users.physics.harvard.eduusers.physics.harvard.edu/...workshop_cyr-racine.pdfSelf-interacting DM and Astrophysics • In the last decade, self-interacting DM has been mostly discussed

Francis-Yan Cyr-Racine Keck Institute for Space Studies Postdoctoral Fellow

JPL/Caltech

Cosmological Signatures of Interacting Dark Matter

Credit: Diana Dragomir

In collaboration with: Roland de Putter Alvise Raccanelli Kris Sigurdson

© 2013 California Institute of Technology. Government sponsorship acknowledged.

Self-interacting DM and Astrophysics •  In the last decade, self-interacting DM has been mostly discussed in

the context of astrophysics.

8/12/13 Francis-Yan Cyr-Racine, JPL/Caltech 2

6 M. Boylan-Kolchin, J. S. Bullock and M. Kaplinghat

Figure 3. Rotation curves for all subhalos with Vinfall

> 30 km s�1 and Vmax

> 10 km s�1, after excluding Magellanic Cloud analogs, ineach of the six Aquarius simulations (top row, from left to right: A, B, C; bottom row: D, E, F). Subhalos that are at least 2� denserthan every bright MW dwarf spheroidal are plotted with solid curves, while the remaining subhalos are plotted as dotted curves. Datapoints with errors show measured V

circ

values for the bright MW dSphs. Not only does each halo have several subhalos that are toodense to host any of the dSphs, each halo also has several massive subhalos (nominally capable of forming stars) with V

circ

comparableto the MW dSphs that have no bright counterpart in the MW. In total, between 7 and 22 of these massive subhalos are unaccounted forin each halo.

of Vcirc

(r1/2) for the bright Milky Way dwarf spheroidals.

As in Fig. 2, we plot only halos with Vinfall

> 30 km s�1

and Vmax

(z = 0) > 10 km s�1. Subhalos that are at least 2�more massive than every dwarf (at r

1/2) are plotted as solidcurves; these are the “massive failures” discussed in BBK,and each halo has at least four such subhalos. Fig. 3 showsthat each halo has several other subhalos with V

infall

> 30that are unaccounted for as well: for example, halo B hasthree subhalos that are not massive failures by our defini-tion but that are inconsistent at 2� with every dwarf exceptDraco. Even ignoring the subhalos that are completely un-accounted for (and are yet more massive than all of the MWdSphs), the remaining massive subhalos do not resemble thebright MW dSph population.

3.3 High redshift progenitors of massive subhalos

To investigate the possible impact of reionization on our re-sults, we show the evolution of the progenitors of all subhaloswith V

infall

> 30 km s�1 in Figure 4. The solid curve showthe median M(z), while the shaded region contains 68% ofthe distribution, centered on the median, at each redshift.

For comparison, we also show Tvir

(z) = 104 K (the tempera-ture at which primordial gas can cool via atomic transitions)and 105 K (dashed lines), as well as the mass Mc(z) belowwhich at least half of a halo’s baryons have been removedby photo-heating from the UV background (Okamoto et al.2008). Subhalos with V

infall

> 30 km s�1 lie above Mc andTvir

= 104 K at all redshifts plotted, indicating that they aretoo massive for photo-ionization feedback to significantly al-ter their gas content and thereby inhibit galaxy formation.

Figure 5 focuses on the z = 6 properties of these sub-halos. It shows the distribution of halo masses at z = 6for “massive failures” (open histogram) and the remainingsubhalos (filled histogram), which are possible hosts of theMW dSphs. The massive failures are more massive at z = 6,on average, than the potentially luminous subhalos. Thisfurther emphasizes that reionization is not a plausible ex-planation of why the massive failures do not have stars: thetypical massive failure is a factor of ten more massive thanthe UV suppression threshold at z = 6. Implications of thisresult will be discussed in Boylan-Kolchin et al. (in prepa-ration).

In a series of recent papers, Broderick, Chang, and

c� 2012 RAS, MNRAS 000, 1–17

Boylan-Kolchin et al. (2012)

Possible Hints of Physics Beyond Cold

Dark Matter

1. Dwarf Galaxy Problem: the number of

detected dwarf galaxies in the Local Group

of the Milky Way appears to be much lower

than predicted by the CDM paradigm. (Moore et al.

1999; Strigari et al. 2007)

2. Mass Profile of Dwarf Spheroidal (dSph)

Galaxies: the inner mass profile of dSphs is

observed to be consistent with a “core” of

constant density while CDM simulations

predict a “cuspy” profile. (de Blok, W. J. G , 2010; Walker, M. G.

