The Cosmic Mach Number: Comparison from Observations, Numerical Simulations and Nonlinear Predictions

0.96 MB·English

by  Shankar Agarwal

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed11December 2013 (MNLATEXstylefilev2.2)
The Cosmic Mach Number: Comparison from
Observations, Numerical Simulations and Nonlinear
Predictions
Shankar Agarwal(cid:63)& Hume A. Feldman†
Department of Physics & Astronomy, University of Kansas, Lawrence, KS 66045, USA.
emails: (cid:63)[email protected]; †[email protected]
3
1
0
2
n ABSTRACT
a
J We calculate the cosmic Mach number M – the ratio of the bulk flow of the
6 velocity field on scale R to the velocity dispersion within regions of scale R. M is
effectivelyameasureoftheratiooflarge-scaletosmall-scalepowerandcanbeauseful

] tool to constrain the cosmological parameter space. Using a compilation of existing
O
peculiarvelocitysurveys,wecalculateM andcompareittothatestimatedfrommock
C cataloguesextractedfromtheLasDamas(aΛCDMcosmology)numericalsimulations.
. We find agreement with expectations for the LasDamas cosmology at ∼ 1.5σ CL.
h
We also show that our Mach estimates for the mocks are not biased by selection
p
function effects. To achieve this, we extract dense and nearly-isotropic distributions
-
o using Gaussian selection functions with the same width as the characteristic depth
r of the real surveys, and show that the Mach numbers estimated from the mocks are
t
s very similar to the values based on Gaussian profiles of the corresponding widths. We
a discuss the importance of the survey window functions in estimating their effective
[ depths. We investigate the nonlinear matter power spectrum interpolator PkANN as

1 an alternative to numerical simulations, in the study of Mach number.
v
Key words: galaxies: kinematics and dynamics – galaxies: can be readily calculated and compared with its measured
9
statistics – cosmology: observations – cosmology: theory – value from the peculiar velocity field catalogues. However,
3
0 largescalestructureofUniverse–methods:N-bodysimula- comparing theoretical predictions with observations is not
1 tions straightforward: (i) one has to correct for the small-scale
. nonlinearitiesinobservationsaswellastakeintoaccountthe
1
factthatobservationsrepresentonlyadiscreetsampleofthe
0
3 1 INTRODUCTION continuous velocity field. This can be remedied by smooth-
1 ing the velocity field on a suitable scale rs (∼ 5h−1Mpc,
: Ostriker & Suto (1990) introduced a dimensionless statistic since on larger scales the matter density field is expected
v ofthecosmologicalstructure-thecosmicMachnumber,as to be linear), before estimating the quantities u(x ;R) and
Xi a way to measure the warmth/coldness of the velocity field σ(x0;R).However,anyresidualnonlinearityinthe0observed
on some scale R. Specifically, the Mach number is defined field can still bias the M estimates; (ii) Non-uniform, noisy
r
a as a ratio and sparse sampling of the peculiar velocity field can lead
(cid:18)|u(x ;R)|2(cid:19)1/2 to aliasing of small-scale power onto larger scales. When
M(x0;R)≡ σ2(x0 ;R) (1) making comparisons with theory, one has to carefully take
0 intoaccounttheselectionfunctionandthenoiseofthereal
whereu(x0;R)isthebulkflow(BF)ofaregionofsizeRcen- dataset.(iii)Peculiarvelocitysurveyshaveonlyline-of-sight
teredatx0,andσ(x0;R)isthevelocitydispersionoftheob- velocity information.
jectswithinthisregion.Theensembleaverageoverx gives
0
the statistic M(R). Since both |u(x ;R)|2 and σ2(x ;R) Over two decades ago, the statistic M has been inves-
0 0
scale equally by the amplitude of the matter density per- tigated in a series of papers: Ostriker & Suto (1990) used
turbation, the statistic M is independent (at least in linear linear theory and Gaussian selection function to show that
approximation) of the normalization of the matter power standard Cold Dark Matter (sCDM) model is inconsistent
spectrum. (predicts M almost twice the observed value) with obser-
In linear theory, given the cosmological parameters, M vations at ∼ 95% CL; Suto, Cen, & Ostriker (1992), using
(cid:13)c 0000RAS
2 Agarwal & Feldman
TophatandGaussianselectionfunctions,studiedthedistri- be defined as
butionofM usingN-bodysimulationstoruleoutthesCDM (cid:90)
scenario at 99% CL; Strauss, Cen, & Ostriker (1993) took u(x0;R) = dx v(x)F(|x−x0|,R), (2)
into account the selection function of real surveys and ex-
where F(|x−x |,R) is the filter used to average the veloc-
tracted mocks from numerical simulations over a range of 0
ity field v(x) on a characteristic scale R. Although Tophat
cosmologies including sCDM and tilted CDM (scalar spec-
andGaussianfiltersarethepreferredchoices,F(|x−x |,R)
tralindex,n (cid:54)=1)amongothers,torejectthesCDMmodel 0
s
can be designed to mimic the selection function of the real
at 94% CL. More recently Ma, Ostriker, & Zhao (2012) ex-
datasets. This is useful when dealing with datasets whose
ploredthepotentialofusingM indistinguishingcosmologi-
selection function depends strongly on the position in the
calmodels,includingmodifiedgravityandmassiveneutrino
sky. In Fourier space, Eq. 2 can be written as
cosmologies.
Inthispaper,(i)weestimatethecosmicMachnumber (cid:90)
u(x ;R) = dk v(k)W(k,R)e−ik·x0, (3)
forvariousgalaxypeculiarvelocitydatasets;(ii)weinvesti- 0
gatehowlikelyitistogettheseMachvaluesinaΛCDMuni-
where v(k) and W(k,R) are the Fourier transforms of the
verse.Toachievethis,westudythestatisticaldistributionof
peculiarvelocityfieldv(x)andthefilterF(|x−x |,R),re-
the expected Mach number by extracting mocks of the real 0
spectively.
cataloguesfromnumericalsimulationsofaΛCDMuniverse.
