Cosmological perturbation effects on gravitational-wave luminosity distance estimates
Daniele Bertacca, Alvise Raccanelli, Nicola Bartolo, Sabino Matarrese
CCosmological perturbation effects on gravitational-wave luminosity distance estimates
Daniele Bertacca, Alvise Raccanelli ⋆ ,
2, 3
Nicola Bartolo,
4, 5, 6 and Sabino Matarrese
4, 5, 6, 7 Argelander-Institut f¨ur Astronomie, Auf dem H¨ugel 71, D-53121 Bonn, Germany Institut de Ci`encies del Cosmos (ICCUB),Universitat de Barcelona (IEEC-UB), Mart´ı Franqu`es 1, E08028 Barcelona, Spain Department of Physics & Astronomy, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218 Dipartimento di Fisica e Astronomia “G. Galilei”,Universita’ degli Studi di Padova, via Marzolo 8, I-35131, Padova, Italy INFN, Sezione di Padova, via Marzolo 8, I-35131, Padova, Italy INAF-Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy Gran Sasso Science Institute, INFN, Viale F. Crispi 7, I-67100 L’Aquila, Italy (Dated: March 23, 2018)Waveforms of gravitational waves provide information about a variety of parameters for the binarysystem merging. However, standard calculations have been performed assuming a FLRW universewith no perturbations. In reality this assumption should be dropped: we show that the inclusion ofcosmological perturbations translates into corrections to the estimate of astrophysical parametersderived for the merging binary systems. We compute corrections to the estimate of the luminositydistance due to velocity, volume, lensing and gravitational potential effects. Our results show thatthe amplitude of the corrections will be negligible for current instruments, mildly important forexperiments like the planned DECIGO, and very important for future ones such as the Big BangObserver.
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I. INTRODUCTION
The existence of gravitational waves (GWs), predicted by Einstein in 1916 [1], was confirmed one century laterwhen gravitational-waves were observed by the LIGO-VIRGO collaborations [2–4]; for a short review of the historyof GWs, see [5].The Advanced LIGO instruments detected GWs from the coalescence of binary Black Holes (BHs); this discoverynot only represents a confirmation of the structure of Einstein’s General Relativity (GR), but also provides new waysto test it, along with other models of gravity, opening up an entirely new window for observational astrophysics.More generally, it will be possible to address important questions about our Universe in a novel way. Measuring thewaveform from a distant coalescing binary provides us, for example, with information on the (redshifted) chirp massof the source and its luminosity distance.Gravitational wave astronomy provides a novel window to investigate the Universe and can be used to test cos-mological (see e.g. [6–10] for the possibility of determining cosmological parameters form GW observations, [11–17]for GW luminosity distance-redshift relation and gravitationally lensed GW sirens, [18–27] for testing general rela-tivity and modified gravity models with GW, [28] for using radio galaxy surveys for GW astronomy) or astrophysical(e.g. [29–35]) models. For some reviews on astrophysical and cosmological studies to be performed using gravitationalwaves, see [36–41].Gravitational waves can also be considered as standard sirens for determining the distance-redshift relation [12].Therefore, using future ground- and space-based gravitational wave detectors such as the Einstein Telescope (ET, [42]),LISA [43–45], the DECI-hertz Interferometer Gravitational-wave Observatory (DECIGO, [46]) and the Big-BangObserver (BBO, [47]), we will be able to measure the cosmological expansion at good precision up to high redshifts.It is therefore timely to develop precise formalisms to use GW observations to perform astrophysical and cosmologicalstudies, not only by understanding the investigations that will be enabled by GW-Astronomy, but also by includingsubtle effects, contaminations and degeneracies (see e.g. [48]). In this paper we take a further step in this direction,and we analyze the effect of cosmological perturbations and inhomogeneities on estimates of the luminosity distanceof black hole binary mergers through gravitational waves. We apply the “Cosmic Rulers” formalism [49], we consider ⋆ Marie Sk(cid:32)lodowska-Curie fellow a r X i v : . [ g r- q c ] M a r the observer frame as reference system and we derive a different expression wrt [50], which is correct for the effect oflarge-scale structures on GW waveforms, accounting for lensing, Sachs-Wolfe, integrated Sachs-Wolfe, time delay andvolume distortion effects, and evaluate their importance for future GW experiments. Finally we are able to connectour results with the initial amplitude of GW signal and interpret the numerical simulations of coalescing BH binaries,which produce the templates, using the observed – rather than the background – frame .This paper is structured as follows. In Section II we setup the formalism to calculate effects of perturbations tothe propagation of gravitational waves, and compute their amplitude and phase shifts, in general and in the Poissongauge. In Section III we numerically evaluate the corrections for different contributions and at different redshifts, andcompare them with predictions for future experiments precision. We discuss our results and conclude in Section IV.Throughout the paper we assume the following conventions: units, c = G =
1; signature (− , + , + , +) ; Greek indices runover 0 , , ,
3, and Latin ones over 1 , , II. GRAVITATIONAL WAVES PROPAGATION IN THE PERTURBED UNIVERSE
Usually, the effect of perturbations on the propagation of gravitational waves has been often neglected so far;however, with the very recent beginning of the so-called gravitational wave astronomy era, it is timely to startdeveloping precise formalisms to investigate the Universe using GWs.There have been initial attempts to investigate the Integrated-Sachs Wolfe (ISW) effect on GWs from supermassiveblack hole mergers and in particular its effect on the system’s parameter estimation in [50], the ISW of a primordialstochastic background [51], and lensing effects [52]; it is worth noting that environmental effects can also affectestimates of the luminosity distance [53]. Recently, [54] analyzed the effect of local peculiar velocities on the relationluminosity distance-redshift and on the chirp mass estimate for LISA, in particular referring to the possibility ofjointly estimating the two with gravitational waves, as originally suggested in [55, 56].In this context, in this work we generalize these studies and develop a formalism to compute the change in theestimation of luminosity distance due to the presence of cosmological perturbations and inhomogeneities. We will usethe “Cosmic Rulers” [49, 57–61] formalism and calculate the correction to the observed luminosity distance due tovolume, lensing and ISW-like effects.
