Precision cosmography with stacked voids
DDraft version October 27, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
PRECISION COSMOGRAPHY WITH STACKED VOIDS
Guilhem Lavaux
Department of Physics, University of Illinois at Urbana-Champaign, 1002 W Green St, Urbana, IL, 61801, USA
Benjamin D. Wandelt
UPMC Univ Paris 06, UMR 7095, Institut d’Astrophysique de Paris, 98 bis, boulevard Arago, 75014 Paris, France
Draft version October 27, 2018
ABSTRACTWe present a purely geometrical method for probing the expansion history of the Universe from theobservation of the shape of stacked voids in spectroscopic redshift surveys. Our method is an Alcock-Paczy´nski test based on the average sphericity of voids posited on the local isotropy of the Universe.It works by comparing the temporal extent of cosmic voids along the line of sight with their angular,spatial extent. We describe the algorithm that we use to detect and stack voids in redshift shells onthe light cone and test it on mock light cones produced from N -body simulations. We establish arobust statistical model for estimating the average stretching of voids in redshift space and quantifythe contamination by peculiar velocities. Finally, assuming that the void statistics that we derivefrom N -body simulations is preserved when considering galaxy surveys, we assess the capability ofthis approach to constrain dark energy parameters. We report this assessment in terms of the figureof merit (FoM) of the dark energy task force and in particular of the proposed EUCLID mission whichis particularly suited for this technique since it is a spectroscopic survey. The FoM due to stackedvoids from the EUCLID wide survey may double that of all other dark energy probes derived fromEUCLID data alone (combined with Planck priors). In particular, voids seem to outperform BaryonAcoustic Oscillations by an order of magnitude. This result is consistent with simple estimates basedon mode-counting. The Alcock-Paczy´nski test based on stacked voids may be a significant additionto the portfolio of major dark energy probes and its potentialities must be studied in detail. INTRODUCTIONThe physical nature of Dark Energy, detected throughsupernovae luminosity distance measurements (Kowalskiet al. 2008), Baryonic Acoustic Oscillations (BAO, Per-cival et al. 2010) and the Cosmic Microwave Background(Komatsu et al. 2011), still evades us. The BOSS sur-vey (Schlegel et al. 2007) has been designed to assesswhether the equation of state of Dark Energy is indeedconstant and equal to minus one. However, observationsbased on baryonic acoustic oscillations are limited by theminimal volume required to estimate the scale of theseoscillations, typically ∼ h − Mpc. With the adventof large galaxy spectroscopic redshift survey, such as theSloan Digital Sky Survey (Abazajian et al. 2009), we nowhave access to a three-dimensional representation of thelarge-scale structure on our light-cone on vastly differentscales.The well known Alcock-Paczy´nski test (Alcock &Paczynski 1979) can be applied to any structure forwhich we know the physical size or, more weakly, the ra- Department of Physics, University of Illinois at Urbana-Champaign, 1002 W Green St, Urbana, IL, 61801, USA Department of Physics and Astronomy, The Johns HopkinsUniversity, 3701 San Martin Drive, Baltimore, MD 21218, USA CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98 bis,boulevard Arago, 75014 Paris, France Department of Astronomy, 1002 N Gregory Street, Universityof Illinois at Urbana-Champaign, Urbana, IL 61801, USA Department of Physics and Astronomy, University of Waterloo,200 University Avenue West, Waterloo, Ontario, Canada, N2L 3G1 Perimeter Institute for Theoretical Physics, 31 Caroline StreetNorth, Waterloo, Ontario, N2L 2Y5 tio of its extent along the line of sight and its angular size.In particular, if we had a population of standard spheresscattered throughout cosmic history we could measurethe cosmological expansion directly. Absent such a pop-ulation, the next best thing is a population of objectswhose average shape is spherical.Cosmic voids are such a population and hence promis-ing candidates for probing the expansion geometry of theUniverse. Even though individual void shapes may becomplicated, the average void is spherical in an isotropicand homogeneous universe. Detecting all voids observedin a galaxy survey and stacking voids of similar sizesand redshifts projects out the details of individual voidshapes. Since the average shape of voids is known to bespherical, the observed, stretched shape in redshift spaceis a direct function of the local Hubble expansion of theUniverse and the angular diameter distance at the red-shift of the void and hence a sensitive function of thecosmological parameters, in particular those parameter-izing the dark energy equation of state.Voids are spatially localized coherent structures withsizes between ∼ a r X i v : . [ a s t r o - ph . C O ] J un G. Lavaux & B. D. Wandeltticed by Ryden (1995) who proposed using the apparentstretching of void shapes in redshift space coordinatesto estimate the local geometry of the expansion. Doingso properly requires selecting voids that have the sameoverall size and density. This work has then yielded anumber of other studies on voids and baryonic acousticoscillations (Ballinger et al. 1996; Ryden & Melott 1996;Schmidt et al. 2001), though the complicated shapes ofindividual voids have made it difficult to extract cosmo-logical information at high signal-to-noise.In a similar spirit, Jimenez & Loeb (2002) proposed atest which uses cosmic clocks as tracers of cosmic timeto which we can compare the measured galaxy redshifts.The Alcock-Paczy´nski test can be thought of as a differ-ential version of this approach—one could say comparingthe radial, temporal extent of voids with their angular,spatial extent amounts to using them as “cosmic stop-watches.” Further, it does not rely on spectral modelingto extract galaxy ages.Park & Kim (2010) proposed using genus statistics tomeasure expansion. Their claim was that the genus is in-sensitive to redshift space distortions while peculiar ve-locities will mildly affect the void technique (Ballingeret al. 1996). However, the genus is not a spatially lo-cal quantity, which may be prone to a number of obser-vational problems like non-trivial edge effects and inho-mogeneous incompleteness corrections. It is also modeldependent.Our method requires spectroscopic survey since red-shift errors in photometric redshift catalogs wash out theline-of-sight information on the scale of all but the mostextreme voids even with ∼ .
7% precision (e.g. Ilbertet al. 2009). An order of magnitude improvement in red-shift precision would likely be required to directly observeany non-linear three-dimensional structures ( cf.
