Simulations of BAO reconstruction with a quasar Lyman-alpha survey
J.-M. Le Goff, C. Magneville, E. Rollinde, S. Peirani, P. Petitjean, C. Pichon, J. Rich, C. Yeche, E. Aubourg, N. Busca, R. Charlassier, T. Delubac, J.C. Hamilton, N. Palanque Delabrouille, I. Paris, M. Vargas
aa r X i v : . [ a s t r o - ph . C O ] J u l Astronomy&Astrophysicsmanuscript no. bao c (cid:13)
ESO 2018August 15, 2018
Simulations of BAO reconstruction with a quasar Ly- α survey J.M. Le Go ff ⋆ , C. Magneville , E. Rollinde , S. Peirani , P. Petitjean , C. Pichon , J. Rich , C. Yeche , E. Aubourg , ,N. Busca , R. Charlassier , T. Delubac , J.C. Hamilton , N. Palanque Delabrouille , I. Pˆaris , and M. Vargas CEA centre de Saclay, irfu / SPP, F-91191 Gif-sur-Yvette, France Institut d’Astrophysique de Paris, UMR7095 CNRS, Universit´e Pierre et Marie Curie, 98 bis bd Arago, 75014 Paris, France APC, 10 rue Alice Domon et L´eonie Duquet, F-75205 Paris Cedex 13, FranceReceived xx xx 2011 / accepted xx xx 2011 ABSTRACT
Context.
The imprint of Baryonic Acoustic Oscillations (BAO) on the matter power spectrum can be constrained using the neutralhydrogen density in the intergalactic medium (IGM) as a tracer of the matter density. One of the goals of the Baryon OscillationSpectroscopic Survey (BOSS) of the Sloan Digital Sky Survey (SDSS-III) is to derive the Hubble expansion rate and the angular scalefrom the BAO signal in the IGM. To this aim, the Lyman- α forest of 10 quasars will be observed in the redshift range 2 . < z < . ∼ ,
000 deg . Aims.
We simulated the BOSS QSO survey to estimate the statistical accuracy on the BAO scale determination provided by such alarge scale survey. In particular, we discuss the e ff ect of the poorly constrained estimate of the unabsorbed intrinsic quasar spectrum. Methods.
The volume of current N -body simulations being too small for such studies, we resorted to Gaussian random field (GRF)simulations. We validated the use of GRFs by comparing the output of GRF simulations with that of the H orizon -4 Π N -body dark-matter-only simulation with the same initial conditions. Realistic mock samples of QSO Lyman- α forest were generated; the3ir powerspectrum was computed and fitted to obtain the BAO scale. The rms of the results for 100 di ff erent simulations provides an estimateof the statistical error expected from the BOSS survey. Results.
We confirm the results from Fisher matrix estimate. In the absence of error on the unabsorbed quasar spectrum, the BOSSquasar survey should measure the BAO scale with an error of the order of 2.3%, or the transverse and radial BAO scales separatelywith errors of the order of 6.8% and 3.9%, respectively. The significance of the BAO detection is assessed by an average ∆ χ = ∆ χ ranges from 2 t o 35. The error on the unabsorbed quasar spectrum increases the error on the BAOscale by 10 to 20% and results in a sub percent bias. Key words. cosmology dark energy - cosmology: LSS - Galaxies: IGM - Galaxies: quasars (absorption lines) - Method: numerical
1. Introduction
Constraining the properties of dark energy that drives the ex-pansion of the Universe is key towards understanding cos-mology. Baryonic acoustic oscillations (BAO) in the baryon-photon fluid of the pre-recombination Universe imprint thesound horizon distance at decoupling as a typical scale in thematter correlation function or power spectrum (Peebles & Yu1970; Sunyaev & Zeldovich 1970; Eisenstein & Hu 1998;Bashinsky & Bertschinger 2002). These oscillations were de-tected both in the cosmic microwave background (e.g. Page et al.2003) and in the spatial distribution of galaxies at low redshift(Eisenstein et al. 2005; Cole et al. 2005; Percival et al. 2009).Their measurements give important and coherent constraints oncosmological parameters (Komatsu et al. 2009).More recently, it was realized that BAOs could be detectedin the Ly- α forest (McDonald 2003; White 2003) used as aprobe of the intergalactic medium (IGM) at intermediate red-shifts ( z ∼ −
