Relativistic effects in Lyman-α forest

Transcription

1 Prepared for submission to JCAP Reativistic effects in Lyman-α forest arxiv: v2 [astro-ph.co] 23 Feb 2016 Vid Iršič, a Enea Di Dio, b,c Matteo Vie b,c a The Abdus Saam Internationa Centre for Theoretica Physics, Strada Costiera 11, I Trieste, Itay b INAF - Osservatorio Astronomico di Trieste, Via G. B. Tiepoo 11, I Trieste, Itay c INFN-Nationa Institute for Nucear Physics, via Vaerio 2, I Trieste, Itay E-mai: virsic@ictp.it, enea.didio@oats.inaf.it, vie@oats.inaf.it Abstract. We present the cacuation of the Lyman-apha Lyman-α) transmitted fux fuctuations with fu reativistic corrections to the first order. Even though severa studies exist on reativistic effects in gaaxy custering, this is the first study to extend the formaism to a different tracer of underying matter at unique redshift range z = 2 5). Furthermore, we show a comprehensive appication of our cacuations to the Quasar-Lyman-α cross-correation function. Our resuts indicate that the signa of reativistic effects are sizeabe at Baryonic Acoustic Osciation BAO) scae mainy due to the arge differences in density bias factors of our tracers. We construct an observabe, the anti-symmetric part of the cross-correation function, that is dominated by the reativistic signa and offers a new way to measure the reativistic terms at reativey sma scaes. The anaysis shows that reativistic effects are important when considering cross-correations between tracers with very different biases, and shoud be incuded in the data anaysis of the current and future surveys. Moreover, the idea presented in this paper is highy compementary to other techniques and observabes trying to isoate the effect of the reativistic corrections and thus test the vaidity of the theory of gravity beyond the Newtonian regime.

2 Contents 1 Introduction 1 2 Ly-α absorption 2 3 Ly-α observabe Quasar density fuctuations Non-inear corrections Biases point correation function Loca term expansion in H/k Symmetric correation function Anti-symmetric correation function Signa-to-noise anaysis Discussion on systematic effects 27 5 Concusions 29 A Anguar power spectrum 30 B CLASS modifications 32 1 Introduction The ast decade has witnessed an enormous progress in the cosmoogica investigation of the intergaactic medium IGM) as probed by the Lyman-α forest see [1, 2] for reviews). In particuar, the Lyman-α forest is now a viabe cosmoogica observabe that can hep in putting constraints on cosmoogica parameters and/or deviations from a standard scenario based on Cod Dark Matter CDM) and a cosmoogica constant. More recenty, the very arge number of quasar spectra of the SDSS-III/BOSS Soan Digita Sky Survey-III/Baryon Osciation Spectroscopic Survey) coaboration using a state-of-the-art anaysis, has aowed to discover Baryonic Acoustic Osciations BAOs) in the transmitted Lyman-α fux at z 2.3 [3, 4]: this spectacuar confirmation of the cosmoogica nature of the Lyman-α forest was mainy made possibe by expoiting the fux correations in three dimensions out to arge scaes. Moreover, the one-dimensiona anaysis of the transmitted fux power has aowed to pace the most stringent bounds on tota neutrino mass and cod dark matter codness by using spectra at high and ow resoution [5 7]. The first steps have aso been taken in measuring the cross-correations between various different observabes in the same voume of the survey. Such exampes incude the cross-correations between the Lyman-α forest and Damped Lyman-α systems [8] 1

3 and the Lyman-α forest and Quasars QSOs) [9, 10]. The first measurements measured the correations up to the BAO scae and confirmed the ocation of the peak, however the error bars remain arge. Nevertheess, the anaysis was done on barey haf of the avaiabe data by the SDSS-III/BOSS survey and future generation of arge-scae structure surveys e.g. SDSS-IV/eBOSS [11], DESI [12], etc.) are going to improve on the precision of the measurements to a percent eve thus forcing the theoretica modes to match unprecedented demanding constraints in terms of accuracy. However, as the observations move out to arger scaes the vaidity of a simpe Newtonian description starts to come into question. Recent years have seen a ot of effort put into understanding and possiby measuring the signa of reativistic effects and/or weak gravitationa ensing in the 2-point statistics of the gaaxy custering, see e.g. [13 21]. Moreover, it has been specuated that the most powerfu too for such an anaysis woud be the cross-correations between different gaaxy popuations [13, 14, 16, 18, 20, 21]. The correation of two differenty biased tracers of the underying matter fied wi, at east to the first order, boost the signa of the beyond-kaiser effects reative to the difference in the density bias factors. Thus the arger the difference the stronger the signa. Unfortunatey the difference in density bias factors among different gaaxy popuations is reativey sma, and the idea has ony received attention very recenty. On the other hand, the tracers such as Lyman-α forest or QSOs that are measured in the same survey voume by aready existing surveys are very differenty biased tracers. Expoiting this difference woud boost the signa of the reativistic effects to a measurabe eve. Given the highy non-inear reation between the observed Lyman-α forest fux fuctuations and the underying density fied, one might question the abiity to use the Lyman-α forest as a inear tracer. However, severa studies have shown that on arge scaes Lyman-α forest behaves as any other oca tracer and any deviations start to show up at smaer scaes [22 24]. In this paper we compute the theoretica predictions for the QSO-Lyman-α crosscorreation function and show how the difference in the density biases of the two tracers boosts the signa of the reativistic effects to a 10% eve above scaes of 40 h 1 Mpc. The paper is structured as foows: in Sections 2 and 3 we introduce the Lyman-α forest observabe and derive the fux fuctuations using fuy reativistic treatment. In Section 4 we appy the cacuations to the QSO-Lyman-α cross-correation function and present the resuts for both symmetric and anti-symmetric parts. We concude in Section 5. 2 Ly-α absorption The Lyman-α forest is an absorption feature seen in the spectra of high redshift QSO and gaaxies and is a unique tracer of the underying matter distribution in the redshift range of z = 2 5. When the ight from a distant QSO passes through the intergaactic medium IGM), the neutra hydrogen aong the ine of sight wi cause absorption of the QSO continuum by the redshifted Lyman-α nm) resonance ine. The coection of this absorption ines bend into one feature caed the Lyman-α forest. In fact it is more correct to think 2

