On the origin of intrinsic alignment in cosmic shear measurements: an analytic argument

0.26 MB·English

by  Giovanni Camelio

Astronomy&Astrophysicsmanuscriptno.Camelio_Lombardi cESO2015
(cid:13)
January14,2015
On the origin of intrinsic alignment in cosmic shear
measurements: an analytic argument
GiovanniCamelio1,2 andMarcoLombardi1,3
1 UniversityofMilan,DepartmentofPhysics,viaCeloria16,I-20133Milan,Italy
2 UniversityofRome“LaSapienza”,DepartmentofPhysics,p.leAldoMoro2,I-00185Rome,Italy
3 Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA
5 Received***date***;Accepted***date***
1
0
2 ABSTRACT

n Galaxyintrinsicalignmentcanbeaseveresourceoferrorinweak-lensingstudies.Theproblemhasbeenwidelystudiedbynumerical
a simulationsandwithheuristicmodels,butwithoutacleartheoreticaljustificationofitsoriginandamplitude. Inparticular,itisstill
J unclearwhetherintrinsicalignmentofgalaxiesisdominatedbyformationandaccretionprocessesorbytheeffectsoftheinstantaneous

3 tidalfieldactinguponthem. Weinvestigatethisquestionbydevelopingasimplemodelofintrinsicalignmentforellipticalgalaxies,
1 basedontheinstantaneoustidalfield.Makinguseofthegalaxystellardistributionfunction,weestimatetheintrinsicalignmentsignal
andfindthatalthoughithastheexpecteddependenceonthetidalfield,itistooweaktoaccountfortheobservedsignal. Thisisan

] indirectvalidationofthestandardviewthatintrinsicalignmentiscausedbyformationand/oraccretionprocesses.
O
Keywords. gravitationallensing:weak–galaxies:kinematicsanddynamics–galaxies:halos
C
.
h
p1. Introduction From the modeling point of view, one of the main difficulties
- isrelatedtotheso-calledintrinsicalignment,thatis,thegalaxy
o
Ithaslongbeenrecognizedthatgravitationallensingisapartic-
r intrinsic shape alignment, which is not caused by gravitational
tularlyrobustmethodtoinvestigatethemassdistributionofmas- lensing, but by the source gravitational field (Heavensetal.
s
siveastronomicalobjects(e.g.,seeRefsdal1964;Fuetal.2008;
a 2000;Croft&Metzler2000;seealsoTroxel&Ishak2014fora
[Jauzacetal. 2012). Gravitational lensing is insensitive to the recentreview).Althoughtheeffectsofintrinsicalignmentcanbe
chemical and physical state of the deflecting matter and there-
reducedbyremovingpairsofgalaxiesthatarephysicallyclose
1foretreatsordinaryanddarkmattercomponentsofthedeflector
to the measurementof the shear correlation(King&Schneider
v
equally. In addition, it dependson a clearly understoodphysi-
4 2002;Heymans&Heavens2003),thereremainsaresidualbias
calprocess, the distortionofthe space-timeinducedbymasses
1 (calledGI mode,see below). Therefore,it isimportanttoclar-
as described by General Relativity, for which it is notrequired
0 ify these effects and to assess their impact on current and fu-
that the deflector be in a state of dynamical equilibrium. As a
3 ture cosmic-shear surveys. Although some heuristic models
0result, it currently represents our best tool to study the matter for intrinsic alignment are available (e.g., Catelanetal. 2001;
1.distributionintheUniverse. Hirata&Seljak 2004), they are unfortunately incomplete; in
0 A particularly important use of weak-lensing methods, particular, they lack an analytic estimate of the bias magnitude
5which are based on the statistical analysis of the weak corre- (in general, such an estimate would require the use of compli-
1lationamongtheobservedshapesofdistantgalaxies,isstudying cated galaxy formation and stellar dynamical models). There-
:the large-scale structure of the Universe. In weak cosmologi- fore,onehastoresorttothebiasmagnitudemeasuredfromsim-
v
callensing,thedeflectingmatterstudied(incontrasttothecase ulations(Croft&Metzler2000;Heavensetal.2000;Jing2002;
i
Xof a cluster of galaxies) is distributed along the entire line of Heymansetal.2006;Tennetietal.2014)foracomparisonwith
rsight. Asaresult,thedistinctionbetweensourceandlensisless theobservations.
aclear: thelightofeverygalaxythatcontributestothedistortion
ofspace-timeisbentbythedistributionofmatteralongitspath. Theexactoriginofintrinsicalignmentisstilldebated(e.g.,
The correlation between the apparent ellipticities of every pair seePereira&Kuhn2005),anditisstillunclearwhetherintrinsic
of galaxies determines the signal; this is called cosmic shear. alignmentof galaxiesis dominatedby formationand accretion
Sincecosmicshearisrelatedtothepowerspectrumoftheden- processesorbytheeffectsoftheinstantaneoustidalfieldacting
sitycontrast(e.g.,seeMiralda-Escudé1991;Kaiser1992),with uponthem. The commonlyacceptedview, supportedby simu-
thistoolonemeasuresthestatisticalpropertiesofthelarge-scale lationsandobservations,isthattheintrinsicalignmentiscaused
structureintheUniverse.Therefore,weakcosmologicallensing by the tidal field acting at the formation epoch (Catelanetal.
isanindependentmethodfortestingcosmologicalmodels(e.g., 2001;Hirata&Seljak2004)andpossiblyonthemergerhistory.
seeKilbingeretal.2013). In this paper we strengthen this view with analytic arguments
Cosmic-shear measurements are particularly challenging: usingareductioadabsurdum:thatis,westartwiththeassump-
the signal is very weak and is affected by many error sources. tion that intrinsic alignment is dominated by the instantaneous
tidal field currently at work on galaxiesand show that the pre-
Sendoffprintrequeststo:[email protected] dictedeffectisseveralordersofmagnitudeweakerthantheob-
Articlenumber,page1of11
servedeffect. In ourmodel, we considera sphericalearly-type where a and a are the semi-major and semi-minor axes of
M m
galaxyandstudythedeformationofitsshapetofirstorderafter the isophotes. If the light paths are bent weakly, the observed
introducinga weak externaltidalfield. We describethe effects ellipticityεofthegalaxyis(e.g.,Bartelmann&Schneider2001)
of the external tidal field on the intrinsic (i.e., unlensed) shape
of the elliptical galaxy, determining the luminous quadrupole ε εs+γ, (4)
≃
of the galaxy by means of its collisionless stellar distribution
function. The galaxy ellipticity is then computed through the whereεs istheintrinsic(i.e.,unlensed)shapeofthegalaxyand
luminousquadrupole. We find the expecteddependenceof the γisthecosmicshear.
ellipticitytothetidalfield,aspostulatedbytheheuristicmodel Incosmic-shearstudiesitisconvenienttodefinethreeellip-
ofCatelanetal.(2001),buttheconstantofproportionalitydeter- ticitycorrelationfunctions:
minedbyourmodelismorethanfourordersofmagnitudelower
C (θ)= ε (θ )ε (θ +θ) , (5)
thanthatobserved(Joachimietal.2011),andconsequentlycan- ++ h + 0 + 0 iθ0
notaccountfortheintrinsicalignment. C (θ)= ε (θ )ε (θ +θ) , (6)
×× h × 0 × 0 iθ0
Thepaperisorganizedasfollows: inSect.2wepresentthe C (θ)= ε (θ )ε (θ +θ) , (7)
problem of intrinsic alignment, summarize the relevant classi- +× h + 0 × 0 iθ0
fications, and outline the properties of different types of align- whereε andε aretherealandimaginarypartofthecomplex
+
ment. In Sect. 3 we introduce the stellar distribution function ellipticity in a ×coordinate system with the real axis along the
andestimatetheellipticityinducedbyanexternaltidalfieldon lineconnectingthetwocorrelatingpoints(thecentersofthetwo
an ellipticalgalaxy. In Sect. 4 we compareourresultwith that galaxies,θ andθ +θ). Thecorrelationfunctionsonlydepend
0 0
ofJoachimietal.(2011)andapplyourmodeltothegalaxydark onthemodulusofθbecauseofisotropy.
