The Local Hubble Flow: Is it a Manifestation of Dark Energy?

0.41 MB·English

by  Yehuda Hoffman

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
Gottl¨ober3
0
0 1Racah Institute of Physics, HebrewUniversity, Jerusalem 91904, Israel
2 2Grupo de Astrof´ısica, Universidad Aut´onoma 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 Macci`o 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 & B¨ohringer 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 Macci`o 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 ha-
los(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,aswaspointedbyMacci`o 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 pa-
rameter.
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. Macci`o 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 simula-
tion proves that the assumption is inapplicable to the LG
systems.
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6 Hoffman et al.
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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
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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.