Mon.Not.R.Astron.Soc.000,000–000 (0000) PrintedJanuary16,2015 (MNLATEXstylefilev2.2) The joint statistics of mildly non-linear cosmological densities and slopes in count-in-cells 5 1 Francis Bernardeau1,2, Sandrine Codis1 and Christophe Pichon1 0 2 1CNRS& UPMC, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France n 2 CNRS& CEA, UMR 3681, Institut de Physique Th´eorique, F-91191 Gif-sur-Yvette,France a J 5 January16,2015 1 ] ABSTRACT O In the context of count-in-cells statistics, the joint probability distribution of the C density in two concentric spherical shells is predicted from first first principle for . sigmasoftheorderofone.Theagreementwithsimulationisfoundtobeexcellent.This h statistics allowsus to deduce the conditionalone dimensionalprobabilitydistribution p - function of the slope within under dense (resp. overdense) regions, or of the density o forpositiveornegativeslopes.Theformerconditionaldistributionislikelytobemore r robustin constrainingthe cosmologicalparametersasthe underlying dynamics is less t s evolved in such regions. A fiducial dark energy experiment is implemented on such a counts derived from ΛCDM simulations. [ Key words: cosmology: theory — large-scale structure of Universe — methods: 1 numerical v 0 7 6 3 1 INTRODUCTION regime is to implement perturbation theory in a highly 0 symmetricconfiguration(sphericalorcylindricalsymmetry) 1. With the advent of large galaxy surveys (e.g. SDSS and in for which the full joint cumulant generating functions can 0 the coming years Euclid (Laureijs et al. 2011), LSST), as- be constructed. Such constructions take advantage of the 5 tronomers have ventured into the era of statistical cosmol- factthatnon-linearsolutionstothegravitationaldynamical 1 ogy and big data. Hence, there is a dire need for them to equations(theso-calledsphericalcollapsemodel)areknown v: build tools that can efficiently extract as much information exactly.Correspondingobservables,suchasgalaxycountsin i as possible from these huge data sets at high and low red- concentricspheresordiscs,thenyieldveryaccurateanalyt- X shift.Inparticular,thismeansbeingabletoprobethenon- icalpredictionsinthemildlynon-linearregime,wellbeyond r linear regime of structure formation. The most commonly whatisusuallyachievableusingotherestimators.Thecorre- a used tools to extract statistical information from the ob- sponding symmetry implies that the most likely dynamical servedgalaxy distribution areN-pointcorrelation functions evolution(amongstallpossiblemappingbetweentheinitial (e.g Scoccimarro et al. 1998) which quantify how galaxies andfinaldensityfield)isthatcorrespondingtothespherical are clustered. In our initially Gaussian Universethematter collapseforwhichwecanwriteanexplicitlineartononlinear density field is fully described by its two-point correlation mapping. This has been demonstrated in the limit of zero function. Howeverdeparturefrom Gaussianity occurs when variance using direct diagram resummations (Bernardeau thegrowthofstructurebecomesnon-linear(atlatertimesor 1992,1994)1 whichwas latershown tocorrespond toasad- smaller scales), providing information that is not captured dle approximation (Juszkiewicz et al. 1993; Valageas 2002; bythetwo-pointcorrelationfunctionbutisrecordedinpart Bernardeau et al.2014).Thekeypointonwhichthiswhole in the three-point correlation