Hierarchical phase space structure of dark matter haloes:Tidal debris, caustics, and dark matter annihilation

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Citation Afshordi, Niayesh, Roya Mohayaee, and Edmund Bertschinger.“Hierarchical phase space structure of dark matter haloes: Tidaldebris, caustics, and dark matter annihilation.” Physical Review D79.8 (2009): 083526. © 2009 The American Physical Society

As Published http://dx.doi.org/10.1103/PhysRevD.79.083526

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Citable link http://hdl.handle.net/1721.1/51762

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Hierarchical phase space structure of dark matter haloes: Tidal debris, caustics,and dark matter annihilation

Niayesh Afshordi,1,* Roya Mohayaee,2,† and Edmund Bertschinger3,‡

1Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5,Canada2Institut d’Astrophysique de Paris, CNRS, UPMC, 98 bis boulevard Arago, France

3Department of Physics and Kavli Institute for Astrophysics and Space Research, MIT,Room 37-602A, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

(Received 12 November 2008; published 24 April 2009)

Most of the mass content of dark matter haloes is expected to be in the form of tidal debris. The density

of debris is not constant, but rather can grow due to formation of caustics at the apocenters and pericenters

of the orbit, or decay as a result of phase mixing. In the phase space, the debris assemble in a hierarchy

that is truncated by the primordial temperature of dark matter. Understanding this phase structure can be

of significant importance for the interpretation of many astrophysical observations and, in particular, dark

matter detection experiments. With this purpose in mind, we develop a general theoretical framework to

describe the hierarchical structure of the phase space of cold dark matter haloes. We do not make any

assumption of spherical symmetry and/or smooth and continuous accretion. Instead, working with

correlation functions in the action-angle space, we can fully account for the hierarchical structure

(predicting a two-point correlation function / �J�1:6 in the action space), as well as the primordial

discreteness of the phase space. As an application, we estimate the boost to the dark matter annihilation

signal due to the structure of the phase space within virial radius: the boost due to the hierarchical tidal

debris is of order unity, whereas the primordial discreteness of the phase structure can boost the total

annihilation signal by up to an order of magnitude. The latter is dominated by the regions beyond 20% of

the virial radius, and is largest for the recently formed haloes with the least degree of phase mixing.

Nevertheless, as we argue in a companion paper, the boost due to small gravitationally-bound substructure

can dominate this effect at low redshifts.

DOI: 10.1103/PhysRevD.79.083526 PACS numbers: 95.35.+d

I. INTRODUCTION

Cosmological N-body simulations show that dark matter(DM) haloes that form in a �CDM Universe contain alarge number of subhaloes of all sizes and masses. Whatremains outside the subhaloes are ungrouped individualparticles whose masses set the resolution limit of thesimulation. If the simulations were to have enough resolu-tion to resolve every single subhalo, then it is expected thatthe smallest subhaloes would be microhaloes of about10�6M� [1–3]. Does all the mass of a given halo resideinside the gravitationally bound subhaloes? As a subhalofalls through the gravitational field of its host halo, itbecomes tidally disrupted. A tidal stream extends alongthe orbit of the subhalo and can contain a large fraction ofthe satellite mass. Therefore, a significant fraction of a DMhalo is expected to be in the form of streams and caustics.Depending on their length, the density of the streams canvary and is relatively not very large. However, as a streamfolds back on itself, zones of higher density, i.e. caustics,form (see e.g. [4–8]. In principle, these are not true causticsbut only smeared-out caustics due to finite DM velocitydispersion, however, it is convenient to refer to them

simply as DM caustics. Hereafter, we shall refer to un-bound streams and caustics jointly as tidal debris.Dark matter tidal debris, so far mostly unresolved in

cosmological N-body simulations, are expected to popu-late our own halo. Many stellar counterparts to such debrishave been detected so far (e.g. [9,10]) and many more areexpected to be detected with future missions like GAIA.The hierarchical growth of the host halo from the disrup-tion of satellite haloes reflects in a hierarchical structure ofthe phase space. The true lowest cutoff to this hierarchy isnot set by the microhaloes but by primordial dark mattervelocity dispersion. The hierarchical phase structure indi-cates that after removing all bound subhaloes from a givenDM halo, its phase space remains still unsmooth due todebris from disrupted subhaloes. The tidal debris are never

smeared out because of conservation of phase space den-sity and volume, although they become less dense as theywrap around the halo. It is this phase structure which westudy here.Secondary infall or self-similar accretion model pro-

vides a solid theoretical base for the study of halo forma-tion, and models the phase structure of DM haloes [11,12].

However, since this model assumes continuous accretion, itcannot capture the hierarchical nature of halo formation.

On the other hand, numerical simulations still lack enoughresolution to fully resolve the hierarchical phase structure,

*nafshordi@perimeterinstitute.ca†roya@iap.fr‡edbert@mit.edu

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although progress is being made in this direction [8,13–15].

Here, we aim at capturing the hierarchical phase struc-ture of dark matter haloes and its intrinsically discretenature, without resorting to any assumption of sphericalsymmetry or smooth and self-similar accretion. We dividethe structure of a dark matter halo into three categories:(1) the primordial and intrinsically discrete phase structure,formed prior to any merger or accretion and entirely due tothe coldness of the initial condition; (2) the hierarchicalphase structure of tidal debris from disrupted satellites, and(3) the hierarchical phase structure of undisrupted subha-loes. We leave the study of the undisrupted substructures toa companion paper [16] and in this work we only studycases (1) and (2).

To study phase structure induced by debris from dis-rupted satellites, we assume that at a given level in thehierarchy, all structures added earlier and which lie atsmaller scales are smooth. This sets the lowest level ofthe hierarchy at the scale determined by the velocity dis-persion of earliest dark matter haloes. However, this is notentirely correct since the earliest dark matter haloes them-selves are not smooth and have a structure that is due to thecoldness of the initial condition. Thus, there is a funda-mental discreteness scale, which is determined by primor-dial dark matter velocity dispersion (see Fig. 1).

