A clear and measurable signature of modified gravity in the galaxy velocity field

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by  Wojciech A. Hellwing

A clear and measurable signature of modified gravity in the galaxy velocity field
Wojciech A. Hellwing∗
Institute for Computational Cosmology, Department of Physics,
Durham University, South Road, Durham DH1 3LE, UK and
Interdisciplinary Centre for Mathematical and Computational Modelling (ICM),
University of Warsaw, ul. Pawin´skiego 5a, Warsaw, Poland
Alexandre Barreira
Institute for Computational Cosmology, Department of Physics,
Durham University, South Road, Durham DH1 3LE, UK and
4 Institute for Particle Physics Phenomenology,
1 Department of Physics, Durham University,
0 Durham DH1 3LE, U.K.
2

n Carlos S. Frenk, Baojiu Li, and Shaun Cole
u Institute for Computational Cosmology, Department of Physics,
J Durham University, South Road, Durham DH1 3LE, UK

0
1 Thevelocity fieldof darkmatterandgalaxies reflectsthecontinuedaction of gravitythroughout
cosmichistory. Weshowthatthelow-ordermomentsofthepairwisevelocitydistribution,v12,area

] powerfuldiagnosticofthelawsofgravityoncosmologicalscales. Inparticular,theprojectedline-of-
O
sightgalaxypairwisevelocitydispersion,σ12(r),isverysensitivetothepresenceofmodifiedgravity.
C Using a set of high-resolution N-body simulations we compute the pairwise velocity distribution
and its projected line-of-sight dispersion for a class of modified gravity theories: the chameleon
.
h f(R) gravity and Galileon gravity (cubic and quartic). The velocities of dark matter halos with a
p wide range of masses would exhibit deviations from General Relativity at the (5−10)σ level. We
-
examinestrategies fordetectingthesedeviationsingalaxyredshiftandpeculiarvelocitysurveys. If
o
detected,this signature would be a “smoking gun”for modified gravity.
r
t
s
a
[ Introduction. Measurements of temperature these observational probes is that they typically rely on

anisotropies in the microwave background radiation and quantities for which we have limited model-independent
3
ofthelarge-scaledistributionofgalaxiesinthelocaluni- information due, in part, to various degeneracies, many
v
6 verse have established “Lambda cold dark matter”, or related to poorly understood baryonic processes associ-
0 ΛCDM,asthestandardmodelofcosmology. Thismodel ated with galaxy formation [e.g. 22–24]. This processes
7 is based on Einstein’s theory of General Relavity (GR) canintroducefurtherdegeneraciesincaseofMGcosmol-
0
and has several parameters that have been determined ogy [25]. In addition, there are numerous statistical and
.
1 experimentally to high precision [e.g. 1–7]. One of these systematic uncertainties in the observationaldata whose
0 parameters is the cosmological constant, Λ, which is re- size can be comparable to the expected deviations from
4
sponsible for the accelerating expansion of the Universe GR.
1
: but has no known physical basis within GR. Modifica- In this Letter we introduce the use of the low-order
v
tions of GR, generically known as “modified gravity” moments of the distribution of galaxypairwise velocities
i
X (MG), could, in principle, provide an explanation (see as a probe of GR and MG on cosmological scales. We
r e.g. [8] for a comprehensive review). In this case, grav- illustrate the salient physics by reference to two classes
a ity deviates from GR on sufficiently large scales so as to of currently popular MG models. The first is the f(R)
give rise to the observed accelerated expansion but on family of gravity models [26–28], in which the Einstein-
small scales such deviations are suppressed by dynam- Hilbert action is augmented by an arbitrary and intrin-
ical screening mechanisms which are required for these sically non-linear function of the Ricci scalar, R. These
theories to remaincompatible with the stringenttests of models include the environment-dependent “chameleon”
gravity in the Solar System [9]. screening mechanism. The second class is Galileon grav-
Significant progress has been achieved in recent years ity [29, 30], in which the modifications to gravity arise
in designing observational tests of gravity on cosmolog- throughnonlinearderivativeself-couplingsofaGalilean-
ical scales which might reveal the presence of MG [e.g. invariant scalar field. These models restore standard
10–12]. Most viable MG theories predict changes in the gravityonsmallscalesthroughtheVainshteineffect[31].