function. Obviously N-point paperisbasedupon,isthatthezerovariancelimitisshown correlation functions are increasingly difficult to measure toprovidearemarkablygood workingmodelfor finitevari- when N increases. They are noisy, subject to cosmic vari- ances (Bernardeau 1994; Bernardeau et al. 2014). anceandhighlysensitivetosystematicssuchasthecomplex This formalism also allows to weigh non-uniformly dif- geometry of surveys. It is thus essential to find alternative ferent regions of the universe making possible to take into estimatorstoextractinformationfromthenon-linearregime account the fact that the noise structure in surveys is not ofstructureformationinordertocomplementtheseclassical probes.This isin particularcritical ifwearetounderstand theorigin of dark energy,which accounts for 70 % of the ∼ 1 Theoriginalderivationswereactuallyderivedfromthehierar- energy budget of our Universe. chicalmodelthataimedatdescribingthefullynonlinearregime, One such method to accurately probe the non-linear (Balian&Schaeffer 1989) 2 F. Bernardeau, S. Codis and C. Pichon homogenous. For instance, low density regions are probed transform of ϕR1R2(λ1,λ2) in thequasi-linear regime. Such by fewer galaxies. Conversely, on dynamical grounds, we a Legendre transform is definedas also expect the level of non-linearity in the field to be in- homogenous: low density regions are less non-linear. Hence ΨR1R2(ρ1,ρ2)=λ1ρ1+λ2ρ2−ϕR1R2(λ1,λ2), (4) it is of interest to build statistical estimators which probe whereρ aredetermined implicitly by thestationary condi- i themildlynonlinearregimeandthatcanbetunedtoprobe tions subsets of the field, offering the best compromise between ∂ theseconstraints. Inthecontextofthecosmic densityfield, λi= ∂ρ ΨR1R2(ρ1,ρ2), i=1,2. (5) theconstruction ofconditionaldistributionsnaturallyleads i totheelaboration ofjointprobabilitydistributionfunctions The fundamental relation is then that, in the limit of zero (PDFhereafter) of the density in concentric cells. variance, this Legendre transforms taken at two different FollowingBernardeau et al.(2014)(hereafterBPC),we times, Ψ(ρ1,ρ2;η) and Ψ′(ρ1,ρ2;η′), takethe same value propose in Section 2 to extend one-point statistics of den- ′ ′ ′ sity profiles and to the full joint probability distribution ΨR1R2(ρ1,ρ2;η)=ΨR′1R′2(ρ1,ρ2;η), (6) function of the density in two concentric spheres of differ- provided that ρ R3 = ρ′R′3, and that ρ′ and ρ are linked ent radii. This is obtained using perturbation theory core i i i i i i together through the nonlinear dynamics of spherical col- results on the cumulant generating function, the double in- lapse. verse Laplace transform of which is then computed from Equation(6),whenappliedtoanarbitrarily early time brute force numerical integration. From that PDF, we will ′ also present the statistics of density profiles restricted to η, yields a relation between Ψ(ρ1,ρ2;η) and the statistical properties of the initial density fluctuations. In particular, underdense (resp. overdense) regions, and the statistics of density restricted to positive (resp. negative) slopes (Sec- for Gaussian initial conditions, Ψ(ρ1,ρ2;ηi) can easily be calculated and expressed in termsof elementsof covariance tion 3). Theoretical predictions will be shown to be in very matrices, good agreement with simulations in the mildly non-linear