This complicated process is studied here through corre-lation functions in the action-angle space whereHamiltonian is only a function of the adiabatic invariants,i.e. the action variables. Their conjugate variables, theangle variables increase linearly in time. The action-anglevariables are extremely useful for studying tidal streams[17–21]. However, working with the action-angle varia-bles, we are restricted to regions within the virial radius(with a quasistatic potential), and hence the phase struc-tures that might arise outside the virial radius (e.g. betweenthe virial and the turnaround radii) cannot be studied in thepresent framework [22]. For direct DM detection andcosmic-ray signal of DM annihilation, only the nearbyphase structure plays a role and our method is valid (seee.g. [23]). However, for lensing experiments and �-rayemission from DM annihilations, for example, from othergalaxies, the structures outside the virial radius can berather important (see e.g. [24]).

We assume that the satellite orbits are integrable in thehost DM potential (although, in Sec. V we remark on chaosand nonintegrable systems). Therefore, the phase spacedistribution can be described in terms of the action-anglevariables, fJi; �ig, so that

_� i ¼ @H@Ji

¼ �i; (1)

_J i ¼ �@H@�i

¼ 0; (2)

where the Hamiltonian, H ¼ H ½J�, is only a function ofaction variables, Ji and �is are the angular frequencies.Figure 2 shows a cartoon picture of phase mixing in theaction-angle space, and its correspondence to the realspace.

FIG. 1 (color online). The top horizontal panel shows thephase space of the merger of two dark matter haloes, each ofwhich has its own hierarchy of phase structure. The times on thetop panel refer to the crossing times. The zooming shows thateach hierarchy contains a lower level and so on. The hierarchy iscut at the scale of the smallest dark matter halo that has beenaccreted to the final halo. However, the phase space is notsmooth below this scale. Indeed, the phase space is intrinsicallydiscrete due to the coldness of dark matter shown by the lastzooming on the left. (Top panel: courtesy of Vlasov-Poissonsimulation [33].)

FIG. 2. A one-dimensional cartoon of the evolution of tidalstreams in both phase and action-angle spaces. As structurewraps around the phase space, more streams cross the sameangle coordinate, which leads to a discrete latticelike structure inthe action space.

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The hierarchical phase structure and its fundamentaldiscreteness set by primordial DM velocity dispersion arecaptured by the correlation function of the phase density.Since, after a long time, the distribution in the angle spaceis uniform, the phase density is only a function of theaction variables. This can be easily seen by writing thecollisionless Boltzmann equation for the equilibrium dis-tribution in the action-angle space

@f

@tþ _�i

@f

@�iþ _Ji

@f

@Ji¼ 0; (3)

which, combining with Eq. (2), implies that the equilib-rium phase space density can only be a function of actionvariables (and is known as the strong Jeans theorem [25]).

This enables us to evaluate the density-density correla-tion function. Our results are only valid statistically fortypical haloes and thus may not agree with results obtainedfor individual haloes in the simulations.

The nature of DM remains a mystery. Supersymmetryand extra-dimensional extensions of the standard electro-weak model provide a natural candidate in the form of aweakly interacting and massive particle (hereafter WIMP).These species should fill up the galactic halo. If DMconsists of WIMP’s, they are expected to strongly annihi-late in the dense regions of our halo and generate, inparticular, gamma rays and charged cosmic rays. Hence,hierarchical structure of phase space can lead to the en-hancement of DM annihilation signal [26]. We also evalu-ate the boost to the annihilation signal due to tidal debrisand discreteness of the phase structure. We show that theboost due to tidal debris is of order one, whereas the boostfrom the discrete phase structure can be up to 1 order ofmagnitude higher.

In Sec. II, we review a few basic relations for action-angle variables. In Sec. III A and III B we describe thecorrelation functions that would account for the phasestructure due to tidal debris and their discreteness. InSec. IVA, we evaluate the boost on the annihilation signaldue to tidal debris. In Sec. IVB, we evaluate the boost ofthe annihilation signal from intrinsic discreteness of thephase structure, and finally Sec. V concludes the paper.

II. STREAMS AND COHERENCE VOLUME OFTHE PHASE SPACE

We use the definition of action-angle variables (2) andassume that the frequencies are not degenerate, i.e. theHessian matrix

H ij � @2H@Ji@Jj

¼ @�i

@Jj(4)

has nonzero eigenvalues, or equivalently, a nonvanishingdeterminant

jH ijj � 0; (5)

with the possible exception of a zero measure region of thephase space. Note that, this implies that the halo potentialcannot be assumed to be exactly spherically symmetric, astwo of the frequencies would be equal.If a satellite galaxy has originally a small spread in the

action variables, �Ji, its spread in the angle variablesincreases as

��i ¼ ðH ij�JjÞtacc;p þ ��0; (6)

where tacc;p is the time since accretion of the progenitor of

the debris into the halo. The last term, the initial extent ofthe debris, is subdominant at large times. We set this termto zero for now, but at the end of Sec. IVB, we discusswhen it can become important and how it could affect ourresults. Therefore, the total volume swept in the anglespace grows as

�3� ¼ ðjH ijj�3JÞt3acc;p ¼ ð�3�Þt3acc;p; (7)

where we used the definition ofH ij in Eq. (4), and�3� is

the volume occupied by the debris of satellite particles inthe frequency space.As the total volume of the angle space is ð2�Þ3, the

number of streams passing through each angular coordi-nate is

Nstream ¼ ð�3�Þ�tacc;p2�

�3: (8)

Thus, the total mass of each stream mstream is the mass ofthe debris m divided by Nstream

mstream ¼ m

Nstream

¼�

m

�3�

��tacc;p2�

��3: (9)

Put another way, the action space is divided into cells ofvolume

-2 -1 0 1

Jx

-2

-1

0

1

2

J y

-2 -1 0 1 2

Jx

-2 -1 0 1

Jx

t=0.1 t=0.3 t=1

FIG. 3. The action-space distribution of debris in a unit 2dtorus with unit particle mass and no potential. The debris isoriginally within 0< x, y < 0:1, and �10< vx, vy < 10. The

figures show a cut through the action space with 0:09< x, y <0:1, which is characterized by ~fpðJ;x; tacc;pÞ [Eq. (14)] in our

formalism.

