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Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed2February2008 (MNLATEXstylefilev2.2) The Local Hubble Flow: Is it a Manifestation of Dark Energy? Yehuda Hoffman1, Luis A. Martinez-Vaquero2, Gustavo Yepes2 and Stefan 8 Gottlöber3 0 0 1Racah Institute of Physics, HebrewUniversity, Jerusalem 91904, Israel 2 2Grupo de Astrof´ısica, Universidad Autónoma de Madrid, Madrid E-28049, Spain 3Astrophysikalisches Institut Potsdam, Ander Sternwarte 16, 14482 Potsdam, Germany n a J 4 2 2February2008 ] h ABSTRACT p To study the local Hubble flow, we have run constrained dark matter (DM) simula- - o tions of the Local Group (LG) in the concordance ΛCDM and OCDM cosmologies, r with identical cosmological parameters apart from the Λ term. The simulations were st performed within a computational box of 64h−1Mpc centred on the LG. The initial a conditions were constrained by the observed peculiar velocities of galaxies and posi- [ tions of X-ray nearby clusters of galaxies. The simulations faithfully reproduce the nearby large scale structure, and in particular the Local Supercluster and the Virgo 2 v cluster.LG-likeobjectshavebeenselectedfromtheDMhalossoastocloselyresemble 9 the dynamical properties of the LG. Both the ΛCDM and OCDM simulations show 8 verysimilar localHubble flow aroundthe LG-like objects.It follows that, contraryto 9 recentstatements,thedarkenergy(DE)doesnotmanifestitselfinthelocaldynamics. 4 . Key words: galaxies: Local Group – cosmology: dark matter – methods: N-body 1 simulations 1 7 0 : v 1 INTRODUCTION by means of constrained simulations (CSs, Kravtsov et al. i X 2002, Klypin et al. 2003, Martinez-Vaqueroet al. 2007 and r It has been recently stated that the cosmological con- Hoffman et al. 2007).The uniquefeature of theCSs isthat a stant (Λ) or its generalisation the dark energy (DE), theirinitial conditions aregenerated as constrained realiza- manifests itself in the dynamics of the local universe tions of Gaussian random fields (Hoffman & Ribak 1991). (Baryshev et al. 2001, Chernin et al. 2004, Teerikorpi et al. Theinitialconditionsareconstrainedbyobservationaldata 2005, Chernin et al. 2006, Chernin et al. 2007b, and and hence they are designed to reproduce the main gross Chernin et al. 2007c). In these papers, the coldness of the featuresofthelocal largescale structure.Assuchtheypro- local Hubble flow around the Local Group (LG) has been vide theoptimal tool for studying the dynamics of the LG, attributed to the existence DE. This has been supported being a given individual butnot an atypical object. In par- by Macciò et al. (2005) who analysed a set of N-body sim- ticular the recent constrained flat Λ dominated ( ΛCDM) ulations and concluded that indeed ...[their] results provide andopen (OCDM) cold dark matterN-bodysimulations of new,independentevidenceforthepresenceofdarkenergyon Martinez-Vaqueroet al. (2007) were designed to study the scales of a few megaparsecs. These results, if correct, would local dynamics in cold dark matter cosmologies with and haveprovidedanindependentcorroborationtotheDEcom- withoutaDEcomponent.Thesesimulations aretobeused ponent whose existence is otherwise inferred from observa- hereasalaboratoryforthestingthehypothesisthatthecold tions of distant objects and the early Universe. These au- local Hubbleflow is a signature of dark energy. We are less thors used the term ’local’ as describing the neighborhood interestedhereintheactualcoldnessoftheflowandmorein of the LG out to a distance of a few Mpc. thepossibilitythattheDEaffectsthelocalflow.Athorough Much of the dynamical implications of the cosmolog- analysisoftheissueofthecoldnessofthelocalflowistobe ical constant for the large scale structure were worked