Jasche& Wandelt (2011) for a possible approach).Our paper is organized as follows. In Section 2, weshow how voids may give us a direct probe of cosmolog-ical parameters through shape stretching. In Section 3,we explain the method we use for finding and stackingvoids on the light-cone and inferring the local expansionfrom the shape of stacked voids. In Section 4, we test ourmethod on N -body simulations. We derive the profiles ofstacked voids in simulation, the sensitivity to contamina-tion of the redshifts by peculiar velocities and the numberdensity of voids found in the simulation. In Section 5,we derive the Hubble-diagram of expansion in the simula-tion from our mock-observation of voids. We do a Fisher-matrix analysis of the measurement of Dark Energy phys-ical parameters using the expansion rate derived fromvoids. We apply this formalism to the survey specifi-cations of the main galaxy sample of the Sloan DigitalSky Survey (SDSS Abazajian et al. 2009), the Baryonicacoustic Oscillation Sky Survey (BOSS Schlegel et al.2007) and the EUCLID survey (Laureijs et al. 2011). InSection 6, we conclude. COSMOLOGY WITH VOIDSIn this Section, we recall the basic equation at the baseof the Alcock-Pasczy´nski test (Alcock & Paczynski 1979)applied on voids. This test comes comes from the relationbetween the comoving angular distance D A and the red-shift z of an event in a Friedmann-Lemaˆıtre-Robertson- Walker (FLRW) cosmology: D A ( z ) = cH f k (cid:18) H c χ ( z ) (cid:19) , (1)with f k ( x ) = √ | k | sinh( (cid:112) | k | x ) if k < x if k = 0 √ | k | sin( (cid:112) | k | x ) if k > , (2)the redshift/comoving distance relation χ ( z ) = cH (cid:90) z d˜ zE (˜ z ) , (3)where E ( z ) = H ( z ) /H , and k = (cid:18) H c (cid:19) (Ω m + Ω Λ − , (4)with Ω m , the mean matter density, and Ω Λ the DarkEnergy density, both normalized to the present criticaldensity. For sufficiently small curvatures k , it is possibleto invert the relation (1) to derive the redshift from thecomoving distance rz = D − A (cid:18) rH c (cid:19) . (5)If we look at a cosmic object, e.g. a galaxy, or a cosmicvoid or the BAO, at redshift z , it has an extent δz in theredshift direction and δr in the angular direction definedas δr ≡ D A ( z ) δθ. (6)Additionally δz corresponds to a comoving distancealong the line-of-sight given by the differentiation ofEq. (1): δl = d D A d z δz = cδzH E ( z ) f (cid:48) k ( χ ( z )) , (7)with f (cid:48) k the first derivative of f k . We have indicated inthe introduction that we assume the Universe is locallyisotropic. Consequently, the large-scale structures mustnot have a preferred direction in average. If we con-sider a void consisting in an infinite average number ofstacked voids of a specific volume, this “stacked void”should have the same extent in all directions. We canthus assume that δl = δr . This equality yields δrδz = cH E ( z ) f (cid:48) k ( χ ( z )) , (8)which in terms of the projected separation δd = czδθ/H gives δzδd = (cid:18) H c (cid:19) D A ( z ) E ( z ) zf (cid:48) k ( χ ( z )) = H c e v ( z ) . (9)We propose observing this quantity through measuringthe shape of stacked voids in redshift space as a functionof redshift. Note that this observable depends directlyon E ( z ) rather than through an integral, as is the casefor methods based on angular diameter distance, probedby observing the angular scale of the BAO peak as arecision cosmography with stacked voids 3 (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:2)(cid:4)(cid:6) (cid:11)(cid:12)(cid:4)(cid:5)(cid:3)(cid:13)(cid:14)(cid:15) (cid:1) (cid:16)(cid:17)(cid:5)(cid:18) (cid:19)(cid:12)(cid:10)(cid:2)(cid:3)(cid:6)(cid:8)(cid:16)(cid:17)(cid:5)(cid:18) (cid:19)(cid:12)(cid:10)(cid:2)(cid:3)(cid:6)(cid:8)(cid:16)(cid:17)(cid:5)(cid:18)(cid:13)(cid:14)(cid:15)(cid:8)(cid:16)(cid:17)(cid:5)(cid:18) (cid:13)(cid:14)(cid:15)(cid:8)(cid:16)(cid:17)(cid:5)(cid:18)(cid:8) (cid:13)(cid:14)(cid:15) (cid:1) (cid:16)(cid:17)(cid:5)(cid:18) (cid:20)(cid:10)(cid:2)(cid:12)(cid:6)(cid:2)(cid:10)(cid:8)(cid:16)(cid:17)(cid:5)(cid:18) (cid:20)(cid:10)(cid:2)(cid:12)(cid:6)(cid:2)(cid:10)(cid:8)(cid:16)(cid:17)(cid:5)(cid:18) Figure 1.
Storing voids in a hierarchical tree.
The spatially adjacent sub-voids are assembled in a single void, the “parent void”, whichcontain at least one additional basin. The identity of the parent void is inherited by the “sub-void” with the smallest core density. Theparent void is itself part of a “greater void”. The tree is ordered in scale. function of redshift, or the luminosity distance which isprobed by supernova surveys. We expect this to enhancethe sensitivity of our method to the physical propertiesof dark energy, which appear directly in E(z). This re-lation is not strictly an image of the Hubble constant atdifferent z as it is modified by D A ( z ) / ( zf (cid:48) k ( χ ( z ))), whichis close to one at low redshift. However, it is a goodproxy for it and it is possible to obtain the equivalent ofan Hubble diagram for voids. Eq. (9) was already de-rived by Ryden (1995) for universes with no curvature.The measurement of the isotropy would be a clean wayto measure finely the cosmic expansion. FINDING AND STACKING VOIDSIn this Section, we present the algorithm that we de-veloped to locate voids and stack voids on an expandingmetric. In the following, we will use the following con-vention. The effective radius of a void corresponds to theradius of the sphere of equivalent volume. So, if V is thevolume of the void, r eff = (cid:18) π V (cid:19) / . (10)First, in Section 3.1, we define the coordinate systemthat we use in this work. In Section 3.2, we give thedetails of the algorithm to find and choose the voids thatare of interest for the stacking procedure. In Section 3.3,we describe the stacking procedure.3.1. The coordinate system
The fundamental point on which we base our methodis the capability to find and stack void structures, evenif they are strongly distorted. For simplicity, we adoptthe following definition, which a posteriori we will showis robust to distortions in the coordinate system. Wewill use the infinite remote observer approximation forthe redshift coordinate and the planar approximation forthe angular coordinate. We consider density tracers, e.g.galaxies, in a hybrid coordinate system ( x, y, z ). For atracer t , e.g. a galaxy, at the sky position ( θ, φ ), θ beingthe sky latitude and φ the sky longitudes, and located at redshift Z , we define: x = cZH cos( φ ) cos( θ ) (11) y = cZH sin( φ ) cos( θ ) (12) z = cZH sin( θ ) (13)In the infinite remote observer approximation θ ∼ π/ z ∼ cZH , but other directions are adequate pro-vided that the extent on the sky is small.3.2. Organizing voids in tree
From the volume sampled by the tracers { t } , we ex-tract a box of side L in the ( x, y, z ) coordinate. Wenow use the Zobov (Neyrinck 2008) algorithm to com-pute and locate local minima in the density of tracers,assuming they are a sample of the underlying matter den-sity field.
Zobov finds local density minima on densityfield sampled by particles and their associate catchmentbasins, in the language of the watershed transform (e.g.Platen et al. 2007). It does so using a Voronoi tessella-tion, derived from the Delaunay tessellation applied on aset of tracers. In addition, basins are assembled in voids,starting from the lowest density basin, such that:- each basin is a void- two basins are assembled in one void if they sharea common boundary, and that the density on thisboundary is the lowest for each of the void. Basinsare always assigned to the voids which have thelowest core density.The whole volume sampled by mass tracers is thus par-titioned into a set of basins. Each basin corresponds to avoid which itself is a collection of basins. Thus, the voidsnaturally acquire a tree-like structure for which voidshave both a single “parent” and possibly many “chil-dren”. We define an order on the tree such that a voidis the immediate parent of another one if it shares thesame zones as the child void and at least one more. We G. Lavaux & B. D. Wandelt −1 Mpc)050100150200250300350400 Y ( h − M p c ) Figure 2.
Void tree from an N -body simulation. We show theconvex hulls of particles within a 40 h − Mpc deep slice, and be-longing to a void. A void is visualized if it has either a meanrelative density less than − .