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�3Jstream ¼�2�

tacc;p

�3jH ijj�1; (10)

as a result of phase space mixing (e.g. [20]).With this picture in mind, we can write the distribution

in the action space as the sum of the contributions fromindividual progenitors

fðJ; �Þ ¼ Xp

fpðJ; �; tacc;pÞ; (11)

where each fp has a cellular structure characterized by

Eq. (10), as shown in Fig. 3, which gets finer and finer withtime. Equation (11) is the phase space analog of the widelyused halo model in cosmology [27,28], where the density isassumed to be the sum of contributions from individualhaloes with given profiles. Correspondingly, fp character-

izes the profile of individual progenitors in our picture.We can now write the real space density as

�ðx; tÞ ¼ Xp

Zd3Jd3�fpðJ; �; tacc;pÞ�3

D½x� ~xðJ; �Þ�

¼ Xp

Zd3J ~fpðJ;x; tacc;pÞ~�ðx; JÞ; (12)

where ~�ðx; JÞ is the density of a distribution of unit mass,with a fixed action variable J, and uniform angle distribu-tion

~�ðx; JÞ �Z d3�

ð2�Þ3 �3D½x� ~xð�; JÞ�; (13)

while

~fpðJ;x; tacc;pÞ � ~�ðx; JÞ�1Z d3�

ð2�Þ3 fpðJ; �; tacc;pÞ� �3

D½x� ~xð�; JÞ�: (14)

An example of ~fp is shown in Fig. 3 for debris in a toy

model of a unit torus. As we will explicitly show inSec. IVA, projecting this discrete structure in the actionspace of the debris into the real space leads to discrete,(nearly) singular, caustics that are only smoothed by theoriginal velocity dispersion of the progenitor.

III. CLUSTERING IN THE PHASE SPACE

Averaging over different possible realizations of thedebris within a halo, the mean phase space density canbe written as an integral�X

p

~fpðJ;x; tacc;pÞ�¼

ZdNpg

ð1ÞðJ� Jp;x; tacc;pÞ; (15)

where gð1Þ and Jp are the profile and mean action of

individual progenitors, while

dNp � dmpd3Jpdtacc;p

dn

dmpd3Jpdtacc;p

(16)

is the differential progenitor number density per units ofprogenitor mass mp, its action-space volume d3Jp, and its

accretion time tacc;p. We now follow an analogy with the

cosmological halo model [27,28] to write the clustering inthe action space as a superposition of one- and two-progenitor terms� X

p1;p2

~fp1ðJ1;x; tacc;p1

Þ~fp2ðJ2;x; tacc;p2

Þ�

¼Z

dNpð1-progÞ þZ

dNp1

ZdNp2

ð2-progÞ: (17)

The one-progenitor term characterizes the self-clusteringof individual progenitor action-space profiles.

ð1-progÞ ¼ gð1ÞðJ1 � Jp;x; tacc;pÞgð1ÞðJ2 � Jp;x; tacc;pÞ;(18)

while the two-progenitor terms characterize the correlationbetween phase space density at different action variables,within different progenitors:

ð2-progÞ ¼ gð2ÞðJ1 � Jp1; J2 � Jp2

;x; tacc;p1; tacc;p2

Þ¼ gð2ÞconðJ1 � Jp1

; J2 � Jp2;x; tacc;p1

; tacc;p2Þ

þ gð1ÞðJ1 � Jp1;x; tacc;p1

Þ� gð1ÞðJ2 � Jp2

;x; tacc;p2Þ: (19)

In the limit that the mean actions of different progenitorsare not correlated, the connected part of the (two-prog.)

term goes to zero: gð2Þcon ! 0, and thus the two-progenitorterm reduces to the correlation within the smooth halo.Note that this limit cannot be strictly realized, as due tophase space conservation, phase streams tend to avoid each

other, leading to gð2Þ < 0 at small separations Jp1� Jp2

.

However, Liouville’s theorem is not valid for coarse-grained phase space density, and thus coarse-grained pro-genitors can overlap in the action space.The connected part of the two-progenitor term originates

from the clustering of the initial conditions of the progen-itors of the host halo, which is generally expected from theclustering of cosmological haloes. However, the structureof the one-progenitor term is more subtle: In addition to thecellular structure described in the previous section (Fig. 3),the internal structure of each progenitor prior to its accre-tion onto the host halo would introduce a hierarchy withineach cell. In fact, in a hierarchical picture of structureformation, one expects the subcellular structure of theone-progenitor term to be inherited from the two-progenitor terms within progenitors prior to their accretiononto the main halo (see Fig. 1). The key difference betweenthe two hierarchies, however, is that phase mixing onlycontinues in the action space of the main halo, and (follow-

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ing the tidal disruption) has stopped in the action spaces ofthe progenitors.

For statistically self-similar initial conditions, we expectthe subcellular and two-progenitor terms to blend into oneroughly self-similar structure in the action space, althoughindividual realizations have periodic structures with thecharacteristic volume given in Eq. (10). We provide ascaling ansatz for this structure in Sec. III A. However,the self-similarity is cutoff by the free streaming of darkmatter particles on small separations, due to their finiteintrinsic velocity dispersion. This is responsible for thefundamental discreteness of the phase space distribution(see Fig. 1), which we model in Sec. III B.