Inlineartheoryofstructureformation,atlowredshifts,
WeshowthataΛCDMuniversewith7-yrWMAP typecos-
the velocities are related to the matter overdensities via
mology is consistent with the Mach observations at ∼1.5σ
k
CL;(iii)wefurthershowthatourM estimatesforthemocks v(k) = ifH δ(k) , (4)
arenotbiasedbytheirselectionfunctions.Towardsthis,we 0 k2
extractdenseandnearlyisotropicdistributionswithaGaus- where H is the present-day Hubble parameter in units of
0
sianprofilef(r)∝e−r2/2R2 withR=10−100h−1Mpc.We km s−1 Mpc−1; δ(k) is the Fourier transform of the over-
showthattheMachnumbersestimatedfromthemocksare density field δ(x); the linear growth rate factor f can be
verysimilartothevaluesbasedonGaussianprofiles(ofsim- approximated as f = Ω0m.55 (Linder 2005). Thus, the ve-
ilar depth R as the mocks); (iv) we use the nonlinear mat- locity power spectrum Pv(k) is proportional to the matter
terpowerspectruminterpolationschemePkANN(Agarwal power spectrum P(k) at low redshifts,
et al. 2012a) to check if we can avoid N-body simulations
P(k)
completely and predict M(R) by only using PkANN’s pre- Pv(k) = (H0f)2 k2 . (5)
diction for the nonlinear power spectrum. This is crucial
Using Eq. 3 and Eq. 5, the mean-squared bulk value of
because high-resolution hydrodynamic N-body simulations
u(x ;R) can be shown to be
arecomputationallyexpensiveandextremelytimeconsum- 0
ing. Exploring parameter space with numerical simulations H2Ω1.1 (cid:90)
σ2(R)≡<u2(x ;R)>= 0 m dk P(k)W2(kR), (6)
withreasonablecomputingresourcesandtimemightnotbe v 0 2π2
possible.AfulluseofastatisticlikeM canonlyberealized
where the average is taken over all spatial positions x .
withaprescriptionforthenonlinearmatterpowerspectrum. 0
The squared velocity dispersion within a region of size
In Sec. 2 we discuss the cosmic Mach number statis-
R centered at x can be similarly defined as
tic. In Sec. 3 we describe the numerical simulations we use 0
(cid:90)
to extract mock surveys. In Sec. 4 we describe the galaxy
σ2(x ;R)= dx |v(x)|2F(|x−x |,R)−|u(x ;R)|2. (7)
peculiarvelocitysurveys(Sec.4.1)andtheprocedurewefol- 0 0 0
low to extract the mock catalogues (Sec. 4.2). In Sec. 5 we
In Fourier space, the ensamble average of Eq. 7 over x
0
review the maximum likelihood estimate (hereafter MLE)
becomes
weighting scheme that is commonly used to analyze pecu-
liar velocity surveys. In Sec. 6, we show our results for the σ2(R)≡<σ2(x ;R)>= H02Ω1m.1 (cid:90) dk P(k)(cid:0)1−W2(kR)(cid:1). (8)
statisticaldistributionoftheMachnumberestimatedusing 0 2π2
variousmockcatalogues.InSec.7,wetesttheperformance Using Eq. 6 and Eq. 8, the cosmic Mach number can now
of PkANN - a nonlinear matter power spectrum interpola- be defined as
tor, in predicting M(R) for the LasDamas cosmology. The
(cid:18)σ2(R)(cid:19)1/2
Machnumberestimatesfromrealsurveysissummarizedin M(R)≡<M2(x ;R)>1/2 = v . (9)
0 σ2(R)
Sec. 8. We discuss our results and conclude in Sec. 9.
As discussed in literature (Ostriker & Suto 1990; Suto,
Cen, & Ostriker 1992; Strauss, Cen, & Ostriker 1993), the
cosmicMachnumberisessentiallyameasureoftheshapeof
2 THE COSMIC MACH NUMBER
the matter power spectrum: The rms bulk flow σ (R) gets
v
Given a peculiar velocity field v(x), one can calculate the most of its contribution from scales larger than R, while
bulk flow (see Feldman & Watkins 1994, for details), which the velocity dispersion σ(R) is a measure of the magnitude
represent the net streaming motion of a region in some di- of velocities on scales smaller than R and gets most con-
rection relative to the background Hubble expansion. The tribution from small scales (for more detailed analyses of
bulk flow u(x ;R) of a region of size R centered at x can velocity dispersion see also Watkins (1997); Bahcall & Oh
0 0
(cid:13)c 0000RAS,MNRAS000,000–000
The Cosmic Mach Number 3
(1996); Bahcall et al. (1994)). Furthermore, the statistic M
Table 1.Thecosmologicalparametersandthedesignspecifica-
isexpectedtobeindependentofthematterpowerspectrum tionsoftheLD-Carmen simulations.
normalization–atleastonlargescales,wheretheperturba-
tionsarestillwelldescribedbylineartheoryandaffectboth Cosmologicalparameters LD-Carmen
σ2(R) and σ2(R) equally. M can be a powerful tool to test
v Matterdensity,Ωm 0.25
not only the ΛCDM scenario, but also a wide range of cos-
Cosmologicalconstantdensity,ΩΛ 0.75
mologies including models with massive neutrinos. Massive Baryondensity,Ω 0.04
b
neutrinossuppressthematterpowerspectruminascalede- Hubbleparameter,h(100kms−1 Mpc−1) 0.7
pendent way, thereby altering the velocity dispersion much Amplitudeofmatterdensityfluctuations,σ8 0.8
moreprominentlythanthebulkflow.MachnumberM pro- Primordialscalarspectralindex,ns 1.0
vides an easy to interpret technique to distinguish between Simulationdesignparameters
various cosmological models.