givenelsewhere(Martinez-Vaqueroetal,inpreparation).In out by Lahav et al. (1991). The LG constitutes a quasi- what follows ’local’ is defined as the region contained in a linear object and therefore its dynamics cannot be mod- sphereof radius R=3 Mpc centred on the LG. eled by the linear theory or the spherical top-hat model. The structure of the paper is as follows. A very brief Consequently, we have recently studied the local universe review of the simulations of Martinez-Vaqueroet al. (2007) 2 Hoffman et al. is presented in Section 2. The selection criteria for LG can- as constraints as if they were linear quantities (Zaroubi, didatesaresummarised inSection3.Theflowfieldsaround Hoffman&Dekel1999). Thisfollows theCSsperformed by thesimulatedLGcandidatesintheΛCDMandOCDMsim- Kravtsov et al. (2002) and Klypin et al. (2003). The other ulationsarepresentedinSection4.InSection5wecompare constraints are obtained from the catalog of nearby X-ray thegravitational field aroundtheLGcandidates.Ageneral selected clusters of galaxies (Reiprich & Böhringer 2002). discussion concludes thepaper (Section 6). Given the virial parameters of a cluster and assuming the sphericaltop-hatmodelonecanderivethelinearoverdensity ofthecluster.Thelargescalestructure,i.e.scalessomewhat larger than 5h−1Mpc, of the resulting density and veloc- 2 CONSTRAINED SIMULATIONS OF THE ity fields are strongly constrained by the imposed data. In LOCAL UNIVERSE particular all the resulting CSs are dominated by a Local OurCSshavealreadybeenusedinMartinez-Vaquero et al. Supercluster(LSC) - like object with a Virgo size DMhalo (2007)andtheyarebrieflysummarisedhere.Thesearedark at its center. The LG is not directly imposed on the initial matter (DM) only simulations employing a periodic cubic conditions, but having reconstructed the actual large scale computational box of 64h−1Mpc on a side using 2563 par- structure of the local universe a LG-like structure is very ticles. Both models use the dimensionless Hubble constant likely to emerge in the right place with dynamical proper- of h=0.7 (whereh=H0/100 km/s/Mpc), thepowerspec- ties similar to the actual ones. The two simulations used trum normalisation σ8 = 0.9 and the cosmological matter herearebasedonthesamerandomrealization oftheinitial density of Ωm = 0.3. The ΛCDM model corresponds to conditions. a flat universe with ΩΛ = 1− Ωm while for the OCDM model ΩΛ = 0. These cosmological parameters correspond to the so-called Concordance Model. We used the parallel 3 SELECTION OF LG-LIKE CANDIDATES TREEPM N-bodycode GADGET2 (Springel,2005) torun these simulations. For the PM part of the algorithm, we The selection of LG candidates is described in detail in used a uniform grid of 5123 mesh points to estimate the Martinez-Vaqueroet al.(2007).Theselectionoftheobjects long-range gravitational force bymeans of FFT techniques. is based on the Macciò et al. (2005) criteria, which consist Thegravitationalsmoothingusedtocomputetheshort-scale of: gravitational forces correspond to an equivalent Plummer i. The group contains two MW and M31 like DM halos smoothing parameter of ǫ=15h−1 kpccomoving. Thenumberofparticlesusedinthesesimulations(2563) with maximum circular velocity in the range of 1256Vc 6 270 km/s. provides a very mild mass resolution (1.3×109h−1M⊙ per ii. The two major DM halos are separated by no more particle) which corresponds to a minimal mass of the DM than 1h−1Mpc. halosof≈2.5×1010h−1M⊙,forobjectsresolvedwithmore iii. The relative radial velocity of the two main halos is than20darkmatterparticles.Atsucharesolutiontheinner negative. structureofthemainhalosoftheLG-likeobjectscannotbe iv. There are no objects with maximum circular velocity resolved, nor can the observed mass distribution of the LG higher than MW and M31 candidates