matterhalo.WedrawourconclusionsinSect.5. InAppendixA Wenowconsiderapairofgalaxiesandusethesuperscripts
we study the dynamics of elliptical and spiral galaxies in the fg and bg to denote foregroundand backgroundobjects. Then
rigid-bodyapproximation.Finally,inAppendixB,wederivean wefind
alternativeintrinsicalignmentellipticitydeterminedbyinstanta-
neoustidalfieldsbymeansoftheequipotentialsurfacesofellip- C = εfgεbg = γfgγbg+γfgεs,bg+εs,fgγbg+εs,fgεs,bg
ticalgalaxies. Throughoutthispaperwetakeastandardcosmo- ij h i j i h i j i j i j i j i
logicalmodelwithHubbleconstantH0 =70kms−1kpc−1,mass =Cij,GG+Cij,GI+Cij,II , (8)
densityparameterΩ =0.3.
m
where[i; j] = +; ,γfg (γbg)isthecosmicshearactingonthe
{ ×}
foreground (background) object, C = γfgγbg is the cos-
ij,GG h i j i
2. Generalcontext micshearcorrelationterm,andC = γfgεs,bg+εs,fgγbg and
ij,GI h i j i j i
2.1.Cosmicshear C = εs,fgεs,bg arethegravitational-intrinsicandtheintrinsic-
ij,II h i j i
intrinsic correlation terms (see below). We note that the dom-
Inprinciple,unbiasedshearmeasurementscanbeobtainedfrom
inant term in C is εs,fgγbg ; the other term, γfgεs,bg , is
the luminous quadrupole tensor Qij of each galaxy in a given ij,GI h i j i h i j i
fieldofview(e.g.,Bartelmann&Schneider2001) only expected to be non-negligibleif there are long-range cor-
relationsin the gravitationaltidal field (e.g., a dark matter fila-
I(θ)θθ d2θ mentalongthelineofsight),andcanbeignoredinmostsitua-
i j
Q = , (1) tions. In thecase ofa flat cosmology,the correlationfunctions
ij R I(θ)d2θ are related to the density contrastpowerspectrum P (e.g., see
δ
Bartelmann&Schneider2001;Catelanetal.2001)
where I(Rθ) is the surface brightness observed at angular posi-
tionθ,definedasthetwo-dimensionalvectorfromtheluminous C (θ)+C (θ)= ∞ ℓdℓP (ℓ)J (ℓθ), (9)
coefnqtueardorfupthoelegmaloamxyentotsaprgoivdeuncepsomineta.sIunrermeaelnittys,wthitihsvdaenfiinsihtiinogn ++,GG ××,GG Z0 2π γ 0
ℓdℓ
signal-to-noiseratiobecausethenoiseatlargeradiiθ= θ dom- C (θ) C (θ)= ∞ P (ℓ)J (ℓθ), (10)
inatesEq.(1). For thisreason,in actualweak-lensingkobkserva- ++,GG − ××,GG Z0 2π γ 4
tionsoneresortstoweightedquadrupoletensorsortoalternative C+ ,GG(θ)=0, (11)
×
meaFsourremseimntptleiccihtny,iquwese(e.ugs.,eKadiesefirneittioanl.19(19)5).here, which Pγ(ℓ)= 9H404cΩ42m wH Wa22((ww))Pδ wℓ,w dw,
has convenient transformation properties. In this case Z0 !
(12)
(Schramm&Kayser 1995; Seitz&Schneider 1997) and under
tihsheinagssauvmerpatgioen,tthheatobthseervuendlecnosmedplseoxuerclleipetlilciiptyti,cditeiefisnehdavaesvan- W(w)= wH p(w′)w′w−wdw′ . (13)
Zw ′
ε= Q11−Q22+2ıQ12 , (2) Here Pγ is the cosmic-shear power spectrum, J0 and J4 are
Bessel functionsof the firstkind, p(w) isthe probabilityof de-
TrQ+2√detQ
tectingagalaxyatcomovingdistancew,andw isthecomoving
H
horizondistance.
providesanunbiasedestimateofthe(reduced)shear.Foralumi-
To obtain relevant cosmological results from cosmic-shear
nosity distribution with constant flattening elliptical isophotes,
measurements, it is important to understand each error source.
themodulusofthecomplexellipticityis
From the theoretical point of view, an important error source
a a is the contribution to the ellipticity correlation induced by the
|ε|= aM−+am , (3) intrinsicalignment,Cij,GI+Cij,II.
M m
Articlenumber,page2of11
G.Camelio,M.Lombardi:Ontheoriginofintrinsicalignmentincosmicshearmeasurements:ananalyticargument
2.2.Intrinsicalignment Here εs is the galaxy ellipticity in the plane perpendicular to
the line of sight, U represents the large-scale potential (at
ext
As noted previously in computational (Heavensetal. 2000;
the galaxy formation epoch) associated with inhomogeneities
Croft&Metzler 2000) and theoretical studies (Catelanetal.
of cosmological origin, and C is a constant that in principle
2001),unlensedellipticitiesneednotbedistributedisotropically.
could be determined by “a complete galactosynthesis model”
Inparticular,closepairsofgalaxiesmayexhibitcorrelatedellip-
(Catelanetal. 2001). Catelanetal. (2001) indirectly estimated
ticitiesasaresultoftheirmutualgravitationalinteractionandof
thisconstantinaheuristicwaybymeansoftherelation
anexternalgravitationalpotential.
The intrinsic alignment (IA) is the contribution to the ob- 8 3 2 1 3J (kR) 2
servedgalaxyellipticitycorrelation,causedbythegravitational hε2i= 15C2 2ΩmH02 2π2 ∞k2Pδ(k) 1kR dk, (16)
tidal field in which galaxies are placed. In the literature it is ! Z0 !
usuallyassumed(Catelanetal.2001;Hirata&Seljak2004)that where R = 1h 1Mpc is the characteristic scale over which
−
the galaxyintrinsic ellipticity is frozen at the formationepoch.
a galaxy forms. An II alignment for elliptical galax-
IAisclassifiedaccordingtothephysicalgeneratingmechanism
ies has indeed been detected in numerical simulations
(Catelanetal.2001),whichisdifferentaccordingtothetypeof
(Croft&Metzler 2000) and observations (e.g. Pereira&Kuhn
galaxythattheexternalgravitationalfieldactsupon.
2005; Agustsson&Brainerd 2006; Hirataetal. 2007;
Inellipticalgalaxies,theexternalgravitationalfieldstretches Faltenbacheretal.2009;Joachimietal.2011).
the galaxy shapes (Croft&Metzler 2000), partly determining For early-type galaxy samples with broad redshift distribu-
theirintrinsicellipticities.1 Incontrast,inspiralgalaxiesthein- tion and for typical foreground mass inhomogeneities, the GI
trinsicellipticityisrelatedtotheirangularmomentum,produced contributionisnon-vanishingandisexpectedtobegreaterthan
bythetorqueprovidedbytheexternalgravitationalfieldduring thatofthe IIcontribution(Hirata&Seljak2004). TheGIterm
galaxyformationandbyprojectioneffects(Heavensetal.2000). is due to the tidal effects of a mass overdensity on foreground
From a geometrical point of view (Hirata&Seljak 2004), galaxies,andthelensingeffectsofthesameoverdensityonback-
the intrinsic correlation (for a second-order statistics, like the groundgalaxies.Therefore,therelevantfactoristheprobability
shear two-point correlation function) is made of two contribu- ofobservinganellipticalgalaxyintegratedalongthelineofsight
tions:intrinsic-intrinsic(II)andgravitational-intrinsic(GI).The fromthe massoverdensitytothe horizon. Incontrast, therele-
II contributionis the correlation of the unlensed ellipticities of vantfactorfortheIItermistheprobabilityofobservingasecond
two physically close2 source galaxies, generated by a corre- galaxyatthesameredshiftasthefirstgalaxy.