A. Cosmic rulers for gravitational waves
We start by assuming Isaacson’s shortwave or geometric optics approximation [62, 63]; in this case, the space-timemetric can be written as the sum of two parts: g µν = ˜ g µν + (cid:15)h µν , where ˜ g µν is usually named “background metric” anddescribes both the FRW metric and first-order perturbations, and h µν is the gravitational wave metric perturbation(where we are using the same notation of [50]). In the shortwave approximation and neglecting the response of matterbackground effect to the presence of h µν , a gravitational wave can be described as¯ h µν = A µν e iϕ / (cid:15) = e µν A e iϕ / (cid:15) = e µν h , (1)where ¯ h µν = h µν − ˜ g µν h / h is the trace of h µν w.r.t. the background metric ˜ g µν . Here e µν is a polarization tensorand A and ϕ are real functions of retarded time and describe respectively the amplitude and the phase of the GW(see e.g. [50]).From now on it is convenient to use the comoving metric ˆ g µν = ˜ g µν / a , where a is the scale factor. Definingthe GW wave-vector k µ = − ˆ g µν ∇ ν ϕ , we have k µ k µ = k µ ∇ µ k ν = k µ ∇ µ e αβ = ddχ ln (A a ) = − ∇ µ k µ , (2)where we have used d / dχ ≡ k µ ∇ µ . The background frame is assumed to be homogeneous and without anisotropies.
We define x µ ( χ ) as the comoving coordinates in the real frame (or real space), where χ is the comoving distance,in real-space, from the source to the detector (the observer). On the other hand, we call Redshift-GW frame (RGW)the “cosmic GW laboratory” where we perform the observations, i.e. the observed frame . In RGW space we usecoordinates which effectively flatten our past gravitational wave-cone so that the GW geodesic from an observed BHhas the following conformal space-time coordinates:¯ x µ = ( ¯ η, ¯ x ) = ( η − ¯ χ, ¯ χ n ) . (3)Here η is the conformal time at observation, ¯ χ ( z ) is the comoving distance to the observed redshift in RGW-space, n is the observed direction of arrival in the sky of the GW, i.e. n i = ¯ x i / ¯ χ = δ ij ( ∂ ¯ χ / ∂ ¯ x j ) . Using ¯ χ as an affine parameterin the observed frame, the total derivative along the past GW-cone is d / d ¯ χ = − ∂ / ∂ ¯ η + n i ∂ / ∂ ¯ x i . Below we will usethe subscripts “ e ” and “ o ” to denote the the observer evaluated at the position where the GW is emitted and at thelocation of the observer where the GWs are received, respectively.It is also useful to define the parallel and perpendicular projection operators to the observed line-of-sight direction.For any spatial vectors and tensors: A ∥ = n i n j A ij , B i ⊥ = P ij B j = B i − n i B ∥ , (4)where P ij = δ ij − n i n j . The directional derivatives are defined as¯ ∂ ∥ = n i ∂∂ ¯ x i , ¯ ∂ ∥ = ¯ ∂ ∥ ¯ ∂ ∥ , ¯ ∂ ⊥ i = P ji ¯ ∂ j = ∂∂ ¯ x i − n i ¯ ∂ ∥ , ∂n j ∂ ¯ x i = χ P ji , dd ¯ χ ∂ i ⊥ = ¯ ∂ i ⊥ dd ¯ χ − χ ∂ i ⊥ , (5)and we have ∂B i ∂ ¯ x j = n i n j ¯ ∂ ∥ B ∥ + n i ¯ ∂ ⊥ j B ∥ + ¯ ∂ ⊥ j B i ⊥ + n j ¯ ∂ ∥ B i ⊥ + χ P ij B ∥ , ¯ ∇ ⊥ = ¯ ∂ ⊥ i ¯ ∂ i ⊥ = δ ij ∂∂ ¯ x i ∂∂ ¯ x j − ¯ ∂ ∥ − χ ¯ ∂ ∥ . (6)Defining ¯ k µ as the null geodesic vector in the redshift frame at zeroth order,¯ k µ = d¯ x µ d ¯ χ = (− , n ) , (7)while for the perturbed case we define the physical k µ evaluated at ¯ χ in the following way k µ ( ¯ χ ) = d x µ d ¯ χ ( ¯ χ ) = dd ¯ χ ( ¯ x µ + δx µ ) ( ¯ χ ) = (− + δν, n i + δn i ) ( ¯ χ ) . (8)Now, let us indicate the apparent positions as ¯ x µ , the observed space (or the redshift-space), while the true positionsare at x µ , real- space, by the displacement ∆ x µ ( ¯ χ ) of the observed position of coalescing BH binaries. In other words,we can set up a mapping between RGW- and real- space (the “physical frame”) in the following way χ = ¯ χ + δχ , x µ ( χ ) = ¯ x µ ( ¯ χ ) + ∆ x µ ( ¯ χ ) , ∆ x µ ( ¯ χ ) = d¯ x µ d ¯ χ δχ + δx µ ( ¯ χ ) = ¯ k µ δχ + δx µ ( ¯ χ ) . (9)From Eq. (8), we obtain explicitly δx ( ¯ χ ) = ∫ ¯ χ d ˜ χ δν ( ˜ χ ) , δx i ( ¯ χ ) = ∫ ¯ χ d ˜ χ δn i ( ˜ χ ) , (10)where we have imposed the boundary conditions at the observer: δx ( n ) o = δx i ( n ) o =
