + z ) M chirp , M chirp = ( M M ) / ( M + M ) − / ). In this particular case weconsider only binaries consisting of two neutron stars of equal masses M = M = 1 . ⊙ , so M chirp = 1 . ⊙ . The signal to noise ratio in the quadruple approximation for merging doublecompact objects is well known 3: ρ ∼ Θ d L ( z ) ( M z ) / p ξ ( z ) , (1)where d L is the luminosity distance, M z is the redshifted chirp mass, z is the redshift, Θ is afunction of sky position and orientation of the source, and ξ is the function that determinesraction of the sensitivity window filled by a signal (it depends on the chirp mass, and for NSNSbinaries its value is close to unity). For a given binary that will be observed in the detector, wecan measure ξ directly, by measuring the cutoff frequency when the inspiral ceases.The function Θ depends on the sky position of the source Ω( ϑ, ϕ ) and on the orientation ofthe orbit with respect to the line of sight Ω p (Ψ , i ):Θ = 2 p (1 + cos i )( F + ) + 4 cos i ( F x ) ,F + = 0 . ϑ ) cos 2 ϕ cos 2Ψ − cos ϑ sin 2 ϕ sin 2Ψ ,F x = 0 . ϑ ) cos 2 ϕ sin 2Ψ + cos ϑ sin 2 ϕ cos 2Ψ . (2)The density of sources in a unit volume can be expressed by: d ndzd Ω d Ω p = n ( z )1 + z dVdz . (3)The comoving volume is dVdz = 4 π cH r ( z ) E (Ω ,z ) , and E (Ω , z ) = p Ω Λ + Ω M (1 + z ) . Then we obtainfor a single detector dndz = Z d Ω d Ω p n ( z )1 + z δ ( ρ − ρ m )= 4 π n ( z )1 + z cH r ( z ) E (Ω , z ) d L r (cid:18) . M z (cid:19) / √ ξ × P ρ m r ( Mz . ) / √ ξ d L ( z ) ! , (4)where n ( z ) is the merger rate, r is the characteristic distance for a given detector (see Table1 in paper by Taylor4 for more details), ρ m is the actual signal-to-noise ratio measured in thedetector. Design of the Einstein Telescope assumes three co-located interferometers lying in the sameplane, so the methods for distance estimation based on triangulation will not be possible. How-ever, a single source will be observed by each of the interferometer with a different orientation.There will be three different measurements of signal to noise ratio. That will provide additionalinformation about the observed source and it allows to constrain the distributions obtained inprevious section.The density of sources per unit volume given by Eq. 3 has to be integrated taking intoaccount that we have three conditions to satisfy.We assume that each signal to noise ratio ismeasured with perfect accuracy: dndz = Z d Ω d Ω p n ( z )1 + z δ ( ρ − ρ m ) δ ( ρ − ρ m ) δ ( ρ − ρ m )= Z d ndzd Ω d Ω p π n ( z )1 + z cH r ( z ) E (Ω , z ) d L r (cid:18) . M z (cid:19) / √ ξ ( ρ m ) × δ Θ − ρ m r ( Mz . ) / √ ξ d L ( z ) ! δ (cid:18) Θ Θ1 − ρ m ρ m (cid:19) δ (cid:18) Θ Θ1 − ρ m ρ m (cid:19) , (5)able 1: Physical parameters and observed quantities of four sources. For first three of them (A,B, C) sky position and orientation were chosen randomly from uniform distributions on ththeobtained signal to noise ratio (the binary is optimally oriented).Physical parameters Observed quantitiesM = M [M ⊙ ] z ϑ [rad] φ [rad] Ψ [rad] i [rad] M z [M ⊙ ] ρ m ρ m ρ m A 1.4 0.1 0.53 π π π π π π π π π π π π π π Distance measurements to merging binaries will be very challenging in the third generationdetectors era. So far, there are no plans for any other detector than Einstein Telescope. Inthis paper, we presented a method that can be used to constrain distance distribution fora given double neutron star observation. We have shown that it is possible to significantlyimprove distance estimates using the measurements of the signal to noise ratio from all threeinterferometers .
Acknowledgments
This work was supported by the following Polish NCN grants DEC-2011/01/V/ST9/03171 and2014/15/Z/ST9/00038.
References
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