lation in the gravitational field felt by the galaxies (which is Finally, we note that accordingto this model, the GI align-
caused by their mutual interaction and by the external large- ment produces an anticorrelation of ellipticities (C < 0):
GI
scalestructure). Incontrast,theGIcorrelationisthecorrelation the foreground elliptical galaxy physical (intrinsic) shape is
between the observed ellipticities of two galaxies that are dis- stretched along the gravitational tidal field, whereas the back-
tant from each other, but angularlyclose in the sky view plane ground galaxy apparent shape is stretched perpendicular to it
(Hirata&Seljak2004).Thiscorrelationiscausedbyforeground because of lensing (Hirata&Seljak 2004). For example, in a
massinhomogeneitiesthatproducetwoeffectsthroughthetidal galaxycluster,clustermemberswouldbepreferentiallyaligned
field: they induce an intrinsic ellipticity in foregroundgalaxies radially,whereasthebackgroundgalaxieswouldbeseenprefer-
throughdirectgravitationalinteractionandmodifytheobserved entiallytangentially.
ellipticities of background galaxies by means of gravitational
lensing.
2.2.2. Spiralgalaxies
Inspiralgalaxiesthe IIcontributionis thoughtto becaused by
2.2.1. Ellipticalgalaxies
thetorqueprovidedby theexternaltidalfield duringformation
Elliptical galaxies are expected to be polarized by the ex- (Heavensetal.2000;Catelanetal.2001;Hirata&Seljak2004,
ternal tidal field they are located in (Croft&Metzler 2000; 2010). Thiscontributionistosecondorderintheexternaltidal
Catelanetal.2001), field because the tidal field first has to generate an anisotropic
momentofinertiaI (asinellipticalgalaxies,thedistributionof
ij
εs =C(∂2 ∂2)U , (14) massisexpectedtobeelongatedalongthegravitationalfieldgra-
+ 1− 2 ext
εs =2C∂ ∂ U . (15) dient),andthentotorqueit(Peebles1969;Doroshkevich1970;
1 2 ext
× White 1984). The torque provided by the external tidal field
generatestheangularmomentumofthespiralproto-galaxy(this
1 Ellipticalgalaxiesmaybenon-sphericalontheirown,independently
is similar to whatwe show in AppendixA, where we compute
ofanytidalstretchingbecauseofanisotropicpressureandangularmo-
mentum. the torque generated by the external tidal field on an elliptical
2 In cosmic-shear measurements, the II signal is usually suppressed galaxynotalignedwithit). Onethenneedstospecifythecorre-
byexcludingpairsofgalaxiesfromtheanalysisthatarethoughttobe lationbetweentheangularmomentumandthegalaxyellipticity,
physicallyclose(withtheirphysicaldistanceestimatedfromphotomet- for example by assuming zero thickness for the (circular) disk
ricredshiftsandangulardistanceinthesky). However,theassumption ofthegalaxyandconsideringtheprojectioneffects,asreported
that only physically close galaxies have II alignment might not hold byHeavensetal.(2000). Finally,sinceacorrelationinthetidal
becausetheIIalignment couldarisealsoinfarawaygalaxiesifthere
fieldinducesacorrelationintheangularmomentaofclosespiral
are long-range correlations in the gravitational field (e.g., those asso-
galaxypairs, the expectedII power spectrum can be computed
ciatedwithadarkmatterfilament,thetypicallengthscaleofwhichis
analytically,asshownbyHirata&Seljak(2004).
numericallyestimatedas100h 1MpcbySpringeletal.2005; seealso
− The II contribution for spiral galaxies is of higher order
Mandelbaumetal.2006;Lee2004;Hirataetal.2007).Moreover,since
thedistancebetweengalaxiesisestimatedfromphotometricredshifts, in the tidal field and thus should be lower than for ellipticals
galaxies thought to be physically distant may actually be physically (Hirata&Seljak2004). Inaddition,littleGIcontributionisex-
closeasaresultoferrorsinthephotometricredshiftestimates. pected for spiral galaxies (Hirata&Seljak 2004). At present,
Articlenumber,page3of11
thereisnoclearobservationalevidenceofIAinspiralgalaxies 1966)forthestellardistributionfunction3
(Hirataetal.2007;Faltenbacheretal.2009;Blazeketal.2012).
In AppendixA we determinethe rigid-bodyprecessionpe- f (x,x˙,t)= A e−U(x)/σ2v−kx˙k2/(2σ2v)−e−Etr/σ2v E < Etr , (19)
riodofspiralgalaxies,whichislongerthanthegalaxydeforma- ⋆ 0 E E ,
 h i tr
tiontime. Thisshowsthatnoprecession-drivenmechanismfor  ≥
abnedaelivginsmede.ntof spiral galaxieswith the externaltidal field can where E = U(x)+ kx˙2k2 is the single-star energy and σv is the
stellarvelocitydispersion,andwestartfromasphericalunper-
turbed(early-type)galaxywithpotentialU . Inotherwords,in
0
the unperturbedgalaxy the DM supplies the exact contribution
3. Distributionfunctionmethod tohavethegivenpotentialU andtheKingdistributionfunction
0
for f .
⋆
Even if galaxies are many-body systems, it might naïvely be
We take anexternaltidalpotentialandadditto the (unper-
thought that their alignment is driven by rigid-body motions
turbed)galaxypotential,thusignoringthechangesinthegalaxy
(e.g., oscillations and precessions) under the effects of exter-
potentialinducedbythedeformationofthegalaxy(forasimilar
nal gravitational forces. However, an analysis of the relevant
not self-consistent approach, see Ciotti&Dutta 1994; see also
timescalesshowsthatrigid-bodymotionsaremuchslowerthan
Bertin&Varri2008). AspointedoutbyCiotti&Dutta(1994),
those characterizing internal dynamics (Appendix A, see also
it is possible to neglect the actual self-force of the galaxy be-
Ciotti&Dutta 1994). ThismeansthatatleastforIApurposes,
cause it decreases from the center of the galaxy (see, e.g., the
the galaxyis beststudiedas a deformingbody. In thissection,
potential(28) of the unperturbedself-field), whereas the exter-
westartfromthehypothesisthatgalaxiesarenotsubjecttoany
naltidalforceincreases. Therefore,wetakeU =U +U (see
intrinsicalignmentduringformationandthatintrinsicalignment 0 tidal
Bertin&Varri 2008; Varri&Bertin 2009 for a self-consistent
is entirely due to tidal fields at the observation time, to which
approachappliedto a globularstellar cluster). We take a weak
galaxies react immediately. As shown in Sect. 4, this assump-
externaltidalfield U /σ2 1andexpandthestellardistri-
tionleadstoanextremelylowIA,completelyinconsistentwith | tidal v| ≪
butionfunctionwithrespecttoit
observations. Thisis anindirectproofthatIA isdrivenbyfor-
matiTohneadnedfomremrgatiinogn.of an elliptical galaxy subjectto an exter- f⋆ = Ae−U0(x)/σ2v−kx˙k2/(2σ2v) 1−σ−v2Φijxixj+... −Ae−Etr/σ2v
nal gravitational field can be modeled in a simplified way by = f⋆(0)+ f⋆(1)+... , (cid:0) (cid:1) (20)
spteunddyiixngBt)h,eanddefionrmaamtioonreofcoitmspelqeuteipmotaennntiearl,sbuyrfamceesan(sseoefAthpe- f⋆(0) = A e−U0(x)/σ2v−kx˙k2/(2σ2v)−e−Etr/σ2v , (21)
stuterlbleadrdspishtreirbiucatilognafluaxnicetsiofno.rIsnimbpoltihcictyasaens,dwsteusdtyartthweiethlliupntipceitry- f⋆(1) = −(cid:16)σ−v2ΦijxixjAe−U0(x)/σ2v−kx˙k2/(2(cid:17)σ2v) , (22)
inducedbytheexternaltidalfield. Thesphericalassumptionis where Φ = 1∂∂ U is the tidal tensor and U =
quitestrong,anditmaynotbestraightforwardtogeneralizeour ij 2 i j ext tidal
1xx ∂∂ U =Φ x x isthetidalfield(potential).