A. Hierarchical phase structure from tidal debris

We first consider the phase structures due to tidal debris.Once again, we emphasize that these are the tidal streamsthat have fallen into the gravitational field of the host haloand are no longer bound to the original satellite.

The first level of approximation that we will use to studyphase space clustering of cold dark matter (CDM) is toassume a (statistically) hierarchical formation history,where any trace of the cold initial conditions has beenwiped out through phase mixing. Furthermore, we ignorethe possibility of gravitationally bound structures in thispaper (see the companion paper [16] on this subject). Theimpact of cold initial conditions will be addressed in sub-sequent sections.

Assuming uniform distribution in angles (or completephase mixing), the density at each point in the halo is givenby

�ðxÞ ¼ ð2�Þ3Z

d3JfðJÞ~�ðx; JÞ; (20)

where fðJÞ is the phase space density, while ~�ðx; JÞ wasdefined in Eq. (13). Note that in (20), the function ~� has thedimension of inverse volume, 1=VðJÞ.

We shall assume adiabatic invariance; the action remainsconstant as new structures are added on larger scales.Hence, the distribution function in the action space fðJÞdoes not change with time, except when new structures areadded due to satellites that are newly accreted inside thevirial radius.

The assumption of uniformity in the angle space allowsus to separate the effect of phase mixing from that ofhierarchical structure formation. While the former is thecause of original caustic formation, too much phase mixing(within a fixed potential) will eventually smooth out thereal space density distribution. The assumption of a smoothfðJÞ distribution, implies that phase mixing is complete.

On the other hand, the effect of hierarchical structureformation is captured in fðJÞ, through the fact that struc-ture in fðJÞ is added on different scales, at different timesin the history of the halo. A statistical measure of thishistory is the two-point correlation function of the action-

space density. We thus hypothesize that the correlationfunction

�fðJ1; J2Þ � hfðJ1ÞfðJ2Þi (21)

should be a power law for statistically self-similar initialconditions

hfðJ1ÞfðJ2Þidebris ’ Aj�Jj��jJ1 � J2j��; (22)

as long as jJ1 � J2j � j�Jjwith �J ¼ ðJ1 þ J2Þ=2. The formof the correlation function uses the symmetry between J1and J2. It also guarantees that small scale structures arecaptured, as structures are added on different scales atdifferent times (see Fig. 1 for a demonstration).Moreover, since actions remain constant in the adiabaticinvariance approximation dA=dt ’ 0 on small scales. Inother words, the correlation function �fðJ1; J2Þ grows in-side out in the J1, J2 space.The mean phase space density of a virialized halo,

assuming a virial overdensity of �200, is given by

ð2�Þ3�3J � r3vir�3vir �

ðGMÞ210H

; (23)

) hfðJÞi � 10H

G2M; (24)

where M is the halo virial mass. The virial action variableis also roughly

Jvir � r�vir � ðGMÞ2=3ð10HÞ1=3 : (25)

Given that M / a6=ðneffþ3Þ and H / a�3=2 when perturba-tions grow during the matter-dominated era, with a beingthe cosmological scale factor, and neff the slope of thelinear power spectrum, we conclude

hfðJÞi / J�½ð3ðneffþ7ÞÞ=ðneffþ11Þ� ) � ¼ 3ðneff þ 7Þneff þ 11

’ 1:6� 0:1; (26)

where we have assumed neff ’ �2:5� 0:5 for cosmologi-cal haloes.An alternative way to derive (26) is to consider the self-

similar collapse models of Fillmore and Goldreich [11],where they calculate the actions and use adiabatic invari-ance to find the outcome of spherical cold secondary infallin an Einstein-de Sitter universe. For the spherical self-similar linear initial condition of

�M

M

��������init/ M�"; (27)

they find the action at the turnaround radius scales as

Jta / M"þ1=3ta tð2=9"Þ�ð1=3Þ; Mta / t2=3" (28)

for " < 2=3, where Mta is the mass within the turnaroundradius. Although (28) is only for the radial action, and the

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two other action variables vanish due to spherical symme-try, one may imagine that for triaxial CDM haloes, thethree actions would become comparable: J � J� �trJr � trJta, where tr characterizes the triaxiality ofthe halo [not to be confused with the self-similar profileindex " in (27)]. Therefore, eliminating time from the twoequations in (28), the phase space density, fðJÞ scales as

fðJÞ �Mta

J3ta� J6=ð3"þ4Þ: (29)

Assuming that the self-similar linear density profile has thesame radial/mass scaling as the variance of the cosmologi-

cal density fluctuations, �ðMÞ / M�ðneffþ3Þ=6 yields " ’ðneff þ 3Þ=6 (< 2=3 for CDM Harrison-Zel’dovich pri-mordial power spectrum), which, plugging into (29), re-produces (26).

We should point out that the correlation function in theaction space [Eqs. (21) and (22)] is, to our knowledge, firstintroduced in the present work. While this function couldbe a very useful tool for statistical description of tidaldebris in (stellar or dark matter) collisionless systems, itsthorough understanding requires further numerical study,and may involve different potential subtleties. For ex-ample, separating the correlations due to gravitationallybound subhaloes, or a numerical definition for ensembleaverage of a multivariable correlation function in a 6Dphase space, may prove difficult in practice. We postponea numerical investigation of clustering in the action space,and its potential applications to stellar systems (e.g. [29])to future studies.