4 of the Lyman-α forest as being caused by continuous absorption of the midy fuctuating density of the IGM. Observed transmission of the fux is thus a tracer of the baryon distribution of the IGM and one of the successes of the Lyman-α in the ast few years has been to estabish it as a precision cosmoogy probe. The photons absorption due to Lyman-α is described by the differentia equation d n N γ = A n HI φe, E α ) dt 2.1) where E α is the Lyman-α absorption ine, N γ the photon occupation number, t the is the proper time in absorption HI rest frame and n HI the number density of neutra hydrogen. A is a physica constant derived from Einstein coefficients and given by A = e 2 f α /4ε 0 m e, where e is the eectron charge, ε 0 is vacuum permittivity, m e eectron mass and f α = osciation strength of the Lyman-α transition. The integra of the absorption ine profie φe, E α )) over the energy is normaized to the unity. We define the optica depth as τ d n N γ. 2.2) ight-cone The above definition agrees with the non-reativistic resut in the appropriate imit. 3 Ly-α observabe On arge scaes we can work under the approximation that φ E, E α ) = δ D E E α ). Hence, we negect therma broadening which affects scaes of 10 h 1 Mpc. This approximation is vaid as ong as we are interested in arge scae correations. Therefore, our observabe is the transmitted fux fraction F, i.e. the photon occupation number observed normaized with the emitted one, which yieds to F = e τ, 3.1) in terms of the measured ange 1 n and redshift z, corresponding to the shift in the Lymanα spectrum i.e. 1 + z = E α /E), namey F n, z). One can define the fuctuations of the transmitted fux as F n, z) δ F n, z) F n, z) 1, 3.2) where. denotes the anguar mean at fixed observed redshift z. On arge scaes, we can inearize the reation between the transmitted fux and the optica depth and express the observabe quantity as δ F n, z) = τ n, z) τ n, z) ) δτ n, z). 3.3) 1 We foow the notation of Ref. [25], where n denotes the direction of propagation of the photon. Hence the measured ange is n, but for sake of simpicity wi refer to n as argument of the observabe quantities. 3

5 The inearization of the exponentia in Eq. 3.1) might question the vaidity of the perturbative approach. Nevertheess, as shown in section 3.2, one can expand perturbativey the transmitted fux F in terms of the optica depth τ by introducing some bias factors which encode the effect of the mode couping. From Eqs. 2.1) and 2.2) we obtain τ n, z) = A = A n HI δ D E E α ) dt = A nhi E obs δ D 1 + z) E α E obs n HI δ D E obs 1 + z) E α ) dt ) dt dz dz = An 1 + z dt HI E α dz 3.4) where a quantities are evauated at 1 + z = E α /E obs and the suffix obs denotes quantities evauated with respect to the observer rest frame and t is the proper time 2 in the photons absorption frame. In this section we aim to derive Eq. 3.3) to first order in perturbation theory. As ong as we describe observabe quantities, we have the freedom to work in any gauge without oss of generaity. We choose to work in Newtonian gauge d s 2 = a 2 ds 2, where a is the scae factor of the universe and ds 2 = 1 + 2Ψ) dη Φ) δ ij dx i dx j 3.5) is the conforma metric, where the metric perturbations Ψ and Φ are the Bardeen potentias. From Eq. 3.4), we need to compute dt dz to first order in perturbation theory dt dz = dt dη dη dz = dt dη dη dz = 1 + Ψ d z 1 + z dη + dδz ) 1 dη = 1 + Ψ d z 1 + δz ) z) d )) δz z dη 1 + z dη 1 + z [ 1 + Ψ = H z) 1 + z) 2 1 δz 1 + z + 1 )] d δz H dη 1 + z [ ) 1 = H 1 + z) Ψ + 1 Ḣ δz H z + 1 d H dη [ ) 1 = H 1 + z) Ψ + 1 Ḣ H 2 ) ] δz 1 + z δz 1 + z + n v + H 1 r n v + H 1 Φ ], 3.6) where H = ȧ/a is the comoving Hubbe parameter and a dot denotes the derivative with respect to the conforma time η. We have aso introduced the unperturbed redshift z by expanding the observed redshift as z = z + δz. In genera, in this work, we wi denote with an overbar background quantities in perturbation theory. In the ast equaity we have assumed that sources move aong geodesic n v + Hn v r Ψ = 0, 3.7) 2 We prefer not to denote the proper time with τ to avoid confusion with the optica depth. 4

6 and used δz ηo ) 1 + z = Ψ + n v + Ψ + Φ) dη, 3.8) η which is derived from geodesic equation. We remind that the physica density of HI, i.e. n HI z) in Eq. 3.4), is reated to the density computed in perturbation theory to first order as n HI z) = n HI z) 1 d n n ) HI δz + δ HI. 3.9) dz The second term in the brackets describes the source evoution and it is often parametrized through the evoution bias fevo Lyα n a 3 n ) HI = z) d n n HI. 3.10) H η dz Finay, we find where τ n, z) = τ z) [ 1 + δ sync HI + H 1 r n v +n v + Ψ + H 1 Φ + f Lyα evo τ z) A E α n HI H 1 + z) 2 + Ḣ H 2 f evo Lyα ] 3 ) Hv ) δz 1 + z. 3.11) 3.12) and v is the gauge invariant) veocity potentia in Newtonian gauge defined through v = v. We have aso expressed the density perturbation in synchronous gauge, through δ sync = δ + 3 f evo ) Hv. The unperturbed optica depth τ can be rewritten as [24]): τ z) = A n HI E α H s 1 ) X 2 ) 0.7 T0 ωb,0 2 H 0 Γ γ,hi K h H 1 + z)6, 3.13) where T 0 is the mean temperature of the gas and we have used the mass fraction of neutra hydrogen through the photo-ionization equiibrium n HI = n 2 α T ) H = X2 ρ 2 b α T ) Γ γ,hi m ) H Γ γ,hi where Γ γ,hi is the photo-ionization rate, X is the hydrogen mass fraction, n H is the hydrogen number density, m H is the hydrogen mass, ρ b is the baryon energy density and the recombination rate αt ) is given by where α 0 = s 1 cm 3. ) T 0.7 α T ) = α , 3.15) K 5