Simulationboxsizeonaside(h−1Mpc) 1000
In previous work (Watkins, Feldman, & Hudson 2009;
NumberofCDMparticles 11203
Feldman, Watkins, & Hudson 2010; Agarwal, Feldman, &
Initialredshift,z 49
Watkins 2012b) we have dealt with peculiar velocity sur- Particlemass,mp (1010 h−1M(cid:12)) 4.938
veys differently. We developed the ‘minimum variance’ for- Gravitationalforcesofteninglength,f(cid:15) (h−1kpc) 53
malism (hereafter MV) to make a clean estimate of the the
bulk, shear and the octupole moments of the velocity field
as a function of scale using the available peculiar velocity to mimic the radial distribution of the real catalogues (de-
data. Higher moments get contribution from progressively scribed in Sec. 4.1), as closely as possible.
smaller scales. In this paper, instead of isolating the higher
moments (shear, octupole etc.), we simply estimate the ve-
locitydispersionσ(R)thatgetscontributionfromallscales
4 PECULIAR VELOCITY CATALOGUES
smallerthanR.Ingeneral,theMLEschemeweemployhere
attempts to minimize the error given a particular survey.
4.1 Real Catalogues
The downside of the MLE formalism is the difficulty to di-
rectlycompareresultsfromdifferentsurveys.Sinceeachsur- Weuseacompilationoffivegalaxypeculiarvelocitysurveys
vey samples the volume differently and although the large tostudytheMachstatistic.Thiscompilation,whichwelabel
scalesignalissimilaracrosssurveys,thesmallscalenoiseis ‘DEEP’, includes 103 Type-Ia Supernovae (SNIa) (Tonry
unique.Thatleadstocomplicatedbiasesoraliasingthatare et al. 2003), 70 Spiral Galaxy Culsters (SC) Tully-Fisher
survey dependent (see Watkins & Feldman (1995); Hudson (TF) clusters (Giovanelli et al. 1998; Dale et al. 1999a), 56
et al. (2000)). The MV formalism corrects for small scale Streaming Motions of Abell Clusters (SMAC) fundamental
aliasingbyusingaminimizationschemethattreatsthevol- plane(FP)clusters(Hudsonetal.1999,2004),50Early-type
umeofthesurveysratherthantheparticularwaythesurvey Far galaxies (EFAR) FP clusters (Colless et al. 2001) and
samples the volume, thus eliminating aliasing and allowing 15 TF clusters (Willick 1999). In all, the DEEP catalogue
for direct comparison between surveys. consists of 294 data points. In Fig. 1, top row, we show the
DEEPcatalogue(left-handpanel)anditsradialdistribution
(right-hand panel). The bottom row shows a typical mock
extractedfromtheLDsimulations.Theproceduretoextract
3 N-BODY SIMULATIONS mocks is described in Sec. 4.2.
In order to study the statistical distribution of the cosmic
Mach number we extract mock surveys from the 41 numer-
4.2 Mock Catalogues
ical realizations of a ΛCDM universe. The N-body simu-
lation we use in our analysis is Large Suite of Dark Mat- Inside the N-body simulation box, we first select a point at
ter Simulations (LasDamas, hereafter LD) (McBride et al. random.Next,weextractamockrealizationoftherealcata-
2009;McBrideetal.2011inprep1).TheLDsimulationisa loguebyimposingtheconstraintthatthemockshouldhave
suite of 41 independent realizations of dark matter N-body asimilarradialdistributiontotherealcatalogue.Wedonot
simulations named Carmen and have information at red- constrain the mocks to have the same angular distribution
shift z=0.13. Using the Ntropy framework (Gardner et al. as the real catalogue for two reasons: (i) the LD simulation
2007), where bound groups of dark matter particles (halos) boxes are not dense enough to give us mocks that are ex-
are identified with a parallel friends-of-friends (FOF) code act replicas of the real catalogue, (ii) the objects in a real
(Davisetal.1985).Thecosmologicalparametersandthede- surveyaretypicallyweighteddependingonlyontheirveloc-
sign specifications of the LD-Carmen are listed in Table 1. ityerrors.Consequently,eventhoughtherealcatalogueand
We extract 100 mock catalogues from each of the 41 itsmockshavesimilarradialprofiles,theirangulardistribu-
LD-Carmen boxes, for a total of 4100 mocks. The mocks tion differ considerably, with the mocks having a relatively
arerandomlycenteredinsidetheboxes.Theyareextracted featureless angular distribution. To make the mocks more
realistic, we impose a 10o latitude zone-of-avoidance cut.
Using the angular position {rˆ ,rˆ ,rˆ }, the true radial
x y z
1 http://lss.phy.vanderbilt.edu/lasdamas distance ds from the mock center and the peculiar velocity
(cid:13)c 0000RAS,MNRAS000,000–000
4 Agarwal & Feldman
vector v, we calculate the true line-of-sight peculiar veloc-
ity v and the redshift cz = d +v for each mock galaxy
s s s DEEP-survey
(in km/s). We then perturb the true radial distance d of
s
the mock galaxy with a velocity error drawn from a Gaus-
sian distribution of width equal to the corresponding real
galaxy’s velocity error, e. Thus, d = d +δ , where d is
p s d p
the perturbed radial distance of the mock galaxy (in km/s)
andδ isthevelocityerrordrawnfromaGaussianofwidth
d
e. The mock galaxy’s measured line-of-sight peculiar veloc-
ity v is then assigned to be v = cz−d , where cz is the
p p p
redshift we found above. This procedure ensures that the
DEEP-mock
weights we assign to the mock galaxies are similar to the
weights of the real galaxies. In Fig. 1 we show the angular
(leftpanels)andradial(rightpanels)distributionofgalaxies
in the DEEP catalogue (top) and its mock (bottom).
