Is the baryon acoustic oscillation peak a cosmological standard ruler?
Boudewijn F. Roukema, Thomas Buchert, Hirokazu Fujii, Jan J. Ostrowski
aa r X i v : . [ a s t r o - ph . C O ] S e p Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed Friday 25 th September, 2015 (MN L A TEX style file v2.2)
Is the baryon acoustic oscillation peak a cosmological standardruler?
Boudewijn F. Roukema , , Thomas Buchert , , Hirokazu Fujii , Jan J. Ostrowski , Toru´n Centre for Astronomy, Faculty of Physics, Astronomy and Informatics, Grudziadzka 5, Nicolaus Copernicus University, ul. Gagarina 11,87-100 Toru´n, Poland Universit´e de Lyon, Observatoire de Lyon, Centre de Recherche Astrophysique de Lyon, CNRS UMR 5574: Universit´e Lyon 1 and ´Ecole NormaleSup´erieure de Lyon, 9 avenue Charles Andr´e, F–69230 Saint–Genis–Laval, France ∗ Departamento de Astronom´ıa, Universidad de Chile, Camino del Observatorio 1515, Santiago, Chile Institute of Astronomy, University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan
Le 25 septembre 2015
ABSTRACT
In the standard model of cosmology, the Universe is static in comoving coordinates; expan-sion occurs homogeneously and is represented by a global scale factor. The baryon acousticoscillation (BAO) peak location is a statistical tracer that represents, in the standard model, afixed comoving-length standard ruler. Recent gravitational collapse should modify the metric,rendering the e ff ective scale factor, and thus the BAO standard ruler, spatially inhomogeneous.Using the Sloan Digital Sky Survey, we show to high significance ( P < . Key words:
Cosmology: observations – cosmological parameters – distance scale – large-scale structure of Universe – dark energy
The choice of a spacetime coordinate system for the Universe thatenables the expansion to be represented via a global scale fac-tor a as a function of just one coordinate (cosmological time t ,Lemaˆıtre e.g. 1927) is extremely convenient. On large enough co-moving length scales, statistical spatial patterns are fixed in this co-ordinate system. Thus, the theory of primordial density fluctuationsleads to that of baryon acoustic oscillations (Eisenstein & Hu 1998;BAOs). The BAO peak in the two-point spatial auto-correlationfunction ξ was clearly detected in the Sloan Digital Sky Survey(Eisenstein et al. 2005) and the Two-Degree Field Galaxy RedshiftSurvey (Cole et al. 2005; earlier surveys may have detected thistoo, Einasto et al. 1997). BAOs now constitute one of the mostimportant tools for making cosmological geometry measurements,especially for upcoming observational projects such as the spacemission Euclid (Refregier et al. 2010) and the ground-based instru-ments DESI (Levi et al. 2013), 4MOST (de Jong et al. 2012), andthe LSST (Tyson et al. 2003). The BAO peak location of about105 h − Mpc (where h is the Hubble constant H expressed in unitsof 100 km / s / Mpc) is commonly expected to be a large enough co-moving length scale for it to provide a fixed comoving ruler in thereal Universe. ∗ BFR: during visiting lectureship; JJO: during long-term visit.
However, the validity of the BAO peak location as a stan-dard ruler depends on the validity of the assumed cosmologi-cal metric (di ff erential rule for measuring lengths) in the contextof the real Universe, which is lumpy (cf. the “fitting problem”,Ellis & Stoeger 1987). Scalar averaging is a general-relativistic for-malism that extends beyond the standard cosmological model, byallowing a spatial section of the Universe at a given time to have aninhomogeneous metric and calculating background-free volume-weighted averages of scalar variables (Buchert 2001, 2008). Theunivariate scale factor a ( t ) is replaced by an e ff ective, environment-dependent volume-based scale factor a D ( t ) ∝ V / D , dependent onboth the choice of compact spatial domain D of volume V D and ontime. Without this extension, the cosmological, comoving metricis forced (by definition) to be rigid in comoving coordinates, i.e.inhomogeneities in the matter distribution are not allowed to “tellcomoving space how to curve” (e.g. Buchert & Carfora 2008). Ap-plying scalar averaging to an observationally standard power spec-trum that statistically represents density fluctuations at an earlyepoch implies that even for spatial domains as large as the BAOpeak length scale, the environment dependence of the scale fac-tor should be observationally detectable, i.e. a M < a E is expected,where M (“Massive”) and E (“Empty”) represent overdense andunderdense spatial regions, respectively. For example, adopting 1 σ initially overdense ( M ) and underdense ( E ) fluctuations in a spher-ical domain of diameter ≈ h − Mpc and using equations (2), c (cid:13) Roukema et al. T w o - po i n t c o rr e l a t i on ξ - ξ Separation s (Mpc/h) supercluster-overlap pairs00.020.040.060.080.140 60 80 100 120 140 160 180 T w o - po i n t c o rr e l a t i on ξ - ξ Separation s (Mpc/h) complementary pairs
Figure 1.