0. In real-space the scalefactor is a = a [ x ( χ )] = a ( ¯ x + ∆ x ) = ¯ a ( + H ∆ x ) , (11) In the case of photons, this prescription has been used for the first time in [57]. where ¯ a = a ( ¯ x ) , H = ¯ a ′ / ¯ a and prime indicates ∂ / ∂ ¯ x = ∂ / ∂ ¯ η . Defining a ¯ a = + ∆ ln a , (12)we have ∆ ln a = H ∆ x = H (− δχ + δx ) . (13)Here we consider the local wave zone approximation to define the tetrads at source position [38]. If we choose thefour-velocity u µ as the time-like basis vector, then u µ = a E ˆ0 µ , u µ = a − E µ ˆ0 , (14)where E µ ˆ α is the tetrad in the comoving frame. At the background level we have E ( ) ˆ0 µ = (− , ) and, perturbing thetetrad, we obtain E ˆ0 µ ( x ν ( χ )) = E ˆ0 µ ( ¯ x ν ( ¯ χ ) + ∆ x ν ) = E ( ) ˆ0 µ ( ¯ χ ) + E ( ) ˆ0 µ ( ¯ χ ) . (15)From k µ , the map from redshift to real-space is given by k µ ( χ ) = d x µ ( χ ) d χ = k µ ( ¯ χ + δχ ) = k µ ( ) ( ¯ χ ) + k µ ( ) ( ¯ χ ) , (16)where k µ ( ) ( ¯ χ ) = ¯ k µ , and at first orderd δk µ ( ¯ χ ) d ¯ χ + δ ˆΓ µαβ ( ¯ x γ ) ¯ k α ( ¯ χ ) ¯ k β ( ¯ χ ) = , (17)where δk µ ( ¯ χ ) = k µ ( ) ( ¯ χ ) and, consequently, δk ( ¯ χ ) = δν , δk i ( ¯ χ ) = δn i . The observed redshift (precisely, if we have its electromagnetic counterpart) is given by ( + z ) = f e f o = a o a ( χ e ) ( E ˆ0 µ k µ )∣ e ( E ˆ0 µ k µ )∣ o , (18)Choosing a o = ¯ a o = a ( ¯ x ) = ( E ˆ0 µ k µ )∣ o =
1, we have1 + z = E ˆ0 µ k µ a . (19)From Eq. (12), ¯ a is the scale factor in redshift-space. Then ¯ a = /( + z ) . From Eqs. (12), (15) and (16), we get1 = + ( E ˆ0 µ k µ ) ( ) + ∆ ln a . (20) For simplicity in the main text we set a o =
1. However, for the sake of completeness, we should include the perturbation of the scalefactor at observation δa o = a o −
1, since we have to assume that the proper time of the observer at observation η is fixed by the choiceof scale factor via the relation a ( η ) = a txt , in the main text, becomes∆ ln a → ∆ ln a = ∆ ln a txt + δa o , where δa o is defined in [49]. The observer “at the emitted position” is within a region with a comoving distance to the source sufficiently large so that the gravitationalfield is “weak enough” but still “local”, i.e. the gravitational wave wavelength is small w.r.t. the comoving distance from the observer¯ χ . Therefore, we can write ∆ ln a as∆ ln a = ( E ˆ0 µ k µ ) ( ) = E ( ) ˆ0 µ k µ ( ) + E ( ) ˆ0 µ k µ ( ) = − E ( ) ˆ00 + n i E ( ) ˆ0 i − δν , (21)Using Eqs. (13) and (21) we obtain δχ = δx − ∆ ln a H = δx − ∆ x . (22)In the rest of this Section we will then use this formalism to compute modifications to amplitude and phase ofgravitational waves due to perturbations around a FRW metric. B. Gravitational waves in the observed frame
We start by calculating perturbations in a general way. Let us write h µν defined in Eq. (1) in Redshift-GW frame.First of all, we will compute the phase ϕ . At first order k µ k µ = − k µ ˜ g µν ∇ ν ϕ = − dd χ ϕ ( x µ ( χ )) = − d ¯ χ d χ dd ¯ χ ϕ ( ¯ x µ + ∆ x µ ) = − ( − d δχ d ¯ χ ) dd ¯ χ [ ¯ ϕ + δϕ ( ¯ x µ ) + ∆ x µ ¯ ∇ µ ¯ ϕ ] = , (23)where ¯ ϕ = ϕ ( ) ( ¯ x µ ) . Defining ¯ k µ = − ¯ ∇ µ ¯ ϕ , d ϕ / d ¯ χ = ¯ k µ ∇ µ ϕ and ϕ ( ¯ x µ ) = ¯ ϕ + δϕ ( ¯ x µ ) , we find − dd ¯ χ δϕ ( ¯ x µ ) = ¯ k µ δk µ ( ¯ χ ) , (24)where ¯ ∇ µ δϕ ( ¯ x µ ) = − δk µ ( ¯ χ ) . Note that d ϕ / d χ = ϕ / d ¯ χ ≠ ( − d δχ d ¯ χ ) dd ¯ χ ln {A( ¯ x µ + ∆ x µ ) ¯ a [ + ∆ ln a ]} = − [( ∂ ¯ x ν ∂x µ ) ∂∂ ¯ x ν ( ¯ k µ + δk µ ) + δ Γ µµν ¯ k ν ] . (25)In particular, following the prescription defined in the previous subsection, we can divide the contributions to Eq. (25)in three partsdd ¯ χ ln [A ( ¯ x µ + ∆ x µ )] = dd ¯ χ ln ¯ A + dd ¯ χ [ δ ln A + ∆ x µ ¯ ∂ µ ln ¯ A] = dd ¯ χ ln ¯ A + dd ¯ χ δ ln A + ( d d ¯ χ ln ¯ A) δχ + ( dd ¯ χ ln ¯ A) ( dd ¯ χ δχ ) + ( dd ¯ χ ¯ ∂ µ ln ¯ A) δx µ + ( ¯ ∂ µ ln ¯ A) δk µ , (26)dd ¯ χ ln [ ¯ a ( + ∆ ln a )] = −H − H ′ (− δχ + δx ) + H (− dd ¯ χ δχ + δk ) , (27) ( ∂ ¯ x ν ∂x µ ) ∂∂ ¯ x ν ( ¯ k µ + δk µ ) = ( δ νµ − ¯ ∂ µ ∆ x ν ) ∂∂ ¯ x ν ( ¯ k µ + δk µ ) = χ ( + δk ∥ ) + ( dd ¯ χ δk ∥ ) + ∂∂ ¯ x ( δk + δk ∥ ) + ¯ ∂ ⊥ i δk i ⊥ − χ ( δχ + δx ∥ ) − χ ¯ ∂ ⊥ i δx i ⊥ , (28)where ¯ ∂ ⊥ i ∆ x i ⊥ = ¯ ∂ ⊥ i δx i ⊥ and P ii =