regime.Dependencewithredshiftwillalsobediscussed.Fi- 1 nallySection4presentsasimplefiducialdarkenergyexper- ΨR1R2(ρ1,ρ2;ηi)= 2iX,j62Ξij(ρi−1)(ρj−1), (7) iment, while 5 wraps up. where Ξ is the inverse of thematrix of covariances, Σ = ij ij τ τ , between the initial density contrasts in the two con- i j h i centricspheresofradiiR .Onecanthenwritethecumulant 2 THE 2-CELL DENSITY STATISTICS i generating function at any time through the spherical col- Forthesakeofclarity,letuspresentandbrieflycommentthe lapsemappingbetweenonefinaldensityattimeηinasphere fcoernmtearleisdmo.nWaegicvoennsildoecrattiwonosopfhsperaecseSxi0o.fOruadriguosaRliis(tio=de1r,iv2e) othferasdaimuseRpioianntdaonndewinitithiarlacdoinutsraRsi′ti=naRisρp1ih/e3re(scoenatserteodeonn- the joint PDF of the density in 1 and 2 denoted ρˆi and compass the same total mass); it can be written formally S S rescaled so that ρˆi =1. as h i 1 2.1 The cumulant generating function ρi =ζSC(η,τi)≈ (1 D+(η)τ/ν)ν , (8) − where, for the sake of simplicity, we use here a simple pre- In the cases we are interested in, the joint statistical prop- ertiesofρˆ1 andρˆ2 arefullyencodedintheirmomentgener- scription, with D+(η) the linear growth factor and ν = 21/13 to reproduce thehigh-zskewness. ating function MR1R2(λ1,λ2) = pX,∞q=0 hρˆp1ρˆq2iλpp1!λq!q2 , (1) TϕRh1eRR2se(tcλaat1il,slλti2tc)ha,aliltsyoonrnelyllyevaacΨcneRts1sRcibu2lm(eρuv1,liaaρn2et)qugiasetnieoeanras(itl4iyn)gtchofrmuonupcguthtioeadnn., inverse Legendre transform which brings its own complica- = hexp(λ1ρˆ1+λ2ρˆ2)i, (2) tions.Inparticularnotethatallvaluesofλi arenotaccessi- that can be related to the cumulant generating bleduetothefactthattheρi–λirelationcannotalwaysbe function, ϕR1R2(λ1,λ2), through MR1R2(λ1,λ2) = inverted.Thisissignaledbythefactthatthedeterminantof exp[ϕR1R2(λ1,λ2)], so that thetransformation vanishes, e.g. Det[∂ρi∂ρjΨ(ρ1,ρ2)]=0. Thisconditionismetbothforfinitevaluesofρ andλ .The i i exp[ϕR1R2(λ1,λ2)] correspondingcontourlines ofϕ(λ1,λ2)wasinvestigated in BPC and successfully compared to simulation. =Z dρˆ1dρˆ2PR1R2(ρˆ1,ρˆ2)exp(λ1ρˆ1+λ2ρˆ2), (3) wherePR1R2(ρˆ1,ρˆ2)isthejointPDFofhavingdensityρˆ1in 2.1.2 Motivation 1 andρˆ2 in 2.Wewillnowexploitatheoreticalconstruc- S S tion that permits the explicit calculation of PR1R2(ρˆ1,ρˆ2). It is beyond the scope of this letter to re-derive equa- tions (4)-(6) - a somewhat detailed presentation can be found in Valageas (2002) and in Bernardeau et al. (2014) 2.1.1 Upshot -butwecangiveahintofwhereitcomesfrom:itisalways Aswewillsketchinthefollowing, thistheoreticalconstruc- possibletoexpressanyensembleaverageintermsofthesta- tion yields the explicit time dependence of the Legendre tistical properties of the initial density field so that we can Statistics of densities and slopes in count-in-cells 3 formally write 2-cellPDF 6 exp[ϕ]=ZDτ1Dτ2P(τ1,τ2)exp λ1ρ1(τ1)+λ2ρ2(τ2) .