within a distance of nearby dwarfs be reconstructed. Yet, the dynamics on the 3h−1Mpc. scale of a very few Mpc is very well resolved. v. Thegroupresideswithinadistanceof5to12h−1Mpc Wesetupinitialconditionsforthesesimulationsinsuch from a Virgo likehalo of 5006Vc 61500 km/s. a way that we can zoom in to any particular object with much more resolution. Thus, we generate the random re- DMhalosarefoundusingboththeBoundDensityMax- alizations of the density fluctuation field for a much larger ima algorithm (Klypin et al., 1999) and the AMIGA Halo number of particles (up to 40963). Then, we substitute the Finder (Gill et al., 2004). In the ΛCDM simulation 26 LG- fourier modes corresponding to the small wavenumber by likeobjectshavebeenfoundand43intheOCDMone.Given those coming from the constrained 2563 density field and thefactthatbothsimulationarebasedonthesamerealiza- make the displacement fields according to the Zeldovich tion of the random Gaussian field we have identified 9 LG- approximation. Thus, we can now resimulate any particu- like objects that appear in both simulations at about the lar zone of the simulated volume with particles of variable sameposition andareverysimilardynamically.Wereferto masses, down to 4096 times smaller than the particle mass these as the ’same’ objects appearing in both simulations. of the simulations used in this work. A comparison of the These ’same’ objects do not form any class by themselves results of ΛCDM 2563 simulation with that from the LG- and are statistically indistinguishable from the other LG- like systems resimulated at 40963 resolution does not yield like objects. These objects are used here to exemplify the any significant differences in their Hubble diagrams (to be effectoftheΛterm onthedynamicsoftheLG,astheyare published). the same object evolving in two identical cosmologies and The algorithm of constrained realizations of Gaussian environmentsthat differ only by their Λ term. randomfields(Hoffman & Ribak,1991)hasbeenusedtoset The fact that there is no one-to-one coincidence of the uptheinitialconditions.Thedatausedtoconstraintheini- LG-like objects of the two simulations should not be sur- tial conditions ofthesimulations is madeof twokinds.The prising. There are two reasons for that. First, the two cos- first data set is made of radial velocities of galaxies drawn mologies arenotidenticalandtheydifferinthelineargrav- fromtheMARKIII(Willick et al.1997),SBF(Tonry et al. itationalgrowthfunction.Second,theLGisasysteminthe 2001) and theKarachentsev (2005) catalogues. Peculiar ve- quasi-linearregimeandisfarfrombeingindynamicalequi- locities are less affected by non-linear effects and are used librium. Had we observed it at a slightly different time it The Local Hubble Flow: Is it a Manifestation of Dark Energy? 3 might not be qualified as a LG-like object according to the r(Mpc) Obs. ΛCDM OCDM selectioncriteriaassumedhere.Thisiscertainlythecasefor our simulated objects. Just a small miss match in the dy- [0.75−2] 65 63 62 namical phase of the objects between the two simulations [0.75−3] 68 72 75 can rule out an object in one or the other simulation from being a LG-like systems. Table1. ThevalueofσH (inunitsofkm/s)oftheLG,compiled Inthepresentpaperweareinterestedincomparingthe fromtheKarachentsev data,andoftheΛCDMandOCDMLG- localHubbleflowaroundLG-likeobjects.Providingthatthe like objects combined together, in the manner of Fig. 3. Two selected systems fulfil all requirements their exact location distancecutsareusedforcalculatingσH. isnotimportantforthepurposeoftheanalysis.. Therefore, wehaveusedalltheobjectsfoundwithinthecomputational InspectionoftheΛCDMHubblediagram(Fig.1)shows box, regardless of their position with respect to the LSC. that indeed the prediction of Chernin et al. (2007b) is con- Some simulated LG-like objects reside close to the actual firmed: only two LG-like groups have, within the range position of the LG but they seem to be dynamically indis- (0.7−3)Mpc,averyfewhaloseachwithapeculiarvelocity tinguishable from theothers. smaller than the ΛCDM escape velocity. However, this be- haviourisreproducedbytheOCDMLG-likeobjectsequally well (Fig. 2). 