& Penarrubia, J., 2011)

Cusp

Wednesday, January 11, 2012 Walker & Penarrubia (2011)

“Core-Cusp” Problem

“Too Big to Fail” Problem “Merging Cluster”

Mahdavi et al. (2007) Dawson et al. (2012)

Self-interacting DM and Astrophysics


6 Rocha et al.

Figure 2. Top: Large scale structure in CDM (left) and SIDM1 (right) shown as a 50!50 h!1 Mpc slice with 10h!1 Mpc thickness through our cosmologicalsimulations. Particles are colored according to their local phase-space density. There are no visible differences between the two cases. Bottom: Small scalestructure in a Milky Way mass halo (Z12) simulated with CDM (left) and SIDM1 (right), including all particles within 200h!1 kpc of the halo centers. Themagnitude of the central phase-space density is lower in SIDM because the physical density is lower and the velocity dispersion is higher. The core of theSIDM halo is also slightly rounder. Note that substructure content is quite similar except in the central regions

cross section runs here because no core density differences were re-solved within the numerical convergence radii of our simulations.

As shown in §3 the self-interaction smoothing length hsi mustbe larger than 20% the inter-particle separation in order to achieveconvergence on the interaction rate. All the work for this paper wasdone with a fixed hsi for all particles carefully chosen for each sim-ulation so that the self-interactions are well resolved at densities afew times to an order of magnitude lower than the lowest densitiesfor which self-interactions are significant. We have run the cosmo-logical boxes with different choices for hsi (changes by factors of2 to 4) and have found that our results are unaffected. We havealso run tests on isolated halos with varying smoothing lengths and

again find that the effects of self-interactions are robust to reason-able changes in hsi.

All of our halo catalogs and density profiles are derivedusing the publicly available code Amiga Halo Finder (AHF)(Knollmann & Knebe 2009).

CDM SIDM

Rocha et al. (2012)

Cosmology

Self-interacting DM and Cosmology


Rocha et al. (2012)

Self-Interacting Dark Matter Simulations I 7

Figure 3. Large-scale characteristics Left: Dark matter two-point correlation functions from our CDM-50 (CDM-25) and SIDM1-50 (SIDM1-25) simulationsin black (grey) and blue (cyan) colors respectively. There are no noticeable difference between the CDM and SIDM1 dark matter clustering over the scalesplotted. Right: Cumulative number density of dark matter halos as a function of their maximum circular velocity (Vmax) at different redshifts for our CDM-50(solid) and SIDM1-50 (dashed) simulations. There are no significant differences in the Vmax functions of CDM and SIDM1 at any redshift.

5 SIMULATION RESULTS

5.1 Preliminary Illustrations

Before presenting any quantitative comparisons between our CDMand SIDM runs, we provide some simulation renderings in order tohelp communicate the qualitative differences.

The upper panels of Figure 2 show a large-scale comparison:two (50!50!10) h!1 Mpc slices from the CDM-50 and SIDM1-50 boxes side-by-side at z = 0. The structures are color-coded bylocal phase-space density (" !/v3rms). It is evident that there areno observable differences in the large-scale characteristics of CDMand SIDM1. We discuss this result in more quantitative terms in§5.2 but of course this is expected. The SIDM models we exploredo not have appreciable rates of interaction for densities outside thecores of dark matter halos. The upper panels of Figure 2 provide avisual reminder that the SIDM models we consider are effectivelyidentical to CDM on larges scales.

The differences between CDM and SIDM become apparentonly when one considers the internal structure of individual ha-los. The lower panels of Figure 2 provide side-by-side images of aMilky-Way mass halo (Z12) simulated with CDM (left) and SIDM1

(right). SIDM tends to make the cores of halos less dense and ki-netically hotter (see §5.3) and these two differences are enhancedmultiplicatively in the phase-space density renderings. The centralregions of the host halo are also slightly rounder in the SIDM case(Peter et al. 2012). Importantly, the difference in substructure char-acteristics are minimal, especially at larger radii. We return to aquantitative description of substructure differences in §5.4.

5.2 Large Scale Structure and Halo Abundances

Figure 3 provides a quantitative comparison of both the clusteringproperties (left) and halo abundance evolution (right) between ourfull-box CDM and SIDM1 simulations. The left panel shows thetwo-point function of dark matter particles in both cosmologicalruns for CDM and SIDM1. There are no discernible differences

between SIDM and CDM over the scales plotted, though of coursethe different box sizes (and associated resolutions) mean that theboxes themselves only overlap for a limited range of scales. Fora given set of initial conditions, however, SIDM and CDM giveidentical results.

The right panel of Figure 3 shows the cumulative numberdensity of dark-matter halos (including subhalos) as a function oftheir peak circular velocity (Vmax) for the CDM-50 (solid) andSIDM1-50 (dashed) simulations at various redshifts. Remarkably,this comparison shows no significant difference either – indicat-ing that SIDM with cross sections as large as 1 cm2/g does notstrongly affect the maximum circular velocities of individual halos.The two panels of Figure 3 demonstrate that for large-scale com-parisons, including analyses involving field halo mass functions,SIDM and CDM yield identical results. The implication is that ob-servations of large-scale structure are just as much a “verification”of SIDM as they are of CDM.