clustering pattern on non-linear and weakly non-linear Our analysis is based on the high-resolution N-body
scales;ongalaxyandhalodynamics[e.g.13–19];onweak simulations of [15], for the Hu-Sawicki f(R) model [32],
gravitationallensingsignalsandontheintegratedSachs- and of [14, 33], for Galileon gravity [30, 34]. These con-
Wolfe effect [e.g. 20, 21]. However, a common feature of sider three flavoursoff(R) gravitycorrespondingto dif-
2
ferent values of the parameter |fR0| (10−4,10−5,10−6), 10 HOD LRGs galaxies σ12
which determine the degree of deviation from standard -0.5v12
0.5vHub
GR[32]. WerefertotheseasF4,F5andF6respectively.
For Galileon gravity we study the so-called Cubic, 3G ,
andQuartic,4G,models,whicharecharacterizedbythe s]
m/
orderatwhichthescalarfieldentersintotheLagrangian k
0
[29]. x10 5
Pairwise velocities. The mean pairwise relative ve- city [
o
locity of galaxies (or pairwise streaming velocity), v12, vel
reflects the “mean tendency of well-separated galaxies
to approach each other” [35]. This statistic was intro-
duced by Davis & Peebels [36] in the context of the
kinetic BBGKY theory [37–40] which describes the dy- 0
1 10
namical evolution of a system of particles interacting
separation [Mpc/h]
through gravity. In the fluid limit its equivalent is the
pair density-weighted relative velocity, FIG. 1: The scale dependence of the pairwise velocity mo-
ments extracted from HOD mock galaxy catalogues. The
v12(r)= v1 v2 ρ = h(v1−v2)(1+δ1)(1+δ2)i , (1) bshlaocwktshoelidF4linmesodshelo.wTthheetGhiRncreadsea,nwdhiblelatchkelirneedsdsahsohwedmliinnuess
h − i 1+ξ(r)
the mean streaming velocity, −v12(r), scaled down by factor
wityhearnedvf1raacntdionδ1al=mρa1t/theρrid−en1sidtyencootnetrtahsetpaetcpuolisairtiovnelorc1-; oσ[3f1]22a(frno)dr;Tc[4lha].reitTsyhh;aetdhdeedoltritneegedsiogwnrierteehnpfirlelilnseeednstchsiroacwnlessiltlsuhhseotwrHatuthibvebedleeirsvrpoeerlrosacisiotniyn,,
r = r1 r2 ; and ξ(r) = δ1δ2 is the 2-point density H0r, also scaled down for comparison.
| − | h i
correlationfunction. The denotes a pair-weighted
ρ
h···i
average,whichdiffersfromtheusualspatialaveragingby
the weighting factor, = ρ1ρ2/ ρ1ρ2 . Note that is
W h i W Since none of these quantities is directly observ-
proportional to the number density of pairs.