3) and the potential of the measurement wasquantified by McDonald & Eisenstein (2007). The structure andcomposition of the IGM has long been studied using the Ly- α forest in QSO absorption spectra (Rauch 1998). The adventof high spectral resolution Echelle-spectrographs on 10 m-classtelescopes has led to a consistent picture in which the absorp-tion features are related to the distribution of neutral hydro- ⋆ e-mail: [email protected] gen (H i ) through the H i Lyman transition lines. The IGM isbelieved to contain the majority of baryons in the Universe atthese redshifts (Petitjean et al. 1993; Fukugita et al. 1998), andis highly ionized by the UV-background produced by galax-ies and QSOs (Gunn & Peterson 1965), at least since z ∼ ff ect the vast majority of the baryons (e.g.Theuns et al. 2002b; McDonald et al. 2005). The relation be-tween the Ly- α forest flux and the underlying matter field is non-linear since fluctuations are compressed to the range 0 < F < ff et al.: simulations of quasar Ly- α survey veys. The shapes and clustering of lines have been extensivelyused to infer the temperature of the IGM (Schaye et al. 1999;Ricotti et al. 2000; Theuns et al. 2000; McDonald et al. 2001),determine the amplitude of the UV-background (Rauch et al.1997; Bolton et al. 2005), trace the density structures aroundgalaxies and quasars (Rollinde et al. 2005; Guimar˜aes et al.2007; Kim & Croft 2008), constrain the reionization historyof the Universe (Theuns et al. 2002a; Hui & Haiman 2003;Fan et al. 2006), measure the matter power spectrum (Croft et al.1999; Viel et al. 2004; McDonald et al. 2006) or constraincosmological parameters (McDonald & Miralda-Escud´e 1999;Rollinde et al. 2003; Coppolani et al. 2006; Guimar˜aes et al.2007; Viel & Haehnelt 2006). Padmanabhan & White (2009)(see also Meiksin et al. 1999) analyzed the amplitude of non-linear e ff ects by comparing perturbative theory with outputs often dark matter numerical simulations. Although they focusedon halos only, they demonstrated that the shift of the recon-structed BAO scale due to non-linearities decreases with redshiftas D ( z ), the square of the linear growth factor. The simple linearbias between flux and matter power spectrum was also predictedby McDonald (2003) at low k -values, and by Slosar et al. (2009)and White et al. (2009) at BAO scales.The observation of BAOs in the Ly- α forest requires a full3-dimensional sampling of the matter density, and therefore amuch higher number density and number of quasars than pre-viously available. The Baryon Oscillation Spectroscopic Survey(BOSS) (Schlegel et al. 2009) of the Sloan Digital Sky Survey-III (SDSS-III) (Eisenstein et al. 2011) aims to identify and ob-serve more than 150,000 QSOs over 10,000 square degrees. TheQSO redshift range useful for BAO reconstruction is limited to z > .
15 on the low side by the requirement that the Ly- α ab-sorption falls in BOSS spectrograph wavelength range. It is lim-ited to about z < . g < α power spectrum, that the den-sity of quasars should be of the order of 20-30 per square de-gree to achieve constraints of the order of 1% in the radialand transverse BAO scales, see also McQuinn & White (2011).Such a high requirement lead to new developments on targetselection (Palanque-Delabrouille et al. 2010; Y`eche et al. 2010;Bovy et al. 2011; Kirkpatrick et al. 2011; Ross et al. 2011).It is important to confirm the predictions on the BAO scalemeasurements with additional work on numerical simulations.Recently, Slosar et al. (2009); White et al. (2009) have studiedthe BAO signature in a typical Ly- α forest survey, using large N -body simulations, but with still higher quasar density. This paperfollows those works and investigates errors in the BAO scale es-timates as a function of the properties of the survey such as thedensity of quasars and the amplitude of the noise. Very recentlyGreig et al. (2011) have published a similar study but assumingall QSO located at the same z , with constant S / N ratio and amuch smaller volume than the BOSS survey (corresponding to79 instead of 10,000 deg ). In Sect. 2 a comparison with theH orizon -4 Π (Prunet et al. 2008) dark-matter-only N -body simu-lation validates the use of linear Gaussian random field (GRF)to study BAO scale reconstruction. The production of realisticmock spectra is described, including quasar unabsorbed spec-trum, noise, and the e ff ect of peculiar velocities. Standard meth-ods to analyze BAO signal in terms of power spectrum are pre-sented in Section 3. The performance of the survey for di ff erentquasar densities and noise amplitudes are presented in Sect. 4and the e ff ect of the error on the estimate of the quasar unab- sorbed spectrum is discussed. The resulting cosmological con-straints are presented in Sect. 5 and we draw conclusions inSect. 6.