4, for clarity of the representation,and an effective radius within [8; 50] h − Mpc, or is within a sub-tree of an drawn void. The color encodes the depth of the void inthe tree as indicated by the top color bar. note that this corresponds to a different tree than definedby Aragon-Calvo et al. (2010). Also, contrary to manygalaxy based void finder, this method does not assumea hard-coded void shape (see Colberg et al. 2008, for areview of void finders). It does not rely on a smootheddensity field but only on the topology of the tracers, suchas particles in N -body simulations or galaxies in observa-tions. This is not the sole technique. More recently thecomplete topology of the cosmic web which can be de-rived through a set of tracers has been formally studiedby Sousbie (2011) in the context of the persistence. Thetree that we have developed helps separating overlappingvoids such that the same volume is not used multipletimes in a statistical analysis.In our approach, there may not be a natural single rootfor this tree because we do not have periodic boundaryfor the box side L . So we introduce an artificial rootnode for which all child nodes correspond to the parent-less voids. Additionally, we compute the mean density ofeach void in the tree, which is a non-monotonous functionof the depth in the tree.3.3. Void stacking
Voids have complicated shapes mainly produced by thegravitational shear field (Park & Lee 2007; Lavaux &Wandelt 2010). It is therefore difficult to use them in-dividually as a probe of the effect of cosmological ex-pansion. However, using the assumption of the isotropyof density fluctuations, we expect that the average voidshape is spherical in physical coordinates. To achievea correct stacking, at least three important features arerequired: 1. for a given stack, the algorithm must only selectvoids within a narrow volume range. This condi-tion will cause a specific average void shape of thegiven size to emerge from the stacking.2. each void of a stack must be unique and must notoverlap with another void. Enforcing this conditionallows the removal of spurious correlation in thevoid shapes, which could systematically affect theresult.3. each void in a stack must be centered precisely onits average lowest density. This is necessary to min-imize the effect of halos, which are expected on theboundaries of voids. These halos may bias the po-sition of the center reducing the amount of signalavailable from the average void shape.The first requirement is satisfied by the
Zobov voidfinder.
Zobov provides a volume for the void which isexact from the point of view of topology. The secondpoint is achieved by storing the voids in the tree struc-ture mentioned at Section 3.2.The center of the void could be derived directly fromthe Voronoi tessellation. However we expect this inferredcenter to be unstable with respect to shot noise in obser-vations, or the Lagrangian grid in N -body simulations.These would cause spurious effects in the stacking pro-cedure. We opt for computing the mean lowest densityposition using the volume-weighed barycenter of the trac-ers attached to a single void of the stack. The Voronoitessellation gives a volume V i surrounding each tracer i .We compute the average position x V of the center of thevoid V by computing x V = 1 (cid:88) i V i (cid:88) i x i V i , (14)where i runs over the tracer, e.g. galaxies, in the void V , and x i its position in the coordinate system given inSection 3.1.Finally, some voids of the stack have large clusters neartheir estimated center. In the limit of an infinite numberof voids, we expect these cases to be vanishingly small.However, we do not have an infinite number of voids inreality. We have decided to avoid these cases by enforc-ing that the core density of the void should be reallyempty. We define the core density as the mean matterdensity within a sphere of fiducial radius r eff /
4. Thiscomes at the cost of a smaller number of available voidsin the stack. A posteriori, in Section 4.1, we see that ouradopted fiducial radius is smaller than the actual size ofthe “core” of the stacked void.To summarize, we proceed as follows:1. We put the tracers in the coordinate system of Sec-tion 3.1.2. We extract a parallelepiped volume of side L × L × L z which resides entirely within the region spannedby the tracers.3. We normalize the coordinates to 1 in each direction.4. We apply the Zobov void finder algorithm.recision cosmography with stacked voids 5 (cid:0) Mpc)05101520 z ( h ✁ M p c ) (cid:0) Mpc)0.20.40.60.81.0 ✁ /¯ ✁ Best fit cubicStacked voidMean density
Figure 3.
Void stacking in the final, z = 0 state of the simulation. We show the result of the stacking procedure given in Section 3.3when it is applied to all voids of size 8 Mpc in the full volume of the simulation without introducing cosmological and peculiar velocitiesdistortions. Left panel: a filled contour plot of the density n ( d, z ). Contours above the density 1 . XY direction (x-axis) andthe axis along the Z direction (y-axis). Right panel: the three-dimensional average density profile, in shells, of the stacked void (solid line)and the fit to a cubic density profile (dashed line). The horizontal solid red line is the mean density. V (cid:0) /¯ (cid:0) R V =4h ✁ MpcR V =8h ✁ MpcR V =10h ✁ MpcR V =14h ✁ Mpc V ✂ /¯ ✂ R V =4h ✄ MpcR V =8h ✄ MpcR V =10h ✄ MpcR V =14h ✄ Mpc
Figure 4.
Stacked void profiles at z = 0 – We show the profiles for voids selected with different effective radii. Left panel: mass densityprofiles in thin shells. Right panel: average mass density profile in the sphere of the given radius. G. Lavaux & B. D. Wandelt5. We store the found voids in a tree according toSection 3.2.6. We walk the tree, starting from the root and stopwhenever the effective radii, r eff , is within a givenrange, between R min and R max . r eff is obtained bycomputing the radius of the sphere which has thesame volume as the void.7. We compute the position of the particles accordingto the volume-weighed barycenter of Eq. (14) foreach void.8. We compute the density of the void within a sphereof radius r eff /
4. We only accept voids that havea core density less than 20% of the mean matterdensity of the universe.9. Around each volume-weighed barycenter, we ex-tract a spherical volume, in the coordinates of Sec-tion 3.1, of radius R cut = 3 × R max . The center ofthe extracted volume is put at the origin. We dothis for all selected voids.The resulting particle distribution gives what we call astacked void. We transform coordinates from ( x, y, z ) to( d = (cid:112) x + y , | z | ). We bin this distribution and dividethe resulting density by the number of stacked voids,the bin width and d , the Jacobian of the transformation.This procedure yields the density n ( d, z ) of tracers perunit volume per void. An example of such a distributionis given in Figure 3. TESTS ON N -BODY SIMULATIONWe have run a series of three pure dark matter N -bodysimulations with different realizations of the initial condi-tions but for the same cosmology. The dark matter parti-cles have different sampling properties than the galaxies.In particular, galaxies are biased tracers of the matterdensity field. As we are relying on topological proper-ties of the large-scale structures, the result should be thesame for the two. We discuss in Section 6 the limits of ourapproach. The volume of each simulation is given by acube of side L = 500 h − Mpc. Each simulation has N =512 particles. We have adopted a ΛCDM-WMAP7 cos-mology with the following parameters: Ω b h = 0 . c h = 0 . H = 71 km s − Mpc − , w = − n S = 1, A S = 2 . × − . This corresponds to Ω b = 0 . M = 0 . σ = 0 .
84. Each particle has a mass m p = 2 .
05 10 h − M (cid:12) . The transfer function for den-sity fluctuations for this cosmology is computed using CAMB (Lewis et al. 2000). The initial conditions aregenerated using
ICgen , which uses the transfer func-tion to generate a density field from the primordial powerspectrum.4.1. Stacking voids in pure comoving coordinates
In this Section, we consider the ideal case of voidsstacked in comoving coordinates, i.e. we purely considerthe distribution of dark matter particles as given by the N -body simulation. We consider voids for which r eff isbetween R min = 8 h − Mpc and R max = 9 h − Mpc. We give the result of the void stacking procedurefor one of the N -body sample realizations in Figure 3.In the left panel, we show the density profile of thestacked void, where on the horizontal axis correspondsto d = (cid:112) x + y , with x and y the first and second co-ordinate of the particle in the stacking, and the verticalaxis to | z | , the third coordinate of the particle. The solidblack contour corresponds to the result given by the like-lihood analysis described in Section 4.3. The filled colorcontours have been chosen equi-spaced from ρ/ ¯ ρ = 0 to ρ/ ¯ ρ = 1 .