B. Fundamental discreteness of the phase spacestructure

In the previous section, we considered the hierarchicaladdition of tidal debris to a DM halo. However, the hier-archy has a lower cutoff set by the velocity dispersion ofthe smallest accreted satellite. Microhaloes of�10�6 solarmass could indeed determine such a cutoff [1–3]. However,this cutoff is far above the primordial velocity dispersion ofDM itself. Therefore, the primordial velocity dispersionintroduces a fundamental discreteness in the phase struc-ture. In other words, the smooth phase space distribution ofthe last section ignores the discrete nature of multiplestreams in the phase space due to the presence of a cutoffin the CDM hierarchy. This discreteness shows up as acellular or lattice structure in the action space, with acharacteristic cell volume given in (10) [see Figs. 2 and 3for 1d and 2d cartoons; More realistic simulated examplesare discussed in [20]]. After averaging over different spac-ings, expected for different accretion times of differentdebris, the discreteness would only show up as the zero-lag of the action-space correlation function

hfðJ1ÞfðJ2Þidis ’ m2stream

�3Jstream�3DðJ1 � J2Þ; (30)

where mstream and �3Jstream were defined in Eqs. (9) and(10), and here we have assumed a zero initial temperaturefor CDM particles. A finite CDM velocity dispersion (cor-responding to less than �10�2ð1þ zÞ cm=s for GeV scaleWIMP dark matter) will smoothen the delta function, asthe phase space density cannot exceed its primordial value(see Fig. 1 for a cartoon). Note that the minimum virialvelocities for first CDM haloes (or microhaloes) withM�10�6M�, is �102 cm=s, which would characterize theminimum spacing between fundamental streams.

IV. EXAMPLE: DARK MATTER ANNIHILATIONMEASURE

A. DM annihilation in tidal debris

In this subsection, we consider the enhancement in theexpectation value of the annihilation measure due to hier-archical structures built in the phase space from tidaldebris.For a uniform distribution in the angle space, the expec-

tation value of the annihilation measure is given by

� ¼Z

d3xh�ðxÞ2i

¼ ð2�Þ6Z

d3J1d3J2�fðJ1; J2Þ

Zd3x~�ðx; J1Þ~�ðx; J2Þ:

(31)

We remark that the above integral can also be relevant forthe direct detection of DM, as it quantifies the variance ofthe density field.In order to investigate the impact of caustics, near the

apocenters and pericenters of the orbits, we make a simpleanalogy with a one-dimensional harmonic oscillator

H ¼ 12ðp2

x þ p2y þ p2

zÞ þ 12!

2x2: (32)

For concreteness, we also assume the other two dimensionsare compact with the length Ly and Lz, although the

Hamiltonian has no explicit dependence on y and z coor-dinates. As the evolution in the three spatial directionsdecouple, we can simply read off three action variablesfrom the areas of phase diagrams for each direction:

Jx ¼ p2x

2!þ 1

2!x2; (33)

Jy ¼Lypy

2�; Jz ¼ Lzpz

2�: (34)

From these relations, we can find ~�ðx; JÞ using its defini-tion in Eq. (13)

~�ðx; JÞ ¼ ðxmax � xminÞ�V

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xminÞðxmax � xÞp ; (35)

where

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xmax ¼ �xmin ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Jx=!

q; (36)

V ¼ LyLzðxmax � xminÞ: (37)

Notice that the square root singularity in the projectionkernel ~�ðx; JÞ is very similar to the singularity expectednear CDM caustics. However, for a smooth distribution inthe action-space fðJÞ, the real space density �ðxÞ is anintegral over the kernel [Eq. (20)], which would lead to asmooth �ðxÞ. Therefore, a discrete distribution in the ac-tion space is necessary to produce caustic singularities inthe real space (otherwise known as fold catastrophes orZel’dovich pancakes).

We then notice that the toy model of Eq. (32) is similarto the motion in a nearly spherical potential, in the sensethat the motion in one direction (x or radial) is limited byrequiring constant action variables, while the two otherdirections (y and z, or angular directions) are compact.Based on this analogy, we will use

~�ðx; JÞ � ½rmaxðJÞ � rminðJÞ�VðJÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½r� rminðJÞ�½rmaxðJÞ � r�p ; (38)

where we have assumed an integrable nearly sphericalpotential, with small angular momentum (and third inte-gral), while VðJÞ is the spatial volume occupied by thestream of action J. The radii rmin and rmax are the minimumand maximum radii of all orbits with the same actionvariable. However, the general structure of the singularityclose to boundaries does not change in other geometries.

Using Eq. (38), we can evaluate the x integral in theemission measure (31). For J1 ’ J2, the integral is loga-rithmically divergent around r ’ rmaxðJ1Þ ’ rmaxðJ2Þ aswell as r ’ rminðJ1Þ ’ rminðJ2Þ. Focussing on the outercaustic rmax we findZ

d3x~�ðx; J1Þ~�ðx; J2Þ ’ 4�r2maxðrmax � rminÞV2ðJ1Þ

cosh�1

���������rmaxðJ1Þ þ rmaxðJ2ÞrmaxðJ1Þ � rmaxðJ2Þ

��������: (39)

This yields

Zd3x~�ðx; J1Þ~�ðx; J2Þ � VðJÞ�1

��������ln�Fi�Jij�Jj

���������; (40)

where Fi � 1=4j�Jj@ lnrmax=@Ji and can be calculated,given the gravitational potential of the host halo.Therefore, the annihilation measure takes the form

�� AZ

d3 �Jj�Jj��Z

d3�Jj�Jj��

��������ln�Fj�Jjj�Jj

���������: (41)

We see that since 3� �> 0, the integral is finite anddominated by large �J’s. The boost to the annihilationsignal, which is introduced by small scale clustering in theaction space (i.e. that �> 0) is thus given by

Bdebris � �

�smooth

’Rd3 �JVð�JÞ�1 �J��

Rd3�J�J��j lnðFj�Jjj�Jj ÞjR

d3 �JVð�JÞ�1 �J�2�Rd3�Jj lnðFj�Jjj�Jj Þj

’ 9

ð�� 3Þ2 F� ¼ Oð1Þ; (42)

given that F� @ lnrmax=@ lnJ is a dimensionless number oforder unity.Therefore, we see that the boost factor obtained here for

the tidal debris is dominated by larger separations, as seenfrom expression (42), whereas our approximation (38) isvalid for small separations. To emphasize, (38) is valid inthe vicinity of caustics, whereas the integral (42) demon-strates that most contributions come from large separationsin the action space.To summarize, while we predict an Oð1Þ boost in anni-

hilation signal due to (finite separation) clustering in actionspace, the main effect comes from large structures (and notcaustics), which are not accurately captured in our frame-work. Following the submission of our paper, this estimatewas recently confirmed by numerical simulations [30]. Inthe next section, we will address the impact of the discrete-ness of the phase space of CDM haloes.