7 In this work we consider the Panck best fit cosmoogica parameters [26], namey ω b,0 = 0.022, h = 0.67 and the astrophysica parameters T 0 = K, X = 0.75 and Γ γ,hi = s 1. With these parameters Eq. 3.13) reduces to z)6 τ z) 7 10 E z) 3.16) with Ez) = Hz)/H Quasar density fuctuations In the foowing of this paper we wi be interested in cross-correating the Lyman-α signa with the quasar density fuctuations. In the ight of that we give an expression with reativistic corrections used for the QSO fuctuations as found in the iterature, and used in this paper. The number counts for discrete tracers has been derived in [25, 27] in gaaxy custering framework. Nevertheess their resuts can be appied as we to QSOs, just considering the appropriate bias factors, Q n, z) = b Q δ sync + 5s 2)Φ + Ψ + 1 [ Φ + r n v)] + fevo Q 3 ) Hv H ) Ḣ + H s rs ) r S H + 5s f evo Q Ψ + n v + dr Φ + Ψ) s rs [ dr 2 r ] S r Ω Φ + Ψ). 3.17) 2r S r 0 Differenty from Lyman-α forest, singe tracer number counts incude a cosmic magnification contribution and a magnification bias parameter s defined as s = 2 n n z, n L) 3.18) 5 n L where L denotes the threshod uminosity of the survey and n is the background number density. Simiary, the evoution bias factor of QSOs is reated to the QSO number density distribution see Eq. 3.10)). 3.2 Non-inear corrections In the previous section we have computed the optica depth τz) to first order in perturbation theory. However, the exponentia 3.1) introduces a mode couping that we negect by inearizing this expression. In the context of the reativistic corrections beyond simpe Kaiser approximation this is vaid if the terms themseves are sma. In this section we give an estimation of the bias for the reativistic terms mainy Dopper term as it is argest). We use the formaism described in [28, 29], and extend it to capture reativistic correction described in this paper. We do not use this inear description to try to describe the density bias. L 6

8 If the terms are not sma, a correction capturing the non-inearity of the F τ transformation is appied in terms of bias factors, as is the case with the redshift-space distortions. The fu cacuation for the veocity-gradient bias factor can be found in the iterature [28, 29]), but we wi review the basic steps here for carity. To the eading order, the reativistic corrections enter our Lyman-α cacuations ony through the change of coordinates, from the rea space time radia coordinate) to a redshift redshift-space coordinate) as can be seen from Eqs Negecting for the moment the reativistic corrections and focusing ony on the veocity-gradient part, we see that the observed optica depth can be written as τn, z) = τ 1 + δ) p 1 + H 1 r n v ), 3.19) where the vaue of p is usuay taken in the range [28], and δ is the non-inear density fuctuations. From here one can define an optica depth bias with respect to the redshift-space distortions as b τ,v 1 τ τ H 1 r n v) = ) This is not a new resut, indeed it tes us that there is no veocity-gradient bias, as one woud expect. However, the observabe of the Lyman-α forest is not the optica depth but the fux, and the non-inear transformation is appied after the redshift-space distortions. One can write the fux bias of the veocity-gradient as b F,v 1 F F H 1 = 1 F r n v) F H 1 = r n v) = 1 F τ F τ H 1 = 1 F r n v) F τ τ = F n F =. 3.21) F This resut has been first obtained by [28] and has been further shown to work extremey we in the absence of therma smoothing [29]). The above review cacuation was done using redshift-space distortions ony, as written in Eq However there is no reason the same cacuation can be carried out in the same manner if one repaces H 1 r n v with any other quantity, e.g. a reativistic dopper term n v). In such a case one woud modify the optica depth reation as ) ) τn, z) = τ 1 + δ) 1 p + H 1 r n v + 3 fevo Lyα + Ḣ H 2 n v. 3.22) Accordingy the optica depth bias, with respect to Dopper term woud now be b τ,d 1 τ = 3 fevo Lyα + Ḣ τ n v) H 2, 3.23) 7

9 and since the b τ,d is a constant it furthermore means b F,D = b τ,d F n F F = b τ,d b F,v. 3.24) Thus any kind of scaing of the redshift coordinate that is ineary added to the redshift space distortions acquires the same fux) bias factor due to non-inear transformation between the fux and the optica depth, with a pre-factor that is nothing ese but the optica depth bias factor for the same quantity. The above cacuation was carried out in the imit of no therma broadening, since our paper focuses ony on the arge scae imit. However there is no reason that therma broadening woud distinguish between veocity-gradient effect or Dopper term effect, thus resuting in the same correction to the overa bias factor. Moreover, in the above mode no second order effects were incuded. In that sense it shoud be tested on numerica simuations, and ony draws on the fact that the eading reativistic effects considered in this paper are nothing ess than sma corrections to redshift-space transformation, compared to the redshift space distortion. Thus they woud behave in the same way under the non-inear transformation. To fuy understand the vaues of the bias factors of the Lyman-α forest a more detaied anaysis is required that is beyond the scope of this paper. Therefore to correcty describe the transmitted fux fuctuation we change the bias factors predicted by the inear theory with the observed bias factors δ F n, z) = b α δ sync + b v H 1 r n v + b R [ 2 + Ḣ H 2 f evo Lyα +n v + Ψ + H 1 Φ + f Lyα evo ) δz 1 + z 3 ) ] Hv. 3.25) The first two terms correspond to a simpe Kaiser approximation for the Lyman-α forest tracer, whie the terms in the square bracket describe the reativistic corrections. Note that the veocity-gradient bias different than unity is the resut of the redshift space distortion acting on the optica depth, before taking the non-inear transformation to the observabe which is the fux, and it is not to be interpreted as a vioation of the equivaence principe. This resut is known and widey used in the iterature [22 24, 28]. According to the mode presented in this section we adopt a universa bias for a reativistic terms, b R, which is to the correct description to inear order. 3.3 Biases In a reativistic framework one has to define the bias parameters with respect the density, and redshift space distortions for Lyman-α forest tracer, in a specific gauge. Indeed, if on sma scaes the differences between different gauges are generay suppressed by a factor H/k) 2, on arge scaes the gauge choice becomes reevant and it might introduce some spurious k-dependences in the bias factors. However they do not affect the main resuts of our work which is determined by terms suppressed ony by H/k. Our choice to appy density bias b α in synchronous 8