5 THE MAXIMUM LIKELIHOOD ESTIMATE
METHOD
Oneofthemostcommonweightingschemeusedintheanal-
ysis of the bulk flow is the maximum likelihood estimate
(MLE) method, obtained from a maximum likelihood anal-
ysis introduced by Kaiser (1988). The motion of galaxies is Figure 1. Top row: DEEP catalogue (left) and its radial distri-
bution(right).Bottomrow:DEEPmockcatalogue(left)andits
modeled as being due to a streaming flow with Gaussian
radialdistribution(right).
distributedmeasurementuncertainties.Givenapeculiarve-
locitysurvey, theMLEestimateof itsbulk flowis obtained
from the likelihood function iteratively.SeeStrauss,Cen,&Ostriker(1993)foradiscus-
L[ui|{Sn,σn,σ∗}]=(cid:89)n (cid:112)σn21+σ∗2 exp(cid:32)−21(Sσnn2−+rˆσn∗2,iui)2(cid:33), bsiyonaTownheeihgeohffwteecdttoisvueemstdiem(cid:80)patwthenortfhnae/s(cid:80)buervswte-nyfitcoafuntihbaeenrrdaoduσiga∗h.lldyisetsatnimceastredn
(10) ofthesurveyobjects,wherew =1/(σ2+σ2).Thisweight-
n n ∗
where ˆr is the unit position vector of the nth galaxy; σ
n n ingschemehasbeenusedbyMa,Ostriker,&Zhao(2012)in
is the measurement uncertainty of the nth galaxy; and σ
∗ their analyses of peculiar velocity datasets. A drawback of
is the 1-D velocity dispersion accounting for smaller-scale using weights w =1/(σ2 +σ2) in estimating the depth of
n n ∗
motions. The three components of the bulk flow u can be
i a survey is that while the weights w take into account the
n
written as weighted sum of the measured radial peculiar
measurement errors σ , they do not make any corrections
n
velocities of a survey
for the survey geometry. A better estimate of the effective
(cid:88) depth can be made by looking at the survey window func-
u = w S , (11)
i i,n n tions W2. Window function gives an idea of the scales that
n ij
contribute to the bulk flow estimates. Ideally, the window
whereS istheradialpeculiarvelocityofthenthgalaxyofa
n function should fall quickly to zero for scales smaller than
survey;andw istheweightassignedtothisvelocityinthe
i,n thatbeingstudied.Thisensuresthatthebulkflowestimates
calculationofu .Throughoutthispaper,subscriptsi,j and
i are minimally biased from small-scale nonlinearities.
k run over the three spacial components of the bulk flow,
Armed with the MLE weights w from Eq. 12, the
i,n
whilesubscriptsmandnrunoverthegalaxies.Maximizing angle-averaged tensor window function W2(k) (equivalent
the likelihood given by Eq. 10, gives the three components to W2(kR) of Eq. 6) can be constructedij(for details, see
of the bulk flow u with the MLE weights
i Feldman, Watkins, & Hudson 2010) as
wi,n =(cid:88)j=31A−ij1σn2rˆn+,jσ∗2, (12) Wi2j(k) = m(cid:88),nwi,mwj,n(cid:90) d42πkˆ(cid:16)ˆrm·kˆ(cid:17)(cid:16)ˆrn·kˆ(cid:17) (14)
where ×exp(cid:16)ik kˆ·(r −r )(cid:17).
m n
A =(cid:88) rˆn,irˆn,j . (13)
ij n σn2 +σ∗2 thebTuhlkefldoiawgocnoamlpeolenmenetnstsuW.Ti2ihaerewtihnedowwinfduonwctifounncgtiivoenssaonf
√ i
The 1-D velocity dispersion σ is 1/ 3 of the 3-D ve- ideaofthescalesthatcontributetothebulkflowestimates.
∗
locity dispersion (see Eq. 8) which we aim to ultimately Ideally, the window function should fall quickly to zero for
measure. Since the weights w (and u ) are themselves a scales smaller than that being studied. This ensures that
i,n i
function of σ , we converge on to the MLE estimate for σ the bulk flow estimates are minimally biased from small-
∗ ∗
(cid:13)c 0000RAS,MNRAS000,000–000
The Cosmic Mach Number 5
nents calculated using MLE weights (see Eq. 12) for the
DEEP catalogue. The x,y,z components are dot-dashed,
short-dashedandlong-dashedlines,respectively.Alsoshown
are the ideal window functions IW2 (see Eq. 17) for scales
ij
R=10−40h−1Mpc (in 5h−1Mpc increments), the window
functions being narrower for larger scales. Comparing the
DEEPandtheidealwindowfunctionsgivestheDEEPcat-
alogue an effective depth of ∼ R = 35h−1Mpc. We note
(cid:80) (cid:80)
thattheweightedsum w r / w givestheDEEPcat-
n n n
alogue a depth of 59 h−1Mpc, an over-estimation by nearly
70%. Estimating the survey depth correctly is crucial when
Figure 2. Left Panel: The window functions W2 of the bulk it comes to comparing the survey bulk flow with theoreti-
ii
flow components calculated using MLE weights for the DEEP cal predictions. One might have a high-quality survey but
catalogue. The x,y,z components are dot-dashed, short-dashed a poorly estimated depth which can introduce substantial
and long-dashed lines, respectively. The solid lines are the ideal errorswhencomparingwiththeory.Throughoutthispaper,
window functions IWi2j for scales R = 10 − 40h−1Mpc (in we define the characteristic depth R of a survey as the one
5h−1Mpc increments), the window functions being narrower for estimated from its window functions. The right-hand panel
largerscales.RightPanel:Thewindowfunctionsforasubsetof
ofFig.2showsthewindowfunctionsforasubsetofthe4100
the 4100 DEEP mocks (solid lines). The characteristic depth of
DEEP mocks (solid lines) extracted from the LD-Carmen
theDEEPcatalogueanditsmocksis∼R=35h−1Mpc(dashed
simulations. The fact that the mock window functions are
line).
nearly centered on the ∼ R = 35h−1Mpc ideal window,
shows that our procedure for mock extraction works well.
scale nonlinearities. See the MLE (Sarkar et al. 2007) and
MV(Watkinsetal.2009)windowfunctionsofthebulkflow
components for a range of surveys.