B. DM annihilation boost due to discreteness of phasestructures and catastrophes

The primordial velocity dispersion of DM induces afundamental discreteness in the hierarchical phase struc-ture and can enhance the annihilation signal. The emissionmeasure due to this discreteness is calculated by inserting(30) in expression (31). Summing over all streams andsubtracting the smooth part, which is obtained by smooth-ing over the fundamental streams, we obtain

��dis ¼X

stream

m2stream

Zd3x½~�ðx; JstreamÞ2

� ~�ðx; JstreamÞ2smooth�’ K2

�t

2�

��3 Zd3�

�dMhalo

d3�

�2 Z

d3x½~�ðx; JÞ2

� ~�ðx; JÞ2smooth� (43)

for the enhancement of the emission measure due to dis-creteness of the phase structure, where ~�ðx; JÞsmooth is thestream density, smoothed to the level that different streamsoverlap, and for the stream mass we assume

mstream � KdMhalo

d3�

�t

2�

��3; (44)

where d3� is the volume element in the frequency space,and factor K is the ratio of the density of debris to that ofthe host halo: �debris ¼ m=v� K�halo.

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The above expression is to be compared to the annihi-lation measure for the smooth halo

�smooth ¼Z

d3�

�dMhalo

d3�

�Zd3�0

�dMhalo

d3�0

�

�Z

d3x~�ðx; JÞ~�ðx; J0Þ; (45)

where � and �0 are functions of J and J0, respectively.Therefore, the boost associated with a given point in theaction space is given by

Bdis½J� � 1 � ��dis

�smooth

¼ K2

�t

2�

��3

� ðdMhalo

d3�ÞR d3x½~�ðx; JÞ2 � ~�ðx; JÞ2smooth�R

d3�0ðdMhalo

d3�0 ÞR d3x~�ðx; JÞ~�ðx; J0Þ :

(46)

In order to estimate the boost factor, we should firstapproximate the density integral

Rd3x~�ðx; JÞ~�ðx; J0Þ. As

we discussed in the previous section for jJ� J0j � jJj theintegral is dominated by the regions around the turnaroundradii (or caustics) and thus grows as

Zd3x~�ðx; JÞ~�ðx; J0Þ � j lnðjJ� J0j=jJjÞj

VðJÞ ; (47)

as seen in Eq. (40). In the opposite limit, jJ� J0j � jJj,assuming that the two density kernels overlap, ~�ðx; JÞ canbe approximated as roughly constant, which yields

Zd3x~�ðx; JÞ~�ðx; J0Þ � V�1ðJÞ: (48)

Now we note that in (46), the integral over �0 (or equiv-alently J0) in the denominator is dominated by the largevalues of jJ�J0j, while the numerator is in the small jJ�J0jregime. Therefore, substituting from (47) and (48), we find

Bdis½J� � 1 ’ K2

�t

2�

��3��������d lnMhalo

d3�

����������������ln

� j�JCDMjj�Jint-streamj

���������¼ K2 1

3

�t

2�

��3��������d lnMhalo

d3�

��������ln�fCDMhfhaloi

�;

(49)

where j�JCDMj and j�Jint-streamj characterize the funda-mental CDM stream width and the interstream spacing,respectively. The stream thickness and spacing in realspace have been calculated within the framework of self-similar model [31]. However, to obtain the ratio, we use afar simpler approximation. To get the last line of expres-sion (49), we have used the fact that the volume in phasespace occupied by the fundamental streams isð2�Þ3�3JCDM ¼ Mhalo=fCDM, while the total volume in

phase space of the entire halo is ð2�Þ3�3Jint-stream ¼Mhalo=hfhaloi.The logarithmic enhancement factor is roughly

1

3ln

�fCDMhfhaloi

�’ 20þ ln

�m�

100 GeV

�(50)

for CDM particles of mass m�. In arriving at (50) we have

used that fCDM ��m�critðT�=TCMBÞ3=�3CDM and hfhaloi �

103�m�crit=�3vir, with TCMB � 10�4 eV, T� ¼

1=3m��2CDM �m�=40 is the CDM kinetic decoupling

temperature.In order to estimate the boost factor in (49), we also need

to find the volume element occupied in the frequencyspace, d3�, by a given mass element. First we shouldnotice that for a Keplerian potential ’ðrÞ / �r�1, thisvolume element vanishes as three frequencies are equal,i.e. �r ¼ � ¼ �3, where �3 is the frequency for the

third integral.For a general potential, ’ðrÞ, we have

�r

�� 1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3þ d ln’0

d lnr

s� 1 (51)

for a nearly circular orbit and where 0 indicates first de-rivative with respect to r, i.e. ’0 ¼ d’=dr. Moreover, theextent in �3 depends on the triaxiality of the halo

d�3 � tr�; (52)

where tr � 10% characterizes the typical triaxiality ofCDM haloes. Therefore, we will approximate the volumeelement as

d3�� 4�tr�2

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3þ d ln’0

d lnr

s� 1

�d�; (53)

where

�2 �GMhalo

r3¼ ’0

r: (54)