2. Description of the simulations
The size of large N-body simulations is typically (2 Gpc / h ) . At z = . which is much smaller thanthe 10,000 deg of the BOSS survey. As will be clear in Sect. 4.1,probing such a volume with the Ly- α forest of 20 quasars perdeg results in a power spectrum or a correlation function wherethe BAO features are hardly seen: some realizations will exhibitthem and some will not. A larger volume is therefore required inorder to study the error on the reconstructed BAO scale.Such a volume can be provided by Gaussian random fieldsimulations, which however do not contain any non-linear ef-fects.The H orizon -4 Π simulation was used to investigate the rel-evance of these e ff ects for the study of BAO scale reconstruction.H orizon -4 Π is a Λ CDM dark-matter-only simulation based oncosmological parameters inferred by the WMAP three-year re-sults, with a box size of 2 h − Gpc on a grid of size 4096 . Thepurpose of this simulation is to investigate full sky weak lensingand baryonic acoustic oscillations. The 70 billion particles wereevolved using the Particle Mesh scheme of the RAMSES codeon an adaptively refined grid (AMR) with about 140 billion cells.Each of the 70 billions cells of the base grid was recursively re-fined up to 6 additional levels of refinement, reaching a formalresolution of 262,144 cells in each direction (roughly 7 kpc / h comoving). The code mpgrafic (Prunet et al. 2008) was usedto generate the initial conditions (ICs).For identical initial conditions, we compared outputs fromH orizon -4 Π simulation with the linear density modified througha lognormal model to incorporate some of the non-linearities.To make this comparison statistically as powerful as possible wedid not implement here all the complications of a realistic sur-vey. Transmitted-flux-fraction spectra were generated withoutany observational noise and extended in wavelength all along thebox, instead of being limited to the Ly- α forest. In addition the x and y positions of those lines were chosen regularly which elim-inates the sampling noise (see Sect. 2.2 and Eq. 3). In this case,we can divide the simulation into eight individual boxes withthe same volume and still see the BAO features in each box. TheFFT of each box was computed to produce the power spectrum,which was Fourier transformed to give the correlation function.The correlation function was fitted to determine the BAO peakposition. This was done both for the full H orizon -4 Π and for thelognormal densities, yielding k A = . ± . / Mpc(LN) and k A = . ± . / Mpc (Horizon), where theerrors are obtained from the rms of the 8 values. The two setsof values are correlated and the di ff erence is 0 . ± . . ± . ff ect on the reconstruction of the BAO scale at this level of ac-curacy. Complex normal Gaussian fields were generated in a box inFourier space with δ ( − k ) = δ ∗ ( k ). The amplitude of each modewas multiplied by the square root of the power spectrum P ( k ) at z = H =
71 km / s / Mpc, Ω m = . Ω b = . Ω Λ = .
73 and w = −
1. An inverse ff et al.: simulations of quasar Ly- α survey k[h/Mpc]0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P [ M p c / h ] =1 HR c =1.2 HR c=0 HR c Fig. 1.
One-dimension power spectrum ( P D ) for our simulationwith c HR = c HR = c HR = . c HR is defined in Eq. 2, comparedto expectation from Eq. 1 (black continuous line).FFT provided a linear simulation of matter density fluctuations, δρ/ρ , at z = × ×
512 pixels of (3.2Mpc / h ) , i.e. 8179 Mpc / h in the transverse directions and 1636Mpc / h in the longitudinal direction. The transverse size covers12,500 deg at z = . at z = .
5, while the longi-tudinal size covers from z = λ = α absorption) to z = .
9. Peculiar velocities in the direc-tion parallel to the line of sights were computed from densitiesin Fourier space using v k k = − ik k k H ( z ) D ( z ) ∂ D ∂ z ( z ) δ k where D ( z ) is thelinear growth factor.The clustering of quasars was neglected and their angularpositions were randomly drawn within the box. They were as-signed a redshift and a magnitude according to the distributionof Jiang et al. (2006). The lines of sight of the quasars were takento be parallel. Real data will have to be analyzed taking into ac-count the angle between the lines of sight, but if we consistentlyproduce the simulation and analyze the resulting spectra withparallel lines of sight, this should have a negligible e ff ect on thestatistical error on the reconstructed BAO scale.The matter density, in pixels located along the line of sightof each quasar, was evolved to the redshift of the pixel by multi-plying δρ/ρ by D ( z ) at the redshift of the considered pixel. Thisprovided us with density spectra in 3.2 Mpc / h bins. We did notinterpolate to regular bins, e.g. in log( λ ), and then interpolateback to regular bins in Mpc to compute the power spectrum.Since we anyway average the flux over (12.8 Mpc / h ) pixels be-fore applying a FFT to compute the power spectrum (Sect. 3.1),we believe this is a minor e ff ect.Because QSOs sample only a very small transverse region,such a simulation misses a significant contribution to the powerspectrum coming from transverse scales smaller than the 3.2Mpc / h pixel size. This is very clear for the 1D power spec-trum, which is an integral of the 3D power spectrum over k ⊥ (Kaiser & Peacock 1991) and therefore includes large k ⊥ : P D ( k k ) = π Z ∞ P ( k k , k ⊥ ) k ⊥ dk ⊥ . (1)This missing small scale contribution is illustrated in Fig. 1,where the 1D power spectrum of the matter density for the sim-ulation with 3.2 Mpc / h pixels (lower blue histogram, C HR =
0) appears significantly lower than the expectation from Eq. 1(black continuous curve). In order to compensate for the miss-ing contribution, we have generated 20 high-resolution (HR)Gaussian random field simulations with a volume correspond-ing to one pixel of our large-volume low-resolution simulations(LR), i.e. (3.2 Mpc / h ) , and a pixel size of 200 kpc. When sim-ulating the matter density along a quasar line-of-sight, for eachlarge pixel we randomly selected one of the HR simulations, andwe defined the density in the small pixels as δρ = ( δρ ) LR + c HR ( δρ ) HR . (2)As illustrated in Fig. 1, if we just add the LR and HR simulations(i.e. c HR = . This isnot surprising since we do not have any correlation between theHR simulations in neighboring large pixels. Using an e ff ectivecorrection factor c HR = . P D ( k ) which fits wellEq. 1, at least in the k k range relevant for BAO, see Fig. 1.McDonald & Eisenstein (2007) showed that the observedpower spectrum, P obs ( k ), is the sum of the true power spectrum,a sampling contribution and a noise contribution : P obs ( k ) = P ( k ) + P DW P D ( k k ) + P N , (3)where, in the absence of pixel weighting, P DW is the inverse ofthe surface density of quasars (in Mpc − ). Eq. 3 indicates thatthe accurate description achieved for P D at k k ≤ .