5.3 Halo Structure

Before presenting statistics on halo structure, we focus on six wellresolved halos that span our full mass range Mvir = 5 ! 1011 #2 ! 1014 M", selected from our full simulation suite, includingour two zoom-simulation halos (Z12 and Z11). Figures 4 through6 show radial profiles for the density, circular velocity and velocitydispersion for all three dark matter cases. In each figure, black cir-cles correspond to CDM, green triangles to SIDM0.1, and blue starsto SIDM1. All profiles are shown down to the innermost resolvedradius for which the average two-body relaxation time roughlymatches the age of the Universe (Power et al. 2003).

We begin with the density profiles of halos shown in the six-panel Figure 4. For each halo in the CDM run we have fit an NFWprofile (Navarro et al. 1997) to its radial density structure:

!NFW(r) =!s r

3s

r(rs + r)2, (11)

and recorded its corresponding scale radius rs. The CDM-fit rs

Rocha et al. (2012)

2 Rocha et al.

In this paper we use cosmological simulations to explore the ob-servational consequences of a CDM particle that is strongly self-interacting, focusing specifically on the limiting case of velocity-independent, elastic scattering.

Dark matter particles with appreciable self-interactions havebeen discussed in the literature for more than two decades(Carlson et al. 1992; Machacek et al. 1993; de Laix et al. 1995;Spergel & Steinhardt 2000; Firmani et al. 2000), and are now rec-ognized as generic consequences of hidden-sector extensions to theStandard Model (Pospelov et al. 2008; Arkani-Hamed et al. 2009;Ackerman et al. 2009; Feng et al. 2009, 2010; Loeb & Weiner2011). Importantly, even if dark sector particles have no couplingsto Standard Model particles they might experience strong interac-tions with themselves, mediated by dark gauge bosons (see Feng2010 and Peter 2012 for reviews). The implication is that astro-physical constraints associated with the small-scale clustering ofdark matter may be the only way to test these scenarios.

Phenomenologically, self-interacting dark matter (SIDM) isattractive because it offers a means to lower the central den-sities of galaxies without destroying the successes of CDMon large scales. Cosmological simulations that contain onlyCDM indicate that dark-matter halos should be cuspy and with(high) concentrations that correlate with the collapse time ofthe halo (Navarro et al. 1997; Bullock et al. 2001; Wechsler et al.2002). This is inconsistent with observations of galaxy rotationcurves, which show that galaxies are less concentrated and lesscuspy than predicted in CDM simulations (e.g. Flores & Primack1994; Simon et al. 2005; Kuzio de Naray et al. 2008; de Blok2010; Dutton et al. 2011; Kuzio de Naray & Spekkens 2011;Oh et al. 2011; Walker & Penarrubia 2011; Salucci et al. 2012;Castignani et al. 2012). Even for clusters of galaxies, the densityprofiles of the host dark-matter halos appear in a number of casesto be shallower than predicted by CDM-only structure simula-tions, with the total (dark matter + baryons) density profile in acloser match to the CDM prediction for the dark matter alone (e.g.Sand et al. 2004, 2008; Newman et al. 2009, 2011; Coe et al. 2012;Umetsu et al. 2012).

One possible answer is feedback. In principle, the expul-sion of gas from galaxies can result in lower dark matter den-sities compared to dissipationless simulations, and thus bringCDM models in line with observations (Governato et al. 2010;Oh et al. 2011; Pontzen & Governato 2011; Brook et al. 2012;Governato et al. 2012). However, a new level of concern ex-ists for dwarf spheroidal galaxies (Boylan-Kolchin et al. 2011a;Ferrero et al. 2011; Boylan-Kolchin et al. 2011b). Systems withM! ! 106 M" appear to be missing ! 5 " 107 M" of dark mat-ter compared to standard CDM expectations (Boylan-Kolchin et al.2011b). It is difficult to understand how feedback from such a tinyamount of star formation could have possibly blown out enoughgas to reduce the densities of dwarf spheroidal galaxies to thelevel required to match observations (Boylan-Kolchin et al. 2011b;Teyssier et al. 2012; Zolotov et al. 2012; Penarrubia et al. 2012;Garrison-Kimmel et al., in preparation).

Spergel & Steinhardt (2000) were the first to discuss SIDM inthe context of the central density problem (see also Firmani et al.2000). The centers of SIDM halos are expected to have constantdensity isothermal cores that arise as kinetic energy is transmit-ted from the hot outer halo inward (Balberg et al. 2002; Colın et al.2002; Ahn & Shapiro 2005; Koda & Shapiro 2011). This can hap-pen if the cross section over mass of the dark matter particle,!/m, is large enough for there to be a relatively high probabil-ity of scattering over a time tage comparable to the age of the halo:

! tage ! 1, where ! is the scattering rate per particle. The rate willvary with local dark matter density "(r) as a function of radius r ina dark halo as

!(r) # "(r)(!/m)vrms(r) , (1)

up to order unity factors, where vrms is the rms speed of dark-matterparticles. Based on rough analytic arguments, Spergel & Steinhardt(2000) suggested !/m ! 0.1$100 cm2/gwould produce observ-able consequences in the cores of halos.