4 THE LOCAL HUBBLE FLOW ToincreasethestatisticalsignificanceoftheHubbledi- agram analysis we have considered all the LG-like objects The local Hubble flow around LG-like objects is probed by in the ΛCDM and OCDM simulations. This is performed meansofHubblediagrams showingtheradial velocitiesrel- byplottingtheradialvelocitiesofallhalosneartheLG-like ative to the objects center of mass within a distance of 3 groupsagainst theirdistancer.Again, theHubblediagram Mpc. In Figs. 1 and 2 we present the Hubble diagrams of of all LG-like objects in both cosmologies looks very simi- 4 randomly chosen candidates out of the 9 LG-like objects lar.Infact,thefractionofhalosbelowtheescapevelocityis which appear in both simulations (Fig. 1: ΛCDM, Fig. 2: somewhatsmallerintheOCDMobjectsthanintheΛCDM OCDM). The figures present all the DM halos around the ones. We conclude that the ΛCDM escape velocity predic- chosenLG-likeobjectsouttoadistanceof3Mpc.Acareful tion is reproduced by theOCDM simulation. comparisonoftheplotsrevealsthattheHubblediagramsof The paper focuses mainly on the possible role of the a given ΛCDM and OCDM simulated LG are very similar. DE in the dynamics of the LG. A thorough analysis of the In particular the r.m.s. value of the scatter around a pure coldnessofthelocal flowwillbegivenelsewhere(Martinez- Hubbleflow(σH,assumingthetruevalueoftheHubblecon- Vaquero et al, in preparation). Here a very brief summary stant of the simulation) does not vary statistically between ofthesubjectisgiven.Theverylocal Hubbleflowhasbeen the ΛCDM and OCDM cases. For the 4 objects shown in recentlystudiedbyKarachentsev et al.(2007)andtheircur- Figs 1 and 2, we find σH =35, 38, 42 and 53 km/s for the rently updated catalog of local peculiar velocities has been ΛCDM objects and 41, 42, 55 and 59 km/s in the OCDM analyzed here (I. Karachentsev, private communication). case. The values of σH for the other 5 common candidates are31, 51, 11, 41and79km/sfortheobjectsintheΛCDM Table I presents the value of σH taken over all the DM halos(simulations)orgalaxies(data)intherangeof[0.75−2] simulation and 61, 38, 18, 63 and 48 km/s in the OCDM Mpc and [0.75−3] Mpc of the Karachentsev’s data and of one. theΛCDMandtheOCDMLG-likeobjects.Thecumulative Much of the theoretical expectations for the possible manifestation of the DE in the local flow is based on the distributionofσH (calculatedovertherange[0.75−3]Mpc) is presented in Fig. 4. The plot shows that more than half model proposed by Chernin et al. (2007c, and references therein).Themodelessentially assumesthatthelocalgrav- the LG-like objects in both models have a σH 6 60 km/s. So, many objects have a flow as cold, or colder, as the ac- ity field around the LG can be decomposed into the con- tualLG.Yet,aswaspointedbyMacciò et al.(2005)thereal tribution of the LG, modeled as a point particle, and the contribution of theDE: problem of the coldness lies with the relation between σH and themean density around theobjects. gPP(r)=−GMr2LG +ΩΛH20r (1) It follows that the local Hubble flow around ΛCDM and OCDM LG-like objects is essentially indistinguish- The zero gravity surface is defined by gPP(RV) = 0. The able. This stands in clear contradiction with previous radius of the zero gravity surface, RV, plays a critical role claimsofBaryshev et al.(2001),Chernin et al.