resultstothecaseofanensembleaverageofellipticalearly-type 2 i j i j ext ij i j
This expansion is valid only inside the galaxy
galaxies. Nevertheless, since we are interested in an order-of-
(Bertin&Varri 2008); in fact, even if the exponential can
magnitudeestimate,thesphericalassumptionissufficientforour
always be expanded (when U /σ 1), this expansion is
purposes. | tidal v| ≪
notgrantedtobemeaningfulunlesstheperturbedself-potential
U + U is not too different from the actual galaxy self-
0 tidal
3.1.Generalcase potential, U. This condition breaks down near the galactic
boundaries because, as stated above, the galaxy self-potential
Agalaxyisacomplexmany-bodysystem,withagravity-driven decreasestowardtheboundaries,whereastheexternaltidalfield
dynamics (Chandrasekhar1942; Bertin 2000). In this context, increases (since Φ is traceless, the isopotential surfaces may
ij
an important tool is the stellar distribution function f⋆(x,x˙,t), notevenbeellipseswhen Utidal U0). FromEq.(1), chang-
which is, essentially, the stellar density distribution in phase ing integration variables to| thos|e≃of| sta|ndard two-dimensional
space. We also define (e.g., Bertin 2000) the (total) distribu- vectors,consideringthatI(x) ρ (x)dx ,andusingEq.(18),
⋆ 3
tionfunction f = f⋆+ fDM asthesumofthestellardistribution weobtainfortheluminousqu∝adrupole
function and the dark matter distribution function. We assume R
thecollisionlessBoltzmannequationandthePoissonequation: 1
Q = f(0)+ f(1)+... xx d3xd3x˙
ij D2N ⋆ ⋆ i j
∂f ∂f ∂f ∂U s Z
∂t + ∂x x˙i− ∂x˙ ∂x =0, (17) (cid:0) (cid:1) = Q(i0j)+Q(i1j)+... , (23)
i i i
where D is the angular-diameterdistance to the galaxy and N
s
isthenumberofstarsofthegalaxy. TheintegrationofEq.(23)
hastobecarriedout(i)intheregion
2U(x,t)=4πGρ(x,t)=4πGm f(x,x˙,t)d3x˙
∇
Z x˙ v (r)= 2 E U (r) (24)
max tr 0
=4πGm f (x,x˙,t)d3x˙+4πGm f (x,x˙,t)d3x˙ , (18) k k≤ −
⋆ DM q
Z Z 3 TheKingmodelis(cid:0)usuallyappli(cid:1)edtostudyglobularclusters; more
complex distribution functions are generally used for galaxies. Here,
wherem is the meanmass ofa galaxystar (andof a DM “par- wechoose theKing model for itssimplicityand abilityto clarifythe
ticle” as well). In the following,we use the Kingmodel(King keyaspectsofthedynamicalmodelweareinterestedin.
Articlenumber,page4of11
G.Camelio,M.Lombardi:Ontheoriginofintrinsicalignmentincosmicshearmeasurements:ananalyticargument
because of the energy truncation in the distribution function
0.1
and (ii) for x rmax < rtr, a condition that mimics a real 0.05
k k ≤
wnqoueuaanskt-qiltueyandrsirnugpoimsleeciahssoucsareernrmiesedonota,uswttbohaeasrveedotihodenraiengwtieoignnrdsaotwiwohnfeurienncttthhioeenel.uxTmtehrie-- 3920s]−− -0-0.0.015
max 1 -0.15
naltidalfieldbecomessimilartoorlargerthanthegalacticfield, U[0 -0.2
thatis,thecondition U (r ) U (r ) holds(seediscus- -0.25
tidal max 0 max
sionabove). Thezer|othorderof|≪the|expansio|noftheluminous -0.3
0.1
quadrupoleQ(0)isproportionaltotheidentitymatrixbecausethe
unperturbedgiajlaxyfieldisspherical: 3pc] 0.01
/k 0.001
⊙
M
Q(i0j) = (34Nπ)D2A2s F1−F2 δij (25) 11ρ[100 01.0e0-0015
1 = rmaxd(cid:16)rr4e−U0(r(cid:17))/σ2v vmax(r)v2e−v2/(2σ2v)dv 1e-060 5 10 15 20 25 30 35
F R[kpc]
Z0 Z0
=σ2v rmaxr4e−U0(r)/σ2v −vmax(r)e−v2max(r)/(2σ2v) Funigp.er1t.urbedSgealfl-apxoyt.enTthiaelv(a2l8u)esaunsdedtoatarel rmass=d1isktrpibc,utrion=(2259)kpocfatnhde
Z0 " M = 1011M . Thedashedverticallineincdoriecatesthetrutrncationradius
+σv π erf vmax(r) dr, (26) rtr. ⊙
2 √2σ
r v !#
2 = e−Etr/σ2v rmaxr4 vmax(r) 3dr. (27) FromEq. (2),consideringthatQ(i0j) ∝ δij (sphericalgalaxy),we
F 3 findforthecomplexellipticity
Z0
(cid:16) (cid:17)
tWotealumseaassgdailsatxryibluotgioanritihnmaicgaploatxeynt(ilaulmUi0no(uFsigp.l1u)stdoamrkomdealtttehre; εs Q(111)−Q(212)+2ıQ(112)
≃ TrQ(0)+2√detQ(0)
seee.g.Bertin2000;Koopmansetal.2006)
1
= F3 Φ Φ +2ıΦ . (34)
r +r r + x − 10σ2 11− 22 12
MG tr core log core k k if x <r , v F1−F2
r2 r +r k k tr (cid:16) (cid:17)
Uρ00((xx))== −4MπMrGtr +rkt2rr1xtrckor−e krx1tkr2!rrc coorreeco++rekkxxkktr2! ififkkxxkk<>rrtrtr,, ((2298)) Fdpsooeurnnilneclftsya-ehslfideildoeyeop,lndpnaw,ercatnaaeicndnpoudsabblraotettharinceifgnautavhclltaeteahorlxoertgyiiczdagiwetlaeyadlnexedfiiyarnierasbelpltdywece,roaoxswnnpiposirteiaenhdxrsetotstserifh:oriennintogasenlfx(soetitpitradspertaacmshlr)teet;afidstoeehsnlld,eldlieiyp.topsttTdeihscnehieizpdtiryseeep,nneiaidcnxtress--t
0 (cid:0) (cid:1) ifkxk>rtr , (e.g., Catelanetal. 2001). The difference with Catelanetal.
where M= 1011M is the galaxy mass, U0(rtr) = 0, rcore = (m2o0m01e)nitsotfhtahteinligEhqt.e(m34is)stihoen,eaxntedrnnaoltfithealdtaiscttihnagtaatctthinegfoartmthae-
1kpc is the core rad⊙ius, and r = 25kpc is the truncation ra-
tr tion epoch. Comparing Eq. (34) with Eqs. (14) and (15), we
dius. Throughout this paper we take for an elliptical galaxy
obtaintheanalyticexpressionfortheconstantofproportionality
σ 200kms 1 andr = 10kpc(seediscussionabove). We
v − max appearinginEqs.(14)and(15)
≃
thenobtain
1
1 6.1 10−5Mpc5(km/s)3 , (30) C =−20σ2 F3 ≃−4.2×1014yr2 . (35)
F ≃ × v F1−F2
4.6 10 5Mpc5(km/s)3 . (31)
F2 ≃ × − Equation(34)isthemainresultofthispaper:itprovidesadirect
way to estimate the intrinsic ellipticity of an early-type galaxy
Thefirst-orderperturbationtotheluminousquadrupoleis
thatissubjecttoanexternaltidalfield.