3, ensures agood description of P obs ( k ) in the range relevant for BAO.The next step was to go from the matter density fluctuations,to QSO transmitted flux fractions F = exp( − τ ), where τ is theoptical depth. Note that F is the traditional notation for the trans-mitted flux fraction and we will use φ for the QSO flux. We usedthe relation F = exp " − a ( z ) exp b δρρ . (4)This means the lognormal approach was used to get the baryondensity from our Gaussian fields (Bi & Davidsen 1997) and thebaryon density was transformed into transmitted flux fractions F using the fluctuating Gunn-Peterson approximation (Croft et al.1998; Gnedin & Hui 1998).We followed the procedure of McDonald (2003) to take intoaccount the e ff ect of the peculiar velocities: we accounted for theexpansion or contraction of cells by translating each cell edge inreal space into redshift space using the average velocity of thetwo cells that the edge separates. The optical depth contributedby each real-space cell was then distributed to multiple redshift-space pixels based on its fractional overlap with each.The value of b in Eq. 4 was fixed to b = − . γ − = . γ = . a ( z ) was fitted to reproduce the experimental 1Dpower spectrum and the resulting mean transmitted flux fraction F ( z ) was checked to be in good agreement with the data, as il-lustrated by Fig. 2. We could alternatively have fitted F ( z ) andchecked P D . More precisely, McDonald et al. (2006) measured P D for k k between ≈ .
14 and ≈ . h / Mpc and 2 . < z < . z we fixed a ( z ) to fit the first four bins in k ,from ≈ ≈ h / Mpc. We did not fit higher k bins becauseour simulations do not include non-linear e ff ects and are not ex-pected to fit data at high k . Note that it would have been more Note that in Eq. 1 we integrated up to k ⊥ = π/ (100 kpc / h ) whichcorresponds to a typical value of the Jeans’ scale. This however is onlya reduction of a few % relative to the integral up to infinity. 3.M. Le Go ff et al.: simulations of quasar Ly- α survey z F Fig. 2.
Mean transmitted flux fraction, F ( z ), as a functionof redshift for the simulation (red histogram) compared toFaucher-Gigu`ere et al. (2008) data (blue circles) and Songaila(2004) data (green squares).natural to use 100 kpc / h pixel size for the HR simulations, a typ-ical value of the Jeans scale. In this case one needs a correctionfactor c HR = F ( z ) and P D .Peculiar velocities introduce a dependence on µ = k k / k inthe redshift-space power spectrum : on large scales we have P ( k , µ ) = (1 + βµ ) P ( k ), where P ( k ) is the isotropic real-spacepower spectrum (Kaiser 1987). For galaxy surveys, β is relatedto the bias and the growth rate of structure, but for Ly- α forestit is an independent parameter (McDonald et al. 2000). Fig. 3shows the ratio of the redshift-space over the real-space powerspectra for our simulation . This ratio follows Kaiser formula inthe k range relevant for BAO, k < .
2. The departure at higher k is due to the fact that our procedure to implement the e ff ect ofvelocities is only valid for scales larger than a few (3 . / h )pixels. The ratio of power spectra is unity in the transverse di-rection ( µ =
0) and about 5 in the longitudinal direction ( µ = β = . β = .
58 according to McDonald (2003) sim-ulations and 0 . < β < .
05, as measured with first BOSSdata (Slosar et al. 2011). Note, however, that the value of β ob-tained by BOSS is contaminated by the presence of damped Ly- α systems and metal lines in the quasar spectra. We also observea bias b = .
19 relative to the matter power spectrum, to be com-pared to 0 . < b < .
25 measured with BOSS data.To get the flux, φ i ( λ ), of quasar i , the transmitted flux frac-tion, F i ( λ ), must be multiplied by the quasar unabsorbed spec-trum, i.e. the quasar spectrum, including the QSO emission lines,if there were no absorption. The principal component analy-sis (PCA) of Suzuki et al. (2005) was used to generate for eachmock spectrum a random PCA unabsorbed spectrum, which wasnormalized according to the g-band magnitude of the quasar.Noise was added according to the characteristics of BOSS spec-trograph, including readout noise, sky noise and signal noise andassuming four exposures of 900 s, for each QSO spectrum. Fig. 4presents the mean signal-to-noise ratio per 1Å bin in the Ly- α forest, which varies from 14 for a quasar magnitude m g = Note that the power spectra were obtained without noise and usingall pixels of the box, not just those along some random QSO lines. Thisremoves the contribution from the noise and sampling terms so that P obs ( k ) = P F ( k ), see Eq. 3. [h/Mpc]k0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 [ h / M p c ] k Fig. 3.