Numerical simulations have confirmed the expectedphenomenology of core formation (Burkert 2000) thoughKochanek & White (2000) emphasized the possibility that SIDMhalos could eventually become more dense than their CDMcounterparts as a result of eventual heat flux from the insideout (much like core collapse globular clusters). However thisprocess is suppressed when merging from hierarchical formationis included (for a discussion see Ahn & Shapiro 2005). We do notsee clear signatures of core collapse in the halos we analyzed for!/m = 1 cm2/g.

The first cosmological simulations aimed at understandingdwarf densities were performed by Dave et al. (2001) who useda small volume (4h#1 Mpc on a side) in order to focus com-putational power on dwarf halos. They concluded that !/m =0.1$ 10 cm2/g came close to reproducing core densities of smallgalaxies, favoring the upper end of that range but not being ableto rule out the lower end due to resolution. Almost concurrently,Yoshida et al. (2000) ran cosmological simulations focusing on thecluster-mass regime. Based on the estimated core size of clusterCL 0024+1654, they concluded that cross sections no larger than! 0.1 cm2/g were allowed, raising doubts that constant-cross-section SIDMmodels could be consistent with observations of bothdwarf galaxies and clusters.

These concerns were echoed by Miralda-Escude (2002) whosuggested that SIDM halos would be significantly more spheri-cal than observed for galaxy clusters. Similarly, Gnedin & Ostriker(2001) argued that SIDM would lead to excessive sub halo evapo-ration in galaxy clusters. More recently, the merging cluster systemknown as the Bullet Cluster has been used to derive the limits (68%C.L.) !/m < 0.7 cm2/g (Randall et al. 2008) based on evapora-tion of dark matter from the subcluster and !/m < 1.25 cm2/g(Randall et al. 2008) based on the observed lack of offset betweenthe bullet subcluster mass peak and the galaxy light centroid. Inorder to relax this apparent tension between what was requiredto match dwarf densities and the observed properties of galaxyclusters, velocity dependent cross sections that diminish the ef-fects of self-interaction in cluster environments have been con-sidered (Firmani et al. 2000; Colın et al. 2002; Feng et al. 2009;Loeb & Weiner 2011; Vogelsberger et al. 2012).

There are a few new developments that motivate us to revisitconstant SIDM cross sections on the order of !/m ! 1 cm2/g. Forexample, the cluster (CL 0024+1654) used by Yoshida et al. (2000)to place one of the tightest limits at !/m = 0.1, is now recognizedas an ongoing merger along the line of sight (Czoske et al. 2001,2002; Zhang et al. 2005; Jee et al. 2007; Jee 2010; Umetsu et al.2010). This calls into question its usefulness as a comparison casefor non-merging cluster simulations. In a companion paper (Peter,Rocha, Bullock and Kaplinghat, 2012) we use the same simulationsdescribed here to show that published constraints on SIDM basedon halo shape comparisons are significantly weaker than previ-ously believed. Further, the results presented below clearly demon-strate that the tendency for subhalos to evaporate in SIDM models(Gnedin & Ostriker 2001) is not significant for !/m ! 1 cm2/g.


While nonstandard dark matter is believed to affect mostly small astrophysical scales, can it also lead to any telltale cosmological signatures (CMB, P(k), weak lensing, etc)?

Key Questions:

•  Which properties does interacting DM need to have in order to affect cosmological observables?

•  What are the relevant parameters describing these properties that can be constrained with cosmological data?



While some people see cosmological scales as constraints that SIDM needs to satisfy, I see them as an

opportunity to unveil new physics in the DM sector.


•  We need at least a fraction of the DM to:

1.  Couple to a light , relativistic particles, or be itself a relativistic particle at early times (e.g. Warm DM). This leads to a non-vanishing sound speed that provides pressure support against gravitational collapse.

2.  Have a relatively late epoch of kinematic decoupling. Such that cosmological scales can be affected.



•  Interacting DM models affect cosmological observables in two ways:

1.  Modifications to the background cosmology.

2.  Modification to the evolution of DM fluctuations.


A Cosmological Limit on Neutrino Self-Interactions

Francis-Yan Cyr-RacineNASA Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA and

California Institute of Technology, Pasadena, CA 91125, USA

Kris SigurdsonDepartment of Physics and Astronomy, University of British Columbia, Vancouver, BC, V6T 1Z1, Canada

(Dated: August 4, 2013)

In the standard model neutrinos are assumed to have streamed across the Universe since theylast scattered at the weak decoupling epoch when the temperature of the standard-model plasmawas ⇠MeV. However, the presence of nonstandard physics in the neutrino sector could alter thissimple picture and delay neutrino free-streaming until a much later epoch. We use observations ofthe Cosmic Microwave Background (CMB) to constrain the strength of neutrino self-interactions G⌫

and put limits on new physics in the neutrino sector from the early universe. The recent improvementin accuracy of CMB measurements made by the Planck satellite and high-l experiments is criticalin obtaining this constraint. We show that cosmological data allows neutrino free-streaming tobe delayed until the Universe has cooled to a temperature close to 35 eV, almost five orders ofmagnitude lower than in the standard cosmological paradigm. Nevertheless, these data constrainneutrino physics at an e↵ective energy scale ⇤⌫ & 30 MeV well above the typical energy scaleof neutrinos when the decouple. While we discuss a specific scenario in which such a late onset ofneutrino free-streaming could occur our constraint on the neutrino visibility function is very general.