Hence, for a general potential we obtain

d lnMhalo

d3�¼ 1

2�trð2þ d ln’0=d lnrÞð’0=rÞ�1=2

�½ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3þ d ln’0=d lnr

q� 1Þð’0=r� ’00Þ��1:

(55)

To evaluate (49) we also need to know the multiplicationfactor K, which is the ratio of the density of disruptedsatellite to that of the host halo. This factor can be eval-uated for a general spherical potential, assuming that thedebris start with the maximum density that allows them tobe tidally unbound:

K ’ 3ð’0 � r’00Þð2’0 þ r’00Þ�1: (56)

Putting (55) and (56) in (49) we obtain

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Bdis � 1 ¼ 36�2ð’0 � r’00Þð2’0 þ r’00Þ�1

� ð ffiffiffiffiffiffiffiffiffiffiffiffiffi3þ ~’

p � 1Þ�1ð’0=rÞ�3=2

�3

8�G�crit

��3=2

� 1

3trln

�fCDMhfhaloi

�; (57)

where ~’ ¼ dðln’0Þ=dðlnrÞ and �crit ¼ 3H20

8�G is the critical

density of the Universe. Moreover, we have used the factthat the product of the present-day Hubble constant and theage of the Universe is unity (H0t ¼ 1:03� 0:04) for thecurrent concordance cosmology.

For a power-law potential

’ðrÞ ¼ �’0r�� � � 1; (58)

one can see that (57) yields

Bdis � 1 ¼ 72�2

3tr

ð�þ 2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1� �Þp

ð ffiffiffiffiffiffiffiffiffiffiffiffiffi2� �

p � 1Þ ln

�fCDMhfhaloi

�

��3�halo

�crit

��3=2; (59)

where �halo, the local halo density for a general potential is

�ðrÞ ¼ 1

4�Gð’00 þ 2’0=rÞ (60)

from the Poisson equation.For the power-law potential (58), we can find the local

density in terms of the critical density and the radius andconsequently obtain the boost factor (57) as a function ofr=rvir, where rvir is the virial radius of the halo, defined asthe radius within which the mean density is 200�crit. Thelocal boost factor for a power-law potential, and our nomi-nal values of tr and fCDM=hfhaloi, is

Bdis � 1 ¼ 36

5�2 �þ 2

ð1� �Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi2� �

p � 1

�r

rvir

�3ð�þ2Þ=2

:

(61)

The above expression shows that the boost is most signifi-cant in the outskirts of the halo, and for large values of �.Indeed, the local boost diverges as � ! 1. We note that, inthe context of a Navarro-Frenk-White (NFW) potential[32]

’NFW ¼ �’0

lnð1þ xÞx

; (62)

i.e. �� 1 corresponds to the outskirts of the halo and ���1 to the central part.

Similarly to the power-law potential, we can obtain thelocal boost factor (57) for an NFW potential (62). Theboost is shown in Fig. 4 as a function of r=rs and fordifferent values of the concentration parameter c �rvir=rs. The boost increases as we go toward the outskirtsof the halo as the number of streams decreases, henceincreasing the density of individual caustics. As we in-

crease the concentration, the central density of the halobecomes large, which in turn decreases the local boostfactor.

10-3

10-2

10-1

100

r/rvir

10-4

10-3

10-2

10-1

100

101

102

103

104

Bdi

s -1

c=40

c=3

c=10

c=20

rs/r

vir

10-4

10-2

100

102

r/rs

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

Bdi

s -1

c=40

c=3

c=10

c=20

FIG. 4 (color online). The local boost factor due to primordialdiscreteness of the phase structure for an NFW potential: Theplots show how the boost in the annihilation measure of a DMhalo changes as we go from the inner part of halo to outer parts,due to the discrete phase space structure of CDM. The localboost increases as we go toward the outskirts of the halo and alsoas we decrease the concentration. The dashed curves on the leftpanel show the boost if we include corrections due to a finiteinitial phase for the debris (68), which become important fornearly degenerate frequencies. The right panel shows that mostof the boost comes from regions beyond 20% of the virial radius.

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Having evaluated the local boost, we can evaluate thetotal boost from the halo, which we defined as

Btotal � �total

�smooth

; (63)

where

�smooth ¼ r3s�2s

Z c

04�x2�ðxÞ2dx; (64)

and

�total ¼ r3s�2s

Z c

0BdisðxÞ�ðxÞ24�x2dx; (65)

where Bdis is given by (57) and the density profile of thesmooth halo is given by �ðxÞ. For the NFW density profile� ¼ �s=½xð1þ xÞ2�, where �s is the scale density, we haveplotted the variation of the total boost with the concentra-tion parameter c in Fig. 5. Again, the boost decreases as weincrease the concentration, since the flux becomes domi-nated by the central part of the halo.

We saw that the boost is mostly in the outskirts of thehaloes, namely, beyond the 20% of the virial radius.However, one has to be cautious, since in the outskirts ofthe halo the gravitational field of the halo approaches aKeplerian potential, causing the frequencies to becomedegenerate and the term ��0 in (6), which we have so farignored, can become important. Thus, we need to study theimportance of this term for our analysis and the boost.

Expression (6) now becomes

�� ¼ t��

�1þ 1

t

��0��

�: (66)

Hence, the volume element in the angle space (7) should bereplaced by

�3� ¼ ð�3�Þt3 det��ij þ 1

t

@�0i@�j

�; (67)

where det stands for the determinant. For circular orbit

approximation we have � ¼ ffiffiffiffiffiffiffiffiffiffi’0=r

pand we use expres-

sions (51) and (52) for the volume element in the frequencyspace. We thus find that the boost (57) has to be multipliedby the inverse of the following determinant:

det

��ij þ 1

t

@�0i@�j

�’�1þ 1

tffiffiffiffiffiffiffiffiffiffi’0=r

p ��1þ 1

trtffiffiffiffiffiffiffiffiffiffi’0=r

p �

��1þ 1

tffiffiffiffiffiffiffiffiffiffi’0=r

p ð ffiffiffiffiffiffiffiffiffiffiffiffiffi3þ ~’

p � 1Þ�

(68)

if the frequencies were to become near degenerate. Inobtaining (68) we have used ��0=��� 1=�. The effectof this factor in reducing the boost is shown for an NFWpotential (62) by the dashed lines in Figs. 4 and 5. The localboost is reduced slightly in the outskirts as expected andthe total boost is reduced by a factor of about 2.