2. Then to the lowest order we have (see also [50])¯ A( ¯ x , ¯ χ ) = Q ¯ a ( ¯ x ) ¯ χ = Q( + z ) ¯ χ , (29)where Q is constant along the null geodesic. Here Q is determined by the local wave-zone source solution andcontains all the physical information on the spiralling binary . At the receiving location it is given by the same The physical meaning of Q is not the same as [50]. Indeed, in this paper, Q is defined directly in the observed frame ¯ x µ = ( ¯ η, ¯ x ) and,in [50], with a unperturbed background metric. solution evaluated at the retarded time. With this result, using Eq. (5) and ( dd ¯ χ ¯ ∂ µ ln ¯ A) δx µ + ( ¯ ∂ µ ln ¯ A) δk µ = H ′ δx + χ δx ∥ − H δk − χ δk ∥ , (30)¯ ∂ ⊥ i δk i ⊥ = ¯ ∂ ⊥ i dd ¯ χ δx i ⊥ = dd ¯ χ ¯ ∂ ⊥ i δx i ⊥ + χ ¯ ∂ ⊥ i δx i ⊥ , (31)we can finally write dd ¯ χ δ ln A = − [ ∂∂ ¯ x ( δk + δk ∥ ) + dd ¯ χ δk ∥ − χ κ + δ Γ µµν ¯ k ν ] , (32)where κ is the weak lensing convergence term κ = −
12 ¯ ∂ ⊥ i ∆ x i ⊥ . (33)The perturbed gravitational waves can then be fully described as h ( η e , x e ) = A( η e , x e ) e iϕ ( η e , x e ) = Q( + z ) ¯ χ ( + ∆ ln A) e i ( ¯ ϕ + ∆ ϕ ) , (34)where the perturbations of amplitude and phase are∆ ln A = δ ln A + ∆ x ¯ ∂ ln ¯ A + ∆ x ∥ ¯ ∂ ∥ ln ¯ A = δ ln A − ( − H ¯ χ ) ∆ ln a + T ¯ χ , (35)∆ ϕ = δϕ + T . (36)with − T = ∆ x + ∆ x ∥ . C. Perturbations in the Poisson gauge
The relations obtained so far are valid in any gauge. We now derive the expressions for the amplitude and phasecorrections in the Poisson gauge, so that we will be able to estimate their practical relevance. The background metric˜ g µν in Poisson gauge reads d s = a ( η ) [− ( + ) d η + δ ij ( − ) d x i d x j ] , (37)where we are neglecting vector and tensor perturbations at first order.For the geodesic equation we obtain, at linear order,dd ¯ χ ( δν − ) = Φ ′ + Ψ ′ , dd ¯ χ ( δn i − n i ) = − ¯ ∂ i ( Φ + Ψ ) , (38)To solve Eq. (38) we need the values of δν and δn i ( ) today. In this case we need all the components of the tetrads E ˆ αµ , which are defined through the following relations (see Appendix A)ˆ g µν E ˆ αµ E ˆ βν = η ˆ α ˆ β , η ˆ α ˆ β E ˆ αµ E ˆ βν = ˆ g µν , ˆ g µν E ˆ βν = E ˆ βµ , η ˆ α ˆ β E ˆ βν = E ˆ βν , (39)where η ˆ α ˆ β is the comoving Minkowski metric [60]. Using the constraints δν o = Φ o + v ∥ o , δn ˆ ao = − v ˆ ao + n ˆ a Ψ o , (40)from Eq.(38) we obtain at first order δν = − ( Φ o − v ∥ o ) + + ∫ ¯ χ d ˜ χ ( Φ ′ + Ψ ′ ) = − ( Φ o − v ∥ o ) + − I , (41) δn i = − v io − n i Ψ o + n i Ψ − ∫ ¯ χ d ˜ χ ˜ ∂ i ( Φ + Ψ ) = n i δn ∥ + δn i ⊥ , (42)where δn ∥ = Φ o − v ∥ o − Φ + Ψ + I , δn i ⊥ = − v i ⊥ o + S i ⊥ . (43)Here I = − ∫ ¯ χ d ˜ χ ( Φ ′ + Ψ ′ ) , S i ( )⊥ = − ∫ ¯ χ d ˜ χ ˜ ∂ i ⊥ ( Φ + Ψ ) (44)where, I is the integrated Sachs-Wolfe (ISW) term [64]. The GW phase in Eq. (24) can be obtained with the followingrelation − d δϕ d ¯ χ = δn ∥ + δν = Φ + Ψ . (49)From Eq. (22) we then have∆ ln a = ( Φ o − v ∥ o ) − Φ + v ∥ + I = ( Φ o − v ∥ o ) − Φ + v ∥ − ∫ ¯ χ d ˜ χ ( Φ ′ + Ψ ′ ) , (50) δχ = − ( ¯ χ + H ) ( Φ o − v ∥ o ) + H ( Φ − v ∥ ) + ∫ ¯ χ d ˜ χ [ + ( ¯ χ − ˜ χ ) ( Φ ′ + Ψ ′ )] − H I . (51)We can therefore write explicitly the components of ∆ x ; from Eqs. (12) and (21) we find∆ x = H [( Φ o − v ∥ o ) − Φ + v ∥ + I ] = H [( Φ o − v ∥ o ) − Φ + v ∥ − ∫ ¯ χ d ˜ χ ( Φ ′ + Ψ ′ )] ; (52)from Eqs. (9) and (45) we have∆ x ∥ = − T − H [( Φ o − v ∥ o ) − Φ + v ∥ + I ]= ∫ ¯ χ d ˜ χ ( Φ + Ψ ) − H [( Φ o − v ∥ o ) − Φ + v ∥ − ∫ ¯ χ d ˜ χ ( Φ ′ + Ψ ′ )] , (53)and ∆ x i ⊥ = δx i ⊥ = − ¯ χ v i ⊥ o − ∫ ¯ χ d ˜ χ ( ¯ χ − ˜ χ ) ˜ ∂ i ⊥ ( Φ + Ψ ) . (54)We can recognize that in Eq. (52) there is an ISW contribution and in Eq. (53) we have both time-delay and ISWcontributions, while Eq. (54) represents the lensing contribution.Finally, from Eq. (32), we can see that δ ln A − δ ln A o = Ψ − Ψ o + κ . Therefore, we can express perturbations inamplitude and phase of Eqs. (35) and (36), in the Poisson gauge, as∆ ln A = δ ln A o + Ψ − Ψ o + κ − ( − H ¯ χ ) ∆ ln a + T ¯ χ , (55)∆ ϕ = δϕ o . (56) To obtain Eq. (49) we used Eqs. (24), and we note that δϕ − δϕ o = T = − ( δx + δx ∥ ) = − ∫ ¯ χ d˜ χ ( Φ + Ψ ) , (45) δx = − ¯ χ ( Φ o − v ∥ o ) + ∫ ¯ χ d˜ χ [ + ( ¯ χ − ˜ χ ) ( Φ ′ + Ψ ′ )] , (46) δx ∥ = ¯ χ ( Φ o − v ∥ o ) − ∫ ¯ χ d˜ χ [( Φ − Ψ ) + ( ¯ χ − ˜ χ ) ( Φ ′ + Ψ ′ )] , (47) δx i ⊥ = − ¯ χ v i ⊥ o − ∫ ¯ χ d˜ χ [( ¯ χ − ˜ χ ) ˜ ∂ i ⊥ ( Φ + Ψ )] . (48) The correction in Eq. (55) can be related to the luminosity distance D L , in the following way∆ D L ¯ D L = − Ψ − κ + ( − H ¯ χ ) ∆ ln a − T ¯ χ , (57)taking into account that h e = Q( + z ) D L e i ¯ ϕ . (58)and ∆ ln A = − ∆ D L ¯ D L , (59)defining ¯ D L = ( + z ) ¯ χ the observed average luminosity distance taken over all the sources with the same observedredshift z , with D L = ¯ D L + ∆ D L .The gravitational wave observed at the detector is red-shifted, hence we find h r ≡ h e ( + z ) = Q( + z )D L e i ¯ ϕ . (60)Now, we have to estimate correctly Q from inspiral of compact binaries [65]. For simplicity, in this work, i) we assumethe Newtonian approximation which agrees with standard weak-field approximation in general relativity (i.e., we areneglecting the post newtonian terms) and ii) we consider only the regime called of “quasi-circular” motion (i.e. theapproximation in which a slowly varying orbital radius is applicable) [38, 66, 67] . Then from the quadrupole formulawe have [65] Q = M e ( πf e M e ) / , (61)where M e , and f e are the intrinsic “chirp mass” and frequency of the binary, respectively. Then ¯ ϕ = ϕ c −( π f e M e ) − / /
16 where ϕ c is the value of the phase at f = ∞ and t ( f ) = t c − ( / )M e ( πf e M e ) − / [67].A note is in order here. The relation in Eq. (61) formally is the same as that of [50], but its physical meaning istotally different. In fact in [50] the authors consider as redshift the inverse of scale factor in the background frame,i.e. a Universe without inhomogeneities and anisotropies. Instead Eq. (61) depends on the measured redshift. Inother words, this value of redshift coincides with that of a hypothetical event where it is possible to measure itselectromagnetic signal, i.e. the photons from the coalescence. Finally, from Eq. (57), we immediately note that wehave recovered the same result obtained by the luminosity distance computed for the photon, see Eq. (51) in ref. [49].Taking into account that Q is computed in the observed frame, defining M r = M e ( + z ) and f r = f e /( + z ) , wehave h r = M r D L ( πf r M r ) / e iϕ r , (62)and considering that ¯ h r = M r ( πf r M r ) / e iϕ r / ¯ D L , we have∆ h r ¯ h r = − ∆ D L ¯ D L . (63)It is worth noticing that the luminosity distance is also related to the signal-to-noise ratio ( σ ) for the detection ofgravitational waves (see e.g. [65, 68]), via σ = ∫ ∞ ∣ ˜ h ∣ S n d f r , (64) Here we have assumed δϕ o =
0, i.e. ϕ e = ¯ ϕ . Moreover, by construction, we have δ ln A o = Ψ o . i.e. f o = f r and ¯ ϕ = ϕ r where ˜ h = (M r /D L )( f r M r ) − / e iψ r is the Fourier transform of Eq. (62), S n is the spectral noise density and ψ r ≡ π f r t o + φ r ( t o ) with t o being the stationary point of the phase [50] . Hence we have [65] σ = ( M r D L ) ∫ ∞ ( f r M r ) − / ( S n /M r ) d ( f r M r ) , (65)and ∆ σ ¯ σ = − ∆ D L ¯ D L . (66)In the rest of the paper we will drop the unobservable constant contribution evaluated at the observer, denoted witha subscript zero. Finally, it is important to point out that using this prescription one can generalize the definitionof Q of Eq. (61) and ψ r by adding all Post-Newtonian corrections to the wave amplitude and phase of gravitationalwaves (see e.g. [65, 66, 69–73]). III. RESULTS
Here we compute the modifications of the value of the luminosity density D L inferred from gravitational waves, dueto perturbations.We can make Eq. (57) explicit, i.e. we can write the correction to the luminosity distance as∆ D L ¯ D L = ( − H ¯ χ ) v ∥ − ∫ ¯ χ d ˜ χ ( ¯ χ − ˜ χ ) ˜ χ ¯ χ △ Ω ( Φ + Ψ ) ++ H ¯ χ Φ − ( − H ¯ χ ) ∫ ¯ χ d ˜ χ ( Ψ ′ + Φ ′ ) − ( Φ + Ψ ) + χ ∫ ¯ χ d ˜ χ ( Φ + Ψ ) , (67)where △ Ω ≡ ¯ χ ¯ ∇ ⊥ = ¯ χ ( ¯ ∇ − ¯ ∂ ∥ − χ − ¯ ∂ ∥ ) = ( cot ∂ θ + ∂ θ + ∂ ϕ / sin θ ) . We can recognize in Equation (67) the presenceof a velocity term (the first r.h.s. term), followed by a lensing contribution, and the final four terms account for SW,ISW, volume and Shapiro time-delay effects.To numerically compute