(9) (cid:0) (cid:1) Asthepresent-timedensitiesρ canarisefrom differentini- i 4 tial contrasts, the above-written integration is therefore a pathintegral (overall thepossible pathsfrom initial condi- tions to present-time configuration) with measure τ1 τ2 D D 2 and known initial statistics (τ1,τ2). Let us assume here P that the initial PDF is Gaussian so that, e (τ1,τ2)dτ1dτ2 = √detΞexp[−Ψ(τ1,τ2)]dτ1dτ2, (10) slop 0 P 2π = s with Ψ then a quadratic form. In the regime where the variance of the density field is -2 small, equation (9) is dominated by the path correspond- ing to the most likely configurations. As the constraint is spherically symmetric, this most likely path should also re- -4 spect spherical symmetry.Itis therefore boundto obey the sphericalcollapsedynamics.Withinthisregimeequation(9) becomes -6 1 2 3 4 5 exp[ϕ] dτ1dτ2 (τ1,τ2)exp λ1ζSC(τ1)+λ2ζSC(τ2) (,11) ≃Z P (cid:0) (cid:1) Ρ=density where the most likely path, ρi = ζSC(η,τi) is the one-to- Figure 1. Joint PDF of the slope (s) and the density (ρ) as one spherical collapse mapping between one final density at time η and one initial density contrast as already de- givenbyequation(13)fortwoconcentricspheresofradiiR1=10 Mpc/handR2=11Mpc/hatredshiftz=0.97.Dashedcontours scribed. The integration on the r.h.s. of equation (11) can corresponds to Log P = 0,−1/2,−1,···−3 for the theory. The now be carried by using a steepest descent method, ap- correspondingmeasurements areshownasasolidline. proximating the integral as its most likely value, where λ1ρ1(τ1)+λ2ρ2(τ2) Ψ(τ1,τ2) is stationary. It eventually leads to the fundam−ental relation (6) when its right hand 0.10 side is computed at initial time (and the fact that (6) is valid foranytimesη andη′ isobtained when thesamerea- 0.08 soning is applied twice, for thetwo different times). compTuhteatpiounrspoofsethoefttwhois-cleeltltPerDFisdteoricvoendfrforonmt ntuhmeeerxipcarellsy- HLPs10000.06 stiioonns.of ϕ(λ1,λ2) with measurements in numerical simula- HL(cid:144)Ps0.04 g o L 0.02 320 2.2 The 2-cell PDF using inverse Laplace 480 transform 0.00 1000 Once the cumulant generating function is known in equa- -3 -2 -1 0 1 2 3 tion (3), the 2-cell PDF, (ρˆ1,ρˆ2), is obtained by a 2D in- s=slope P verse Laplace transform of ϕ(λ1,λ2) Figure 2. Dependence of the PDF of the slope on the number P=Z−ii∞∞d2λπ1i Z−ii∞∞d2λπ2i exp(−iX=1,2ρˆiλi+ϕ(λ1,λ2)), (12) PotofDptFhoeiinsrtcesosuumsltepduotfientdhteuhseniunnmgum1er0ei0cr0iac2laiplnoitnientgetrgsart(aidotainorknwbihnleun(e1)u2as)in.ndTghi3se2c0roe2mfe(prbaelnruecede) and4802 (lightblue)points. with ϕ given by equations (4)-(6). From this equation, it is straightforward to deduce the joint PDF, ˆ(ρˆ,sˆ), for the P density, ρˆ = ρˆ1 and the slope sˆ (ρˆ2 ρˆ1)R1/∆R, ∆R step∆λhasbeensetto0.15.Themaximumvalueofλ used ≡ − i being R2 R1, as here is 75 resulting into a discretisation of the integrand − i∞ i∞ on 10002 points. Fig. 1 compares the result of the numer- dλ dµ ˆ = exp( ρˆλ sˆµ+ϕ(λ,µ)), (13) ical integration of equation (12) to simulations. The corre- P Z−i∞ 2πiZ−i∞ 2πi − − spondingdarkmattersimulation (carried outwith Gadget2 with λ=λ1+λ2, µ=λ2∆R/R1. Following this definition, (Springel 2005)) is characterized by the following ΛCDM ϕ(λ,µ) is also the Legendre transform of Ψ(ρˆ1,sˆ = (ρˆ2 cosmology: Ωm = 0.265, ΩΛ = 0.735, n = 0.958, H0 = 70 ρˆ1)R1/∆R). − kms−1Mpc−1 and σ8 = 0.8, Ωb = 0.045 within one stan- · · Inordertonumericallycomputeequation(12),wesim- dard deviation of WMAP7 results (Komatsu et al. 2011). ply choose the imaginary path (λ1,λ2) = i(n1∆λ,n2∆λ) The box size is 500 Mpc/h sampled with 10243 particles, where n1 and n2 are (positive or negative) integers and the the softening length 24 kpc/h. Initial conditions are gener- 4 F. Bernardeau, S. Codis and C. Pichon Figure3.Densityprofilesinunderdense(solidlightblue),over- Figure 4.DensityPDFinnegativeslope(solidlightblue),pos- dense (dashed purple) and all regions (dashed blue) for cells of itiveslope(dashedpurple)andallregions(dashedblue)forcells radii R1 = 10 Mpc/h and R2 = 11 Mpc/h at redshift z =0.97. ofradiiR1=10Mpc/handR2=11Mpc/hatredshiftz=0.97. Predictions aresuccessfully compared to measurements insimu- Predictions are successfullycompared to measurements insimu- lations(pointswitherrorbars). lations(points witherrorbars). 