PACS numbers: 98.80.-k

INTRODUCTION Neutrinos are the most elusivecomponents of the standard model (SM) of particlephysics. Their tremendously weak interactions withother SM fields render measurements of their fundamen-tal properties very challenging. At the same time, theexistence of neutrino mass [? ] constitutes one of themost compelling evidence for physics beyond the SM, andmakes the neutrino sector a prime candidate for searchesof such new physics. In recent years, cosmology has pro-vided some of the most stringent constraints on neutrinoproperties, most notably the sum of their masses andtheir e↵ective number [? ? ? ]. Can cosmological datacan inform us about other aspects of neutrino physics?

GF

(~c)3 =

p2

8

g2

m2W

= 1.16637(1)⇥ 10�11 MeV�2 (1)

�W / n⌫G2FT

2⌫ / G2

FT5⌫ (2)

�W ⇠ H (3)

z⌫,dec ⇠ 1010 (4)

R⌫ =⇢⌫

⇢� + ⇢⌫' 0.403 (5)

Ne↵ ' 3.046 (6)

H0 ' 70 km/s/Mpc (7)

⌦⇤ ' 0.71 (8)

ns ' 0.935 (9)

⌦bh2 ' 0.0223 (10)

⌦ch2 ' 0.118 (11)

⌧ ' 0.09 (12)

ln(1010As) ' 3.03 (13)

Ge↵ ' 0.016 MeV�2 (14)

�DM = 3�� ✓DM (15)2

✓DM +a

a✓DM � c2DMk2�DM � k2 = �DM(✓REL � ✓DM)

One assumption that is rarely challenged is the free-streaming nature of cosmological neutrinos (for excep-tions, see [? ? ? ? ? ? ? ]). Within the standard modelthis assumption is justified since SM neutrinos are ex-pected to have decoupled from the primeval plasma in thevery early Universe at a temperature T ' 1 MeV. Yet,this assumption is not a priori driven by any cosmolog-ical observations, but is the results of a particle-physicsprior on the choice of cosmological models that we chooseto compare with data. Abandoning this prior allows usto answer the important question: How does cosmologi-cal data inform us about possible interactions in the neu-trino sector? Free-streaming neutrinos create anisotropicstress which, through gravity, alters the evolution of theother particle species in the Universe [? ? ]. As cosmo-logical fluctuations in the photon and baryon fluids areparticularly sensitive to the presence of a free-streamingcomponent during the radiation-dominated era, we ex-pect the recent measurements of the CMB to provide astrong constraint on the onset of neutrino free-streaming.

In this Letter, we compute the first purely cosmologicalconstraints on the strength of neutrino self-interactions.In the following, we model the interaction as a four-fermion vertex whose strength is controlled by a dimen-sional constant, analogous to the Fermi constant, G⌫ . Inthis scenario, the onset of neutrino free-streaming is de-layed until the rate of these interactions fall below theexpansion rate of the Universe, hence a↵ecting the evo-lution of cosmological fluctuations that enters the causalhorizon before that epoch. As we discuss below, the cos-mological observables are compatible with a neutrino vis-ibility function peaking at a temperature orders of mag-nitude below that of the standard picture.

In earlier investigations of neutrino properties [? ?? ? ? ? ], neutrinos were modeled as a fluid-like [?] and constraints were placed on the phenomenologicalparameters ce↵ and cvis, the rest-frame sound speed andthe viscosity parameter of the neutrino fluid respectively.These analysis found consistency with the free-streaminglimit. However, by modeling these parameters as con-stant throughout the history of the Universe they couldnot capture the realistic physics of neutrino decoupling.We incorporate here the physics necessary to follow indetail the dynamics of the transition of neutrinos froma tightly-coupled fluid to particles free-streaming acrossthe Universe.

NEUTRINO INTERACTIONS In addition totheir regular SM interactions, we assume that all ofthe neutrinos have non-negligible self-interactions due totheir interaction with a new heavy mediator X. We takeX to be a singlet under all SM interactions and assumethat it only interacts with neutrinos through a couplingconstant gX . When the temperature of the neutrinosfalls significantly below the mediator mass, one can inte-

grate out the heavy mediator and model the interactionas a four-fermion vertex controlled by a dimensionfullcoupling constant G⌫ / g2⌫/M