(2007c)and in that simple model. A central prediction of the model is Chernin et al. (2007a). Also, both the ΛCDM and OCDM that thelocal Hubbleflow should not contain galaxies with LG-likegroupsobeyequallywelltheescapevelocitypredic- radial velocities smaller than the escape velocity (see the tion of the flat -Λ cosmology, as if they are not affected by Appendix),calculated undertheassumption that the grav- theΛ term. itational field is given by the point particle approximation (Chernin et al.2007b).Thispredictionexcludesgalaxiesre- siding within theLGitself, namely within 0.7 Mpc. Totest 5 THE LOCAL GRAVITATIONAL FIELD thepredictiontheradialescapevelocityprofiles(Eq.6)have been plotted in both Figs. 1 and 2 as solid (ΛCDM model) To understand thepossible reason for the discrepancies be- and dashed (OCDM model) lines. From here on the term tweenthepresentresultsandtheandthemodelpredictions ’escape velocity’ refers to the one calculated under the as- we have studied the nature of the local gravitational field. sumption of thepoint particle approximation. The prime motivation here is to check the validity of the 4 Hoffman et al. Chernin et al. (2007c) model of the gravitational field (Eq. 6 DISCUSSION 1),whichcorrespondstothefullgravitationalfieldexpressed ThemoststrikingresultofthispaperisthatthelocalHub- in physical, and not co-moving, coordinates. The relation ble flow around LG-like objects in the OCDM model is al- betweenthepeculiargravity(outputofGADGET)andthe most indistinguishable from theΛCDM flow. To theextent physicalonewasderivedbyMartinez-Vaquero et al.(2007), that the models do differ it is the OCDM model that has but it is repeated herefor thesake of completeness. somewhat colder Hubble flow than the ΛCDM one. It fol- The physical r and comoving x coordinates are related lows that the local flow is not affected by the DE and does by: r = ax. The gravitational field equals the physical ac- not manifest thepresent epoch dominance of theDE. celeration of an object ¨r=g. The GADGET code provides Oneshouldnotbesurprisedbythedepartureofthesim- an acceleration-like term definedas: ulated local Hubble flow from the prediction of the simple fp = 1 d (a·vp)=v˙p+H·vp, (2) model proposed by Chernin et al. (2007c). First, theactual adt gravitational field deviates considerably from the predicted where vp = r˙ −Hr is the peculiar velocity, H is Hubble’s one.Second,thegravitationaldynamicsisnotlocalandthe tidalfieldplaysanimportantroleinthequasi-linearregime constant and a is the expansion scale factor. It follows that (Hoffman 1986, Hoffman 1989, Zaroubi & Hoffman 1993, ¨r=fp+raa¨ =fp+„−21ΩM +ΩΛ«H2r. (3) vfoalnlowdestWhaeytgtaheerdty&naBmaibcuslde1p99en4d,sDneoltPoonplyoloonetthael.lo2c0a0l1fi)e.ldIt, butisalsoaffectedbytheshear,namelythetidalfield.The Namely, the linear term corresponds to the unperturbed shear breaks the simple linear relation of the density and Friedman solution and fp to thefluctuating component1. velocity fields of the linear regime and therefore the local The gravitational field is taken with respect to the LG density field cannot account for thelocal Hubbleflow. reference frame. So that, one finally obtain: The comparison of theΛCDM and OCDM simulations showsthattheyyieldverysimilarLG-likeobjectswithvirtu- g=(fp−fpLG)rr +„−12ΩM +ΩΛ«H2r (4) ally identicallocal Hubbleflows.Itfollows thatthedynam- icalpropertiesofLG-likeobjectsandtheirenvironments,in where r is the distance from the center of mass of the LG. thelinearandquasi-linearregime,dependmostlyonthecold This is the field acting on each dark matter particle. The mattercontentoftheuniverse,namelyΩm,andonlyweakly total acceleration of halos was computed by averaging this on the DE. This is another manifestation of the fact that quantityoverallparticlesbelongingtoeachhalo.Theaccel- the properties of the cosmic web, expressed in co-moving eration isscaled byH02×1 Mpc.Insuchscaling theunper- coordinates, depend mostly on Ωm and hardly on the DE turbedgravitational acceleration of ashell of radius 1 Mpc (Hoffman et al. 2007). equalsto−q0,whereq0 isthecosmological deceleration parameter. In Figs. 5 and 6 we compare the radial component of ACKNOWLEDGEMENTS theexact (i.e.in thesense of thesimulations) gravitational Fruitful discussions with A. Chernin, A. Macciò and fieldwiththeChernin et al.