The large scale structure in the 3D luminosity-distance space and its cosmological applications
aa r X i v : . [ a s t r o - ph . C O ] O c t The large scale structure in the 3D luminosity-distance space and its cosmologicalapplications
Pengjie Zhang
1, 2, 3, 4 Department of Astronomy, School of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai, 200240, China ∗ Tsung-Dao Lee institute, Shanghai, 200240, China IFSA Collaborative Innovation Center, Shanghai Jiao Tong University, Shanghai 200240, China Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240, China
Future gravitational wave (GW) observations are capable of detecting millions of compact starbinary mergers in extragalactic galaxies, with 1% luminosity-distance ( D L ) measurement accuracyand better than arcminute positioning accuracy. This will open a new window of the large scalestructure (LSS) of the universe, in the 3D luminosity-distance space (LDS) , instead of the3D redshift space of galaxy spectroscopic surveys. The baryon acoustic oscillation and the APtest encoded in the LDS LSS constrain the D L - D co A (comoving angular diameter distance) relationand therefore the expansion history of the universe. Peculiar velocity induces the LDS distortion,analogous to the redshift space distortion, and allows for a new structure growth measure f L σ .When the distance duality is enforced (1 + z = D L /D co A ), the LDS LSS by itself determines theredshift to ∼
1% level accuracy, and alleviates the need of spectroscopic follow-up of GW events.Buta more valuable application is to test the distance duality to 1% level accuracy, in combination withconventional BAO and supernovae measurements. This will put stringent constraints on modifiedgravity models in which the gravitational wave D GW L deviates from the electromagnetic wave D EML .All these applications require no spectroscopic follow-ups.
PACS numbers: 98.80.-k; 98.80.Es; 98.80.Bp; 95.36.+x
Introduction .— Discoveries of gravitational wave(GW) produced by black hole (BH)/neutron star (NS)-BH/NS mergers [1–4] have opened the era of gravita-tional wave astronomy. These GW events can serve asstandard sirens to measure cosmological distance fromfirst principles [5, 6] and therefore avoid various system-atics associated with traditional methods. It will thenhave profound impact on cosmology. However, to fulfillthis potential, usually it requires spectroscopic follow-upsto determine redshifts of their host galaxies or electro-magnetic counterparts. This will be challenging, for thethird generation GW experiments such as the Big BangObserver (BBO, [7, 8]), and the Einstein Telescope[28],which will detect millions of these GW events. Variousalternatives have been proposed to circumvent this strin-gent need of spectroscopic follow-ups [9–12].We point out a new possibility to circumvent this chal-lenge. These GW events are hosted by galaxies and aretherefore tracers of the large scale structure (LSS). Witharcminute positioning accuracy and 1% level accuracy inthe luminosity distance D L determination achievable byBBO, we are able to map the
3D large scale struc-ture in the luminosity-distance space (LDS) . Itis analogous to the redshift space LSS mapped by theconventional spectroscopic redshift surveys of galaxies( D L ↔ z ). Therefore it also contains valuable informa-tion of baryon acoustic oscillation (BAO), both across thesky and along the line of sight. As BAO in the redshift ∗ Email me at: [email protected] space measures the comoving angular diameter distance D co A and H ( z ) = dz/dχ at given redshift bins, BAO inLDS measures D co A and H L ≡ dD L /dχ at given D L bins.Here χ is the comoving radial distance. Both the D L - D co A relation and the D L - H L relation constrain cosmology(Fig. 1), without the need of redshift. Furthermore, both D co A and H L can be converted into cosmological redshiftthrough the distance duality relation 1 + z = D L /D co A .Similar to the redshift space distortion (RSD), peculiarvelocity also induces the luminosity-distance space dis-tortion (LDSD). This will enable a new measure of struc-ture growth rate f L σ , which differs from f σ measuredin RSD by a redshift dependent factor. The luminosity-distance space LSS .— Each GWevent provides a 3D position ( D obs L , ˆ n ). With millions ofthem, arcminute positioning accuracy, and O (1%) accu-racy in D L , we are able to measure the number densityfluctuation δ GW over effectively the entire cosmic vol-ume. This LSS is statistically anisotropic, since D obs L differs from its cosmological value D L , D obs L = D L (1 + 2 v · ˆ n − κ + · · · ) . (1)Here κ is the lensing convergence, describing the effectof gravitational lensing magnification. This effect is ahighly valuable source of cosmological information (e.g.[8]). v is the physical peculiar velocity [13] and ˆ n is theline of sight unit vector. If an object is moving awayfrom us ( v · ˆ n > z > ∼ κ ∼ O (10 − ) and v · ˆ n ∼ O (10 − ). Naively one wouldthink the lensing effect overwhelms the peculiar velocityeffect. This is indeed the case if we can subtract D L FIG. 1: The D GW L - D co A and D GW L - H L relations. Like the z - D co A relation and z - H relations constrained by galaxy spec-troscopic redshift surveys, the new set of relations is also sen-sitive to dark energy, demonstrated by the cases of variousdark energy equation of state. A more unique applicationof these relations is to constrain modified gravity models inwhich D GW L = D EM L . We show two such cases, parameterizedby ǫ a = ± .