the magnitude of this effect, we calculate the mean fluctuation of the effect, at any givenredshift, as C D L (cid:96) = ⟨ ∆ D L ¯ D L ∆ D L ¯ D L ∗ ⟩ = π ∫ d k k [ I D L (cid:96) ( k )] P Ψ ( k ) , (68)where I D L (cid:96) = T m ( k ) ∫ d ¯ χ ¯ χ W χ ⎧⎪⎪⎨⎪⎪⎩ − j (cid:96) ( ¯ χk ) [ G Ψ ( ¯ a, k ) ¯ a + ( − H ¯ χ ) G Φ ( ¯ a, k ) ¯ a ]− ( − H ¯ χ ) [ (cid:96) ¯ χk j (cid:96) ( ¯ χk ) − j (cid:96) + ( ¯ χk )] G v ( ¯ a, k ) + ∫ ¯ χ d ˜ χ j (cid:96) ( ˜ χk )⎡⎢⎢⎢⎢⎣ χ ( G Ψ ( ˜ a, k ) + G Φ ( ˜ a, k ) ˜ a )− ( − H ¯ χ ) ˜ a H( ˜ a ) dd˜ a ( G Ψ ( ˜ a, k ) + G Φ ( ˜ a, k ) ˜ a ) + (cid:96) ( (cid:96) + ) ( χ − ˜ χ ) χ ˜ χ ( G Ψ ( ˜ a, k ) + G Φ ( ˜ a, k ) ˜ a ) ⎤⎥⎥⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ . (69)Here W χ represents the normalized object selection function (the normalization convention is ∫ d χ χ W χ = , j (cid:96) ( x ) are spherical Bessel functions of order (cid:96) and argument x and the quantities G Φ , G Ψ G m and G v are defined inAppendix B. Following [50], we have used the stationary-phase approximation and neglected the antenna-pattern functions. W χ can be easily related to the redshift distribution W z = χ W χ / a H , where ∫ d z W z = LensingDopplerISW z=0.1z=0.3z=0.5z=0.7z=0.9 C ℓ − − − − − − − − ℓ
10 100 1000
LensingDopplerISW z=1.1z=1.3z=1.5z=1.7z=1.9 C ℓ − − − − − − − − ℓ
10 100 1000
LensingDopplerISW z=3.1z=3.3z=3.5z=3.7z=3.9 C ℓ − − − − − − − − ℓ
10 100 1000
LensingDopplerISW z=4.1z=4.3z=4.5z=4.7z=4.9 C ℓ − − − − − − − − ℓ
10 100 1000
FIG. 1: Different contributions to the modification of D L , for a variety of redshift ranges. For the expression of different terms,see Eq. (67). In Figure 1 we show the relative importance of different contributions to Eq. (67): velocity (dotted lines and squaresymbols), lensing dashed lines and circles), and gravitational potentials (as the sum of SW, ISW, volume and Shapirotime-delay effects; dot-dashed lines and diamonds). We plot results computed by calculating the C (cid:96) of Eq. (68) forsources on a variety of redshift ranges, z ∈ {[ , ] , [ , ] , [ , ] , [ , ]} ; in the plots, colors indicate different values of z within the range (in ascending order, black, red, blue, green, purple).We can see how the velocity terms are the most important on large scales and low redshift (in analogy with thevelocity contributions to galaxy clustering, see [74]). The gravitational potentials contributions are always very small;they become more important than velocity ones at higher- z , but they stay almost two orders of magnitude below themost important contribution at each redshifts and scales, at maximum. Lensing terms are in general the dominantones; being an integrated (along the line of sight) quantity, their importance is lowest al lowest z , but they quicklyoverpower velocity terms (for (cid:96) > z > . (cid:96) , in Figure 2 we show how the luminosity distance correction dependson the maximum multipole used. The plot shows that the results are only mildly dependent on the scale selected; inthe following we will assume (cid:96) max = ∼ . z magnitudes, see e.g. [75, 76]), giventhat non-linear lensing contributions are dominant over linear perturbations [77]. However, in the non-linear regime,there are large effects on the luminosity distance-redshift relation and on the chirp mass estimate [54], due to localpeculiar velocities of the merging binary systems. For this reason we limit our analysis to large, linear, scales.The final result is then computed by calculating: ד ל = ⎛⎜⎝ ∑ (cid:96) (cid:96) + π C DL (cid:96) ⎞⎟⎠ / . (70)1 z=5z=3z=1z=0.1 δ D L / D L ℓ max
200 500 1000 2000
FIG. 2: Total correction to the luminosity distance as a function of the maximum (cid:96) used, for different redshifts.