AΦ Tobetterappreciatethisresult,considerathinlenssuchasa
Q(i1j) =−ND2lσm2 xlxmxixje−U0(x)/σ2v−kx˙k2/(2σ2v)d3xd3x˙ galaxycluster. Inthiscase,wecanwritethelensshearas
s v Z
= (4π)2AF3 2Φ +δ TrΦ , (32) γ=c−2DlsDl ∂11 ∂22+2ı∂12 dx3U(x)
−15σ2vND2s ·(cid:16) ij ij (cid:17) c 2DDlsDs lL(cid:16)cl Φ− Φ +2(cid:17)ıΦZ , (36)
whereTrΦ=Φ11+Φ22+Φ33,and ≃ − Ds 11− 22 12
(cid:16) (cid:17)
rmax wherethederivativesare givenwith respectto thephysicalco-
F3 =σ2v r6e−U0(r)/σ2v −vmax(r)e−v2max(r)/(2σ2v) ordinates, Ds, Dl and Dls are the angular distance of a source
Z0 " galaxy,ofthelens,andbetweenthelensandthesourcegalaxy,
π v (r) respectively,and L is the galaxycluster typicalsize. Interest-
+σ erf max dr (33) cl
v 2 √2σ ingly, the expression for γ has the same form as Eq. (34), that
r v !# is, it dependson the same combinationof second-orderpartial
4.9 10−9Mpc7(km/s)3 . derivativesofthetidaltensorΦ.Theconstantsinvolved,instead,
≃ ×
Articlenumber,page5of11
whereRisthedistanceofthegalaxyfromtheDMstraightline
1
DMfilamentfield,equi (the filament), δ˜ = diag(1,1,0) and λ is the linear mass dis-
DMfilamentfield,f
0.1 Keplerianfield,equi tribution of the dark matter filament. Obviously, a3,tidal,λ = 0.
Keplerianfield,f We now need to estimate the quantity λ, the linear density of
0.01 theDM filament. DM filamentsare thoughtto extendbetween
massive galaxy clusters (e.g., Springeletal. 2005). We there-
0.001 fore consider the number density of massive clusters and esti-
sǫ|| matetheaveragedistancedclbetweentwoclusters. Todoso,we
0.0001 use the cumulative distribution of galaxy clusters measured by
Bahcall&Cen(1993):
1e-05
1e-06 n(> Mcl)=4×10−5MM∗e−MMc∗lh3Mpc−3 , (42)
cl
1/3
1e-07 d (M )= n(> M ) − , (43)
100 500 1000 cl cl cl
R[kpc] (cid:16) (cid:17)
where M = (1.8 0.3) 1014h 1M and h = 0.7. Setting
∗ −
Fig. 2. Radial dependence of the intrinsic galaxy ellipticity in the M = M leads4to± × ⊙
cl ∗
particularcases(KeplerianandDMfilamentfields)consideredinthis
paper, computed through the equipotential approximation (equi, see d 58Mpc, (44)
cl
≃
Appendix B) and through the distribution function formalism (f, see
Sect. 3). Note the different radial dependence of the Keplerian (R 3) which is consistent with the typical value reported by
−
andtheDMfilament(R−2)cases;theequipotentialapproximationand Springeletal. (2005), that is, 100Mpc. We assume that 34%
thedistributionfunctionformalismyieldthesameradialdependencein of the mass in the Universeis concentratedin filaments, as the
the twocases, but different multiplicative factors(see the text for de- numerical results of Hoffmanetal. (2012) indicates. We then
tails).
havethat
λ(M ) 0.34ρ d2 4.67 107M pc 1 . (45)
are clearly different: in particular, for a typical galaxy cluster cl ≃ m cl ≃ × ⊙ −
with L 1Mpcatz = 0.5andfora sourceatz = 1wefind
cl ≃ l s Ifweplacethefilamentinsuchawaythatitcrossesthesource
thatthecouplingconstantbetweentheshearγandthetidalfield
plane at R = (R,0), with R = 500kpc, and use Eq. (34), we
isabout 20C. Asaresult,inthiscontext(foragalaxycluster)
− obtain(Fig.2)
weexpectthattheintrinsicalignmentproducedbytheinstanta-
neoustidalfieldisnegligible.
Gλ
εs =+ F3 +1.4 10−3 . (46)
5σ2R2 ≃ ×
v F1−F2
3.2.Particularcases
We observe that in this case the dependence on the distance is
We nowconsideranexternalKeplerianfield, thatis, thepoten- εs =O(R 2).
−
tial generatedby a second galaxyof mass M , assumed to be
ext
sphericalandcenteredatR. Thetidalaccelerationandthetidal
potentialactingonastarofthefirstgalaxyare 4. Discussion
GM RR 4.1.Comparisonwithliterature
a = ext δ 3 i j x , (37)
i,tidal,M − R3 ij− R2 j It is common (e.g., Joachimi&Bridle 2010; Joachimietal.
!
GM RR 2011) to consider the galaxy number density contrast-intrinsic
Utidal,M =+ 2Re3xt δij−3 Ri2j xixj . (38) ellipticitycorrelation,thatis,
!
C (x)= δ (x )εs(x +x) , (47)
IfweplacetheexternalgalaxyatR = (R,0,0),usingEq.(34), gI h n 0 + 0 ix0
weobtain(Fig.2)
wherethe correlationfunctiononlydependsonthemodulusof
εs =+230GσM2Rex3t F3 ≃+4.6×10−6 , (39) εxs+beiscathueseriesaoltrpoapryt,oδfntihsethceomgaplalexxy(niunmtribnesricd)eenlsliitpyticcoitnytriansta,acnod-
v F1−F2 ordinatesystemwiththerealaxisalongthelineconnectingthe
whereinthelaststepweused M =1011M andR=500kpc. two correlatingpoints x0 and x, as in Eqs. (5), (6), and (7). In
ext
Wenotethatεs =O(R−3). ⊙ contrasttoEqs.(5)andfollowing,thecorrelationinEq.(47)is
Wenowexamineadifferentcase:theexternalfieldisgener- carried on in the three-dimensionalreal space. We now define
theFouriertransform f˜(k)ofafunction f(x)as
atedbyadarkmatter(DM)filament,andthegalaxyisoutsideit.
Weareconsideringatidalfieldgeneratedbyan(infinite)linear
distributionofmatteralongthelineofsightxˆ f˜(k)= f(x)eıkxd3x. (48)
3 ·
Z
2Gλ RR
a = δ˜ 2 i j x , (40) 4 Animportantpointtokeepinmindisthatthelarge-scalestructureis
i,tidal,λ − R2 ij− R2 j
stronglytime-dependent andthatthemeanhigh-mass-cluster distance
!
Gλ RR changeswithtime(Springeletal.2005). Here,weareconsideringthe
Utidal,λ =+R2 δ˜ij−2 Ri2j xixj , (41) local Universe, that is, regions where the redshift ismuch lower than
! unity.