Two-dimension power spectrum P ( k ⊥ , k k ) in redshiftspace divided by the 2D power spectrum in real space. g m
19 19.5 20 20.5 21 21.5 22 S / N Fig. 4.
Mean signal-to-noise ratio in the Ly- α forest for 1Å bins,as a function of the quasar magnitude in the g band.to 1.6 for m g =
22. Fig. 5 shows an example of such a mockspectrum. The pdf of the transmitted flux fraction is presentedin Fig. 6. For high resolution bins this pdf exhibits peaks at zeroand unity but for low resolutions bins, the flux is averaged andthere is a single peak around F . In addition, due to noise thetransmitted flux fraction can be larger than unity and also nega-tive.
3. From spectra to BAO signal
The mock spectra, produced as described above, were analyzedto reconstruct the power spectrum and the BAO scale.
The flux φ i in mock spectrum i must be divided by the product of F ( z ) and the quasar unabsorbed spectrum c i in order to providethe fluctuation of the transmitted flux fraction, δ F ( λ ), as δ F ( λ ) = φ i ( λ ) F ( z ) c i ( λ ) − . (5)This requires estimates of F ( z ) and c i ( λ ). In Sect. 4.2 we willdiscuss various methods of doing this. In this section we use the ff et al.: simulations of quasar Ly- α survey ]Å[ λ Å / s / e r g / c m - F Fig. 5.
Mock spectrum of a quasar with redshift z = .
02 andmagnitude m g = .
30. The continuous red line is the input PCAunabsorbed spectrum.
F0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.20.40.60.811.21.41.61.8
Fig. 6.
Probability distribution function of the transmitted fluxfraction in the 3.8 Mpc / h simulation bins.true values of F ( z ) and c i , a procedure which, as we will see,only slightly overestimates the precision of the determination ofthe BAO signal. A grid with (12.8 Mpc / h ) cells was filled withthe part of the δ F spectra which corresponds to the Ly- α for-est (we selected 1041 < λ < δ F was calculated for all pixels of allLy- α forest spectra lying in a considered grid cell. Some gridcells do not contain any Ly- α forest spectra. The average of allother cells was computed and this average was subtracted fromthe content of all filled cells, while the content of unfilled cellsremained zero. This was done in order to avoid including theFourier transform of the quasar spatial distribution in the powerspectrum. This is analogous to subtracting a synthetic catalogin the case of galaxy surveys, as advocated by Feldman et al.(1994).A Fourier transform of the resulting grid was performed andthe modulus squared computed for each mode ( k x , k y , k z ), whichgives the power for the considered mode. The angular-averagedpower spectrum was then obtained as the average of the powerfor all the modes with | k | in a given bin. Note, however, thatwe did not include purely radial (0 , , k z ) and purely transverse( k x , k y ,
0) modes. The former correspond to Fourier transformsalong quasar lines of sight. These are severely a ff ected by the k[h/Mpc]0 0.2 0.4 0.6 0.8 1 ( M p c / h ) P Fig. 7.
Power spectrum in the simulation (red upper histogram)compared to the expectation of Eq. 3 (blue, just below red). Thelatter is the sum of the input power spectrum (black, decreasingquickly with k ), the noise contribution (black, constant with k )and the sampling contribution (green, decreasing slowly with k ).uncertainty on the quasar unabsorbed spectrum estimate. Thelatter have a very large sampling term contribution due to thelarge value of P DF ( k k = P ( k ) isobtained as the rms of the power in all modes in the considered k bin, divided by the square root of the number of modes. Thecorrelations between the power in di ff erent k bins are expectedto be small and we neglect them.The resulting power spectrum exceeds Eq. 3 by about 10%,as illustrated in Fig.7. We see that the sampling and noise contri-butions are of similar sizes and much larger than the input (LSS)power spectrum, which means that the BAO oscillations will beconsiderably diluted, as can be seen in Fig. 8. To infer the BAO scale, the angular-averaged power spectrumwas first fitted by a polynomial. The power spectrum divided bythis polynomial was then fitted with1 + A ( k exp " − k τ ! p sin " π kk A , (6)as suggested by Blake & Glazebrook (2003). This provides the(isotropic) BAO scale k A , as illustrated in Fig. 8. This figure cor-responds to an average realization with ∆ χ = . / h FFT cells which result in a maximum k of 0.245 h / Mpc in x , y or z direction. The lower limit and theorder of the polynomial were set so as to get a good χ to thepower spectrum obtained for some additional simulations whichdid not include the BAO features. These simulations were gen-erated using the power spectrum with baryons but without BAOby Eisenstein & Hu (1998). We noted that using a polynomial ff et al.: simulations of quasar Ly- α survey / ndf χ ± ± ± ± k[h/Mpc]0 0.05 0.1 0.15 0.2 0.25 P ( k ) / po l y no m i a l ( k ) / ndf χ ± ± ± ± Fig. 8.