We evaluate the ד ל contribution for a variety of future GW experiments, such as the Einstein Telescope (ET),DECIGO and the Big-Bang Observer.In Figure 3 we show the total correction to luminosity distance estimates due to perturbations, as a function of z . The dotted, dashed and dot-dashed lines show velocity, lensing and ISW-like contributions, respectively, whilethe solid line shows the total effect. Points show the predicted precision in measurements of the luminosity distance,at any redshift, for the Einstein Telescope (green points), DECIGO (red points) and the Big Bang Observer (blackpoints), taken from [9, 14].We can see that the additional D L uncertainty due to the inclusion of perturbations has a peak at low- z dueto velocity contributions, that surpasses the predicted measurement errors for all the experiments considered here.However, velocity effects rapidly decrease and lensing takes over. The integrated lensing effects increase with z , andtheir amplitude is of a factor ∼ D L uncertainty is consistently twice the predicted errors, making it a very relevant correction, thatone will need to take into account. IV. CONCLUSIONS
In this paper we investigated the effect of including perturbations in the estimate of luminosity distances as inferredfrom gravitational wave observations. While the usual calculation is performed in a homogeneous and isotropic FLRWUniverse, we show that the observed
GWs have a different luminosity distance D L than the real-space ones, in the sameway observed (redshift) -space galaxies are in a different location than the ones in real space. We derive expressions forthe luminosity distance in what we call Redshift-GW frame and the difference between that and the unperturbed case,in analogy with what has been done for photons (see e.g. [78, 79]), using the Cosmic Rulers formalism to generalizethe results of [50] by including all velocity, lensing and gravitational potentials (Sachs-Wolfe, Integrated Sachs-Wolfe,2
BBODECIGOETtotal correction DopplerlensingISW δ D L / D L − − − − − FIG. 3: Total correction to luminosity distance estimates due to perturbations (computed as in Eq. (70)), as a function of z .The dotted, dashed and dot-dashed lines show Doppler, lensing and ISW-like contributions, respectively, while the solid lineshows the total effect. Points show the predicted precision in measurements of the luminosity distance, at any redshift, for theEinstein Telescope (green points), DECIGO (red points) and the Big Bang Observer (black points). volume distortion and Shapiro time-delay effects).The main difference w.r.t. [50] is that the amplitude of the GW at the initial condition is not evaluated in background frame, but in Redshift-GW frame. In this way we are able to connect our results with the initial amplitude, usingdirectly the output from simulations of coalescing binary BHs, which produce the templates.The inclusion of deviations due to perturbations therefore causes an additional uncertainty in the determinationof D L from GW observations; we then computed the root-mean square fluctuations of this effect for a wide range ofredshifts.Our results show that, as expected, the dominant source of correction is due to lensing magnification effects, againin analogy with the case of photons and galaxy number counts (see e.g. [80]). The amplitude of this effect is howeversmall, and it will not constitute a relevant additional uncertainty for near future and next generation GW experiments.However, for future GW detectors such as the planned Big Bang Observer, the uncertainty we compute is predicted tobe as much as twice the normally predicted error in D L , making it necessary to be included in any realistic analysis.Finally, the inclusion of perturbations and the distinction between real space and Redshift-GW frame might affectother aspects of GW astronomy; we leave this investigation to a future work. Acknowledgments:
We thank Ilias Cholis for useful suggestions during the development of this project, Donghui Jeong, Julian Mu˜nozand Helvi Witek for interesting discussions.During the preparation of this work DB was supported by the Deutsche Forschungsgemeinschaft through the Tran-sregio 33, The Dark Universe.AR has received funding from the People Programme (Marie Curie Actions) of the European Union H2020 Programme3under REA grant agreement number 706896 (COSMOFLAGS). Funding for this work was partially provided by theSpanish MINECO under MDM-2014-0369 of ICCUB (Unidad de Excelencia “Maria de Maeztu”) and the TempletonFoundation.