2X . In this scenario, the

possible emission of X particle by neutrinos in the fi-nal state of kaon and W decay leads to upper boundson the value of g⌫ . For a vector boson, we must haveg⌫ < 8 ⇥ 10�5(MX/MeV) [? ], while for a scalar Xwe have g⌫ < 0.014 (90%-C.L.) [? ]. In comparison,SN1987A places a much weaker constraint on neutrinoself-interaction, leading to G⌫ . 144MeV�2 [? ]. In thefollowing, we focus on the case where X is a scalar.The key quantity characterizing the interactions in

the neutrino sector is the thermally-averaged neutrinoself-interaction cross section h�⌫iT⌫ ⌘ G2

e↵T2⌫ , where all

the order unity numerical factors have been absorbed inGe↵ / G⌫ , and T⌫ is the temperature of the neutrinobath. The X-mediated self-interactions render the neu-trino medium opaque with an opacity ⌧⌫ = an⌫h�⌫iT⌫ ,where n⌫ is the number density of neutrinos and a isthe scale factor describing the expansion of the Universe.In this work, we focus our attention on the case whereG⌫ > GF, where GF is the Fermi constant. Therefore, itis justified to neglect the contributions from electroweakprocesses to the neutrino opacity.The opacity of the neutrino medium e↵ectively defines

a neutrino visibility function given by f⌫(z) ⌘ �⌧⌫e�⌧⌫ .This visibility function can be thought of as a probabil-ity density function for the redshift at which a neutrinobegins to free-stream. For neutrino self-interacting withthe cross section given above, the visibility function isalways sharply peaked with a nearly Gaussian shape ex-cept for a long tail extending toward lower redshifts. Weplot the neutrino visibility function for di↵erent values ofGe↵ in Fig. 1. We observe that the main e↵ect of neu-trino self-interaction is to considerably delay the onset offree-streaming.EVOLUTION OF FLUCTUATIONS To deter-

mine the impact of neutrino self-interaction on cos-mological observables, we evolve the neutrino fluctua-tion equations from their early tightly-coupled stage totheir late-time free-streaming solution. By prohibitingfree-streaming, neutrino self-interaction severely dampsthe growth of anisotropic stress associated with thequadrupole and higher moments of the neutrino distribu-tion function. Indeed, while the equations for the densityand velocity fluctuations of the neutrinos are una↵ectedby the self-interaction, the moments with l � 2 are cor-rected by a damping term proportional to ⌧⌫ which ef-fectively suppresses their growth,

F⌫2 =8

15✓⌫ +

8

15k� � 3

5kF⌫3 �

9

10⌧⌫F⌫2, (16)

F⌫l =k

2l + 1

⇥lF⌫(l�1) � (l + 1)F⌫(l+1)

⇤� ⌧⌫F⌫l, (17)

where we follow closely the notation of [? ] in syn-chronous gauge. We solve these equations numerically

Sound speed

Pressure term gravitational potentials

Collision term Momentum transfer rate

2

✓DM

+a

a✓DM

� c2DM

k2�DM

� k2 = �DM

(✓REL

� ✓DM

)

H2 =8⇡G

3⇢tot

(16)

One assumption that is rarely challenged is the free-streaming nature of cosmological neutrinos (for excep-tions, see [? ? ? ? ? ? ? ]). Within the standard modelthis assumption is justified since SM neutrinos are ex-pected to have decoupled from the primeval plasma in thevery early Universe at a temperature T ' 1 MeV. Yet,this assumption is not a priori driven by any cosmolog-ical observations, but is the results of a particle-physicsprior on the choice of cosmological models that we chooseto compare with data. Abandoning this prior allows usto answer the important question: How does cosmologi-cal data inform us about possible interactions in the neu-trino sector? Free-streaming neutrinos create anisotropicstress which, through gravity, alters the evolution of theother particle species in the Universe [? ? ]. As cosmo-logical fluctuations in the photon and baryon fluids areparticularly sensitive to the presence of a free-streamingcomponent during the radiation-dominated era, we ex-pect the recent measurements of the CMB to provide astrong constraint on the onset of neutrino free-streaming.

In this Letter, we compute the first purely cosmologicalconstraints on the strength of neutrino self-interactions.In the following, we model the interaction as a four-fermion vertex whose strength is controlled by a dimen-sional constant, analogous to the Fermi constant, G⌫ . Inthis scenario, the onset of neutrino free-streaming is de-layed until the rate of these interactions fall below theexpansion rate of the Universe, hence a↵ecting the evo-lution of cosmological fluctuations that enters the causalhorizon before that epoch. As we discuss below, the cos-mological observables are compatible with a neutrino vis-ibility function peaking at a temperature orders of mag-nitude below that of the standard picture.

In earlier investigations of neutrino properties [? ?? ? ? ? ], neutrinos were modeled as a fluid-like [?] and constraints were placed on the phenomenologicalparameters c

e↵

and cvis

, the rest-frame sound speed andthe viscosity parameter of the neutrino fluid respectively.These analysis found consistency with the free-streaminglimit. However, by modeling these parameters as con-stant throughout the history of the Universe they couldnot capture the realistic physics of neutrino decoupling.We incorporate here the physics necessary to follow indetail the dynamics of the transition of neutrinos froma tightly-coupled fluid to particles free-streaming acrossthe Universe.