able, following [42] we also consider the centred
Gravitational instability theory predicts that the am-
line-of-sight pairwise velocity dispersion, σ2 (r) =
plitude ofv12(r) isdeterminedbythe 2-pointcorrelation R ξ(R)σ2(R)dl/R ξ(R)dl. Here r is the projecte1d2 galaxy
function, ξ(r), and the growth rate of matter density p
perturbations, g dlnD+/dlna (where D+(a) is the separation, R = √r2+l2, and the integration is taken
linear growing mo≡de solution and a is the cosmological along the line-of-sight within l 25h−1Mpc. The quan-
scalefactor)throughthepairconservationequation[35]. tity σp2 is the line-of-sight cen±tred pairwise dispersion,
Juszkiewiczet al.[41]providedananalyticexpressionfor defined as in [42]:
Eqn. (1) that is a good approximation to the solution of
r2σ2/2+l2(σ2 v2 )
the pair conservation equation for universes with Gaus- σ2 = ⊥ k − 12 . (2)
sian initial conditions: v12 = −32H0rgξ¯¯(r)[1 + αξ¯¯(r)], p r2+l2
where ξ¯(r) = (3/r3)R0rξ(x)x2dx ≡ ξ¯¯(r)[1+ξ(r)]. Here Fig. 1 shows the scale dependence of the lower-order
α is a parameter that depends on the logarithmic slope moments of the pairwise velocities measured in our N-
of ξ(r) and H0 =100hkms−1 Mpc−1 is the presentday body simulations in the GR case (black lines and sym-
value of the Hubble constant. It is clear that v12(r) is a bols) and in the F4 model (red lines and symbols). We
strong function of ξ(r) and g, both of which will differ choose the F4 model for illustration because this model
in MG theories from the GR values. This dependency is the one for which the chameleonscreening mechanism
motivates the use of the low-order moments of the pair- is the least effective [20].
wise velocity distribution as tracers of MG and of the For the purposes of this comparison, and to allow for
fifth-force it induces on galaxies and dark matter halos. a better connection to observations, we construct mock
Specifically, we will consider the following quantities: galaxy catalogues for these two models by performing
a halo occupation distribution (HOD) analysis [e.g. 43].
• the mean radial pairwise velocity, v12; Our HOD catalogues are tuned to resemble a sample of
Luminous Red Galaxieswith a satellite fractionof 7%
• tvheelodciitsipeesr,sσiokn=(novt122ce1n/t2r;ed)ofthe(radial)pairwise anda totalgalaxynumber density of4×10−5(h/M∼pc)3.
h i This number density is roughly consistent with that of
the mean transverse velocity of pairs, v ; the SDSS DR7 sample presented in [44]. We do this by
⊥
• followingasimilarprocedureasdescribedin[45,46]. The
the dispersion of the transverse velocity of pairs, shadedregioninthefigureshowsanillustrativeerrorthat
• σ⊥ =hv⊥2i1/2. reflects the accuracy of σ12 measurements form galaxy
3
R=1Mpc/h R=5Mpc/h 10 R=1Mpc/h R=5Mpc/h
103
GR s]
F4 m/
ξ(r) 102 3FFG56 100 k 5
101 4G [x2
1
v
-
100
0
0.5 0.5
2
ξ 1
∆ 0 ∆v 0
-0.5 -0.5
10 R=1Mpc/h R=5Mpc/h 10 R=1Mpc/h R=5Mpc/h
m/s] m/s]
σ(r) [x100 k|| 5 σ(r) [x100 k12 5
0 0
0.5 0.5
∆σ|| 0 ∆σ21 0
-0.5 -0.5
1011 1012 1013 1014 1011 1012 1013 1014 1011 1012 1013 1014 1011 1012 1013 1014
M200 [MO• /h] M200 [MO• /h] M200 [MO• /h] M200 [MO• /h]
FIG. 2: Comparison of absolute values (top panel in each pair) and therelative deviation form the GR case (bottom panel in
each pair) of: the 2-point correlation function, ξ2(r) (top-left panels); minus the mean streaming velocity, −v12(r) (top-right
panels); thepairwise velocity dispersion, σk(r) (bottom-left); and the projected pairwise velocity dispersion, σ12(r). The data
are binned in halo mass, M200, and shown at two different pair separations: 1 and 5h−1Mpc. The legend in the panel for
ξ2(5h−1Mpc) gives thecolours and symbols that weuse to distinguish thedifferent models. Top panels show only theLCDM
and f(R) cases; theQCDM and Galileons were omitted for clarity.