Power spectrum for a realization of the nominal setup(15 QSO / deg ) divided by a 7 th order polynomial and fitted withBlake & Glazebrook (2003) formula, see text.of too low a degree could result in a significant bias in the re-constructed BAO scale, while on the other hand increasing thepolynomial degree beyond what is needed to get a good fit coulddegrade the performance (i.e. increase the rms of k A ) because thepolynomial fit starts to fit out the BAO features. Depending onthe studied scenario, the lower limit was set between 0.02 and0.05 h / Mpc and the order of the polynomial was either 6 or 7.The range for oscillation fitting was set to 0 . < k < . h / Mpc,somewhat narrower to avoid edge e ff ects in the polynomial fit.We anyway do not want to include in this fit high k values forwhich non-linear e ff ects are important.Ten Gaussian random field simulations were produced. Eachof them was used ten times, with di ff erent random seeds to gen-erate the quasar positions. One hundred values of k A were thenobtained and their rms provides an estimate of the statistical pre-cision on the BAO scale. This was done for several di ff erentscenarios and the results are presented in Sect. 4.1. The meanvalue of the error on k A returned by the Minuit fitting pack-age (James & Roos 1975) is found to be compatible with therms of k A , which confirms that our estimate of the error on P ( k )and the fact that we neglected correlations are both reasonable. In addition to the anisotropy of P ( k ) due to peculiar velocities,the reconstructed power spectrum, P obs ( k ), involves a strongeranisotropy due to the sampling contribution in Eq. 3 which de-pends on k k . It is di ffi cult to find a functional form to fit the 2Dpower spectrum P ( k ⊥ , k k ). Instead P ( k ⊥ , k k ) was divided by thepolynomial obtained from the 1D fit (previous section) to pro-vide a “reduced” power spectrum, which was then smoothed us-ing algorithms implemented in the CERN ROOT package. Theratio of the reduced-power-spectrum to the smoothed reduced-power-spectrum was fitted with1 + A k exp " − k τ ! p sin π s k ⊥ k A ⊥ + k k k A k . (7)The rms from the 100 resulting values of k A ⊥ and k A k provides anestimate of the error on the transverse and radial BAO scales.Note however that this 2D fitting is quite delicate. It does notwork in all cases and a more sophisticated procedure would beneeded to analyze real data.
4. Results
In this section we discuss the performance of BAO reconstruc-tion in the ideal case where in Eq. 5 we divided by the true F ( z ) and c i ( λ ). By default the simulation was performed for 15QSO / deg in the redshift range 2 . < z < . < m g <
22. The results were slightly scaled to corre-spond to the 10,000 deg of the BOSS survey. Di ff erent scenar-ios in terms e.g. of number of quasars per deg or noise levelwere studied. Table 1 presents, for each scenario, the rms in per-cent for the isotropic ( k A ), radial ( k k ), and transverse ( k ⊥ ) BAOscales. To assess the significance of the BAO detection, we alsogive the di ff erence of χ relative to the case with no BAO ( A = ff ect of veloc-ities (Table 1, line 1). In this case, the transverse scale is muchbetter reconstructed than the radial one because there are moretransverse than radial modes (there are two transverse direc-tions and only one radial). Including peculiar velocities ampli-fies the power spectrum in the radial direction and dramaticallyimproves the radial scale reconstruction, as can be seen on line2. The nominal case is obtained when we add noise accordingto BOSS setup, which results in rms of 2 . ± .
16, 6 . ± . . ± .
27 per-cent for the isotropic, transverse and radialBAO scales, respectively (line 4 of Table 1). On average over the100 simulations, ∆ χ is 17.4, which means a significant detec-tion of BAO features. This is, however, just an average and thedi ff erence of χ ranges from 2 to 35, as illustrated by Fig. 9. Atthis point, we note that the (2Gpc / h ) Horizon simulation has a17 times smaller angular coverage than our simulation. The χ di ff erences for Horizon simulation would therefore be of orderunity for BOSS Ly- α survey, so, as announced in Sect. 2.1, theHorizon simulation is clearly not large enough to study the re-construction of the BAO features with BOSS survey.For the same nominal setup, an updated version(P. McDonald, private comm.) of the analytic estimate ofMcDonald & Eisenstein (2007) gives an error of 1.8% onthe isotropic BAO scale when weighting pixels according tosignal-to-noise ratio. Without weighting the computed errorincreases to 1.91%. Our result, 2 . ± .
16, is larger by a factor1 . ± .
08, which is consistent with the fact that we find a powerspectrum about 10% larger than predicted by formula 3, asillustrated in Fig.7. Greig et al. (2011) very recently publishedsimulation results; they get an error of 1.38% on the BAOscale for a fixed ratio S / N =
5, whereas we have in average S / N = .