05. Using GW alone, w DE and ǫ a constraints arelargely degenerate. Combination witt electromagnetic waveobservations can break this degeneracy straightforwardly. with cosmological redshift from spectroscopic follow-up.However, what affects the LDS LSS is the gradient of κ and v · ˆ n along the line of sight. Under the distanceobserver approximation and up to leading order, δ LDSGW ≃ δ GW + α ∇ κ · ˆ n + β ∇ ( v · ˆ n ) · ˆ n . Since κ is lack of variationalong the line of sight, its contribution is sub-dominantcomparing to the velocity gradient contribution[29]. TheLDS power spectrum then resembles the Kaiser [14] plusFinger of God formula in RSD, P LDS ( k ⊥ , k k ) = P g ( k ) (cid:18) f L b g u (cid:19) F ( k k ) . (2)Here u ≡ k k /k and k ≡ q k ⊥ + k k . k ⊥ ( k k ) is thewavevector perpendicular (parallel) to the line of sight. b g is the density bias of GW host galaxies. F ( k k ) de-scribes the FOG effect. There are two majo differencesto RSD. First, f L ≡ (cid:18) D L / (1 + z ) d ( D L ) /dz (cid:19) × f . (3)It differs from f ≡ d ln D/d ln a in RSD by a redshiftdependent factor. This arises from the different effects FIG. 2: The forecasted measurement errors on D co A , H L and f L σ for a number of D GW L bins, assuming a ten year obser-vation with a BBO-like experiment. These measurements canconstrain dark energy (Fig. 1), or determine redshift adoptingthe distance duality. The distance measurement error σ D is amajor limiting factor. BBO can reach σ ln D ∼ .
01 for NS-NSmergers and ∼ .
001 for BH-BH mergers. Therefore we showthe cases of σ ln D = 0 . , . , . , . σ D degrades mea-surement of Fourier modes with k k > ∼ . . /σ ln D ) h/ Mpc. of peculiar velocity on the luminosity distance ( D L → D L (1 + 2 v · ˆ n )), and on redshift ( z → z + v · ˆ n (1 + z )).The prefactor in Eq. 3 is zero at z = 0 and increaseswith z . It becomes larger than unity at z > ∼ .
7, wherethe peculiar velocity induced distortion is larger in LDSthan in redshift space. The second difference is that the H factor shown up in FOG should be replaced by H L . Cosmological applications .— Now we proceed toconstraints on D co A , H L and f L σ using the LDS powerspectrum measurement. Assuming Gaussian distributionin the power spectrum measurement errors, the Fishermatrix is F αβ = X k ∂P LDS ( k ) ∂λ α σ − P ∂P LDS ( k ) ∂λ β . (4)The sum is over k bins. Instead of directly fitting D co A ,1 /H L and f L σ , we fit their ratios ( A ⊥ , A k , A v ) with re-spect to the fiducial cosmology, along with b g . Namely λ = ( A ⊥ , A k , A v , b g ). A ⊥ ( A k ) scales the pair separationperpendicular (parallel) to the line of sight. Under suchscaling, P LDS ( k ⊥ , k k ) → A − ⊥ A − k P LDS (cid:18) k ⊥ A ⊥ , k k A k (cid:19) . (5)Statistical error σ P in the power spectrum measurementis σ P = r N k (cid:20) P LDS ( k ) + 1¯ n GW W − k ( k ) W − ⊥ ( k ) (cid:21) . (6) N k is the number of independent Fourier modes in the k bin, proportional to the survey volume V survey . W k ( W ⊥ )is the window function parallel(perpendicular) to the lineof sight, due to statistical errors in the D L measurementand angular positioning.We adopt the fiducial cosmology as the ΛCDM cos-mology with Ω m = 0 . Λ = 1 − Ω m , Ω b = 0 . h = 0 . σ = 0 .
83 and n s = 0 .
96. We are tar-geting at BBO or experiments of comparable capabil-ity. BBO has a positioning accuracy better than 1arc-minute for all NS/BH-NS/BH mergers in the hori-zon [8]. Since we are only interested at large scale( k < ∼ . h/ Mpc), W ⊥ = 1 to excellent approximation.In contrast, W k = exp( − k χ σ D /
2) and the distancemeasurement error σ D has a significant effect. For typ-ical z ∼ σ ln D ∼ .