NEUTRINO INTERACTIONS In addition totheir regular SM interactions, we assume that all ofthe neutrinos have non-negligible self-interactions due totheir interaction with a new heavy mediator X. We take

X to be a singlet under all SM interactions and assumethat it only interacts with neutrinos through a couplingconstant gX . When the temperature of the neutrinosfalls significantly below the mediator mass, one can inte-grate out the heavy mediator and model the interactionas a four-fermion vertex controlled by a dimensionfullcoupling constant G⌫ / g2⌫/M

2

X . In this scenario, thepossible emission of X particle by neutrinos in the fi-nal state of kaon and W decay leads to upper boundson the value of g⌫ . For a vector boson, we must haveg⌫ < 8 ⇥ 10�5(MX/MeV) [? ], while for a scalar Xwe have g⌫ < 0.014 (90%-C.L.) [? ]. In comparison,SN1987A places a much weaker constraint on neutrinoself-interaction, leading to G⌫ . 144MeV�2 [? ]. In thefollowing, we focus on the case where X is a scalar.

The key quantity characterizing the interactions inthe neutrino sector is the thermally-averaged neutrinoself-interaction cross section h�⌫iT⌫ ⌘ G2

e↵

T 2

⌫ , where allthe order unity numerical factors have been absorbed inG

e↵

/ G⌫ , and T⌫ is the temperature of the neutrinobath. The X-mediated self-interactions render the neu-trino medium opaque with an opacity ⌧⌫ = an⌫h�⌫iT⌫ ,where n⌫ is the number density of neutrinos and a isthe scale factor describing the expansion of the Universe.In this work, we focus our attention on the case whereG⌫ > G

F

, where GF

is the Fermi constant. Therefore, itis justified to neglect the contributions from electroweakprocesses to the neutrino opacity.

The opacity of the neutrino medium e↵ectively definesa neutrino visibility function given by f⌫(z) ⌘ �⌧⌫e�⌧⌫ .This visibility function can be thought of as a probabil-ity density function for the redshift at which a neutrinobegins to free-stream. For neutrino self-interacting withthe cross section given above, the visibility function isalways sharply peaked with a nearly Gaussian shape ex-cept for a long tail extending toward lower redshifts. Weplot the neutrino visibility function for di↵erent values ofG

e↵

in Fig. 1. We observe that the main e↵ect of neu-trino self-interaction is to considerably delay the onset offree-streaming.

EVOLUTION OF FLUCTUATIONS To deter-mine the impact of neutrino self-interaction on cos-mological observables, we evolve the neutrino fluctua-tion equations from their early tightly-coupled stage totheir late-time free-streaming solution. By prohibitingfree-streaming, neutrino self-interaction severely dampsthe growth of anisotropic stress associated with thequadrupole and higher moments of the neutrino distribu-tion function. Indeed, while the equations for the densityand velocity fluctuations of the neutrinos are una↵ectedby the self-interaction, the moments with l � 2 are cor-rected by a damping term proportional to ⌧⌫ which ef-

SIDM: Historical Perspective


1992ApJ...398...43C

1992ApJ...398...43C

SIDM: Cannibal Dark Matter


•  Chemical equilibrium keeps DM warm, providing it with pressure support against gravitational growth.

1994ApJ...431...41M

1994ApJ...431...41M

Machacek (1994); De Laix et al.(1995).

“Sticky Neutrino” Model


Interacting hot dark matter

Fernando Atrio-BarandelaFısica Teorica, Universidad de Salamanca, Plaza de la Merced s/n, 37008 Salamanca, Spain

Sacha DavidsonMax-Planck-Institut fur Physik, Fohringer Ring 6, D-80805 Munchen, Germany

~Received 17 June 1996!

We discuss the viability of a light particle ~;30 eV neutrino! with strong self-interactions as a dark mattercandidate. The interaction prevents the neutrinos from free-streaming during the radiation-dominated regime sogalaxy-sized density perturbations can survive. Smaller scale perturbations are damped due to neutrino diffu-sion. We calculate the power spectrum in the imperfect fluid approximation, and show that it is damped at thelength scale one would estimate due to neutrino diffusion. The strength of the neutrino-neutrino coupling isonly weakly constrained by observations, and could be chosen by fitting the power spectrum to the observedamplitude of matter density perturbations. The main shortcoming of our model is that interacting neutrinos can-not provide the dark matter in dwarf galaxies. @S0556-2821~97!06110-9#