10 comoving gauge is we justified from the assumption that the tracer and the underying density perturbation experience the same gravitationa fied and they move with the same veocity. Hence in their rest frame we can appy the inear bias prescription. In the standard approach the veocity-gradient bias is appied to the Kaiser term, and is derived using Newtonian reation between time derivative of density perturbation and pecuiar veocity. If Cod Dark Matter CDM) is the ony species which contributes to energy-momentum tensor perturbations, this reation is equivaent to the inearized Einstein equation provided that the density perturbation is expressed in the synchronous and veocity in the newtonian gauge. Therefore, to be consistent with standard anaysis we used veocity-gradient bias in newtonian gauge. In this work we consider the Lyman-α density bias evoution as suggested by the observations [3, 4, 23, 30]) b α z) = b α z = 2.5) ) 1 + z ) 3.5 with b α z = 2.5) = For the veocity-gradient bias of the Lyman-α forest we foow the recent work using simuations [22]) which gives: b v β ) αz)b α z) 1 + z 0.86, β α z) = β α z = 2.5), 3.27) fz) 3.5 with the vaue at the mean redshift β α z = 2.5) = 1.36, and f = d n δ/d n a is the growth rate factor. In addition we have a non-vanishing evoution factor for Lyman-α forest, i.e. fevo Lyα 0 in Eq. 3.25). Foowing Eq. 3.10) the vaue of the Lyman-α evoution bias can be predicted from the redshift evoution of the neutra hydrogen fraction. Using the resut of Eq. 3.13), we see that n HI z) Γ 1 γ,hi T ρ 2 b. For τ, we have foowed the cacuations of [24]), and assumed that both UV background radiation Γ γ,hi ) and the mean temperature of the gas T 0 are constant with redshift, thus resuting in a simpe evoution of neutra hydrogen n HI 1 + z) 6. In turn this impies a vaue of fevo Lyα = 3. This is the vaue used in the cacuations in the remainder of this paper, uness stated otherwise. Moreover, for the Lyman-α forest, we aso have bias of the reativistic terms. Given the definition of such a bias Eq and 3.25) we use an approximation for its vaue and say b R z) = b v z). 3.28) For the QSO density bias factor b Q ) we use the semi-empirica reation derived by [31, 32]: b Q z) = z) ) The empirica reation fits the observed data very we at ower redshifts and can be safey extrapoated to our mean redshift of z = 2.5, giving a vaue of b Q z = 2.5) 4, which is a bit high, but in agreement with independent observations at higher redshifts [9, 10, 33]. 9

11 To derive both the evoution f Q evo) and magnification s) bias parameters we have used a fitting mode for Quasar Luminosity Function used in BOSS DR9 data anaysis [34]. The assumed threshod absoute magnitude was M = The derived vaues for the magnification and evoution bias factors are s = and f Q evo = point correation function Having derived the Lyman-α forest observabe and defined the QSO observabe we are now interested in studying the 2-point function of the correation between Lyman-α and QSOs. Namey ξ Qα z 1, z 2, θ) Q n 1, z 1 ) δ F n 2, z 2 ) 4.1) where cos θ = n 1 n 2 and Q n, z) is the QSO number counts. Equivaenty one can define the cross-correation where the positions of Ly-α and QSO have been exchanged: ξ αq z 1, z 2, θ) δ F n 1, z 1 ) Q n 2, z 2 ). 4.2) Ly! F x r 2 QSO r 1 O Figure 1. The figure iustrates the correation between Quasars and Lyman-α forest. The comoving distances of the sources from the observer are denoted by r 1 and r 2, whie x refers to the comoving separation between them. From Fig. 1 we remark that, by assuming a fiducia cosmoogy, we can reate the anges β and θ. Hence we can express the correation functions 4.1, 4.2) in terms of the tripet z 1, z 2, β), and we denote them as ˆξ αq and ˆξ αq. We see directy that the transformation of coordinates z 1 z 2, β π β) maps from one to the other crosscorreation functions, ˆξ Qα z 2, z 1, π β) = ˆξ αq z 1, z 2, β). Whie the interchange of the redshifts is obvious, one can see from Fig. 1 that the ange that needs to be fipped is β, 10

12 the ange between the QSO and the pixe in Lyman-α forest, whie the ange θ describes their anguar separation. The anges are of course connected and either can be used to fuy describe the cross-correation function, however θ is a more natura choice to use with the anguar power spectrum expansion see beow). In genera, it is aways possibe to express the 2-point correation function in terms of anguar power spectra C z 1, z 2 ) as ξ z 1, z 2, θ) = 1 4π 2 + 1) C z 1, z 2 ) P cos θ), 4.3) where P denote the Legendre poynomias. In appendix A we introduce the corresponding anguar power spectra. Since we are mainy interested in the 1-dimensiona correation function aong the radia direction a direct evauation of Eq. 4.1) is simper and more intuitive than passing through Eq. 4.3). Nevertheess, we prefer to use Eq. 4.3) to hande the integrated terms cosmic magnification, integrated Sachs-Wofe, Shapiro time-deay) and take advantages of cass code [35, 36], whose modified transfer functions are shown in appendix B, than rey on approximations. For the 1-dimensiona radia correation, Eq. 4.1) reduces to a function of two redshifts z 1 and z 2. It is then convenient to re-expressed them as a redshift separation between the two source positions z = z 1 z 2 and a mean redshift z mean = z 1 + z 2 ) /2. By assuming a fiducia mode we can aways map the redshift separation in terms of comoving separation as x z 1, z 2 ) r z 1 ) r z 2 ), where rz) = z 0 dz Hz ) 4.4) is the unperturbed comoving distance and H 2 z) = H0 2 a 3 Ω m + a 2 ) Ω K + Ω Λ. 4.5) We remark that to first order in perturbation theory it is enough to use the background unperturbed redshift-distance reation Eq. 4.4). Because, in genera, sources are not homogeneousy distributed aong the redshift, the 1-dimensiona correation function wi not depend ony on the absoute vaue of z, or x z 1, z 2 ), but on its sign as we. This change of coordinates simpifies the reation when switching the paces of Lyman-α and QSO, ξ αq x, z mean ) and ξ Qα x, z mean ), since the ony transformation required is x x. Ideay, since the pure radia correation is described by a singe ine of sight, the ight emitted by a QSO at high redshift woud be partiay absorbed by the IGM at ower redshift and it wi generate the Lyman-α forest absorption ines. This situation is ceary a vioation of the photon number conservation assumption on which the derivation see for instance Ref. [25, 27]) of number counts observabe for discrete tracers is based. Nevertheess in our work we never consider this situation and we assume that the two tracers, Lyman-α forest and QSO are correated gravitationay ony. This assumption is consistent with what is done in observations, where the correation between the Lymanα forest and the QSO, which produces the absorption ines, is not considered. Another 11