6 COSMIC MACH NUMBER STATISTICS
Having constructed the survey window functions W2,
ii
the effective depth of the survey can be defined to be the 6.1 Mach statistics for DEEP mocks
oneforwhichW2 isaclosematchtothewindowfunctionfor
ii
Using the MLE weighting scheme (Sec. 5), we estimated
an idealized survey. In order to construct the ideal window
the bulk flow moments {u ,u ,u }, the velocity dispersion
functions, we first imagine an idealized survey containing x y z
σ and the cosmic Mach number M for each of the 4100
radial velocities that well sample the velocity field in a re-
DEEPmockrealizations.InFig.3,weshowtheprobability
gion. This survey consists of a large number of objects, all
distribution for the 4100 DEEP mocks: bulk flow u (left-
withzeromeasurementuncertainty.Theradialdistribution
hand panel), dispersion σ (middle panel) and the cosmic
of this idealized survey is taken to have a Gaussian profile
of the form f(r) ∝ e−r2/2R2, where R gives a measure of MachnumberM (right-handpanel).Wefoundtherms bulk
flowtobeσ =222±86kms−1withavelocitydispersionof
the depth of the survey. This idealized survey has easily in- v
σ=511±98kms−1.TogetherthisimpliesM =0.43±0.17
terpretable bulk flow components that are not affected by
at1σCL.SincetheDEEPmockshaveacharacteristicdepth
small-scale aliasing and which reflect the motion of a well-
of R = 35h−1Mpc, we can say that for the LD cosmology,
defined volume.
the expected Mach number on scales of R = 35h−1Mpc is
The MLE weights of an ideal, isotropic survey consist-
ing of N(cid:48) exact radial velocities vn(cid:48) measured at randomly M =0.43±0.17.
selected positions r(cid:48) are
n(cid:48)
w(cid:48) =(cid:88)3 A−1rˆn(cid:48)(cid:48),j, (15) 6.2 Mach statistics for Gaussian Realizations
i,n(cid:48) ij N(cid:48)
InordertofindtheexpectedMachnumberasafunctionof
j=1
scale R for the LD cosmology, we went to the same cen-
where
tral points for each of the 4100 DEEP mocks and com-
A = (cid:88)N(cid:48) rˆn(cid:48)(cid:48),irˆn(cid:48)(cid:48),j. (16) putedtheweightedaverageofthevelocitiesofallthegalax-
ij N(cid:48) ies in the simulation box, the weighting function being
n(cid:48)=1 e−r2/2R2. We repeated this for a range of scales between
Similar to Eq. 14, the window functions IWi2j for an R = 10−100h−1Mpc in increments of 5h−1Mpc. We sum-
idealized survey of scale R can be constructed as marizetheexpectedvaluesforthebulk,dispersionandMach
IWi2j(k;R) = (cid:88)wi(cid:48),m(cid:48)wj(cid:48),n(cid:48)(cid:90) d42πkˆ(cid:16)ˆr(cid:48)m·kˆ(cid:17)(cid:16)ˆr(cid:48)n·kˆ(cid:17)(17) nFuigm.b4e,rwfeorshsocawletsheRex=pe1c0te−d v1a0l0uhe−s1fMorpcthienbTualkb,ledi2sp.eIrn-
m,n sionandMachnumber(dashedline)togetherwiththeir1σ
×exp(cid:16)ik kˆ·(r(cid:48) −r(cid:48) )(cid:17). CL intervals. The corresponding values for the 4100 DEEP
m n
mocksareshownbyasolidcircleatthecharacteristicscale
In Fig. 2, left-hand panel, we show the diagonal win- R=35h−1Mpc.
dow functions W2 (see Eq. 14) of the bulk flow compo- The expected bulk (σ =234±94 km s−1), dispersion
ii v
(cid:13)c 0000RAS,MNRAS000,000–000
6 Agarwal & Feldman
10
8
6
4
2
0
0 200 400 600 200 400 600 800 1000 0 0.5 1 1.5 2
Figure3.Histogramsshowingthenormalizedprobabilitydistributionforthe4100DEEPmocks:bulkflowu(left-handpanel),dispersion
σ (middle panel) and the cosmic Mach number M (right-hand panel). We also superimpose the best-fitting Maxwellian (for bulk and
Mach) and Gaussian (for dispersion) distributions with the same widths as the corresponding histograms. The rms values and the 1σ
CLintervalsarementionedwithineachpanel.TheseresultscorrespondtotheLDcosmology(seeTable1).
600
1
400
200
0
20 40 60 80 100 20 40 60 80 100 10 100
Figure4.Therms valuesofthebulkflow(left-handpanel),dispersion(middlepanel)andthecosmicMachnumber(right-handpanel)
areplottedasafunctionofscaleR.Ineachpanel,thedashedlinecorrespondstomeasurementsfromtheGaussianrealizations,whereas
the shaded region is the 1σ CL interval. The solid circle at R = 35h−1Mpc is the result for the DEEP mocks. The error bar is the
statistical variance of the mean calculated from the 4100 DEEP mocks. Linear theory predictions are shown by the solid line. These
results correspond to the LD cosmology. The nonlinear contributions to the dispersion are clearly seen in both the center and right
panels.