redshift surveys as in [3] and [4]. Firstly, we note that galaxypairdoes nothaveanetcontributionfrommodes
the stable clustering regime [35] (the scales over which with wavelengths larger than the pair separation since
the mean infall velocity exceeds the Hubble expansion, those modes make the same contribution to the veloci-
v12 > Hr) extends to larger separations for the F4 ties of both galaxies. Hence, on the scale of the typical
−
model than for the GR case. However, v12 in F4 dif- interhalo separation (at which the galaxies in a pair in-
fers significantly from GR only in the mildly non-linear habit different halos), the distribution of v12 factorises
regime, 2 < r < 10h−1Mpc. The maximum difference into two individual peculiar velocity distributions, one
∼ ∼
between the two models occurs at r 3.5h−1Mpc and foreachgalaxyorhalo,andthese are alwayssensitiveto
∼
is 30%. The situation is quite different when we con- non-linearitiesdrivenbyvirialmotionswithinthegalaxy
∼
sider σ12. While the F4 values are also roughly 30 to host halo (see [47] for more details). In most MG theo-
35% larger than in GR, the signal now is noticeable on riestheeffects ofthefifthforceonthedynamicsareonly
all scales plotted. Now, if we compare σ12 for F4 with significant on small nonlinear or mildly nonlinear scales
the GR case with errors obtained as in [3, 4], we can see (< 10h−1Mpc) which are probed by the pairwise veloc-
∼
thattheamplitudeofthisstatisticsinF4is(2 4)σaway ity dispersion. Because of this, the amplitude of σ12 is
−
from the GR case. potentially a powerful diagnostic of MG.
The differences between F4 and GR are driven by the Theeffectofthefifthforceonσ12isillustratedinFig.2
fact that the distribution of v12 never reaches the Gaus- whereweplotξ(r) δ1δ2 ,v12,σk andσ12 asafunction
≡h i
sian limit, even at large separations. This is because, at ofM200 [66]for the MGmodels we consider. Results are
a given separation, r, the velocity difference between a shown at pair separations r = 1h−1Mpc and 5h−1Mpc.
4
Heretheerrorbarsshowthevarianceestimatedfromthe formation and observational errors,as illustrated by our
ensembleaverageofsimulationsfromdifferentphasereal- HOD analysis. However, the quality of the data as used
isationsoftheinitialconditions. Wealsoplottherelative by[3,4]wouldalreadybe enoughtodistinguishbetween
deviation, ∆X = X /X 1, from a fiducial model GR and F4, F5 and 3G at the 2σ level, and these are
MG GR
−
whichhasthesameexpansionhistory,butincludesafifth relatively older datasets. With current and future sur-
force. Thishelpsidentify changesdrivenbythemodified veyslike SDSS-II, BOSSandPan-STARRS1[e.g. 49–53]
force law rather than by the modified expansion dynam- one can hope to do better, since the new data provide
ics. For clarity, we only show results for the Galileon already 30% improved accuracy.
model in the relativedifference panels. In the 4G model, ∼
The remaining important question is whether the MG
although gravityis enhanced in low-density regions,it is
footprint we have identified is actually observable in the
suppressedinthehigh-densityregionsofinterestbecause
realUniverse. As mentioned above, the σ12(r) value can
the Vainshtein mechanism does not fully screen out all
be estimated from galaxy redshift survey data but only
of the modifications to gravity [33, 48]. This is the rea-
in a model-dependent way. Specifically, one can obtain
son why the results for this model point in the opposite
the line-of-sight dispersion by fitting the 2D galaxy red-
direction to those for the other models (F4, F5, F6 and shift space correlation function to a model, ξs(r ,π) =
3G),forwhichgravitycanonlybeenhancedbyapositive ′ ′ p
fifth force. For models other than 4G, Fig. 2 shows posi- R ξ (rp,π−v/H0)h(v12)dv, where ξ is the linear theory
model prediction (which depends on coherent infall ve-
tiveenhancementsrelativetoGRinv12,σk andσ12 buta locities) and the convolution is made with the assumed
smallreductioninthe amplitude ofξ2. Furthermore,the distribution of pairwisevelocities, h(v12) [35, 47, 54, 55].