13 issue is due to the fact that ight emitted by a QSO at high redshift woud be affected aso by the gravitationa potentia generated by the Lyman-α forest through the cosmic magnification effect. Being the Lyman-α forest and the QSO on the same ine of sight, the impact parameter of the ight defection vanishes and we can not anymore considered it in the weak ensing regime. Since, in any case, exact radia correation are not taken into consideration we negect this effect and we describe it as a weak ensing phenomenon. Let us aso stress that these possibe issues are competey reieved once we consider the dipoe with respect to the ange β defined in Fig. 1. Indeed by averaging over a anges the contribution of one singe direction has not weight in the integra and its contribution shoud be negigibe. We eave the anaysis of the dipoe of the correation function to a future work. We compute then the fu reativistic 2-point correation function ξ Qα = ξ newt Qα + ξ magnification Qα + ξ reativistic Qα 4.6) where we dropped the dependence on z and z mean for simpicity. We start by introducing the primordia curvature power spectrum through R in k) R in k ) = 2π) 3 δ 3) D k + k ) P R k) 4.7) and we write any first order variabe X in terms of transfer functions normaized with the primordia curvature as ξ newt X k, η) = T X k, η) R in k). 4.8) Then, by computing the different contributions we find Qα [ b Q δ + H 1 r n v) ] n, z 1 ) [ b α δ + b v H 1 r n v) ] n, z 2 ) dk = b Q z 1 ) b α z 2 ) 2π 2 k2 T δ k, η 1 ) T δ k, η 2 ) P R k) j 0 kx) dk k 4 +b v z 2 ) 2π 2 H z 1 ) H z 2 ) T v k, η 1 ) T v k, η 2 ) P R k) [ 1 5 j 0 kx) 4 7 j 2 kx) + 8 ] 35 j 4 kx) [ dk bq z 1 ) b v z 2 ) + 2π 2 k3 P R k) T δ k, η 1 ) T v k, η 2 ) + b ] α z 2 ) H z 2 ) H z 1 ) T v k, η 1 ) T δ k, η 2 ) [ 1 3 j 0 kx) + 2 ] 3 j 2 kx) 4.9) where x = x z 1, z 2 ), and we aow for redshift dependent bias factors, whie we have impicity assumed that these are scae independent. Nevertheess a generaization to scae dependent bias factor woud be straightforward. We define the correation function induced by cosmic magnification as foows ξ magnification Qα [ 5s 2 2r r 0 dr [ r r r Ω ] Φ+Ψ) ] n, z 1 ) [ b α δ + b v H 1 r n v) ] n, z 2 ). 4.10) 12

14 Because of the integra aong the ine of sight this expression can not written in a simpe form ike Eq. 4.9), uness we adopt Limber approximation [37]. We therefore prefer to compute expicity this contribution through Eq. 4.3). We define the reativistic contribution as foows ξ reativistic Qα Q n, z 1 ) δ F n, z 2 ) ξ newt Qα ξ Dopper Qα where we have a Dopper term 3 + ξ potentia Qα ξ magnification Qα + ξ non-oca Qα 4.11) ξ Dopper Qα [ [b Q δ + H 1 r n v) + R Q n v ] n, z 1 ) [ b α δ + b v H 1 r n v) + R α n v ] n, z 2 ) with = ξqα newt + + dk 2π 2 k2 P R k) j 1 kx) [b Q z 1 ) R α z 2 ) T δ k, η 1 ) T v k, η 2 ) R Q z 1 ) b α z 2 ) T v k, η 1 ) T δ k, η 2 )] [ dk 2π 2 k3 P R k) 3 5 j 1 kx) + 2 ] [ 5 j Rα z 2 ) 3 kx) H z 1 ) R ] Q z 1 ) b v z 2 ) T v k, η 1 ) T v k, η 2 ) H z 2 ) [ dk 1 2π 2 k2 P R k) 3 j 0 kx) 2 ] 3 j 2 kx) R Q z 1 ) R α z 2 ) T v k, η 1 ) T v k, η 2 ) 4.12) R Q = R α = b R Ḣ H s r S H + 5s f evo Q, 4.13) 3 + Ḣ H 2 f evo Lyα where we use a biased expression for R α. And the potentia term is ), 4.14) ξ potentia Qα [ b Q δ + H 1 r n v) + R Q n v + P Q ] n, z1 ) [ b α δ + b v H 1 r n v) + R α n v + P α ] n, z2 ) ξqα newt ξ Dopper Qα dk = 2π 2 k2 P R k) j 0 kx) [ b Q z 1 ) T δ k, η 1 ) T Pα k, η 2 ) + b α z 2 ) T PQ k, η 1 ) T δ k, η 2 ) ] [ dk + 2π 2 k3 P R k) 1 3 j 0 kx) + 2 ] 3 j 2 kx) [ 1 H z 1 ) T v k, η 1 ) T Pα k, η 2 ) + b ] v z 2 ) H z 2 ) T P Q k, η 1 ) T v k, η 2 ) dk + 2π 2 k2 P R k) j 1 kx) [ R Q z 1 ) T v k, η 1 ) T Pα k, η 2 ) R α z 2 ) T PQ k, η 1 ) T v k, η 2 ) ] dk + 2π 2 k2 P R k) j 0 kx) T PQ k, η 1 ) T Pα k, η 2 ) 4.15) 3 For the veocity transfer function T v k, η) we foow the notation of Ref. [25, 38], namey it refers to vector potentia V defined through v = iˆkv. 13