(2007c)model(gPP),astraced A. Tikhonov are gratefully acknowledged. We thank I. bytheDMhalosaroundtheLG-likeobjects. Onlythefluc- Karachentsev for providing us his updated catalog of pe- tuatingcomponentofthegravitational fieldisshown inthe culiar velocities of galaxies in the Local Volume. We thank figures,namelyforthenumericallyexactfielditistheradial DEISA consortium for granting us computing time in the icsomfpp,PoPnen=t−ofGfpMaLnGd/fro2r+thΩeMpoHin2tr/p2a.rtWicleesahpopwroaxlilmDaMtionhait- MALaTreINXossutrpuemrcosmuppeurtceormaptuLteRrZat(GBeSrCma(Snyp)aitnh)roaungdhththeeSEGxI-- los in the distance range of 0.7 6 r 6 3.0 Mpc from the tremeComputingProject(DECI)SIMU-LU.Wealsothank same LG-like objects that were presented in Figs. 1 and 2. NIC Ju¨lich (Germany) for the access to the IBM-Regatta SinceDM halos closer that r=0.7 Mpc are affected by the p690+ JUMP supercomputer and CesViMa (Spain) for ac- two-body dynamics of the LG, they are excluded here. Fig. cess to the Magerit IBM-BladeServer supercomputer. GY 7 shows the scatter plot of the gravitation field of all the would like to thank also MEC (Spain) for financial sup- LG-likeobjects of theΛCDM andOCDMsimulations. The port underproject numbersFPA2006-01105 andAYA2006- plots show that the numerically exact value of the radial 15492-C03.LAMVacknowledgesfinancialsupportfromCo- component of fp and the point particle prediction (fp,PP) munidaddeMadridthroughaPhDfellowship.Thesupport areverypoorlycorrelated.Alinearregressionanalysisfinds of theISF-143/02 and theSheinborn Foundation (YH)and a correlation coefficient of 0.40 (0.23) and a slope of 0.35 theEuropean Science Foundation through the ASTROSIM (0.16) for the ΛCDM (OCDM) model. It is clear that the ExchangeVisits Programme (SG) is also acknowledged. point particle model fails to reproduce the actual gravity field, and therefore it cannot account for the dynamics of theLG. APPENDIX Assuming the local gravitational field around the LG is in- deedgiven bythepoint particleapproximation (Eq.1),the 1 Note the typo in Eq. A3 of Martinez-Vaqueroetal. (2007) effectiveNewtonian potential is given by ownhleyrethΩeΛfluwcatsuaotminitgtetde.rmHoowfegvewr,asthciosndsiiddenroedtatffheecrte.theresult,as φ(r)=−GMLG − ΩΛH20r2. (5) r 2 The Local Hubble Flow: Is it a Manifestation of Dark Energy? 5 The effective Newtonian energy (per unit mass) is simply Teerikorpi, P., Chernin, A. D., & Baryshev, Y. V. 2005, given as ǫ=v2/2+φ(r).In the presence of the Λ term the A&A,440, 791 potential reaches a maximum at RV at which the potential Tonry, J. L., Dressler, A., Blakeslee, J. P., Ajhar, E. A., peaks at ǫV ≡φ(RV). It follows that a particle is unbound Fletcher,A.B.,Luppino,G.A.,Metzger,M.R.,&Moore, totheLGifitsenergyislargerthanǫV.Theescapevelocity C. B. 2001, ApJ,546, 681 is therefore given by van deWeygaert, R. & Babul, A.1994, ApJ,425, L59 Willick, J. A., Courteau, S., Faber, S. M., Burstein, D., v2e2sc =GMLG“1r − R1V ”+ ΩΛ2H20“r2−R2V”. (6) ZDareokuebl,i,AS..,&&HStorffamusasn,,MY..A1.919939,7A,pAJp,J4S1,4,10290, 333 In the case of a vanishing cosmological constant the classi- calexpression ofEq.5and6isrecovereduponsubstituting RV =∞ and ΩΛ =0. It should be reemphasised here that the present derivation is done under the assumption of the point particle approximation. The analysis of the simulation proves that the assumption is inapplicable to the LG systems. REFERENCES Baryshev, Y. V., Chernin, A. D., & Teerikorpi, P. 2001, A&A,378, 729 Chernin, A. D., Karachentsev, D. I., Kashibadze, I. D., Makarov, O. G., Teerikorpi, P., Valtonen, M. J., Dol- gachev, V. P., & Domozhilova, L. M. 2007a, ArXiv e- prints,0706.4171 Chernin, A.D., Karachentsev, I. D., Teerikorpi, P., Valto- nen,M.J.,Byrd,G.G.,Efremov,Y.N.,Dolgachev,V.P., Domozhilova, L. M., Makarov, D. I., & Baryshev, Y. 