Appendix A: Perturbation terms in the Poisson Gauge
From Eq. (37), the perturbations of ˜ g µν , ˜ g µν and the comoving metric ˆ g µν , ˆ g µν are˜ g = a ˆ g = − a ( + ) , ˜ g = a − ˆ g = − a − ( − ) , ˜ g i = a ˆ g i = , ˜ g i = a − ˆ g i = , ˜ g ij = a ˆ g ij = a ( δ ij − δ ij Ψ ) , ˜ g ij = a − ˆ g ij = a − ( δ ij + δ ij Ψ ) . (A1)For Christoffel symbols ˜Γ µρσ = ˜Γ µ ( ) ρσ + ˜Γ µ ( ) ρσ (using ˜ g µν ) and ˆΓ µρσ = ˆΓ µ ( ) ρσ + ˆΓ µ ( ) ρσ (using ˆ g µν ), we obtain˜Γ ( ) = H , ˜Γ ( ) i = , ˜Γ i ( ) = , ˜Γ i ( ) = , ˜Γ i ( ) j = H δ ij , ˜Γ i ( ) jk = , ˆΓ µ ( ) ρσ = , (A2)˜Γ ( ) = ˆΓ ( ) ˜Γ ( ) i = ˆΓ ( ) i ˜Γ ( ) ij = ˆΓ ( ) ij − H ( Φ + Ψ ) δ ij ˜Γ i ( ) = ˆ˜Γ i ( ) ˜Γ i ( ) j = ˆΓ i ( ) j ˜Γ i ( ) jk = ˆΓ i ( ) jk , ˆΓ ( ) = Φ ′ , ˆΓ ( ) i = ∂ i Φ , ˆΓ ( ) ij = − δ ij Ψ ′ , ˆΓ i ( ) = ∂ i Φ , ˆΓ i ( ) j = − δ ij Ψ ′ , ˆΓ i ( ) jk = − δ ik ∂ j Ψ − δ ij ∂ k Ψ + δ jk ∂ i Ψ , (A3)For four-velocity u µ , we find u = − a ( + Φ ) , u i = av i u = a ( − Φ ) , u i = a v i . (A4)For the tetrad: E ( ) ˆ00 = − Φ , E ( ) ˆ0 i = v i , E ( ) ˆ a = − v ˆ a , E ( ) ˆ ai = − δ ˆ ai Ψ , (A5) Appendix B: Power Spectra relations
In order to encode all possible DE models let us define in Fourier space the following relations v ( a, k ) = − T m ( k ) k G v ( a, k ) Ψ p ( k ) , (B1) δ Cm ( a, k ) = − T m ( k )G m ( a, k ) Ψ p ( k ) , (B2)Ψ ( a, k ) = T m ( k ) G Ψ ( a, k ) a Ψ p ( k ) , (B3)Φ ( a, k ) = T m ( k ) G Φ ( a, k ) a Ψ p ( k ) , (B4)4where G are suitable transfer functions and depend on the model that we are considering and T m ( k ) is the EisensteinHu transfer function [81] (or BBKS [82]).Here Ψ p ( k ) is the primordial value set during inflation, whose power spectrum ⟨ Ψ ∗ p ( k ) Ψ p ( k )⟩ = ( π ) δ D ( k − k ) P Ψ ( k ) . (B5)reads P Ψ ( k ) = π A δ H [ Ω m ( a = )G Ψ ( a = , k ) ] k − ( kH ) n − . (B6) G Φ , G Ψ G m and G v are, in principle, all functions of space and time. For ΛCDM and Dark energy + Dark mattermodels, we can write them as G Φ = G Ψ = D m (B7) G m = k Ω m H D m (B8) G v = f H k G m . (B9)where D m ( a ) is the growth function [defined as in 83] and f m ( a ) = d ln δ Cm d ln a = d ln G m d ln a = d ln D m d ln a (B10)is usually referred to as the growth factor. Here at background level, H = a H ( Ω m a − + Ω Λ0 ) , (B11) H ′ H = ( −
32 Ω m a − ) , (B12)and, at the first perturbative order, we have δ Pm ′ + ∇ v − ′ = , (B13) v ′ + H v + Ψ = , (B14) ∇ Φ = πG ¯ ρ m δ Cm , (B15)Ψ = Φ , (B16)where we have defined δ Pm as the matter overdensity in Poisson gauge and δ Cm = δ Pm − H v (B17)the gauge-invariant comoving density contrast. Appendix C: Dependence on maximum multipole
Here we show how the results change as a function of the maximum multipole (cid:96) max used in our calculations.In Figure 4 we show the results, for the lensing component and the total contributions, in the fiducial case of (cid:96) max = (cid:96) max = (cid:96) max = (cid:96) max ( z ) cases. In the latter, we assume (cid:96) max = , , , , ∼ In particular, D m is normalised as D m ( a DM ) = a DM , where a DM = a ( τ DM ) and Ω DM = ℓ max =1000 ℓ max = ℓ (z) ℓ max =200 δ D L / D L − − z0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 FIG. 4: Total correction to luminosity distance estimates due to perturbations, as a function of z , for three different cases of (cid:96) max . Solid lines show the total effect, while the dashed one refers to the lensing contributions. 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