(σ = 517±56 km s−1) and Mach number (σ = 0.44±17) Fig.5forarangeofGaussianwidthsR.Forclarity,weonly
for Gaussian window with R = 35h−1Mpc are in excel- show scales R=15, 35, 55, 75 and 95h−1Mpc.
lentagreementwiththecorrespondingvaluesfortheDEEP
mocks. This shows that the DEEP catalogue probes scales As expected, the rms bulk flow (dispersion) is a de-
up to ∼ R = 35h−1Mpc, and not R = 59h−1Mpc as one clining (increasing) function of scale R (see Figs. 4 and 5).
(cid:80) (cid:80)
would have inferred from wnrn/ wn using the weights Thiscanbereadilyunderstoodfromtheidealwindowfunc-
wn =1/(σn2 +σ∗2). tions in Fig. 2. Larger scales have narrower window func-
tions in Fourier space. Only small scale modes (k ∝ 1/R)
Linear theory predictions for the LD cosmology are
contribute to the rms bulk flow integral in Eq. 6, resulting
shown by the solid lines in Fig. 4. The onset of nonlinear
in smaller bulk flow on larger scales. The dispersion inte-
growth in structure formation at low redshifts boosts the
gral (see Eq. 8) gets most of its contribution from higher
velocitydispersion,causinglineartheorytoover-predictthe
k−values (k >1/R) and gradually increases with narrower
Mach values.
windows.SimilarhistogramtrendswerefoundbySuto,Cen,
The probability distributions for u, σ and M from the &Ostriker(1992)fromnumericalsimulationsofaCDMuni-
Gaussian realizations in the LD simulations are plotted in verse.
(cid:13)c 0000RAS,MNRAS000,000–000
The Cosmic Mach Number 7
30
20
10
0
0 200 400 600 200 400 600 800 1000 0 0.5 1 1.5 2
Figure 5.ThesameasFig.3,butforGaussianwindowwithR=15h−1Mpc(dotted),R=35h−1Mpc(solid),R=55h−1Mpc(short-
dashed), R = 75h−1Mpc (long-dashed) and R = 95h−1Mpc (dot-dashed). For clarity, instead of the histograms, only the best-fitting
Maxellian/Gaussian distributions with the same widths as the corresponding histograms are shown. The rms values and the 1σ CL
intervalsforR=35h−1Mpcarelistedwithineachpanel,andareingoodagreementwiththecorrespondingvaluesfortheDEEPmocks
(showninFig.3).Table2summarizestheresultsforGaussianwidthsR=10−100h−1Mpc.
6.3 Mach statistics for other mocks EAR (Early-type Nearby Galaxies) (da Costa et al. 2000;
Bernardi et al. 2002; Wegner et al. 2003), SFI++, SNIa
In the following we extend our analysis to include various and SC peculiar velocity surveys. Note that the SC and
different peculiar velocity surveys, specifically to show that SNIa surveys are also part of our DEEP compilation. The
our results are not dependent on any radial or angular dis- SFI++(SpiralFieldI-band)catalogue(Mastersetal.2006;
tributions, nor any distinct morphological types. We com- Springobetal.2007,2009)isthedensestandmostcomplete
paredtheGaussianrealizationswithmocks(4100each)cre- peculiarvelocitysurveyoffieldspiralstodate.Weusedata
ated to emulate the radial selection function of the SBF from Springob et al. 2009. The sample consists of 2720 TF
(Surface Brightness Fluctuations) (Tonry et al. 2001), EN- field galaxies (SFI++f) and 736 groups (SFI++g).
Table2.Therms valuesofthebulkflow(2ndcolumn),velocity
dispersion (3rd column) and cosmic Mach number (4th column)
together with their 1σ CL intervals for Gaussian windows with
width R (1st column). These values are calculated for the LD
cosmology(fortheLDparameters,seeTable1).
√ √ √
R <u2> <σ2> <M2>
(h−1Mpc) (kms−1) (kms−1)
10 341±133 379±108 0.85±0.33
15 308±120 433±89 0.68±0.27
20 286±111 464±76 0.59±0.23
25 267±104 487±68 0.53±0.21
30 248±96 504±62 0.48±0.19
35 234±91 517±56 0.44±0.17
40 218±85 526±50 0.41±0.16
45 204±79 535±47 0.38±0.15
50 194±75 541±43 0.35±0.14
55 182±71 547±40 0.33±0.13
60 173±67 551±37 0.31±0.12
65 163±63 556±35 0.29±0.11
70 154±60 560±33 0.27±0.11
75 145±57 562±31 0.26±0.10
80 137±53 565±29 0.24±0.09
85 130±51 567±27 0.23±0.09
90 125±48 569±26 0.22±0.08
95 118±46 571±25 0.21±0.08
100 113±44 572±23 0.20±0.07
(cid:13)c 0000RAS,MNRAS000,000–000
8 Agarwal & Feldman
Figure6.SimilartoFig.2,fortheSBF,ENEAR,SFI++g,SNIa,SFI++f,DEEPandSCcatalogues(toptobottomrow,respectively).