Articlenumber,page6of11
G.Camelio,M.Lombardi:Ontheoriginofintrinsicalignmentincosmicshearmeasurements:ananalyticargument
Since 4.2.Halocase
δ˜(k)=b δ˜(k), (49) Untilnow,wehaveonlyconsideredthegalaxywithitstotal(lu-
n g
minousplusdark)mass. Wehavestoppedtheintegrationofthe
ε˜s(k)= C(k2 k2)U˜(k)=4πCGk12−k22ρ Ω δ˜(k), (50) luminous quadrupole at rmax < rtr. However, the ellipticity of
+ − 1− 2 k2 cr m the whole DM halo remains unclear. We remark that it is not
obvious that Eq. (2) is the correct choice for the halo elliptic-
wherebgisthegalaxybiasandinthesecondequalityofEq.(50) itybecausethehaloluminousquadrupolecannotbemeasuredin
we used Eq. (14), the Poisson equationand the relationρ˜(k) = real surveys. For the same reason, it is unclear how to choose
ρcrΩmδ˜(k) (we neglectthe three-dimensionalDirac delta in the rmax, butforthecondition U0(rmax) Utidal(rmax). Still, itis
origin).Ifweintroducetheellipticalgalaxyfraction fell(because interesting to consider the|extreme|c≫ase|of a dark |matter halo
onlyellipticalgalaxiesaligninourmodel),wefind M = 2 1013M ,r = 560kpcandσ = 250kms 1,asfound
tr v −
byGava×zzietal.⊙(2007)fromanensembleaverageover22mas-
(2π)3δ(3)(k k)P (k,z)= δ˜(k)ε˜s(k) sive halos. Even now, we obtain (taking r = 0.01 r and
− ′ gI h n + ′ i core · tr
r =0.80 r )
(2π)3δ(3)(k k′)4πbgfellCGρcrΩmPδ(k,z), (51) max · tr
≃ −
C 1.3 1017yr2 , (57)
where P (k,z) is the density contrast power spectrum at the halo
δ ≃− ×
galaxy redshift z and where we have approximated the term 4πC Gρ =0.0042, (58)
halo cr
−
(k2 k2)/k2 to unity. We may now consider Eqs. (6) and (19)
1 − 2
ofJoachimietal.(2011),thatis, which is still almost a factor 20 lower than the value of
∼
Eq. (53), that is, even if instantaneous tidal fields produce a
Ω 1+z η L β greaterIAondarkmatterhalosthanongalaxies,theeffectisstill
P(JMAB)(k,z,L)= AC b ρ m P (k,z) , (52)
gI − 1 g crD(z) δ 1+z L muchtooweaktoaccountfortheobservedsignal.Arguably,the
0! 0! galaxiesobservedbyJoachimietal.(2011)are,onaverage,less
whereD(z)isthelineargrowthfactornormalizedtounitytoday, massive objects, and therefore the value of Eq. (57) should be
P is the (nonlinear) power spectrum of the density contrast, z taken as an upper bound. In addition, we stopped the integra-
δ
and L are the redshiftand the absoluteluminosityof the early- tionnearthehaloboundaries(rmax =0.80 rtr),andtheresulting
·
typegalaxy,andthearbitraryreferencevaluesarez = 0.3and ellipticity is thereforegreaterthan thatwe would obtainwith a
0
L0, thelattercorrespondingtoanabsoluter-bandmagnitudeof smallerrmax.Thisshowsthatourmodelisessentiallyinadequate
22, passively evolved to z = 0. The parameters AC , η, and toexplaintheobservedamountofIA.
1
−
β have been measured by Joachimietal. (2011) for comoving
transverseseparationsgreaterthan6h 1Mpc,
−
5. Conclusions
AC ρ =0.077 0.008, (53)
1 cr ± Theexactoriginofintrinsicalignmentofgalaxiesisstillunclear,
β=1.13+0.25, (54) andthere are noanalytic estimatesof the amountofIA caused
0.20
η= 0.2−7+0.80. (55) by different possible processes. We here support the standard
− −0.79 view on IA (i.e., it is caused by formation and accretion pro-
cesses)byareductionadabsurdum. Toarriveatthisresult,we
ComparingEq.(51) toEq.(52)andneglectingtheredshiftand
estimated the amount of IA in elliptical galaxies due to an ex-
luminosity dependence,we obtain that Eq. (53) correspondsto
ternalinstantaneoustidalfieldbyconsideringthegalaxystellar
theterm
distributionfunctionandtheluminousquadrupole. Inaddition,
4πCGρ 3.2 10 6 (56) inAppendixA,wedeterminedthetypicaloscillationtime-scale
cr −
− ≃ × foranellipticalgalaxymodeledasarigidbody,subjecttoanex-
ofEq.(51),whereweset f =1becauseJoachimietal.(2011) ternaltidalfield,andinAppendixBwedeterminedtheellipticity
ell
ofagalaxy(subjecttoanexternalinstantaneoustidalfield)ina
consideredonlyearly-typegalaxiesfortheshapemeasurements.
different way by studying its equipotentialsurfaces. The main
Theresultisclearlyinconsistent.Itisunlikelythatthisinconsis-
resultsarethefollowing:
tencyisduetoouruseofthesimpleKingmodelbecausethedif-
ferentapproachadoptedinAppendixB,wherewedonotmake
useofthestellardistributionfunction,yieldssimilarresults(cf. 1. Thedistributionfunctionapproachallowsustoanalytically
Eqs.(39), (46), (B.8)and(B.10)). Then,atleastatlargescales determine the dependence of the intrinsic ellipticity on the
(r >6h 1Mpc),ourmodeldoesnotreproducetheobservations. tidalfield, heuristicallyformulatedbyCatelanetal.(2001),
p −
in terms of the properties of the galaxy (i.e., its mass,
The incompatibility of Eq. (56) with Eq. (53) means that
size, velocity dispersion, and stellar distribution function),
the galaxy deformation due to the instantaneous external tidal
Eqs.(34)and(35).
field cannot yield the observed IA signal. A possible explana-
tionisthattheIAsignaliscausedbythegalaxyformationpro- 2. The intrinsic alignment signal obtained when our model
cessand/oritsmerginghistory. Toobtainanalyticresults,these is applied to an elliptical galaxy is negligible with respect
processesthereforeneedto belinkedto the externaltidalfield. to the observed one (at least at large scales, > 6h−1Mpc,
Inparticular,atleastforthemerginghistory,thevelocityshear Joachimietal. 2011), cf. Eqs. (53) and (56). Thusone has
field needstobe consideredbecauseitwas recentlydiscovered toconsiderthegalaxyformationprocessand/oritsmerging
(Hoffmanetal. 2012; Libeskindetal. 2014; Lee&Choi 2014) history.
thatmergerspreferentiallyoccuralongthevelocityshearminor 3. When our model is applied to the whole galaxy halo, the
eigenvector. A detailed analysis of this process is beyond the intrinsicalignmentsignalincreases(butitisstillinconsistent
aimsofthispaper. withtheobservedsignal),Eq.(58).
Articlenumber,page7of11
Theworkwedescribedhereisasteptowardasimplephys- Miralda-Escudé,J.1991,ApJ,380,1
icalunderstandingof thebias introducedin weak-lensingmea- Peebles,P.J.E.1969,ApJ,155,393
Pereira,M.J.&Kuhn,J.R.2005,ApJ,627,L21
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ofconcernforcosmic-shearmeasurements.Morerecently,ithas Schramm,T.&Kayser,R.1995,A&A,299,1
Seitz,C.&Schneider,P.1997,A&A,318,687
beenregardedasanopportunitytoinvestigatethephysicalprop-
Springel,V.,White,S.D.M.,Jenkins,A.,etal.2005,Nature,435,629
ertiesofgalaxies,theirDMhalos,andtheirformationhistory.In Tenneti, A., Mandelbaum, R., DiMatteo, T., Feng, Y., &Khandai, N.2014,
thisperspective,itisimportanttodevelopanalyticmodelsofIA, MNRAS,441,470
Troxel,M.A.&Ishak,M.2014,arXiv:1407.6990[astro-ph.CO]
suchasthosepresentedhere,tobeabletointerprettheresultsof Varri,A.L.&Bertin,G.2009,ApJ,703,1911
futureweak-lensingsurveys. Inthis respect, a completemodel White,S.D.M.1984,ApJ,286,38
of IA could even be used directly to reconstructthe local tidal
fieldactingonellipticalgalaxies. AppendixA: Rigid-bodyapproximation
Thetheoreticalanalysiscouldbeimprovedbyapplyingthe
techniques we presented (i.e., the stellar distribution function Inthisappendixwecomparethetimescaleofmotionsofgalax-
method) to the study of the formationand accretion processes, iesmodeledasrigidbodieswiththetimescaleoftheirinternal
inordertodeterminetheircontributestotheIAsignal. Itwould dynamics.Inparticular,weconsiderthe(rigid)oscillationsofel-
alsobeinterestingtostudythecontributiontoIAduetocontin- lipticalgalaxiesandtheprecessionofspiralgalaxies. Weshow
uouslyappliedtidalfields(insteadofonlyconsideringthetidal belowthatinternaldynamicsismuchfasterthanrigid-bodydy-
fieldactingattheemissionepoch,aswehavedidhere),andthe namics,thusconfirmingtheresultsofCiotti&Dutta(1994)for
evolutionofIAwithredshift. ellipticalgalaxies.