15 where we have used the foowing definitions ) P Q = 5s 2) Φ Ḣ H s r S H + 5s f evo Q Ψ + H 1 Φ + f Q evo 3 ) Hv, 4.16) [ ) P α = b R 3 + Ḣ H 2 f evo Lyα Ψ + H 1 Φ + f Lyα evo 3 ) ] Hv, 4.17) and T PQ and T Pα denote the respective transfer functions normaized with the primordia curvature perturbation. The ast term in Eq. 4.11) denotes the contribution of the non-oca terms, and their cross-correation with a the other terms. In particuar this incudes the correation between QSO magnification with a oca and non-oca) Lyman-α reativistic effects and the correation between reativistic non-oca terms e.g. ISW, Shapiro time deay) with everything ese. Again, as we did for the cosmic magnification we prefer to use Eq. 4.3) to compute them. 6 x 2 ξqαx, z = 2.5) [h 2 Mpc 2 ] Newt. Newt. αq) Newt. + RE Newt. + RE αq) Newt. + RE + ens. Newt. + RE + ens. αq) x [h 1 Mpc] Figure 2. The figure shows the Quasar-Lyman-α cross-correation function, aong the ine of sight. Different coours correspond to different contributions: red - Newtonian terms Newt.), bue - Newtonian and reativistic effects RE), green - Newtonian, reativistic and ensing terms. The fu ine and dashed ine correspond to the cross-correations ξ Qα and ξ αq respectivey. The asymmetry due to the bias evoution, reativistic effects and ensing produce a spit of the cross-correation function around the symmetric part. The dominant effect due to the pecuiar veocities of the tracers Dopper effect) is arge on a scaes above 20 h 1 Mpc whie the ensing and redshift evoution effects grow significanty towards arger scaes. 14

16 Fig. 2 shows different contributions to the overa cross-correation function. The red ine shows the contribution from the Newtonian terms ξαq Newt ), which are the most standard ones considered in the arge-scae correation anaysis. The argest correction to the signa comes from the reativistic effects ξαq RE ), more specificay from the Dopper terms see the subsection on anti-symmetric part for detais). The signa from the reativistic effects is neary constant for a scaes above 40 h 1 Mpc, and is of around 10% compared to the pure Newtonian terms. Further we show that the ensing contribution ony gives a sma correction at arge scaes. Dashed ines correspond to the cross-correation ξ αq, where the positions of Lyman-α and QSO have been exchanged see Fig. 1). This exchange is equivaent to the transformation x x. The correction due to reativistic effects is much arger than anticipated from the gaaxy surveys, which is due to two effects: firsty, the difference in biases is much arger for QSO and Lyman-α tracers than for two gaaxy popuations, and secondy, the overa signa from the Newtonian terms is arger as we. The cacuations for two gaaxy popuations have aready been shown in [18]. 4.1 Loca term expansion in H/k In our derivation we have been as genera as possibe and our resuts are vaid for any metric theory of gravity provided that photons move aong geodesic and photon number is conserved. Since we describe geometricay our observabe, the theory of gravity can modify the transfer functions ony 4, as ong as the previous assumptions are vaid. We have aso assumed that sources move aong geodesics, but this assumption can be easiy reieved. In order to give the favor of the ampitude in different regimes of the effects previousy computed in a reativistic framework, we repace the transfer functions with their counterparts in a newtonian framework. This oversimpification wi give us the physica understanding of the effects shown in the next sections. Therefore, we appy the foowing substitutions T v k, η) H k ft δ k, η) 4.18) T Ψ k, η) 3 H 2 2 k 2 Ω MT δ k, η) 4.19) H 2 T Φ k, η) 3 2 k 2 Ω MT δ k, η) 4.20) T Φ k, η) 3 H 3 2 k 2 Ω M f 1) T δ k, η) 4.21) where the growth factor can be approximated in perturbation theory in ΛCDM by fz) = Ω M z) 0.6. By evauating a the cosmoogica parameters and biases at the mean redshift z mean 4 We remark that expressions for bias factors impicity assume a theory of gravity. 15

17 we find dk ξqα newt b Q b α 2π 2 k2 P k) j 0 kx) [ dk 1 +b v 2π 2 k2 f 2 P k) 5 j 0 kx) 4 7 j 2 kx) + 8 ] 35 j 4 kx) [ dk 1 + b Q b v + b α ) 2π 2 k2 fp k) 3 j 0 kx) 2 ] 3 j 2 kx), 4.22) ξ Dopper dk Qα b Q R α + R Q b α ) 2π 2 k2 fp k) j 1 kx) H k [ dk 3 + R α + R Q b v ) 2π 2 k2 f 2 P k) 5 j 1 kx) 2 ] H 5 j 3 kx) k [ dk 1 +R α R Q 2π 2 k2 f 2 P k) 3 j 0 kx) 2 ] ) H 2 3 j 2 kx), 4.23) k b Q Pα + P ) ) dk H 2 Q b α 2π 2 k2 P k) j 0 kx) k + Pα + P ) [ dk 1 Q b v 2π 2 k2 fp k) 3 j 0 kx) 2 ] ) H 2 3 j 2 kx) k + R Q Pα + P ) ) dk H 3 Q R α 2π 2 k2 fp k) j 1 kx) k ) + P dk H 4 α PQ 2π 2 k2 P k) j 0 kx) 4.24) k ξ potentias Qα where P k) = T δ k, η) 2 P R k) denotes the matter power spectrum at the mean redshift. P Q and P α are the pre-factors defined through the transformations ), namey and they can be expicity computed P Q = 3 2 Ω M 2 f + fevo Q Ḣ H P α = b R 2 Ω M 2 + fevo Lyα Ḣ H 2 f + T PQ k, η) P H 2 Q k 2 T δ k, η), 4.25) T Pα k, η) P H 2 α k 2 T δ k, η), 4.26) ) 2f ) 3 f Q 5s 2 3Ω evo 10s +, 4.27) M Hr 2f ) ) 3 f Lyα evo. 4.28) 3Ω M We remark that the transformation ) has been performed by using Newtonian equations. Nevertheess, it is we known that, for CDM, inearized Einstein equations agree with Newtonian dynamics if perturbations are considered in specific gauges, namey the density perturbation in synchronous gauge and the veocity in newtonian gauge. Since we have chosen to express perturbations in this combination of 16