In Fig. 6, left-hand panels, we show the window func- SBFmocksaredeeperthantherealSBFsurveybecausethe
tionsW2 ofthebulkflowcomponentsfortheSBF,ENEAR, LDsimulationsarenotdenseenoughtoextractmockswith
ii
SFI++g, SNIa, SFI++f, DEEP and SC catalogues (top to depths less than ∼ R = 12h−1Mpc. This explains why the
bottom row, respectively). The right-hand panels show the SBF window functions for the mocks (see Fig. 6, first row,
window functions for a subset of the corresponding mocks. right-hand panel) are narrower than the one for the SBF’s
Comparingthewindowfunctionsoftherealcatalogueswith depth of R = 10h−1Mpc. Narrower window functions de-
those of the ideal ones (solid lines in the left-hand panels), crease(increase)ourbulkflow(dispersion)estimatesforthe
we estimate the characteristic depths of the SBF, ENEAR, SBF mocks. For the SC mocks, the bulk flow (dispersion)
SFI++g, SNIa, SFI++f, DEEP and SC catalogues to be getsexcess(suppressed)contributionfromsmallerscalesdue
R=10, 19, 20, 23, 30, 35and40h−1Mpc,respectively.The to the extended tails of the window functions (see Fig. 6,
window functions for these depths are shown in the right- row seven). The SC catalogue, with only 70 clusters, does
hand panels (dashed lines, top to bottom row). nothaveagoodskycoverage.TheDEEPcompilation,how-
ever, has a much better sky coverage, and the results (see
In Table 3 we summarize the results for the various
Fig.7,solidcircle)matchthosefromR=35h−1MpcGaus-
surveys (column 1), where the surveys are listed in order of
sianmocks.WehaveincludedtheresultsfortheSBFandSC
increasing characteristic depth R (column 2) (based on the
cataloguestospecificallyshowthatiftheselectionfunction
windowfunctionsinFig.6,asdescribedinSec.5).Theerror-
weighted depths (cid:80)w r /(cid:80)w , where w = 1/(σ2 +σ2) of the real survey is not properly modeled, the predictions
n n n n n ∗
(in our case, based on Gaussian selection function) can be
are listed in column 3, and are typically ∼75% larger than
misleading.
R. For the mock surveys, the expected values for the bulk,
dispersion and Mach number and their 1σ CL intervals are
summarizedincolumns4−6.Columns7−9arecomputed Forreasonablydenseandwellsampledvelocitysurveys,
using the real surveys. The quoted errors are calculated us- likeDEEP,SFI++f,andSFI++g,aclosematchbetweenthe
ing the measurement uncertainties σn of the nth galaxy of mockandtheGaussianresultsshowsthattheMachanalysis
a survey. Comparing columns 6 and 9, the Mach estimates for such catalogues is not overly sensitive to the selection
forallcataloguesagreeat∼1.5σ CLfortheLDcosmology. functionsoftheindividualmocks.Assuch,onecanskipthe
SimilartoFig.4,weshowresultsfortheSBF,ENEAR, stepofextractingmockrealizationsoftheobservationsfrom
SFI++g,SNIa,SFI++fandSCmocksinFig.7.Exceptfor N-bodysimulations,andsimplyuseMachpredictionsbased
the SBF and SC catalogues, the results for the other cata- on Gaussian selection function e−r2/2R2 with R set to the
loguesareaclosematchtotheirGaussiancounterparts.Our characteristic depth of the survey being studied.
(cid:13)c 0000RAS,MNRAS000,000–000
The Cosmic Mach Number 9
Table3.Peculiarvelocitystatisticsforvarioussurveys(1stcolumn).Foreachsurvey,4100mockswereextractedfromtheLDcosmology
(for the LD parameters, see Table 1). The characteristic depth R (2nd column) of the mock catalogues is estimated from the effective
width of their window functions shown in Fig. 6. For reference, the error-weighted depth (cid:80)wnrn/(cid:80)wn where wn =1/(σn2 +σ∗2), is
listedinthe3rdcolumn.Therms valuesofthebulkflow(4thcolumn),velocitydispersion(5thcolumn)andcosmicMachnumber(6th
column)togetherwiththeir1σ CLintervals.Columns7−9correspondtotherealsurveyswiththequotederrorscalculatedusingthe
radialdistanceuncertainties.
Mocks Real
Survey R (cid:80)(cid:80)wwnnrn √<u2> √<σ2> √<M2> u σ M
(h−1Mpc) (h−1Mpc) (kms−1) (kms−1) (kms−1) (kms−1)
SBF 10 19 322±125 415±100 0.74±0.29 354±66 428±32 0.83±0.15
ENEAR 19 34 262±102 490±104 0.53±0.21 292±46 528±24 0.55±0.09
SFI++g 20 35 280±101 473±66 0.59±0.18 221±57 436±27 0.29±0.08
SNIa 23 42 275±107 465±73 0.58±0.21 430±87 478±47 0.90±0.18
SFI++f 30 52 240±86 510±81 0.47±0.15 320±44 503±22 0.42±0.06
DEEP 35 59 222±86 511±65 0.43±0.17 312±61 446±27 0.70±0.14
SC 40 75 227±88 485±43 0.47±0.15 116±123 520±74 0.22±0.23
600
1
400
200
0
20 40 60 80 100 20 40 60 80 100 10 100
Figure 7.SimilartoFig.4,includingresultsfortheSBF(opentriangle),ENEAR(solidtriangle),SFI++g(opensquare),SNIa(solid
square),SFI++f(opencircle),DEEP(solidcircle)andSC(cross)mocks.TheDEEPcompilationincludestheSC,SNIa,SMAC,EFAR
andWillicksurveys.
7 MOVING BEYOND N-BODY theory.However,forlineartheoryresultstobeapplicable,as
SIMULATIONS: MACH PREDICTIONS mentionedinSec.1,oneneedstocorrectforthenonlineari-
USING PKANN tiesintheobservedvelocityfield.Anyresidualnonlinearity
can still bias the Mach predictions.