Acknowledgements. WewishtothankGiuseppeBertinandBenjaminJoachimi
forinvaluable suggestionsandveryusefuldiscussions,whichsignificantlyim- AppendixA.1:Generalcase
provedthepaper. Wealsothanktheanonymousrefereeforhisorherhelpful
suggestionsconcerningthepresentationofthispaper. G.C.thanksMalegori’s We assume that the distribution of mass of the galaxyonly de-
family("FrancaErba"scholarship) andBCCCarugateeInzagoBank("Gildo pends on the elliptical radius r˜ and that its principal axes are
Vinco"award)forsupportinghisstudies.
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ρ(x,x˙,t)=ρ(x)=ρ˜(r˜), (A.1)
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Articlenumber,page8of11
G.Camelio,M.Lombardi:Ontheoriginofintrinsicalignmentincosmicshearmeasurements:ananalyticargument
xˆ3 Mext tude,wefindthattheperiodtosc,M ofanoscillationis
θ
DM R3
R t 2π
osc,M
≃ sGMext
xˆ2 3.0 1011 R 23 Mext −21yr. (A.14)
≃ × × 1Mpc × 1011M
φ ! ⊙!
xˆ We remark that Eq. (A.14) is reminiscent of the expression of
1
the free-fall time t (Gρ) 1/2 t (D/2R)3/2, being D
ff − osc,M
∝ ∝ ×
thediameterofthegalaxy.Usingthevaluesadoptedhereforthe
externalmass M =1011M andforthedistanceR=500kpc,
ext
theperiodist 1.0 10⊙11yr,whichisgreaterthantheage
osc,M
of the Universe5 (t≃he num×erical integration of Eq. (A.13) is in
Fig.A.1. Sphericalcoordinatesystemadoptedinthetext,Keplerian
accordancewiththeapproximateresult).
(M )andDMfilamentcases.
ext
Thisperiodhastobecomparedwiththetimescaleonwhich
the shape of a galaxychanges, because of the externalgravita-
AppendixA.2:Ellipticalgalaxies:oscillationperiod tionalfield. Aroughestimateofthisisthetimenecessaryfora
startotravelacrossthegalaxy(e.g.,Fleck&Kuhn2003)
ForaKeplerianpotential,thegalaxyenergyis
D
t = , (A.15)
W = GMextM GMextQij 3RiRj δ . (A.11) cross,ell σv
M − R − 6R3 R2 − ij where σ is the stellar velocity dispersion and D is the diame-
! v
ter of the galaxy. For D 40kpc and σ 200kms 1, we
v −
Withoutlossofgenerality,let bediagonalandletθ andϕbe obtaintcross,ell 2 108yr≃ tosc,M. Inother≃words,anellipti-
theanglesbetweenthexˆ axisaQndRandbetweenthexˆ axisand calgalaxy,not≃align×edwith≪thegravitationaltidalfield,deforms
3 1
R R xˆ ,inasphericalcoordinatesystem(seeFig.A.1).Inother itselfbeforecompletingarigidoscillation.
3
wo−rds·,weconsideracoordinatesystemwhoseaxesareparallel If we consider a DM filament along the xˆ1 axis, using
totheprincipalaxesofthegalaxy. Havingkept diagonal,we Eq.(41),wecanwritethegalaxyenergyas
Q
havetorotatethetidaltensor.ConsideringthatTr =0,wefind
Q R Gλ RR
W =+2MGλlog Qij 2 i j δ˜ , (A.16)
λ ℓ − 3R2 R2 − ij
W = GMextM GMext cos2(ϕ)sin2(θ) ′! !
M − R − 2R3 Q11 whereℓ isanarbitraryconstantintroducedtohaveanadimen-
′
+ sin(cid:16)2(ϕ)sin2(θ)+ cos2(θ) . (A.12) sional logarithmic argument, and δ˜ = diag(0,1,1). As before,
22 33
Q Q we rotate the coordinate frame in such a way that be diag-
ij
(cid:17) onal6 (see Fig. A.1), we use spherical coordinatesQand assume
We can in principle determine the oscillation equations of the
that the galaxy is prolate, that there are no proper rotation nor
galaxy via the Euler-Lagrange equations, using the correct ki-
precessionmotions,andthatthepositionofthegalaxyisfixed.
neticenergyexpression. Thisapproachleadstocomplexequa-
Wethenobtain
tions and therefore we assume that (i) the elliptical distribu-
tion of mass is prolate (i.e., r = r = r < r ), from which Gλ
1 2 eq 3 I θ¨ = sin(2θ) . (A.17)
I11 = I22 = Ieq and 11 = 22 = eq,and(ii)therearenoinitial eq 3R2 · Qeq−Q33
Q Q Q
intrinsicrotation(ϕ˙(0) = 0)or precessionmotions; there could
beonlynutation(θ˙(t),0).UsingtheEuler-Lagrangeequations, Usingthesameappr(cid:0)oximation(cid:1)sasbeforeandEq.(45),wemay
estimatetheperiodofasmallrigidoscillationofthegalaxy
wethenobtain
3
GM
Ieqθ¨ = 2Re3xt sin(2θ)· Qeq−Q33 . (A.13) tosc,λ ≃2πRr2Gλ
R
(cid:0) (cid:1) 1.0 1010 yr, (A.18)
We explicitly note that this equation is not self-consistent: in ≃ × × 1Mpc
fact, in principle it should be completed with the equations of !
the relative motion between the galaxy center of mass and the whichinthecaseconsideredthroughoutthepaper(R=500kpc)
monopole. However,we are notinterested in a completetreat- correspondsto 5.0 109yr. This is shorter than the period of
×
mentoftheproblem,butonlyinananalyticestimateofthetime
5 Itisinterestingtoconsider thecaseofstellarglobular clusters: as-
scaleofanoscillation. Thereforeweassume,withoutclarifying
suming a distance from the host galaxy of R = 30kpc, we obtain a
thephysicalmechanismthatwouldpermitthis,thattherelative
period t 1.5 109yr. We emphasize that this value has to be
positionbetweentheexternalmonopoleandthegalaxyisfixed, osc,cl ≃ ×
increasedbythemultiplicativefactor I /( ).
andthatonlytheorientationofthegalaxycanvary. Withthese 6 Thisprocedure ismore complicated tehqanQi3n3t−heQKeqeplerian casebe-
assumptions,Eq.(A.13)issufficienttodescribethedynamicsof causeδ˜isnotathree-dimensionaldeltpa. Itcanbewrittenasδ˜ =δ
thesystem. Toobtainthetimescaleofanoscillation,wemake xˆ xˆ ,wherexˆ =(1,0,0)istheversorofthefirstaxis,paraliljeltoitjh−e
i,1 j,1 1
thesmall-angleapproximation(θ 1)andrecallthat 33 > eq DMfilament.Whenwechangethecoordinateframe,wehavetorotate
≪ Q Q
(prolategalaxy),obtainingthependulumequation. Considering Randxˆ ;wethenobtainW (θ,φ) = const 3 1GλR 2( cos2φ(1+
thatthecomponentsofIand areofthesameorderofmagni- sin2θ)+1 sin2φ(1+sin2θλ)+ (1+cos−2θ)−). − Q11
Q Q22 Q33
Articlenumber,page9of11
oscillationfoundintheKepleriancase,butneverthelessgreater AppendixB: Equipotentialapproximation
than t . We thereforehave to drop the rigid-bodyapprox-
cross,ell In Sect. 3 we have obtained the expression of the intrinsic el-
imation and deepen the description of elliptical galaxies to ac-
lipticity of an early-type galaxy subjected to an external tidal
countfortheinternaldegreesoffreedom,asdoneinSect.3and
field. To do so, we have calculated the luminous quadrupole,
AppendixB.
makinguseofthestellardistributionfunction. Inthisappendix
wepresentanotherapproach,whichislesscompletebuthasthe
AppendixA.3:Spiralgalaxies:precessionperiod advantageofhavingaclearandsimplephysicalunderstanding.