18 gauges, the expressions here derived incude correcty the GR dynamics, as ong as the approximation that CDM is the ony species which contributes to perturbations is vaid. According to this scheme, we note that a the newtonian terms contribute at the same parametrica order. The Dopper term, Eq. 4.23), contains the eading correction of order H/k to the newtonian approximation. This term is non-vanishing ony if we use two different tracers or probes. With a singe tracer the eading correction woud be of the order H/k) 2, and the ampitude of these terms woud be much more suppressed. The eading corrections have aso a different parity under the exchange x x. Because of that, they contribute ony to the antisymmetric part of the correation function, whie newtonian terms with biases and cosmoogy evauated at the mean redshift z mean ) ead to a symmetric correation function ony. This argument does not ony show the importance of using different probes to try to measure reativistic correction, as aready shown in [13, 14, 16, 18], but it aso indicates that when we consider cross-correations between different probes we need to carefuy consider effects which might be negected with a singe tracer, as shown in [14, 19]. For simpicity, in this counting scheme we did not incude integrated effects ike cosmic magnification, ISW and Shapiro time deay effects. It is known that cosmic magnification can be of the same parametrica order of density and redshift space distortion perturbations, see e.g. [17, 39, 40] and we do incude it in the fu reativistic anaysis in next sections. In the fu anaysis there is aso a contribution to the antisymmetric correation function due to newtonian terms because of the biases and cosmoogica parameter evoutions, that have been negected in this simpe approach. 4.2 Symmetric correation function The correation function 4.1) can decompose in a symmetric and anti-symmetric part. In this section we are interested in the symmetric correation function ξ Qα) z 1, z 2, θ) = ξ Qα z 1, z 2, θ) + ξ αq z 1, z 2, θ) ) Up to the redshift dependence of transfer functions or bias factors, ony the correation functions which invove even spherica Besse functions contribute to the symmetric part of the correation function. Figure 3 shows the symmetric contribution to the QSO-Lyman-α cross-correation function. As expected the Newtonian terms density and RSD) dominate the symmetric signa, whie the Reativistic terms Dopper and potentia) ony sighty contaminate the resut. Simiar effect coud be observed when one considers ony auto-correations of one tracer. It is interesting to note that whie both the Dopper and potentia terms in the symmetric form are of the same order of magnitude in the expansion of k H/k) 2 ), the vaues of the cosmoogica and bias prefactors are different - mainy the arge contribution to the Dopper term comes from the arge vaues of the QSO fevo) Q and Lyman-α evoution bias fevo Lyα ). Aso, whie Dopper terms are in genera arger than the potentia terms, the ater become the eading sub-dominant signa at sma scaes. At very arge scaes, on the other hand, the ensing terms of the QSO start to become 17

19 10 1 x 2 ξ Qα) x, z = 2.5) [h 2 Mpc 2 ] Newt. n v Ψ, Φ ensing non-oca x [h 1 Mpc] Figure 3. The figure shows the symmetric part of the Quasar-Lyman-α cross-correation function, aong the ine of sight. Different coours correspond to different contributions: red - Newtonian terms Newt.), bue - Dopper terms, green - potentia terms, magenta - ensing terms and orange - non-oca terms. The fu ine and dashed ine correspond to the positive and negative vaues of the cross-correations respectivey. Newtonian terms dominate the signa in the symmetric part of the cross-correation function. Note that non-oca terms were not computed at very sma scaes due to numerica issues, however they are very sma. The ensing contribution magenta) is increasing towards arge scaes which has been subject to many discussions in the iterature see text for detais). the dominant contaminant. This is the main idea behind the recent papers caiming that using a arge window function in the radia direction wi boost the ensing signa [17]. In fact, it is more about decreasing the overa signa of the density perturbation, which remain more or ess constant in radia direction, compared to the ensing term which is aways increasing with the distance. The non-oca terms are aways sma compared to the other effects in the symmetric part of the cross-correation function. As for the ensing part of the non-oca terms, the reative contribution of it grows for arge redshift) separations. 4.3 Anti-symmetric correation function Anaogousy to the symmetric part, we can define the anti-symmetric part of the correation function ξ [Qα] z 1, z 2, θ) = ξ Qα z 1, z 2, θ) ξ αq z 1, z 2, θ) ) 18

20 A non vanishing anti-symmetric correation function wi indicate a vioation of the exchange symmetry of the tracers z 1 z 2 ). Thus, the interchange of the tracers at the same position woud produce different resuts, and which tracer is in the background and which in the foreground woud matter. However, this is ony true when one is deaing with differenty biased tracers. If, for instance, one computes the auto-correation function of one singe tracer, the correation function wi be by definition symmetric. On the other hand two different effects can break the symmetry when deaing with different tracers. The asymmetry can come either from different vaue and redshift evoution of the bias factors or from the intrinsic evoution of the transfer functions. In the case of the gaaxy surveys that is the case, and severa sources of the asymmetry have been identified and suggested in the iterature e.g. gravitationa redshift, Dopper effects, etc.) [13, 16, 18]. These effects refect the fact that there is not a transation symmetry aong the redshift direction, and the asymmetry is generated aong the radia direction. Whereas in the anguar direction, if statistica isotropy and anguar) homogeneity are not broken, any asymmetry can not be sourced. x 2 ξ [Qα] x, z = 2.5) [h 2 Mpc 2 ] Newt. Newt. + RE Newt. + RE + ens. ens. RE x [h 1 Mpc] Figure 4. The figure shows different contributions to the anti-symmetric part of the crosscorreation function, aong the ine of sight. Fu red ine shows the contribution due to the often negected redshift evoution of the bias factors, dashed magenta ine shows the contribution due to the reativistic effects and dashed green ine contribution of the ensing terms. The fu bue and green ines represent summing up various different terms - redshift evoution and Dopper terms fu bue ine), and a the effects fu green ine). The main contributor to the signa are the reativistic effects mosty Dopper terms), whie the bias redshift evoution and ensing form the sub-dominant corrections of order of 10% of the signa in the anti-symmetric cross-correations. 19