01, the induced damping issignificant at k > ∼ . h/ Mpc. This limites the powerspectrum measurement to the linear regime. On onehand, it reduces the constraining power. On the otherhand, it simplifies the theoretical modeling, and allowsus to neglect the FOG term in Eq. 2. ¯ n GW is the av-erage number density of GW events in the survey vol-ume. The local NS-NS merger rate is constrained to R = 1540 +3200 − Gpc − year − [4]. The BH-BH mergerrate is a factor of ∼
10 smaller [15]. Therefore ¯ n GW is dominated by NS-NS mergers. For the evolution ofNS-NS merger rate, we adopt the model in [7, 8]. Forthe bestfit R , the total number of GW events per yearis 0 . , . , . × at z < , , D co A , H L and f L σ in multiple D L binsto a few percent accuracy (Fig. 2). These estimationsadopt ∆ t = 10 years and b g = 1. Since the power spec-trum measurement error is shot noise dominated, the sta-tistical errors roughly scale as ( R ∆ t ) − b − g . But theirdependence on σ D is more complicated. Fig. 2 showsthe cases of σ ln D = 0 . , . , .
02, within the reach ofBBO capability. σ D has major impact on cosmology, bysignificantly affecting the number of accessible Fouriermodes. For σ ln D = 0 .
001 which may be achieved byBH-BH merger observations of BBO or NS-NS mergersobservations of more advanced experiments, cosmologi-cal constraints can be significantly improved, especiallyfor H L and f L σ .These constraints alone are able to constrain dark en-ergy, demonstrated in Fig. 1. One way to under its con-straining power is that, when the distance duality holds(1 + z = D L /D co A ), the D L - D co A relation is equivalent tothe more familiar z - D L relation in the supernovae cos-mology. It indeed contains valuable information of darkenergy. However, due to lower number density and largererror in the D L measurement, these constraints are sig-nificantly worse than what will be achieved by stage IV FIG. 3: Constraints on D GW L /D EM L , which is parametrized bya physically motivated parameter ǫ a . This will put uniqueand powerful constraints on modified gravity models. redshift surveys such as DESI [16] and Euclid [17].Nevertheless, these measurements are unique in con-straining modified gravity (MG) models. In these mod-els, GW propagation may differ from electromagneticwave propagation and D GW L = D EM L . This has beenproposed and been applied to constrain gravity (e.g.[18, 19]). There are two degrees of freedom to modify theGW propagation equation [20]. One allows for deviationbetween the GW speed and the speed of light. However,GW170817 [4] has constrained the relative difference tobe within O (10 − ) [21], and ruled out a large fraction ofMG models (e.g. [22]). In contrast, the other degree offreedom is essentially unconstrained. This is to modifythe friction term in the GW propagation equation. [20]parametrizes this modification as H ( t ) → H ( t )(1 − δ ( t )).To avoid confusion of δ ( t ) with the commonly used LSS δ symbol, we adopt a different notation ǫ GW . ǫ GW = 0leads to η ≡ D GW L D EM L = exp (cid:18) − Z z dz z ǫ GW ( z ) (cid:19) = 1 . (7)Usually we expect no deviation from GR in the earlyepoch ( ǫ GW ( a → → ǫ GW ( a ) = ǫ a a . Under thisparametrization, η = exp( − ǫ a (1 − a )) = exp( − ǫ a z/ (1 + z )).Combining the z - D co A and/or z - D EM L measurementsfrom electromagnetic wave telescopes, and the D GW L - D co A measurements here, we can measure D GW L /D EM L . Com-bining the z - H and D GW L - H L measurements can also con-strain this ratio. BBO can measure this ratio and con-strain ǫ a to percent level accuracy (Fig. 3). It will thenbe sensitive to MG models such as the RR model with m R ✷ − R correction in the action [20, 23]. Since thistest of gravity is on the tensor part of space-time metric,it is highly complementary to tests on the scalar part.The statistical error here is dominated by the GW ob-servations. σ ln D ≃ .
001 will allow for better than 1%accuracy in ǫ a , and longer observations can further help. Further applications .— We point out that fu-ture GW experiments will map LSS in a new space,namely the luminosity-distance space (LDS), through theluminosity-distance determined using NS/BH-NS/BHmergers. We present a proof of concept study on itsmajor LSS patterns (BAO and LDSD), and list a fewcosmological applications (constraining dark energy, de-termining cosmological redshift and probing gravity). Ithas other applications. One is to probe the primordialnon-Gaussianity. Another is to probe the horizon scalegravitational potential, since it alters the luminosity dis-tance and generate a relativistic correction to the numberdensity distribution of GW events. Both require the LSSmeasurement near the horizon scale. The LDS LSS is inparticular suitable since it naturally covers the whole 4 π sky and can extend to z ≫