1. When we decrease the S / N ratio by a factor 1.7,we observe an increase of the rms by a factor 1.75, so it is notunlikely that an increase of S / N by a factor 5 / = / = / N QS O . So, if we could completely neglect P F ( k ) in Eq. 1, wewould expect the rms of the BAO to scale as 1 / N QS O . We ob-serve a decrease of the rms with the quasar density, which is notas strong as 1 / N QS O but statistically compatible with the N − . QS O dependence found by Greig et al. (2011). We investigated the de-pendence on the noise level. Line 6 shows that increasing thenoise by a factor 1.7 results in a significant degradation of theperformance. As is clear from Fig. 7, increasing the noise power ff et al.: simulations of quasar Ly- α survey χ ∆ Fig. 9.
Nominal simulation (15 QSO / deg ). Di ff erence betweenthe χ for the nominal fit (Eq. 6) and the χ for no oscillations( A = . makes it the dominant contribution tothe total power spectrum.We also considered the case of the BigBOSS project, assum-ing 60 QSO / deg up to a g -magnitude of 23, over 14,000 deg (last line of Table. 1). With a 4 meter telescope and 5 exposuresof 900 s instead of 2.5 m and 4 exposures, the noise is reducedby nearly a factor two relative to BOSS case. The resulting rmsof the BAO scale is improved by a factor 4.2 relative to BOSSnominal case.Finally, we note that naively combining the rms on the trans-verse and radial scales in Table 1 results in an error which issignificantly larger than the rms on the isotropic scale. Thisis (partly) due to a significant anticorrelation between the re-constructed transverse and radial scales. When the rms is quitesmall, e.g. in the nominal case with 20 QSO / deg or in theBigBOSS case, taking into account a correlation coe ffi cient oftypically − .
40, results in a combined rms in agreement with therms on the isotropic case. In other cases, the combined rms isstill larger by a factor on the order 1.2, up to a factor 1.4 for thecase without velocities. This confirms that fitting the BAO fea-tures in two dimensions is di ffi cult, in particular when the quasardensity is low. Using Eq. 5 to obtain δ F requires an estimate of F ( z ) × c i ( λ ) foreach spectrum and the above results were obtained using the truevalue of this product. One can get a fair estimate of F ( z ) fromthe observed spectra, up to a normalization factor which is com-pletely degenerate with the normalization of the c i . On the otherhand, with the resolution and the S / N ratio of the BOSS survey,the unabsorbed spectra, c i ( λ ), cannot be accurately determinedfrom the observed spectra (see Fig. 5). A possible approach isto fit a power law in λ on the red side of the Ly- α emissionline, to extrapolate it in the Ly- α forest, possibly multiplyingit by some average shape of the unabsorbed spectrum as a func-tion of λ r f , the quasar rest-frame wavelength (e.g. Slosar et al.2011). We could not do that because our mocks are based onPCA by Suzuki et al. (2005) which extend only up to 1600 Å inthe quasar rest frame and therefore do not allow for a reliablepower law fit. Reliable fits will be possible with the PCA pro- Table 1.
Rms of the BAO scale and significance of BAO featuredetection, for di ff erent scenarios. QSO a Area b Noise c Velo d k Ae k ⊥ f k k g ∆ χ h (deg − ) (deg ) (%) (%) (%)15 10,000 0 - 2.67 5.26 10.5 15.815 10,000 0 + + + + + + Notes. ( a ) assumed number of QSO / deg b ) survey size ( c ) scaling factorapplied to the noise level, i.e. 0 means no noise and 1 noise correspond-ing to BOSS nominal setup. ( d ) indicates whether the e ff ect of peculiarvelocities was taken into account or not. ( e ) rms for the isotropic BAOscale ( f ) transverse BAO scale ( g ) radial BAO scale ( h ) significance ofBAO feature detectionWith 100 simulations, the statistical error on the rms, is justrms / √ = . ∗ rms, e.g. 0.16, 0.40 and 0.28% for the nominalcase (line 4). vided very recently by Pˆaris et al. (2011), extending up to 2000Å. Instead, we computed the average spectrum in the forest, asa function of λ r f . Then we divided each spectrum by this aver-age spectrum, fitted the result with a power law in λ , and finallydivided by this power law. This is dividing the mock spectrumby an estimate of F ( z ) × c i ( λ ). For the nominal setup, this resultsin an rms of (2 . ± . . ± .
04 larger than what is obtained with the true unabsorbedspectrum. We also note that, while we did not observe any biason the reconstructed BAO scale when using the true unabsorbedspectrum, with this approximate unabsorbed spectrum, there is abias of − . ± . .
55% statistical error on the BAO scale. In this procedure, sincewe do a power law fit along the quasar spectra, we remove largescale power in the radial direction, so all modes with low valueof k k are strongly reduced and P ( k ) becomes very anisotropic.In this case, the smoothing used in the 2D fit procedure does notmake sense. So we do not get results for the transverse and radialBAO scales, separately. A more sophisticated procedure wouldneed to be developed.Another possible approach is to use PCA to predict the un-absorbed spectrum in the forest from the flux redward of theLy- α emission line. However, we cannot use mock spectra gen-erated with PCA to study how precisely PCA reconstruct theunabsorbed spectrum. Instead we used 78 observed unabsorbedspectra, c obs ( λ ), and their corresponding PCA estimates, c PCA ( λ )provided by Pˆaris et al. (2011). The observed unabsorbed spectrawere manually estimated from high S / N SDSS II spectra, whilethe PCA unabsorbed spectra were obtained for each spectrumby a PCA analysis of the 77 other spectra. For each of our spec-tra, with true unabsorbed spectrum c i ( λ ), we randomly selectedspectrum j within the 78 spectra from Pˆaris et al. (2011), andused as an unabsorbed spectrum c i ( λ ) × c jPCA ( λ ) / c jobs ( λ ). Sincethere is much less uncertainty on F ( z ) we used the true valueof this function. This results in an rms of (2 . ± . . ± .