Inparticular,wemodelthedeformationofthegalaxybymeans
The key feature that distinguishes spiral galaxies from ellipti-
ofthe equipotentialsurfacesof the totalgravitationalpotential.
calonesisthedominanceoforderedmotionsoverchaoticones,
In this approach, we “assume[s] that the local galaxy density
thatis,thecharacterizingpresenceoftheangularmomentum.If
isproducedapproximatelybystarsneartheirzero-velocitysur-
we place a rigid body with an angular momentum in an exter-
faces”(Ciotti&Dutta1994). AsinSect.3,(i)westartwithan
nalfield,itstartsprecessing. Inthissubsectionweestimatethe
unperturbedsphericalgalaxy,(ii)wetakeanexternaltidalpoten-
precessionperiodforaspiralgalaxy,takenasarigidbody.
tialandaddittothe(unperturbed)galaxypotential,thusignor-
Theprecessionperiodt ofarotatingrigidbodyis
prec ingthechangesinthegalaxypotentialinducedbythedeforma-
2πLsin(θ) tionofthegalaxy(seealsoCiotti&Dutta1994;Bertin&Varri
tprec = τ(θ) , (A.19) 2008) , and (iii) we assume that the galaxy immediateylreacts
toa changeofthe externalgravityfield bymodifyingits shape
where L is theangularmomentumof the spiralgalaxy,θ is the accordingly(seeSect.3fordetails).
anglebetweenLandtheexternalforce,andτ(θ)istheamountof
themomentumoftheexternalforce.Inspiralgalaxiesthelumi-
nousmassdistributionisverydifferentfromthatofthedarkmat- AppendixB.1:Generalcase
ter ; in ourcalculationwe candetachthetwo contributionsbe-
Inthe absenceof externalfields, the galaxypotentialU obeys
0
causetheDMdistributionis(approximately)sphericalanddoes
thePoissonequation
not generate a torque on the visible mass, in which we are in-
terested. To determine the total stellar angular momentum L⋆, 2U0 =4πGρ0 , (B.1)
wecanusetherigid-bodyformulaL =I ω . Inspiralgalax- ∇
⋆ ⋆33 ⋆
iesstarsatdifferentdistancesfromthecenterofthegalaxyhave where ρ is the unperturbedgalaxymass distribution. We now
0
differentangularvelocities(ω =ω (ℓ),whereℓisthedistance introduce a (weak) external potential U , so that U U =
⋆ ⋆ ext 0 7→
fromthesymmetryaxisofthespiral).Thereforeitissensibleto U + U . The introduction of the external field changes the
0 ext
useaweightedangularvelocityω¯ ,definedas equipotential surfaces of the galaxy. Given a volume en-
⋆ 0
V
closedbyaparticularequipotentialsurface∂ = xU (x) =
ω¯ = L /I V0 {∀ | 0
⋆ ⋆ ⋆33 E at energy E of the unperturbedpotential, we consider the
0 0
}
correspondingequipotentialsurface∂ forU
= ℓ2v (ℓ)Σ (ℓ)dℓ ℓ3Σ (ℓ)dℓ, (A.20) V
⋆ ⋆ ⋆
Z ,Z ∂ = xU(x)=U0(x)+Uext(x)= E0+δE = E . (B.2)
V {∀ | }
where Σ (ℓ) is the projected stellar surface density at distance
⋆
TheenergyshiftδEischoseninsuchawaythatthemassinside
ℓ from the spiral symmetry axis and v (ℓ) is the stellar tan-
⋆
thesurfacedoesnotchange:
gential velocity. In spiral galaxies, the stellar tangential veloc-
ity is approximately constant, v (ℓ) v , and the stellar col-
⋆ ⋆
umn density follows an exponential≃law with length scale ℓ , Θ E U (x) ρ (x)d3x= Θ E U(x) ρ (x)d3x. (B.3)
0 0 0 0 0
Σ (ℓ)=Σ exp( ℓ/ℓ ). Wethenobtain − −
⋆ 0 0 Z Z
− (cid:16) (cid:17) (cid:16) (cid:17)
A Taylor expansionof the right-handside of Eq. (B.3) for low
ω¯⋆ = v⋆0R∞0∞ℓ3ℓ2exepx(p−(−ℓ/ℓℓ/0ℓ)0d)ℓdℓ = 3vℓ⋆0 . (A.21) δE−Uextgives
δ E U (x) ρ (x) δE U (x) d3x=0. (B.4)
UsingthRisexpressioninEqs.(A.13)and(A.17),wefindforan 0 0 0 ext
− −
oblategalaxy(r <r =r =r ) Z
3 1 2 eq (cid:16) (cid:17) (cid:16) (cid:17)
Since ρ has spherical symmetry, ρ and U are uniform on
0 0 0
t = 2πv⋆ I⋆33 R3 1 , (A.22) ∂ 0,andwefinallyobtain ∇
prec,M 3Gℓ M cos(θ) V
0 ⋆eq ⋆33 ext
tprec,λ = Gπvℓ⋆0QQ⋆eqI⋆−3−3QQ⋆33Rλ2cos1(θ) . (A.23) δE = H∂V0∂UVe0xtρ(0x()xρ)0(∇xU)(cid:12)(cid:12)(cid:12)∇0(Ux)0−(x1d)(cid:12)(cid:12)(cid:12)2−x1d2x = H∂V0∂UVe0xtd(x2x)d2x . (B.5)
Assumingv⋆ =200kms−1andℓ0 =10kpc,weobtain IfU UH = Φ(cid:12)(cid:12) xx ,th(cid:12)(cid:12)atis,theexternaHlpotentialisatidal
ext ≡ tidal (cid:12)ij i j (cid:12)
tprec,M =1.2×1013 I33 cos(θ) −1yr, (A.24) sopnhee,rδicEalissyemqumaelttroy.zero, because Φij is traceless and ∂V0 has
11 33
tprec,λ =2.7 1010Q I−33Q (cid:16)cos(θ)(cid:17)−1yr. (A.25) introWdeucueseantheexgtearlnaaxlytliodgalarpitohtmenitciaplo,ttehnetiaelquoifpEotqe.n(t2ia8l).suIrffawcee
×
Q11−Q33 becomesanellipsoid(seeFig.B.1). Weplacethegalaxycenter
(cid:16) (cid:17)
Thereforethe time scale is longer than or similar to the age of at the origin and align the coordinate axes along the eigenvec-
theUniverse,anditisalsolongerthanthedeformationtimeof torsofthetidaltensorΦ . Then,wecanevaluatethedeviation
ij
spiralgalaxies(similarlytothatoftheellipticalgalaxy). fromthecircularshapeofaparticularequipotentialsurfacewith
Articlenumber,page10of11