3.2 Density in regions of given slope atedusingmpgrafic(Prunet et al.2008).AnOctreeisbuilt tocountefficiently allparticles within concentricspheresof Conversely,onecanstudythestatisticsofthedensitygiven radii between R = 10 and 11Mpc/h. The center of these constraints on the slope. For instance, the density PDF in spheres is sampled regularly on a grid of 10Mpc/h aside, regions of negative slope reads leading to 117649 estimates of the density per snapshot. 0 dsˆ ˆ(ρˆ,sˆ) NoteTthheatcotnhveecrgelelnscoevoefrloauprfnourmraedriiciallasrcgheermtheainsi1n0veMstpigca/the.d P(ρˆ|sˆ<0)= 0∞Rd−ρˆ∞−0∞PdsˆPˆ(ρˆ,sˆ). (16) by varying the number of points. Fig. 2 shows that the nu- R R Fig.4displaysthepredicteddensityPDFinregionsofpos- merical integration of theslope PDFhas reached 1% preci- itive or negative slope. As expected, the density is higher sionforthedisplayedrangeofslopes.Obviously,theintegra- in regions of negative slope. An excellent agreement with tion is very precise for low values of the slope and requires simulations is found. a largest numberof points for the large-slope tails. 3.3 Redshift evolution Fig. 5 displays the density profiles in underdense and over- 3 CONDITIONAL DISTRIBUTIONS dense regions as measured in the simulation for a range of redshifts.Thisfigureshowsthatthehighdensitysubsetfor 3.1 Slope in sub regions moderately negative slopes is particularly sensitive to red- Once the full 2-cell PDF is known, it is straightforward to shift evolution, which suggests that dark energy investiga- derive predictions for density profiles restricted to under- tions should focus on such range of slopes and regions. dense 1dρˆ ˆ(ρˆ,sˆ) P(sˆ|ρˆ<1)= ∞Rd0sˆ 1Pdρˆ ˆ(ρˆ,sˆ), (14) 4 FIDUCIAL DARK ENERGY EXPERIMENT −∞ 0 P R R Let us conduct the following fiducial experiment. Con- and overdense regions sider a set of 10,000 concentric spheres, and measure for each pair the slope and the density, ρˆ,sˆ . Recall (sˆρˆ>1)= 1∞dρˆPˆ(ρˆ,sˆ) . (15) that the cosmology is encoded in the para{mietrii}zation of P | ∞Rdsˆ 1dρˆ ˆ(ρˆ,sˆ) the spherical collapse on the one hand (ν), and on the R−∞ R0 P linear power-spectrum, Plin, (via the covariance matrix, k Fig.3displaysthesepredicteddensityprofilesinunderdense Σ = Plin(k)W(R k)W(R k)d3k/(2π)3 with W(k) = ij k i j andoverdenseregionscomparedtothemeasurementsinour 3(sin(k)R/k cos(k))/k2) on the other hand. For scale in- − simulation.Averygoodagreementisfoundwithsomeslight variant power spectra with power index n, given equa- departures in the large slope tail of the distribution. As tion (8), we have a three parameter (n,ν,σ) set of mod- expected, the underdense slope PDF peaks towards posi- els. For a parametrized PDF, P (ρˆ,sˆ) given by equa- n,ν,σ tiveslope,whiletheoverdensePDFpeakstowardsnegative tion (13), we can compute the log likelihood of the set as slope. The constrained