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V., Governato, F., & Horellou, C. 2005, MN- RAS,359, 941 Martinez-Vaquero, L. A., Yepes, G., & Hoffman, Y. 2007, MNRAS,378, 1601 Reiprich, T. H. & Böhringer, H. 2002, ApJ, 567, 716 Springel, V.2005, MNRAS,364, 1105 6 Hoffman et al. 600 600 400 400 200 200 0 0 -200 -200 0 1 2 3 0 1 2 3 r (Mpc) r (Mpc) 600 600 400 400 200 200 0 0 -200 -200 0 1 2 3 0 1 2 3 r (Mpc) r (Mpc) Figure 1. TheHubblediagramsaroundfourLG-likeobjects intheΛCDMsimulation.Thescatter plotsrepresenttheradialpeculiar velocity of the DM halos vs. the distance from the MW and M31-like DM halos. The solid curve corresponds to the escape velocity profile of the ΛCDM model, calculated under the assumption of the point particle approximation. For reference the escape velocity of thecorrespondingOCDMmodelisgivenaswell(dashedline),namelyitiscalculatedasiftheΛtermismissing.ThevalueofσH,taken overtherange0.756r63MpcandRV ofeachobjectisgiven. The Local Hubble Flow: Is it a Manifestation of Dark Energy? 7 600 600 400 400 200 200 0 0 -200 -200 0 1 2 3 0 1 2 3 r (Mpc) r (Mpc) 600 600 400 400 200 200 0 0 -200 -200 0 1 2 3 0 1 2 3 r (Mpc) r (Mpc) Figure 2. The Hubble diagrams around four LG-like objects in the OCDM simulation. The four LG-like objects shown here are the OCDMcounterparts oftheΛCDMonesshowninFig.1.Thestructureandnotations oftheplotsarethesameasinFig.1. 600 600 400 400 200 200 0 0 -200 -200 0 1 2 3 0 1 2 3 r (Mpc) r (Mpc) Figure 3. Combined Hubble diagram of 26 LG candidates in the ΛCDM (left panel) and 43 candidates in the OCDM (right panel) models. The radial distance r is scaled by the value of RV of each object. The values of σH within 2 and 3 Mpcare given inTable 1. The escape velocity curves are plotted inthe same manner as inFigs. 1 and 2. <RV > is the mean RV of all the LG-likeobjects for eachsimulation. 8 Hoffman et al. Figure4. ThefractionalcumulativeσH functionofLG-likeobjects,namelythefractionofobjectswithσH lowerthanacertainvalue. Fulllinecorrespondstothe26ΛCDMobjectsandthedashedonetothe43OCDMobjects.ThedispersionaroundapureHubbleflow, σH,iscalculatedovertherange0.756r63Mpc. The Local Hubble Flow: Is it a Manifestation of Dark Energy? 9 4 4 2 2 0 0 -2 -2 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 4 4 2 2 0 0 -2 -2 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 Figure 5. TheΛCDMgravitational field:Ascatter plotoftheexact gravitational fieldg (Eq. 4)vs. theapproximatedonegPP (Eq. 1)experiencedbyDMhalosaroundLG-likeobjects.ThefourobjectsandpanelscorrespondtotheonesinFig.1.OnlyDMhalosinthe rangeof0.756r63.0Mpcareplottedhere.ThegravitationalfieldisscaledbyH02×1 Mpc. 10 Hoffman et al. 4 4 2 2 0 0 -2 -2 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 4 4 2 2 0 0 -2 -2 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 Figure 6. SameasFig.5butfortheOCDMsimulation.TheLG-likeobjectsandpanelscorrespondtotheones inFig.2. Figure7. Scatterplotofthefluctuatingcomponentofthegravitationalfieldpredictedbythepointparticlemodel(fp,PP)againstthe exactvaluecalculatedbythesimulations.TheleftpanelshowsthedistributionoftheDMhalosinthevicinity(0.756r63.0Mpc)of the26LG-likeΛCDMobjectsandtherightpanelexhibitsthe43OCDMobjects. ThegravitationalfieldisscaledbyH02×1Mpc.