InSec.6,weshowedthatforvelocitysurveyswithlowcon-
tamination from small scales, reasonably accurate predic- In this section, we attempt to predict M(R) using
tionsfortheMachnumbercanbemadebyextractingmocks PkANN(Agarwaletal.2012a)–aneuralnetworkinterpola-
havingaGaussianradialprofilee−r2/2R2,Rbeingthechar- tionschemetopredictthenonlinearmatterpowerspectrum
acteristic depth of the survey being studied. uptok<∼0.9hMpc−1betweenredshiftsz=0−2.Although
A further simplification in the Mach analysis one can PkANNaccuracyworsens(startsunder-predictingthenon-
hope to achieve is to be able to predict M(R) as a function linear spectrum for k>∼0.9hMpc−1), we do not attempt to
ofscaleRwithoutresortingtoN-bodysimulations.Running correctthisbysmoothingthevelocityfieldovertherelevant
high-resolution N-body simulations, even in the restricted spatial scale. In Fig. 8, we replace linear theory predictions
parameterspacearound7-yrWMAP (Komatsuetal.2011) shown in Fig. 7, with the ones calculated using PkANN
central parameters, is beyond present day computing capa- for the LD cosmology. PkANN (solid lines) gives an excel-
bilities. It would be much easier and faster to explore the lent match with the N-body results (dashed lines) on all
parameter space using a prescription for the matter power scales, showing PkANN can substitute numerical simula-
spectrum, and using Eq. 6 and Eq. 8 to predict the cosmic tionsforthepurposeofcalculatingthe Machnumber given
Machnumber.Sofar,thishasbeenpossiblebyusinglinear a set of cosmological parameters. Although, we have shown
(cid:13)c 0000RAS,MNRAS000,000–000
10 Agarwal & Feldman
660000
1
440000
220000
00
2200 4400 6600 8800 110000 20 40 60 80 100 10 100
Figure 8.SimilartoFig.7,butinsteadofshowinglineartheorypredictions,weplotpredictionsbasedonthenonlinearmatterpower
spectrumfortheLDcosmologyestimatedusingPkANN.
PkANN’s performance for only the LD cosmology, it is ex- full3Dvelocitymeasurements.Theline-of-sightcomponent
pected to perform satisfactorily for cosmologies around 7- extends the tails of the survey window functions in k-space
yr WMAP central parameters for which PkANN has been (see Grinstein et al. 1987; Kaiser 1988). This is the reason
specificallytrained.SeeAgarwaletal.(2012a)fordetailson why in our analysis, we do not use W2(kR) = e−K2R2; in-
the parameter space of PkANN’s validity. stead,wecomputetheidealwindowfunctionsusingonlythe
line-of-sight information (see Eq. 17). The extended tails of
theidealwindowfunctionscanbeseeninFig.6andshould
8 MACH NUMBER ESTIMATES FROM REAL be contrasted against W2(kR)=e−K2R2.
CATALOGUES Ma, Ostriker, & Zhao (2012) estimated the character-
(cid:80) (cid:80)
istic depth of these surveys using w r / w where
Ma, Ostriker, & Zhao (2012) measured Mach number for n n n
w = 1/(σ2 + σ2). Specifically, they found depths of
four peculiar velocity surveys (SBF, ENEAR, SNIa and n n ∗
16.7, 30.5, 30.7and50.5h−1MpcfortheSBF,ENEAR,SNIa
SFI++f ) and found that the ΛCDM model with 7-yr
andSFI++f,respectively.However,fromFig.6andTable3
WMAP parametersismildlyconsistentwiththeMachnum-
(rows one, two, four and five), we show that these surveys
ber estimates for these four surveys at 3σ CL. However,
probe scales of ∼ R = 10, 19, 23 and 30h−1Mpc, respec-
as the authors mention in their work, their estimates are
tively. Using linear theory with Tophat filters, and neglect-
based on using linear approximation for the power spec-
ing the survey window functions while estimating the effec-
trum. Given the fact that at low redshifts structure forma-
tivedepths,makesthebulkflow(andanyderived)statistic
tion has gone nonlinear on small scales, it is necessary to
highly complicated to interpret.
considernonlinearitieswhenmakingtheoreticalpredictions.
InSec.6.3,weusednumericalsimulationstostudythe
Comparing Figs. 7 and 8 (middle panels), one can see that
Mach statistic for SBF, ENEAR, SFI++g, SNIa, SFI++f,
dispersionissignificantlyboostedbynonlinearities,lowering
DEEP and SC mocks. In this section, we calculated the
the Mach predictions (third panels) by 1σ level.
Mach number using the real catalogues themselves. The
Further, they work with Tophat window functions in
results are shown in Fig. 9 and summarized in Table 3,
theiranalysis.ATophatfilterassumesavolume-limitedsur-
columns 7−9. We find the Mach observations lie within
vey with a sharp edge in real space. However, the number
∼1.5σ interval for a ΛCDM universe with LD parameters.
density of objects sampled in a real survey typically fall at
The high uncertainty in the Mach number for the SC cata-
large distances. Real surveys thus have a narrower depth
logue we attribute to its poor sky coverage.
than what a Tophat would suggest. The sharp edge of a
Tophat creates both ringing and extended tails in k-space.
Sinceitisthesmallscalemodesthataremostcontaminated
by nonlinearities at low redshifts, a Tophat filter leads to
9 DISCUSSION AND CONCLUSIONS
aliasing of small-scale power onto larger scales. As such, a
Tophat filter is a poor choice if one wants to isolate the The estimates of bulk flow and dispersion on scale R are
contribution from small scales. subjecttoobservationalerrorsstemmingfromtheaccuracy
ItisworthmentioningherethatusingaGaussianwin- levels of distance indicators used, and the survey geometry.
dow function W2(kR) = e−K2R2 over-damps the high-k Typically, the velocity power spectrum is smoothed using
tails associated with a Tophat. The reason being that we Tophat or Gaussian filters, with results depending on the
onlyobservetheline-of-sightcomponentofthevelocityfield, exact smoothing procedure used. Often, bulk flow results
whereas the equations presented in Sec. 2 are based on the are quoted and inferences drawn about our cosmological
(cid:13)c 0000RAS,MNRAS000,000–000