1. The solid line in the right panel shows thethree-dimensional density profile, in thin shells, of thestacked void.The one-dimensional density profile shown in the rightpanel of Figure 3 is similar to the one shown in Figure2 of Ceccarelli et al. (2006), though we use pure dark-matter simulation in place of mock/real galaxy samples.The matter shell around the void is clearly visible forradii greater than the chosen R min .The inspection of both panels shows a number of in-teresting features. It is clear that the voids stack coher-ently and form a region of lower density for √ d + z < h − Mpc. The mass density inside the stacked void, inboth panels, is featureless. Outside the expected bound-ary of the stacked void, at √ d + z > h − Mpc, thedensity profile continues upward to ρ (cid:39) . ρ and thenfalls back to homogeneity. This shell is clearly seen in theone-dimensional averaging of the density profile shown inthe right panel of the same Figure. Our density profilehas similar features to the ones presented by Benson et al.(2003), Padilla et al. (2005) and Ceccarelli et al. (2006).In the two-dimensional mass density diagram of the leftpanel of Figure 3, it is clear that this is a real spheri-cal shell, and that it harbors clumps highlighted by thewhite regions at ρ/ ¯ ρ > . ρ ( r )¯ ρ = A + A (cid:18) rR V (cid:19) (15)with R V the radius of the stacked void, A i the parametersagainst which the fit is computed. As we have voidswith 8 h − Mpc ≤ r eff ≤ h − Mpc, we have set R V =8 h − Mpc. The best fit gives A = 0 . ± . A =0 . ± .
03 for voids of 8 h − Mpc.First, we note that the value of A attributes a signif-icantly non-zero density to the center of the void. Thismay be due to a resolution effect as putting one par-ticle within (1 h − Mpc) yields a density fluctuation of δ = − . − A ) /A is not strictly equalto one. This can probably be explained by the size of theinterval of accepted effective radii. We expect that largebin sizes, such as in Colberg et al. (2005), may signifi-cantly change the profile of the stacked void, notably at r ∼ R V owing to the non-commutativity of the operationof re-scaling and averaging. This problem also affects allthe quantities that we may derive from this profile, likepeculiar velocities. The result of the fit (dashed line) andthe actual void profile (solid line) is shown in the rightpanel of Figure 3. Visual inspection shows a good agree-ment between those two profiles within the bounds of thevoids. We will use this profile for the remainder of thiswork. For our method the choice of the profile is onlyimportant insofar as it does not bias (or remove signalto noise from) the shape measurement.In Figure 4, we show the density profiles, both in shellsand cumulative, with radii normalized to the sought ef-fective radii of the voids in each stack. We have consid-ered voids with r eff between 4 − h − Mpc, 8 − h − Mpc,10 − h − Mpc and 14 − h − Mpc. We clearly see thatall voids with r eff greater than 8 h − Mpc have nearly ex-actly the same density profile, even at radii larger than R V . While the universality of void profiles was previ-ously noted by Colberg et al. (2005), they did not showthat this universality also extends outside the void, inthe shell region. Our density profiles compares well alsowith that in the seminal paper by van de Weygaert &van Kampen (1993). Note that their void size definitionis radically different from ours. Finally, from the leftpanel of Figure 4, we note the presence of a structurelooking like a “core”, which does not extend above halfthe effective radii of the individual voids. This justify aposteriori our choice for the mean density within r eff / Stacking with redshift distortion
In this section we consider the possibility of applyingthe algorithm for void detection of Section 3. We haveselected five snapshots for each of the three simulations.They correspond to the simulated universes at z b = 0, z b = 0 . z b = 0 . z b = 0 .
81 and z b = 1 .
0. We haveconsidered that the first two dimensions of the positionsof the particles in the snapshots correspond to the an-gular coordinates and the last, z C , to the distance alongthe line of sight, relative to the face with z C = 0. Asour simulated universes are flat, we have converted thecomoving positions z C in redshift positions z using z = χ − (cid:18) H c ( z C + χ ( z b )) (cid:19) + v z c , (16)with χ as defined in Eq. (3), v z the peculiar velocity ofthe particles in the z direction, c the speed of light. Wehave not included the additional D A ( z ) /z term presentin e v ( z ) (Eq. 9) which adds an additional small effectat low redshift. In each case, we have extracted a boxof 400 h − Mpc × h − Mpc × − from thedistribution of particles in redshift coordinates. We haverun the void identification and stacking algorithm on theparticles of this box. Additionally, we have consideredthe stacked particles when peculiar velocities are eitherincluded (dubbed “mock catalog with full redshifts”) or excluded (dubbed “mock catalog with pure expansionredshifts”).In Figure 5, we show the result of the stacking for oneof the simulations, for the snapshot at redshift z = 1.For visualization purposes, the effects of expansions wereremoved after having stacked the voids. In the left panel,we show the result of the stacking algorithm for the mockcatalog with pure expansion redshifts. In the right panel,we show the same test but for mock catalogs with fullredshifts. We highlight with black contours the fittedthree-dimensional density profile of Section 4.1. The plotis corrected for the inferred expansion, so the contoursare perfectly circular. Qualitatively, they match the colorcoded density in the left panel.We note in the right panel of Figure 5 that there isa non-trivial deformation of the void. At low ( d, z ), thevoid is emptier and slightly elongated in the redshift di-rection. This is expected because of a finger-of-god effectin the void. At high ( d, z ), the void is flattened. Thisis clearly seen by considering, e.g. , the yellow iso-densityin the right panel. The decrease of the peculiar veloc-ities are not sufficient to explain the amplitude of thiseffect. We have tested our fitting procedure on mockvoids whose shaped have been transformed by the aver-age peculiar velocity field both expected and measuredin voids. We have found that they are not introducing asignificant pancaking effect.We have investigated the origin of this systematic ef-fect, which happens to be non-trivial. It may be ex-plained by a two-step process. First, large-scale redshiftspace distortions induce a selection bias on cosmic voids:cosmic voids with collapsed structures along the angularcoordinate direction are slightly disfavoured when theyare binned by void sizes. This discrimination is generatedby large scale flows that induces a small modification ofthe void size, which is sufficient to displace voids fromone void size bin to another. However, because voids donot have a flat size distribution, it is statistically muchless likely to displace a large void into the bin than todisplace it out to a bin corresponding to larger voids.The stretching is far less present when large clusters arepresent along the line of sight, which means that thesevoids stays in the same void size bin. This selection ef-fect makes the distribution of large-scale structure out-side the void slightly anisotropic. Additionally, as halosare preferentially located at small angular distance andhigh redshift distance from the void center, the finger-of-gods that they produce cause a thickening of the voidwall in the redshift direction, which in turns cause thepancaking of the voids.At smaller radii, like ∼ h − Mpc, the stacked voidis seemingly spherical because it is not contaminated byany of the two above effects. By considering the voidshape near the half-radius from the center therefore min-imizes possible biases due to peculiar velocities. This is arobust procedure, but it is not lossless. Further modelingof the profile could improve signal to noise in our shapeinference, described in the next section. In Section 5.1,we propose a simple alternative de-biasing scheme.4.3.