sizeoftheMGeffectinbothσk andσ12 isapproximately Alternatively,one canuse the redshift spacepowerspec-
independentofhalomass,althoughthereisaweaktrend
trum of the galaxy distribution to derive a quantity in
in σ12 for the most massive halos (M200 ∼> 1013M⊙/h). Fourier space, σ12(k), which is not an exact equivalent
The most striking result of this Letter is the ampli-
of the configuration space dispersion, but is closely re-
tude of the halo mass-binned σ12 both at r = 1h−1Mpc latedto it [e.g. 56–58]. To apply either ofthese methods
and 5h−1Mpc. Relative to GR, the deviations in the
one needs a self-consistent model of the redshift-space
F4 model range from 30% to 75%. For the F5 and 3G
clustering expected in a given MG theory. In particu-
models, the deviation is smaller, but still visible at the
lar,suchamodelneedstodescribethelineargalaxybias
∆σ12 ∼0.25level. The strong signalin the amplitude of parameter, b; the linear growth rate of matter, g; and
σ12 isacombinationofthecontributionsfrom∆v12,∆σk the pairwise velocity distribution in configuration space,
and∆σ thatareincorporatedinσ asshowninEqn.(2)
⊥ p h(v12), or, equivalently, the damping function in Fourier
and from ∆ξ2 which appears in the line-of-sight inte- space, D[kµσ12(k)]. Fortunately, all these quantities can
grals for σ12. Together, their combined effect results in be derived self-consistently for MG theories using linear
a prominent fifth force-like signature. The amplitude of
perturbation theory complemented with N-body simula-
σ12 isthestrongestobservabledeviationfromGRoncos- tions. Such a programme is currently being developed.
mologicalscalessofaridentified,apotentialsmokinggun
Instead of using redshift data, it is possible, in princi-
forMG.Thissignal,however,isnotentirelygeneric. For
example,theF6modelisvirtuallyindistinguishablefrom ple, to estimate v12 and σ12 directly from measurements
of galaxy peculiar velocities. The advantage of this ap-
GR:thefifthforceinthisflavouroff(R)gravityismuch
proach is that it is model independent. The disadvan-
too weak to produce a detectable effect in the dynamics
tageisthatpeculiarvelocitiescanonlybemeasuredwith
of galaxies and halos.
sufficient accuracy for a small sample of local galaxies
Summary. Using dark matter halo catalogues ex-
(z <0.05) and even then there are potentially large sys-
tracted from high-resolution N-body simulations of the
tematic errors in the estimates of redshift-independent
formation of cosmic structure in two representative
distance indicators [e.g. 59, 60]. A further complication
classes of modified gravity theories we have computed
is that only the radial component of a galaxy peculiar
the mean pairwise streaming velocity and its dispersion
velocity is observable (but see [61]), so it is necessary to
(radial and projected along the line-of-sight). Our simu-
construct special estimators for pairwise velocities such
lations show that there is a strong MG signal contained
as those proposed by [62–65].
intheline-of-sightprojectedpairwisevelocitydispersion.
For the F5,3G and 4G models, deviations from GR are Thereisalreadyalargebodyofvelocitydataofpoten-
at the > 5σ level for all masses. The deviation is even tially sufficient quality for the test we propose (cf. the
more pronounced for the F4 model, where it is at the size of the velocity error bars in Fig. 23 of [3]). Further
> 10σ level and higher. This is the clearest footprint of theoretical work is required to refine the redshift-space
modified gravity found to date in quantities that are, in probes and further observational work to exploit direct
principle, observable. Nonetheless,in a realisticobserva- peculiar velocity measurements. It is to be hoped that
tionalsituationonecanexpectthesignificanceoftheMG the presence of a fifth force, if it exists, will be revealed
signaltobe reduceddue toambiguitiesrelatedtogalaxy in measurements of the galaxy velocity field.
5
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