negative tails are more sensitive to (n,ν,σ) = logP (ρˆ,sˆ). Fig. 6 displays the corre- L i n,ν,σ i i the underlying constraint, providing improved leverage for spondinglikePlihoodcontoursatone,threeandfivesigmasin measuring the underlyingcosmological parameters. thesimplecaseinwhichonlyoneparameter(σ here)varies. Statistics of densities and slopes in count-in-cells 5 allowustoprobedifferentiallytheslopeofthedensityinre- gionsofloworhighdensity.Itcanserveasastatisticalindi- cator to test gravity and dark energy models and/or probe key cosmological parameters in carefully chosen subsets of surveys.Thetheoryofcount-in-cellscouldbeappliedto2D cosmicshearmapssoastopredictthestatisticsofprojected density profiles. Velocity profiles and combined probes in- volvingthedensityandvelocityfieldsshouldalsobewithin reach of this formalism. Acknowledgements:Thisworkispartiallysupported by the grants ANR-12-BS05-0002 and ANR-13-BS05-0005 of the French Agence Nationale de la Recherche. The sim- ulations were run on the Horizon cluster. We acknowledge support from S.Rouberol for runningthecluster for us. Figure 5. Same as Fig. 3 for a range of redshifts as labeled. Onlytheunderdense(ρ<1)andtheoverdense(ρ>1)PDFsare References shown. Balian R., SchaefferR., 1989, Astr. & Astrophys. , 220, 1 Bernardeau F., 1992, Astrophys. J. Letter, 390, L61 logL 3690 Bernardeau F., 1994, Astr. & Astrophys. , 291, 697 Bernardeau F., Pichon C., Codis S., 2014, Phys. Rev. D , 1-Σ 90, 103519 3688 3-Σ Juszkiewicz R., Bouchet F. R., Colombi S., 1993, Astro- 5-Σ phys. J. Letter, 412, L9 3686 KomatsuE.,SmithK.M.,DunkleyJ.,BennettC.L.,Gold B.,HinshawG.,JarosikN.,LarsonD.,etal.2011,Astro- phys. J. Suppl. Ser. , 192, 18 3684 Laureijs R., Amiaux J., Arduini S., Augu`eres J. ., Brinch- mannJ.,ColeR.,CropperM.,DabinC.,DuvetL.,Ealet 3682 A.,et al. 2011, ArXiv e-prints PrunetS.,PichonC.,AubertD.,PogosyanD.,TeyssierR., 3680 Σ Gottloeber S.,2008, Astrophys. J. Suppl. Ser. , 178, 179 0.220 0.225 0.230 0.235 0.240 ScoccimarroR.,ColombiS.,FryJ.N.,FriemanJ.A.,Hivon Figure 6. Log-likelihood for a fiducial experiment involving E., Melott A.,1998, Astrophys. J. , 496, 586 10,000 concentric spheres of 10 and 11 Mpc/h measured in our Springel V., 2005, Mon. Not. R. Astr. Soc. , 364, 1105 simulation. The model here only depends on the variance σ (ν Valageas P., 2002, Astr. & Astrophys. , 382, 412 and n arefixed). The contours at 1, 3 and 5sigmas centered on thetruevalue0.23aredisplayedwithdarkbluedashedlines.The same experiment can be carried out when the three parameters vary. This experiment mimics the precision expected from a sur- veyofusefulvolumeofabout(350h−1Mpc)3 whichisfound to be at the percent level. This work improves the findings ofBPC which relieson alow-densityapproximation for the joint PDF. 5 CONCLUSION Extending the analysis of BPC, predictions for the joint PDF of the density within two concentric spheres was straightforwardly implemented for a given cosmology as a function of redshift in the mildly non-linear regime. The agreement with measurements in simulation was shown on Figs. 1, 3 and 4 to be very good, including in the quasi- linear regime where standard perturbation theory normally fails.Afiducialdarkenergyexperimentwasimplementedon countsderived from ΛCDM simulations. Suchstatisticswillproveusefulinupcomingsurveysasthey