Void shape inference in redshift space
Stacked voids have an isotropic cubic density profile incomoving coordinates, as found in Section 4.1. We modelthe redshift space distortion by fitting the density n ( d, z ) G. Lavaux & B. D. Wandelt (cid:0) Mpc)012345678 z ( h ✁ M p c ) ✂ Mpc)012345678 z ( h ✄ M p c ) Figure 5.
Impact of peculiar velocities.
Density in stacked voids in redshift coordinates, in the snapshot at z = 1, for a400 h − Mpc × h − Mpc slice with a thickness of 10,000 km s − . Left panel: expansion only, without the peculiar velocities in theredshift positions of particles. Right panel: we have introduced the distortion due to peculiar velocities. The black contours are the samein the two panels and highlight the fitted density profile to the left, expansion-only case. estimated using the stacking procedure of Section 3.3, inredshift/angular coordinates this time, to the function n ( d, z ) = min (cid:16) n + (cid:0) ( d/a d ) + ( z/a z ) (cid:1) / , n max (cid:17) (17)with n the density at the minimum in ( d, z ) = (0 , a d the semi-axis along the angular coordinate direction, a z the semi-axis the redshift direction, n max a maximumdensity value. We expect n max ought to be near unityto show convergence to the mean density. However, weleave it as a free parameter because of the presence of theshell around the void and the limited accessible volumearound the stacked void which could bias this value.To fit the model to the estimated density we assumethat the fluctuations according to the ellipsoidal modelare Gaussian but with two different variances dependingon the location in the void. The likelihood χ takes thusthe following shape: χ ( n , n max , a d , a z , σ , σ ) = N d (cid:88) i =1 N z (cid:88) j =1 S i,j (cid:32) ( n ( d i , z j ) − n i,j ) σ ( d i , z i ) + 2 log( σ ( d i , z i )) (cid:33) , (18)with ( i, j ) the i -th and j -th bin, for which the ( d, z ) takethe value ( d i , z j ), n i,j the value estimated in the bin( i, j ), N d the number of bins in the d direction, N z thenumber of bin in the z direction. S i,j is either one orzero depending whether we want to include the bin ( i, j )in the optimization. We limit the fit to the disc of radius R cut , as there is no stacked void data at distance larger than this. S i,j takes the form S i,j = (cid:26) (cid:112) d i + ( z j /E ) ≤ R cut , (19)where we may correct for the expansion in the pixel se-lection through the coefficient E . In practice, we keep E = 1 in all the following.At the position ( d, z ), and within the void, we enforcethat the Gaussian part, σ , of the distribution R scales as1 / √ d . This follows from the cylindrical averaging whenbuilding the stack of voids. Outside the void, we takea fixed standard deviation to account for uncertainty inapproximating the outside profile by a single constant.The standard deviation is spatially varying according to: σ ( d, z ) = (cid:40) σ (cid:113) h − Mpc d if n ( d, z ) < n max σ otherwise (20)We find the parameters ( n , n max , a d , a z , σ , σ ) and theirerror bars by running a Monte-Carlo Markov-Chain ex-ploration of the four parameters on the sub-sample ofpixels for which (cid:112) d i + ( z j /E ) ≤ R cut . All the measure-ments of ratios shown in the Section 5 are made usingthis technique.The error model contains two components—the Pois-son error due to the number of tracers in each pixel, and acorrelated error due to dense clumps which occurs aroundindividual voids and which have not been completely re-moved by the averaging procedure. Such clumps of scale (cid:46) h − Mpc are visible in Figure 5. A crude way tomodel such correlated fluctuations is to choose the pixelrecision cosmography with stacked voids 9 −1 Mpc)02468101214 z ( h − M p c ) Figure 6.
Result of the likelihood analysis on the binned parti-cles of the stacked void – We give in solid black line the iso-densitycontour of the stretched cubic model (Eq. 17) fit using the likeli-hood analysis of Section 4.3. For this Figure, we have consideredvoids with r eff = 8 h − Mpc. We plot the underlying density fieldbetween the null density and 1.1¯ ρ , with ¯ ρ the mean density of theslice. Density pixels are 2 h − Mpc. The error is modeled as inde-pendent from pixel to pixel. size to be large enough that such fluctuations mostly af-fect single pixels only. We conservatively choose a pixelsize of 2 h − Mpc to satisfy this criterion. For pixels ofthis size the Poisson error due to individual tracers isentirely negligible.We illustrate in Figure 6 the outcome of the likelihoodanalysis on a stretched stacked void, obtained from amock light-cone at z = 1. We show both the densityfield of the stacked void and the iso-density contours ofthe fitted profile using the likelihood analysis. The iso-density contours follow the outer edge of the void. Thefluctuations of the binned density field looks clearly ran-dom and uncorrelated at this resolution. This was notthe case in Figure 5. We conclude that the likelihoodthus behaves as designed.We tested the robustness of this procedure to changesin the number density of tracers by re-running the en-tire pipeline on a subsample of the N -body particles.The results essentially did not change if the number ofparticles was reduced by a factor 5 (to ∼ . h Mpc − ),which corresponds to a typical galaxy density expectedfor the EUCLID survey at low redshift ( z (cid:46) . Number of voids
We give in Figure 7 the averaged number density ofvoids, in comoving coordinates, for the three simulations,at each redshift and each r eff that we have considered forestimating the Hubble constant. −1 Mpc)10 -8 -7 -6 -5 -4 -3 N u m b e r d e n s i t y n ( R ) ( h M p c − ) Fitz=1z=0.818z=0.538z=0.25z=0
Figure 7.
Comoving number density of voids.
We show resultsfor purely expanding universes, the result when peculiar velocitiescontaminates redshifts is the same. The dashed lines give the lowestand the highest value of the density when the three simulations areconsidered.
The dependence of the number of voids with redshiftis as expected. Small voids should be more abundantat high redshifts and large voids more abundant at lowredshifts (Sheth & van de Weygaert 2004). The rela-tion pivots about a radius of ∼ h − Mpc. As notedby these same authors, the number density of voids de-pends principally on their volume and then on redshiftsthrough growth of structures. The number of voids de-tected, when peculiar velocities contamination is addedin our mock catalogs, is roughly the same and with thesame dependence with radii. We note a small but sys-tematic destruction of voids of 4 h − Mpc. It is plausiblethat the original topology is lost at these small scales be-cause of the contamination by fingers of god, which canbe effectively as deep as 10 h − Mpc.In Figure 7, we note that whatever the dependence ofthe physical number density of voids with redshifts, forvoids as we define them in redshift space , this dependenceis small within the redshift range z = 0 −
1. Thus for thepurpose of this work, we neglect the time dependence ofthe void abundances and focus on the scale dependence.For each radius, we average the densities at all redshiftsand fit an exponential law, which seemed most suited forrepresenting this set of curves, on the number density asa function of r eff , for the range 4 − h − Mpc: n ( r eff )1 h Mpc − =(3 . ± . − exp (cid:18) − (0 . ± . r eff h − Mpc (cid:19) . (21)We show this relation in Figure 7 with a thick blacksolid line. This relation is used for the Fisher-Matrixanalysis of the Section 5.2 as an approximation of the0 G. Lavaux & B. D. Wandelt E x p a n s i o n e v ( z ) Simulation R=6 h (cid:0) MpcTheory E x p a n s i o n e v ( z ) Simulation R=14 h ✁ MpcTheory E x p a n s i o n e v ( z ) Simulation R=8 h ✂ MpcTheory E x p a n s i o n e v ( z ) Simulation R=8 h ✄ MpcTheory
Figure 8.