Compression of the baryon acoustic oscillation peak.
Upperpanel:
BAO peak (cubic-subtracted correlation function) for pairs of lu-minous red galaxies whose paths either overlap with superclusters by ω ≥ ω min = h − Mpc or are entirely contained within the superclusters.The overlap ω is the chord length defined (Roukema et al. 2015, Sect. 2.3,Fig. 1) by the intersection of the path joining two LRGs and the superclus-ter modelled as a sphere. Individual curves represent 32 bootstrap resam-plings of the supercluster catalogue (235 objects) and the “random” galax-ies (484,352 selected from 1,521,736). The real galaxies (30,272) are notresampled. Most of the curves peak sharply at 95 h − Mpc; a few peak at85 h − Mpc. The high amplitude (in comparison with the lower panel) isconsistent with biasing that modifies the amplitude of ξ . Lower panel:
BAOpeak for the complementary subset of galaxy pairs. The peak occurs at thestandard value of about 105 h − Mpc. (13), (32), (50), and (54) of Buchert, Nayet & Wiegand (2013) tointegrate the Raychaudhuri equation (9) of the same paper gives arelativistic Zel’dovich approximation estimate of a M / a E ≈ . . (1)In other words, in the scalar averaging approach, one way in whichmatter inhomogeneities are expected to a ff ect the large-scale geom-etry and dynamics is to shrink the curved-space volume of overden-sities on the BAO scale by about 1 − ( a M / a E ) ≈ BA O pea k s h i ft ∆ s ( M p c / h ) Minimum overlap ω min (Mpc/h) observ. fitlower lim Figure 2.
Overlap dependence of the BAO peak shift. BAO peak shift ∆ s : = s non-sc − s sc , where s sc and s non-sc are the median estimates of thecentres of the best-fit Gaussians to ξ − ξ for LRG pairs that overlap su-perclusters (sc) and those that do not (non-sc), respectively. The error barsshow a robust estimate of the standard deviation, σ ( ∆ s ), defined here as1.4826 times the median absolute deviation of ∆ s . At each ω min , thesestatistics are calculated over 32 bootstrap resamplings of the observationaldata. Four out of the 320 Gaussian fits in the supercluster-overlap case failedand were ignored in calculating these statistics; no failures occurred for the320 non–supercluster-overlap cases. A linear least-squares best fit relation ∆ s = . h − Mpc + . ω min is shown with a red line. The most signifi-cant individual rejection of a zero shift is ∆ s = (6 . ± . h − Mpc for ω min = h − Mpc, a 3.1 σ (Gaussian) rejection. Since bootstraps are used,this estimate is conservative: σ ( ∆ s ) is expected to be an overestimate of thetrue uncertainty (Fisher et al. 1994, Sect 2.2). A 9% shift [Eq. (1)] wouldgive ∆ s = . ω . Since ω ≥ ω min , a scalar-averaging lower expected limit ∆ s > . ω min is shown as a green line. An environment-dependent e ff ect has recently been detectedas a six percent compression of the BAO peak location for spatialpaths that touch or overlap superclusters of luminous red galax-ies (LRGs) in the Sloan Digital Sky Survey (Roukema et al. 2015;SDSS; environment dependence of ξ at smaller scales has also beendetected in the SDSS, Chiang et al. 2015).In this Letter , we check whether the compression is dependenton the minimum overlap between spatial paths and superclusters,as it should be if the e ff ect is induced by the statistically overdensenature of the superclusters. We modify the previous method (Roukema et al. 2015, Sect 2) inorder to allow stronger overlaps. As in the original method, we cal-culate the correlation function ξ of the “bright” sample of LRGsin the SDSS Data Release 7 (DR7) for pairs of LRGs selected foroverlap (Roukema et al. 2015, Sect. 2.3, Fig. 1) (or non-overlap)of superclusters in the survey (Nadathur & Hotchkiss 2014), us-ing the Landy & Szalay estimator (Landy & Szalay 1993) on realand “random” (artificial) catalogues (Kazin et al. 2010). Comovingseparations s are calculated assuming the standard Λ CDM model(Spergel et al. 2003; Ade et al. 2014) with matter density parame-ter Ω m0 = .