21 Our cacuations in the previous sections show that the main source of asymmetry of the correation functions comes from the Dopper terms in the reativistic expansion see Fig. 4). In our simpe expansion, see Eqs. 4.22, 4.23, 4.24), the Dopper term, Eq. 4.23) contains terms with odd spherica Besse functions mutipied by ony one factor H/k. In the newtonian term, Eq. 4.22), a the terms are mutipied by even Besse function, and the asymmetry is generated by the redshift dependence ony, whie in potentia term, Eq. 4.24), the terms with odd Besse function are suppressed by H/k) 3. The argest singe contribution is the Dopper - density correation which is boosted by the difference in bias factors see Eq. 4.23). This resut is known in the iterature and has been commented on at various times, mosty in the context of two gaaxy popuations [13, 18]. In this paper we have focused on two tracers whose bias factors are as different as possibe: quasar bias being very arge, and Lyman-α forest fux bias being even negative. Thus ensuring that the signa from the Dopper effect is boosted as much as possibe. The first of the sub-dominant signas is the redshift evoution of the bias factors see Fig. 4). Usuay one approximates the redshift dependent quantities within a redshift bin with a constant vaue at the mean redshift. However, redshift evoution of the bias factors gives a fairy arge contribution to the asymmetry. This contribution mainy comes from the standard Kaiser terms - indeed one can easiy repicate the resuts by using we known Eq. 4.22) and requiring that the bias factors for quasar and Lyman-α are evauated not at the mean redshift z mean but at z 1 and z 2 respectivey. Stricty speaking, this effect shoud not be considered for biases ony, but for any redshift dependent prefactors e.g. cosmoogica prefactors of the reativistic effects, growth factors, etc.), however in our cacuations the effects of growth of structure and evoution of the cosmoogica parameters most notaby Hz)) are very sma compared to the other effects, and are we beow the 1% of the fina resut. Thus one can safey use the approximation of constant cosmoogica parameters evauated at the mean redshift. The signa of the bias redshift evoution gives an important contribution, that needs to be modeed if one wants to measure the effects to beow 10% accuracy. Whie it can be regarded as a nuisance signa to the reativistic effects, it requires a precise knowedge of the bias redshift dependence of both tracers. However, this information can be acquired from independent and compementary resuts, such as auto-correation functions of each of the tracers, and can be we constrained. Fig. 5 shows the contributions of the reativistic terms to the anti-symmetric part of the ξ Qα, broken down into its components. It is cear that at a scaes the crosscorreations due to Dopper terms dominate the reativistic signa, whie the potentia and non-oca parts are sma. The next to eading order corrections in the anti-symmetric part come from the redshift evoution of the bias factors in the Newtonian terms, an effect often considered to be too sma to be of any importance. In the case of QSO-Lyman-α cross-correation it is an important sub-dominant contribution to the anti-symmetric part. Compared to the symmetric part of the ξ Qα see Fig. 3) the potentia terms are aso suppressed at a scaes, even compared to the non-oca terms. A specia note deserves the fact that a the contaminating effects are sma at the 20

22 10 0 x 2 ξ [Qα] x, z = 2.5) [h 2 Mpc 2 ] Newt. n v Ψ, Φ ensing non-oca x [h 1 Mpc] Figure 5. The figure shows different contributions to the reativistic effects in anti-symmetric part of the cross-correation function, aong the ine of sight. The bue ine shows contributions due to the Dopper effects, red ine due to potentia terms, and green ine due to a non-oca terms. The difference between fu and dashed ine is whether the contribution to ξ [Qα] is positive or negative respectivey. Compared to the symmetric part the potentia and Newtonian terms are strongy suppressed. BAO scae. This resut does not carry any deeper significance, and is a resut of the vaues of bias factors for the tracers used in this study. Nevertheess, for this particuar case it does aow for an easier estimation of the reativistic effects at the BAO scae - which is aready a target of many surveys and being a robust feature, very accuratey measured. Figure 6 shows that the signa of the reativistic effects is argest at the BAO scae, compared to the cross-correation function computed just with Newtonian terms. The size of the asymmetry is around 10%. One shoud note that even though the ratio reach around 30% at the BAO scae this is an artificia enhancement of the signa since the cross-correation function drops neary to zero. The reativistic effects are neary constant on the scae arger than 40 h 1 Mpc and contribute to around 10% distortion on a scaes arger than that. The second contaminating effect is that of the weak ensing signa of quasars. In our specific case the ensing signa can be as arge as the effects due to the bias redshift evoution. Simiary to the bias redshift evoution signa, the ensing signa becomes more important on arger scaes, and requires carefu modeing if one is to measure the overa asymmetry effects to a high precision. 21

23 0.1 x, z = 2.5) ξ [Qα] x, z = 2.5)/ξ Qα Newt Newt. Newt. + RE Newt. + RE + ens. ens. RE x [h 1 Mpc] Figure 6. The figure shows the reative size of the anti-symmetric part of the cross-correation function, aong the ine of sight, for different terms. Fu red ine shows the contribution due to the redshift evoution of the bias factors, dashed magenta ine shows the contribution due to the reativistic effects and dashed green ine contribution of the ensing terms. The fu bue and green ines represent summing up various different terms - redshift evoution and Dopper terms fu bue ine), and a the effects fu green ine). The reativistic effects can be as arge as 10% at a scaes above 40 h 1 Mpc, compared to the pure Newtonian cacuation incuding the bias redshift evoution). However, the weak ensing effect is proportiona to the magnification bias and on density bias, since the eading term is < κδ >), and thus its signa is strongy dependent on this nuisance parameter. We caution that magnification bias needs to be we known, and possiby constrained from independent resuts, to avoid the contamination of the signa. The imprecise knowedge of the bias factor affects arger scaes the most, it remains important contaminant even at the BAO scaes, where it affects the ampitude of the BAO peak, but not its position. Figure 7 shows the effects on the asymmetry produced if we change the measured parameters of the Quasar Luminosity Function used to derive the evoution and magnification bias parameters. Whie the effect of the evoution bias parameter is smaer, it is aso degenerate with the effect of the magnification bias at arger scaes. However, the change of parameters by 5% shown in the pot is conservative, and future surveys and current surveys wi be abe to estimate the quasar uminosity function more accuratey and hence provide a better estimate on the parameter fevo. Q In a standard singe tracer) arge scae observabe, reativistic effects except cos- 22