Hubble diagram derived from the voids.
Hubble diagrams derived from voids with three effective radii, without includingdistortions due to peculiar velocities: 6 h − Mpc (top-left), 8 h − Mpc (top-right), 14 h − Mpc (bottom-left). The bottom-right panelcorresponds to r eff = 8 h − Mpc and including distortions due to peculiar velocities. We show the actual expansion in the mock catalogs(black line) and the recovered average expansion from stacked void shapes (colored dashed lines) for the three N -body simulations. Thecolored error bars show the standard deviation inferred using our statistical model described in Eq. 18. In the bottom right panel, we useda 40,000km s − thick slice instead of 10,000km s − . recision cosmography with stacked voids 11behavior of the number of voids as a function of scale.We establish this empirical relation without fitting to anyvoid formation models, such as in the one in Sheth & vande Weygaert (2004). This relation only reflects how voidsare defined by our algorithm described in Section 3. ESTIMATING THE EXPANSION HISTORYUSING VOIDSWith what precision can we obtain e v ( z ), which isclosely related to H ( z ), by only considering the redshiftshape deformation of stacked voids? We use the algo-rithms and results of Section 3 and 4. In Section 5.1,we present and discuss the quality of the derived Hubblediagrams. In Section 5.2, we construct a Fisher-matrixanalysis of the constraints that can be obtained on DarkEnergy with this method, and compare it to expectedbaryonic acoustic oscillations constraints from the Dark-Energy Task Force (Albrecht et al. 2006).5.1. Results from simulations
We show in Figure 8 the complete “Hubble” diagramobtained from voids of radius 6 h − Mpc (top-left panel),8 h − Mpc (top-right panel) and 14 h − Mpc (bottom-leftpanel). These voids were obtained from the simulationspresented in Section 4 using the algorithm described inSection 3. For all these diagrams, we have not contam-inated the cosmological redshifts by peculiar velocities.Thus they simulate the measurements in a purely ex-panding universe. We show in the bottom right panelthe Hubble diagram for voids of 8 h − Mpc, when pecu-liar velocities are included in redshifts. We show the ac-tual measurements on the three N -body samples, along-side with the 68% error bar as derived from the Bayesianredshift shape inference. Each color corresponds to thesame simulation in all panels.We have used five snapshots of the simulation, cor-responding to time, expressed in redshift, z = 0, z =0 . z = 0 . z = 0 .
818 and z = 1. For each ofthese snapshots, we have applied cosmological expan-sion as explained in Section 4.2. For voids of 6 h − Mpcand 8 h − Mpc, we divide the volume in four slicesof 10,000 km s − according to the redshift direction.For voids of 14 h − Mpc, we keep the full volume of400 h − Mpc × h − Mpc × − .In black solid line, we plot the average local stretching¯ E ( z, ∆ z ), derived from Eq. (8),1¯ E ( z ) = (cid:28) δrδz (cid:29) = 1∆ z (cid:90) z +∆ zz d˜ zE (˜ z ) , (22)where z is the minimum redshift of the slice and ∆ z itsthickness. Assuming that voids are uniformly distributedin the volume, the stacked voids should be stretched by¯ E . The theoretical expectation and the measurement in N -body simulation are in agreement for the three voidsizes 6 h − Mpc, 8 h − Mpc and 14 h − Mpc, as clearlyshown by the comparison in Figure 8 (black solid line).We note that the standard deviation derived from theposterior are quite larger than the residual for the voidswith r eff = 6 h − Mpc. This is linked to our binningchoice and the need for a better model of variations ofthe density at scales smaller than 2 h − Mpc.Considering the Hubble diagram obtained when thedistortion due to peculiar velocities are included, we note E x p a n s i o n e v ( z ) Simulation R=8 h (cid:0) MpcTheory
Figure 9.
De-biased Hubble diagram – Same as Figure 8.We show the Hubble diagram derived from voids with r eff =8 h − Mpc, including peculiar velocities distortions and after cor-rection of the systematic bias. an important systematic effect in the bottom right panelof Figure 8. This distortion is a consequence of the pan-caking effect that has been discussed in Section 4.2. It isunfortunately non-trivial to model.The pancaking effect does not affect the overall stretch-ing but it transforms the structure of the density fieldoutside the void, which may bias our statistical estimatorof the shape. It is also expected to be weakly dependenton cosmology, essentially σ and Ω m which both affectsthe amplitude of peculiar velocities inside and outsideclusters. We have estimated using mock catalogs fromwhich finger-of-gods were removed that it correspondsto a ∼
5% effect on the stretching. The thickening of thewalls by the finger-of-god corresponds to the rest of theapparent flattening ( ∼ E ( z ) ( e v inobservations) should be multiplied by the constant de-biasing factor of 1 . ± .
04. We have checked that thisconstant is not strongly dependent on void radii. Wenote that this multiplicative factor is unimportant forthe actual determination of the equation of state of DarkEnergy. This statement is correct as long as the pancak-ing effect is not redshift dependent, which could alter theslope of the relative between redshift and the stretching.Consequently, though we adopt a factor of 1 . ± .
04 for2 G. Lavaux & B. D. Wandeltthis section, this factor should be left as a free parameterin any attempt to fit the observations of void ellipticities.We show the resulting diagram in Figure 9. We notethat the effect has now disappeared except at the lowestredshift where this leads to a slight overstretching. Thisoverstretching is due to a competing effect only presentat small redshift and for small voids.5.2.