32 and dark energy parameter Ω Λ = .
68. A best-fitcubic ξ ( s ) over separations s ≤ h − Mpc and s ≥ h − Mpc(i.e. excluding the peak), is found for the tangential signal (pairs c (cid:13) , 000–000 s the BAO peak a standard ruler? L3 ≤ ◦ from the sky plane). The procedure is repeated, bootstrap re-sampling the supercluster catalogue (Nadathur & Hotchkiss 2014)and the “random” LRG catalogue, several times. The BAO peaklocation is estimated from the medians and median absolute de-viations of the centres of the best-fit Gaussians to ξ ( s ) − ξ ( s ).In this work, minimum overlaps ω min in the range 10 h − Mpc ≤ ω min ≤ h − Mpc rather than ω min = h − Mpc are consid-ered. In order that ξ be defined for s ≤ ω min for these high valuesof ω min , we consider a pair of LRGs joined by a comoving spatialpath entirely contained within a supercluster to satisfy the overlapcriterion. Figure 1 shows that for a minimum overlap ω min = h − Mpc,the BAO peak is shifted to lower separations s (panel a) thanfor the complementary set of LRG pairs (panel b). The shift isclearer than for the earlier analysis, which had ω min = h − Mpc(Roukema et al. 2015, Fig. 8). Requiring a stronger overlap yieldsa stronger shift.Figure 2 shows the dependence of the shift ∆ s on ω min .The BAO peak shift for supercluster-overlapping LRG pairs ap-pears to increase from ∆ s ≈ h − Mpc for ω min < ∼ h − Mpc to ∆ s ≈ h − Mpc for greater overlaps. The Pearson product-momentcorrelation coe ffi cient of ∆ s and ω min is 0.87, with a probability of P ≈ . . ± .
040 and 4 . ± . h − Mpc, respectively.As a rough guide to what is expected from scalar averaging,we can use the 9% shift estimate from Eq. (1), which would give ∆ s = . ω , where the overlap path lengths are approximated ascorresponding to 1 σ overdense regions on the BAO scale, eventhough in reality, the overdense regions are superclusters, somesmaller and some larger than this scale. Since ω ≥ ω min , this impliesa rough scalar-averaging lower expected limit of ∆ s > . ω min ,shown as a green line in Fig. 2. Since the BAO peak location serves as a major tool for cosmo-logical geometry measurements, it is clear that its environmentdependence will need to be observationally calibrated and cor-rectly modelled theoretically. It is possible that the e ff ect couldalso be interpreted within the standard Λ CDM model, as is thecase for many large-scale phenomena. For example, observedsupervoids on the 200–300 h − Mpc scale (Nadathur & Hotchkiss2014; Szapudi et al. 2015) can be interpreted within the Λ CDMmodel (Hotchkiss et al. 2015), although their occurrence is ex-pected to be rare (Szapudi et al. 2015). In contrast, the studyof SDSS DR7 “dim” (or “bright”) LRGs via Minkowski func-tionals on scales ranging up to the BAO peak scale, within a500 h − Mpc (or 700 h − Mpc, respectively) diameter region, shows3–5.5 σ (or 0.5–2.5 σ ) inconsistencies with Λ CDM simulations(Wiegand, Buchert & Ostermann 2014, Table 1). Minkowski func-tionals have more statistical power than lower order statistics thatare commonly used in analysis of large-scale structure, such as the two- and three-point correlation functions, the correlation di-mension or percolation (“friends-of-friends”) analyses. This is be-cause all the n -point correlation functions would be needed inorder to represent the statistical geometrical information that theMinkowski functionals contain.A possible avenue to studying the environment-dependentBAO shift within the standard approach would be to carry out aFourier analysis rather than using the two-point correlation func-tion. This would require the development of a supercluster-overlap–dependent Fourier analysis method. Another alternative, to avoidhaving to determine the position of the peak itself, would be com-parison of radial to tangential correlation functions directly. Thiswould require correcting for peculiar velocity e ff ects, which arehighly anisotropic with respect to the observer.Interpreting the environment dependence of the BAO peak lo-cation reported in this Letter within the standard Λ CDM modelwould require the comoving length scale at which the Universe isrigid in comoving coordinates to be pushed up to a scale greaterthan 