V. CONCLUSIONS

Working in phase space and with action-angle variables,we have shown that the density-density correlation func-tion can capture the hierarchical phase structure of tidaldebris and also the fundamental discreteness of the phasestructure due to the coldness of the CDM initial conditions.The study presented here assumes no spherical symmetry,no continuous or smooth accretion, and no self-similarinfall for the formation of dark matter haloes. It is thus ageneral scheme for quantifying the statistical properties ofthe phase structure of the virialized region of cosmologicalhaloes.As an application, we have obtained the significance for

dark matter annihilation signal due to the hierarchicalphase structure of tidal debris and have shown that thisstructure boosts the annihilation flux by order unity. On theother hand, the total boost due to the primordial discrete-ness of the phase space can be 1 order of magnitude higherfor low-concentration (or recently formed) dark matterhaloes.While this paper dealt with unbound debris and caustics

in dark matter haloes, in a companion paper [16], wecalculate the boost to the annihilation signal due to thegravitationally bound substructure or subhaloes.Combining the results of both papers, we can write downa concise and approximate formula for the local boost dueto all substructures:

10 100

concentration

10-4

10-3

10-2

10-1

100

101

tota

l B

dis -

1

FIG. 5. The estimated total boost in the annihilation measureof a DM halo, due to the discrete distribution in the CDM phasespace is shown for an NFW halo. The lower dashed curve showsthe total boost when we include corrections due to a finite initialphase for the debris (68), which become important for nearlydegenerate frequencies.

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Boost� 1 � h�2ðxÞih�ðxÞi2 � 1

¼ BdebrisBdisBsub � 1

�Oð1Þ þ 3� 105�

�crit

�haloðxÞ�3=2

þ 106�

�crit

�haloðxÞ��H0

H

�2; (69)

which should be valid within a factor of 3 in the virializedregion of the haloes. �halo is the local coarse-graineddensity of the halo at redshift z, while �crit is the criticaldensity of the Universe at redshift z. The first term in (69) isdue to the hierarchical structure of CDM debris that weestimated in Sec. IVA. The second term is due to thediscrete nature of the CDM phase space, where we used(59) with �� 0:5 and our nominal values for other pa-rameters. Lastly, the third term is the contribution due togravitationally bound subhaloes [16]. We take H2=H2

0 ¼�mð1þ zÞ3 þ� and plot (69) in Fig. 6.

One may wonder whether the discreteness of the phasespace of subhaloes could lead to an additional boost in theannihilation signal. In other words, should we add Bsub � 1and Bdis � 1 to get the total boost, or rather should they bemultiplied? To answer this question, we notice that themain contribution to Bsub is due to the smallest subn haloes(or microhaloes), which have the highest densities [16],while the Bdis is mainly due to the lowest density regions ofthe haloes, which have the lowest degree of phase mixing.Therefore, we expect the two terms Bsub � 1 and Bdis � 1to simply add incoherently, as the cross correlation be-tween the two sources of substructure should be small.

We should point out that the results here apply to phasestructure within virial radius, and those outside the virialradius, which we have not studied here, might yield abigger boost factor. It is reasonable to study the streamsand caustics that lie between the virial and the turnaroundradii by using the secondary infall model [11], as radialapproximation can be reasonably applied to regions be-yond the virial radius (see Fig. 4).

Finally, we remark on the most instrumental assumptionin our framework, which was the integrability of orbits inthe CDM potential, since one cannot define action-anglevariables in a nonintegrable system. This is characterizedby the appearance of chaotic orbits in parts of the phasespace. First, we should point out that as CDM haloes have atriaxial structure, a significant fraction of halo particlescannot be on chaotic orbits. Moreover, the difference be-tween chaotic and integrable orbits only becomes impor-tant after many orbital times, which are only possible in the

inner parts of the halo. As most of the boost to the annihi-lation caused by the discreteness in the action space comesfrom the outskirts of the halo (Fig. 4), we do not expect asignificant difference due to chaotic orbits. Nevertheless,the implications of chaos for the structure of the CDMphase space correlation function remains an intriguingquestion.

ACKNOWLEDGMENTS

N.A. is supported by Perimeter Institute for TheoreticalPhysics. Research at Perimeter Institute is supported by theGovernment of Canada through Industry Canada and bythe Province of Ontario through the Ministry of Research& Innovation. N.A. is grateful to IAP for hospitality. R.M.thanks French ANR (OTARIE) for grants and PerimeterInstitute for hospitality. E. B. acknowledges support fromNSF Grant No. AST-0407050 and NASA GrantNo. NNG06GG99G.

102

103

104

105

106

107

108

ρhalo

(r,z)/ρcrit

(z)

100

101

102

103

104

Boo

st -

1 z=0z=1z=10

FIG. 6. The estimated local boost including contributions fromthe debris, discreteness and the subhaloes, given by (69) with thefirst term set at its lowest value of unity, is shown for differentredshifts. At high redshifts (e.g. z ¼ 10), the primordial causticsdominate over all other effects. However, at low redshifts thediscreteness effect due to caustics is only important in theoutskirts of the haloes [notice that the discreteness/debris con-tribution remains independent of redshift in the units used in thisplot, while the subhalo contribution evolves as �H�2ðzÞ].

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