Fisher-matrix analysis for Dark Energy
To assess the power of the Alcock-Paczy´nski test usingstacked voids, we derive the Fisher matrix for the Darkenergy properties that can be derived from the Hub-ble constant. We consider a Chevalier-Polarski-Linder(CPL) parametrization (Chevallier & Polarski 2001; Lin-der 2003) of Dark energy equation of state: w ( z ) = w + w a z z (23)For sufficiently low redshift the reduced Hubble constantis thus E ( z, w , w a ) = (cid:18) ω m h (1 + z ) + Ω Λ exp (cid:18) − (cid:90) z d z (cid:48) w ( z (cid:48) )1 + z (cid:48) (cid:19)(cid:19) / , (24)with w ( z ) as defined in Eq. (23), and ω m = Ω m h . Wedo not assume that the cosmology is flat.We consider a hypothetical measurement of thestretching constant e v ( z ) from the study of voids. Thuswe have an estimate of e v ( z ) at different redshifts z i thatwe label e v ,i . Each of the estimates has an independentrandom error variance V i . The likelihood L of the cos-mological parameters p is thus simply described by: L ( p ) = N z (cid:88) i =1 ( e v ,i − e v ( z i , p )) V i . (25)In p , we include a parameter, b v , which corresponds tothe overall flattening of the voids due to effects of redshiftspace distortions. We use the value and the uncertaintyas determined in Section 5. This prior is of no conse-quence on the Fisher-Matrix derived for the EUCLIDsurvey as it is sufficiently dense to constrain its value atlower redshift. The definition of the Fisher matrix is F k,l = (cid:28) ∂ L ∂p k ∂ L ∂p l (cid:29) , (26)with k, l ∈ p and the averaging is taken over all the pos-sible realizations of the data-sets { e v ,i } given the cosmo-logical parameters p and the noise determined by { V i } .In our case, this definition simplifies to: F k,l = N z (cid:88) i =1 V i ∂e v ∂p k ∂e v ∂p l . (27)To complete the evaluation it is required to have anestimate of V i = (cid:104) ( e v ,i − e v ( z i )) (cid:105) , the variance of theestimated Hubble constant in each redshift slice. Thisvariance may have the following typical dependence: V i = (cid:88) j ( (cid:15) ( R j , N ( R j , z ))) − , (28) with (cid:15) ( R, N ) the standard deviation of the estimator ifwe stack N voids with a size R , ¯ n ( R, z ) the number den-sity of voids of size R at redshift z . In Eq. (28), we aresumming over all the void effective radii that are observ-able in the slice i . If we bin void by size, they may formstatistically independent stacks. It is thus possible to im-prove the variance of the local expansion factor ˜ E ( z ) byconsidering all possible sizes at once. From the tests thatwe have run in Section 4, we know that binning voids inbins of ∆ = 1 h − Mpc in effective radii gives adequateresults. Thus, in Eq. (28), we use R j = R min obs. ( z ) + j ∆ , (29)with j < ( R max obs. − R min obs. ) / ∆. Of course, it is notpossible to observe void which have a size larger thanhalf the width of the slice i . Consequently, R max obs. = cδz H , (30)with δz the width of the slice, expressed in redshift units.We note that this is only an approximation.We have chosen the following scaling for the minimalobservable void size R min obs. used in Eq. (29) R min obs. ( z ) = min (cid:16) h − Mpc; s r n ( z i ) − / (cid:17) , (31)with n ( z ) the mean comoving density of galaxies at red-shift z . n ( z ) typically depends on φ ( M ), the luminosityfunction of galaxies in the observed band. For a magni-tude limited catalog, as the SDSS Main galaxy sample,assuming magnitudes are corrected for evolution n ( z ) isrelated to φ ( M ) as n ( z ) = (cid:90) m − ( d ( z )) −∞ φ ( M ) d M (32)Our choice of R min obs. ensures that the size of the struc-tures that are observed is limited by the number densityof tracers. The Fisher matrix, and of course the fig-ure of merit for the determination of Dark Energy, isgoing to depend on s r . We discuss in Section 5.3, theimpact of varying s r . We do not accept voids smallerthan 6 h − Mpc because they may be easily disruptedand strongly contaminated, in redshift coordinates, bydistortions due to peculiar velocities and are more diffi-cult to identify. We leave the exact determination of thelower limit of observable voids for future work.We approximate the number of voids in the slice ofthickness δz at redshift z by N = 4 π f sky (cid:18) cH (cid:19) (cid:0) χ ( z + δz ) − χ ( z ) (cid:1) ¯ n void ( R, z ) . (33)We determine (cid:15) , ¯ n from the three simulations that wehave run for a ΛCDM-WMAP7 cosmology. As this cos-mological model gives a good fit to CMB and large-scalestructures observations (Komatsu et al. 2011; Reid et al.2010), it should give a fair representation of the statisti-cal structure of the galaxy redshift survey.For the three void sizes that we have considered in thiswork, 6 h − Mpc, 8 h − Mpc, and 14 h − Mpc, we showin Figure 10 the function (cid:15) ( R, N ). We may approximaterecision cosmography with stacked voids 13
Survey Fraction Luminosity Limiting z max Numberof sky function magnitude of galaxiesSDSS-DR7 24% φ ∗ = 1 .
46 10 − h Mpc − r = 17 .
77 0 . . M ∗ = − . α = − . φ ∗ = 2 .
63 10 − h Mpc − r = 19 . .
45 10 M ∗ = − . α = 3 . r = 20 0 . . EUCLID 36% φ ∗ = 1 .
16 10 − h Mpc − H = 24 1 . ∼ . M ∗ = − . α = − . Table 1
Survey parameters this function by: (cid:15) ( R, N ) (cid:39) (cid:18) . N (cid:19) − . , (34)which is independent of the radius R . The reduction inthe variance is scaling roughly as a Poisson distribution.This corresponds to the expectation that the noise in theshape estimation comes essentially from the void-to-voidfluctuations. For future surveys we expect to observe ahuge number of small voids which would make our esti-mate sensitive to the low error cases, corresponding tothe red points Figure 10. Conservatively, we thus takethat the standard deviation is scaling exactly as (cid:15) ( N ) = 1 √ N . (35)This slightly overestimates the errors for the shape mea-surement and should give a conservative estimate of theexpected constraints.5.3.
Application to present and future surveys
In this section, we move on to apply the formalism ofSection 5.2 to three surveys: the SDSS-DR7 (Abazajianet al. 2009), the BOSS survey (Schlegel et al. 2007) andthe EUCLID survey (Laureijs et al. 2011). We use slicesof δz = 0 .
03, which corresponds approximately to a co-moving thickness of 90 h − Mpc. This limits the effectiveradius of the void to ∼ h − Mpc.Table 1 lists the parameters of those surveys which areuseful for the Fisher-Matrix analysis. These parameterscan be translated, using Eq. (32), in a number density oftracers at a given redshift. This determines the minimalsize of observable voids through Eq. (31). Using Eq. (35),we obtain the number density of voids and with the voidshape standard-deviation reduction (cid:15) ( R, N ).For deriving Fisher-matrix constraints, we assume thepriors that should come out from the
Planck mission.We use the Dark Energy Task Force (Albrecht et al. 2006)prescription for deriving these priors. In addition, weapply constraints from Stage II experiments as expressedin the DETF report. The figure of merit (FoM) is definedas FoM = 1 σ ( w a ) σ ( w p ) (36) Number of voids10 -4 -3 -2 -1 S h a p e s t a n d a r d d e v i a t i o n (cid:0) Mpc 8h (cid:0) Mpc 12h (cid:0) Mpc ResidualStatistical modelFit
Figure 10.
Standard deviation of void apparent shape vs thenumber of stacked voids.
Shape errors inferred by our statisticalmodel Eq. (18) (solid lines), the residual between the expectedshape and the actual best estimate (thick cross markers). We alsoshow with a dashed line the best fit of a power-law on the residual(Eq. 34). The results for the three N -body simulations are shownhere. The colors correspond to voids with an effective radii of6 h − Mpc (red), 8 h − Mpc (blue) and 12 h − Mpc (green). with w p = w + (1 − a p ) w a (37)and 1 − a p = − (cid:104) δw a δw (cid:105)(cid:104) ( δw a ) (cid:105) . (38) w p is the best estimate we can obtain on the equation ofstate of Dark Energy, evaluated at the scale factor a p ofour Universe.We consider either an experiment consisting of onlyBAO analysis, only Voids analysis or the two combined.We do this analysis for the SDSS main galaxy sample,the BOSS survey and the EUCLID survey. We give theresults in Table 2, Figure 11 and Figure 12.4 G. Lavaux & B. D. Wandelt −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4δw −1.0−0.50.00.51.0 δ w a BOSSSDSS Voids (Main+BOSS), s r =2Both, s r =2SDSS Voids (Main+BOSS), s r =5Both s r =5 Figure 11.