105 h − Mpc. The environment dependence (e.g. the ∆ s ( ω min )relation) would have to be modelled within a rigid comoving back-ground that can only exist at larger scales, at which no sharp sta-tistical feature that can function as a standard ruler is presentlyknown. This leads to a Mach’s principle type of concern that atrecent epochs, it is di ffi cult to have confidence that the standardcomoving coordinate system is correctly attached to an observa-tional extragalactic catalogue (peculiar velocity flow analyses indi-cate similar concerns, Wiltshire et al. 2013). Interpretation withinthe scalar averaging approach should be easier because its descrip-tion of fluctuation properties and the cosmological expansion rateis environment-dependent.Nevertheless, within the rigid comoving background frame-work (i.e. the standard model), small shifts in the BAO peak lo-cation have been predicted analytically and from N -body simula-tions (Desjacques et al. 2010; Sherwin & Zaldarriaga 2012), whileBAO reconstruction techniques (Padmanabhan & White 2009;Padmanabhan et al. 2012; Schmittfull et al. 2015) have been devel-oped to attempt to evolve galaxies’ positions backwards in cosmo-logical time, using a blend of theoretical calculations and N -bodymodels. The expected mean shift in the BAO peak location is lessthan one percent, i.e. an order of magnitude less than what we findfor the shift conditioned on ω min ≥ h − Mpc. The amplitudeof the shifts found in these calculations is constrained by the as-sumption that curvature averages out on the assumed background,i.e. that a conservation law for instrinsic curvature holds globally(Buchert & Carfora 2008).Theoretical work is underway in the scalar averaging ap-proach, which general-relativistically extends the standard model,allowing the restrictive assumption of a conservation law for in-trinsic curvature to be dropped (Buchert & Carfora 2008). Thephysical origin of curvature deviations from the background onscales as large as the BAO scale can then be thought of as fol-lowing from the non-existence of a conservation law for intrin-sic curvature. To reconstruct the primordial comoving galaxy po-sitions more accurately than in the standard model, i.e. to allowflexible comoving curvature that varies with the matter densityand the extrinsic curvature tensor across a spatial slice, relativis-tic Lagrangian perturbation theory (Buchert & Ostermann 2012;Buchert, Nayet & Wiegand 2013; Alles et al. 2015) is available foranalytically guided calculations. N -body simulations in which thegrowth of inhomogeneities is matched by inhomogeneous metricevolution will most likely also be needed to develop numerical con-fidence in what could be called “relativistic BAO reconstruction”. c (cid:13) , 000–000 Roukema et al.
No matter which approach is chosen, analytical, numerical and ob-servational work will be required if the BAO peak location is tocorrectly function as a standard ruler for cosmological geometricalmeasurements, since the evidence is strong ( P < . ff ected by structure formation. Moreover, the formationof superclusters—in reality, filamentary and spiderlike distributionsof galaxies (Einasto et al. 2014) rather than the spherically symmet-ric objects assumed here for calculational speed—can now be tieddirectly to a sharp statistical feature of the primordial pattern ofdensity perturbations. ACKNOWLEDGMENTS
Thank you to Mitsuru Kokubo for useful comments. The workof T.B. was conducted within the “Lyon Institute of Origins”under grant ANR-10-LABX-66. T.B. acknowledges financialsupport from CONICYT Anillo Project (ACT–1122) and UMI–FCA (Laboratoire Franco-Chilien d’Astronomie, UMI 3386,CNRS / INSU, France, and Universidad de Chile) during a lec-turing visit. A part of this project was funded by the NationalScience Centre, Poland, under grant 2014 / / B / ST9 / / / M / ST9 / . We gratefully acknowledge use ofthe Kazin et al. (2010) version of SDSS DR7 real and randomgalaxies at http://cosmo.nyu.edu/˜eak306/SDSS-LRG.html and of v11.11.13 of the Nadathur & Hotchkiss (2014) su-percluster catalogue at http://research.hip.fi/user/nadathur/download/dr7catalogue . REFERENCES
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