11 larger than what is ob-tained when using the true unabsorbed spectrum, and the bias is + . ± . ff et al.: simulations of quasar Ly- α survey w -1.2 -1.1 -1 -0.9 -0.8 a w -1-0.500.51 Fig. 10.
Confidence level contours in the ( w , w a ) parameterplane for Planck + BOSS LRG + BOSS Ly- α .pected if a criteria can be found to identify a-priori the spectra forwhich PCA will fail so that a di ff erent approach can be used forthese few spectra. Finally, note that the PCA unabsorbed spec-trum estimates of Pˆaris et al. (2011) were obtained for high S / N spectra, but PCA estimates might not be very sensitive to addi-tional noise.
5. Constraint on cosmological parameters
The observation of the transverse and radial BAO scales providesa measurement of d T ( z ) / s and d H ( z ) / s , respectively, where d T ( z )is the comoving angular distance, d H ( z ) the Hubble distance and s the sonic horizon at decoupling. For a flat Λ CDM universe d T and d H read d T = Z z ( c / H ) dzE ( z ) , d H = c / H E ( z ) , (8)where E ( z ) = p Ω Λ + Ω M (1 + z ) .To estimate the sensitivity to parameters describing the darkenergy equation of state, p = w ρ , we follow the procedure ex-plained in Blake & Glazebrook (2003). We introduce the z de-pendence of w as w ( z ) = w + w a · z / (1 + z ) and replace Ω Λ in E ( z ) in Eq. 8 by: Ω Λ −→ Ω Λ exp " Z z + w ( z ′ )1 + z ′ dz ′ . (9)Using the relative errors on the transverse (6.79%) and radial(3.86%) BAO scales, obtained for the nominal setup (Table 1,line 4), and taking into account the anticorrelation, we can com-pute the Fisher matrix for the five cosmological parameters Ω m , Ω b , h , w and w a . We also use the Fisher matrix for Planck mis-sion computed for the Euclid proposal (Laureijs 2009) whichassumes a flat universe and involves the 8 parameters: Ω m , Ω b , h , w , w a , σ , n s (spectral index of the primordial power spec-trum) and τ (optical depth to the last-scatter surface). Combining BOSS and Planck Fisher matrices allows us to compute the er-rors on dark energy parameters. If we define the factor of meritas the inverse of the 1- σ uncertainty ellipse in the ( w , w a ) plane,we get 34 for Planck and BOSS LRG survey. When we addBOSS Ly- α survey this increases to 48. The corresponding con-fidence level contours are plotted in Fig. 10.
6. Summary and conclusion
In this paper we have investigated the possibility to constrainthe BAO scale from the quasar Ly- α forest and in particular theprecision that will be reached by a survey such as BOSS. Tothis aim, we have simulated realistic mock quasar spectra mim-icking the survey. The volume of the largest N-body simulationsbeing too small for such a study, we resorted to Gaussian randomfields, combined with lognormal approximation and FGPA. Weinvestigate the e ff ect of noise, peculiar velocity and random un-absorbed quasar spectra generated using a principal componentanalysis (PCA).The power spectrum of the transmitted flux fraction in theLy- α forest was thus computed and was either fitted in two di-mensions to reconstruct the radial and transverse BAO scales, oraveraged over angles and fitted in one dimension to reconstructthe isotropic BAO scale. This was done over 100 realizations,resulting in an rms of 2.3% on the isotropic BAO scale, or 6.8%and 3.9% on the transverse and radial BAO scales determina-tions separately. This is compatible with analytical estimates byMcDonald & Eisenstein (2007). The BAO features are detectedwith an average ∆ χ of 17 and the FOM for dark energy pa-rameter determination improves from 34 for Planck and BOSSLRG survey to 48 when the Ly- α survey constraint is included.We note, however, that ∆ χ varies significantly between realiza-tions, ranging from 2 to 35.These above estimates were obtained assuming a perfectknowledge of the quasar unabsorbed spectrum. Errors on the es-timate of the unabsorbed spectrum will increase the errors on theBAO scale by 10 to 20% and result in sub percent biases, overallquite small compared to the statistical error. Note that the e ff ectof the presence of damped Ly- α and metal lines on the BAOmeasurement was not included in our mocks. Acknowledgements.
We would like to thank Rupert Croft, Andreu Font-Ribera,Patrick McDonald and Anze Slosar for useful discussions and the last two forcomments to the paper.
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