Fisher matrix forecasts for SDSS/BOSS survey.
Pre-dicted 95% confidence regions, assuming
Planck prior and the de-termination of the Hubble constant H = 72 ± − Mpc − .The Planck priors were obtained using the procedure in the re-port of the DETF. Please note that we plot w vs w a and not w p vs w a . The black solid line gives the constraints derived from theanalysis of baryonic acoustic oscillations in the BOSS survey. Thedotted lines gives the constraints derived from the shapes of stackedvoids using SDSS main galaxy sample and the BOSS sample. Thedashed line shows the obtained constraint by combining both Bary-onic Acoustic Oscillations and void shape measurement. The con-straints are either derived assuming s r = 2 (black) or s r = 5 (red). We note that voids and BAO have roughly the sameconstraining power when considering the SDSS/BOSSexperiment (FoM of 71 for BAO only vs. 68 for voidsonly). The combination of the two slightly improves theconstraints (FoM of 75). On the other hand, voids arefar superior at constraining the equation of state of DarkEnergy with the EUCLID survey. We have found a FoMof ∼ ∼ h − Mpc scale which corresponds to BAO.The results depend on several parameters that we haveadopted for the analysis. For example, we have chosen R max = 45 h − Mpc. We have barely found any changein the results by limiting to r eff (cid:39) h − Mpc or bychanging the thickness δz of the slice up to 0 .
1. How-ever, changing R min have a lot more impact on the FoM.This indicates that, as expected, most of the informationcomes from the smallest observable voids because of theirhuge abundance relatively to bigger ones. Similarly, wehave tried varying s r between 1 and 10 to check the sta-bility of the constraints for EUCLID. We have found thatthe Hubble diagram for voids always give a significantadditional information. As an example, in Figure 12, weshow in black (in red respectively) the constraints for s r = 2 ( s r = 5 respectively). The figure of merit may ei-ther strongly improve whenever s r is reduced, up to onehundred times improvement, or diminish by a factor ofa few when s r is increased. We also tried to check theinfluence of the sky coverage on the FoM. Interestingly,reducing the EUCLID survey to 100 deg yields a figureof merit of ∼
380 for voids, while the one we expect from −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4δw −1.0−0.50.00.51.0 δ w a BAO EUCLIDVoids EUCLID, s r =2Voids EUCLID, s r =5 Figure 12.
Fisher matrix forecasts for the EUCLID wide survey.
Same as Fig. 11 but for the EUCLID survey. The solid (dotted)line gives the constraints derived from the analysis of BaryonicAcoustic Oscillations (shapes of stacked voids, assuming s r = 2 inblack and s r = 5 in red).Method Data FoMBAO BOSS 71Voids ( s r = 2) SDSS+BOSS LRG 69Voids ( s r = 5) 68BAO+Voids ( s r = 2 or 5) SDSS+BOSS 75Voids ( s r = 2) EUCLID ∼ s r = 5) ∼ s r = 2) ∼ s r = 5) ∼ Note : The figure of merit is computed as the non-normalized1 / ( σ ( w p ) × σ ( w a )) as in the DETF report. We have included theprior from Stage II dark energy experiments, prior on H from theHubble space telescope and Planck prior. Table 2
Comparison of figure of merits (FoM)
BAO with BOSS is ∼
71. Voids could thus yield verygood constraints on Dark Energy from deep redshift sur-veys with smaller sky coverage.Finally, we note that in all this work we have onlyused the shape of the stacked voids as a probe. Addi-tionally, there is information in the distribution of voidorientations Jones & Fry (1998). The distribution of theintrinsic shapes of individual voids contains a great dealof complementary information, at the potential cost ofrequiring to assume a galaxy bias (Park & Lee 2007;Biswas et al. 2010; Lavaux & Wandelt 2010). We thinkthat the intrinsic shape could increase substantially theconstraints that only comes from void shape analysis.This could be done in two steps: the first pass would usethe method that we have developed in this paper to con-strain the local geometry, the second pass would use theshape distribution to constrain the growth of structures.recision cosmography with stacked voids 15 DISCUSSION AND CONCLUSIONWe showed that by identifying and stacking cosmicvoids in redshift shells and size bins, and then measur-ing their shapes in redshift space we can directly con-strain the cosmological expansion through a purely ge-ometric approach. Several steps are required to use thestacked voids technique to connect a spectroscopic sur-vey to the expansion geometry and hence to dark energyphenomenology. In this paper we proposed methods foreach one of these steps which, together, amount to a firstanalysis pipeline for the stacked voids technique.We use a modified
Zobov (Neyrinck 2008) algorithmfor finding and stacking voids on a light-cone, extendedto produce non-overlapping voids selected according totwo criteria: an effective radius within a given range anda central density sufficiently low to mark the region as avoid.In Section 4, we applied this algorithm to mock light-cone catalogs obtained from three N -body simulations.We have tested the method in the original comoving co-ordinates of the simulation, which has provided us with amodel of the density profile of the stacked void. Then, wehave simulated cosmological expansion and distortionsdue to peculiar velocities, which allowed us to qualita-tively estimate the impact on our measurement of thestretching of voids. We find that even a crude de-biasingprescription of the (mild) peculiar velocity systematicsyields a powerful method. By generating void stacks fordifferent void sizes and redshift shells, we project out thedetails of individual void shapes.We have developed and tested a Gaussian statisticalmodel able to estimate the stretching of the stackedvoids. Combining the results obtained from void stacksat all redshift shells results in an estimate of the expan-sion history.We are aware that each one of these analysis steps canlikely be improved significantly. There are parameters tooptimize, such as the widths of the redshift shells, andthe size bins. We only scratched the surface of possiblemethods in spatial statistics and computational geome-try when it comes to defining voids in realistic surveys.Our method for measuring the redshift space shapes ofstacked voids is only one of many that one could imag-ine. In particular, we have not yet taken advantage of theability to model the systematics due to peculiar veloci-ties, present at 10-15% of the expansion signal, except tosuggest a crude de-biasing approach which gives a con-sistent result on our simulations.Based on these first results we extrapolated and per-formed Fisher-matrix forecast of the constraints on DarkEnergy equation of state we expect from the SDSS, BOSSand EUCLID spectroscopic surveys.We have found that cosmic voids have the potential toprovide a far more powerful constraint on dark energythan measurements of the Baryonic Acoustic Oscillationscale, by an order of magnitude. This large increase ofinformation is easily understood in comparing the num-ber of modes probed by voids compared to BAOs, whichscales roughly as the third power of the ratio of the BAOscale to the scale of the smallest usable voids ∼ ∼
30. When projected intothe w a , w p plane using the Fisher matrix formalism for the EUCLID wide survey, we find the improvement overBAO on those parameters by a factor of ∼ REFERENCESAbazajian, K. N., et al. 2009, ApJS, 182, 543Albrecht, A., et al. 2006, ArXiv Astrophysics e-printsAlcock, C., & Paczynski, B. 1979, Nature, 281, 358Aragon-Calvo, M. A., van de Weygaert, R., Araya-Melo, P. A.,Platen, E., & Szalay, A. S. 2010, MNRAS, 404, L89Ballinger, W. E., Peacock, J. A., & Heavens, A. F. 1996,MNRAS, 282, 877Benson, A. J., Hoyle, F., Torres, F., & Vogeley, M. S. 2003,MNRAS, 340, 1606 G. Lavaux & B. D. Wandelt