PACS number~s!: 95.35.1d, 14.60.St, 98.80.Cq

I. INTRODUCTION

Observations on many scales imply that most of the mat-ter in the universe is not emitting electromagnetic radiation@1#. The ratio of the energy density in this unseen mattercomponent @usually referred to as dark matter ~DM!# to thecritical density today, is defined as V0 and is under debate.Various dynamical determinations of V0 on the scales ofgalaxies and clusters give V0.0.260.1 @1#. Observations oflarge velocity flows suggest that V0 could be larger than this,possibly V0.1 with large uncertainties @2,3#. Theoreticalprejudice favors V051, because this is more natural thatV0,1, and because it is predicted by inflation. However, inthis case, one needs V;0.8 smooth on galaxy scales.Primordial nucleosynthesis implies that the energy density

in baryons satisfies 0.003,VB,0.06 @4#, which might justbe consistent with the observations of V0 . However, it isvery difficult to build models involving only baryons thatproduce the large scale structure and the cosmic microwavebackground ~CMB! temperature fluctuations we see today@5,6#, so it is usually assumed that the dark matter is made ofsomething other than baryons. The list of potential candi-dates is long; the popular possibilities are massive neutrinos,axions, and supersymmetric particles.In the standard picture, the dark matter ~DM! is in thermal

equilibrium with the rest of the Universe at some early ep-och, but effectively noninteracting when length scales of in-terest for structure formation come into the horizon. In thiscase, the only free parameter is the mass of the DM particle.If it had m*1 GeV, it would be nonrelativistic at all times ofinterest for structure formation, and it would behave like apressureless fluid. If it was light, it would be relativistic untilcluster scales came within the horizon, and perturbations onsmaller scales would be damped by free-streaming. As a re-sult, in this second type of model @called hot dark matter~HDM! models# clusters are produced first and must frag-ment into galaxies. This ‘‘top-down’’ galaxy formation sce-nario is difficult to reconcile with observations @7# whichindicate that galaxies and small scale structure formed first.

A model where the dark particles behave nonrelativisticallyis therefore preferred. The standard cold dark matter ~CDM!

model was found to have too much power on small scaleswhen normalized to the amplitude of the measured CosmicBackground Explorer ~COBE! Differential Microwave Radi-ometer ~DMR! temperature anisotropies and several modifi-cations were introduced. Another possibility is a ‘‘warm’’dark matter particle with a mass (;keV) intermediate be-tween that of ‘‘hot’’ (;eV) and ‘‘cold’’ (.GeV) @9#. Cur-rently, models with a mixture of hot and cold dark matter~MDM! are favored @8#.It is usually assumed that during structure formation (T

g

,100 eV) the evolution of the Universe was controlled byelectromagnetism and gravitation, while new particle physicstakes place at accelerator energies. However, it is possiblethat the dark matter particles could have some unexpectedinteraction during the epoch of structure formation, despitethis occuring at a low-energy scale. Some examples of thishave been previously studied @10–13#. In this paper, we con-sider a simple interacting dark matter model; we assume thatthe dark matter consists of light particles with a large crosssection for scattering off each other ~we discuss later themeaning of ‘‘large’’!. For definiteness, we take the light par-ticle to be a neutrino with mass ;20–30 eV ~as for HDM!.Our interest is to determine whether, in the presence of hotdark matter self-interactions, density perturbations on scalessmaller than galaxy clusters remain undamped. In normalHDM, they are washed out by free streaming of relativisticparticles; in an interacting model, one would rather expectthe perturbations to be damped on a smaller scale by diffu-sion ~Silk damping!. We will show that this is in fact thecase, and compute on what scale they are damped.The question of whether ‘‘sticky neutrinos behave like

cold dark matter’’ has been previously addressed by Raffeltand Silk @10#. They argue that light (m;20 eV) interactingneutrinos will dissipate small scale perturbations(,galaxy sized) by ‘‘Silk damping,’’ but there will be nofree streaming. This result was briefly questioned by Dicuset al. in their paper about neutrino interactions in supernovae

PHYSICAL REVIEW D 15 MAY 1997VOLUME 55, NUMBER 10

550556-2821/97/55~10!/5886~9!/$10.00 5886 © 1997 The American Physical Society

See also, FYCR & Sigurdson, arXiv:1306.1536


“Sticky Neutrino” Model

approximation is still valid, then we know that the viscosityand heat conduction coefficients are proportional to the pres-sure @25#, and the pressure is negligible. So we expect thenonrelativistic, interacting neutrinos to behave like standardneutrinos, if the interaction is short range. A longer rangeinteraction ~exchanged boson mass !m

n

! is potentially morecomplicated, particularly if the interactions are still strongtoday. Dark matter with long-range interactions has beenconsidered in @13#; we neglect it here. Constraints on darkmatter self-interactions from galaxy formation and evolutionin the nonlinear regime have been estimated in @12#. Theauthors argue that the average interaction length of dark mat-ter today must be longer than 100 Mpc. Sticky neutrinos whointeract via the exchange of a m;MeV boson and who areconsistent with the supernova bound are essentially noninter-acting today ~the interaction rate scales as a25 or a27

!, sothey easily satisfy the constraints listed in @12#.Based on our analytic results, we expect the power spec-

trum of interacting HDM’s to be similar to standard CDM onlarge and intermediate scales, but damped on small scales byneutrino diffusion. We are working in the fluid approxima-tion, so to compute the power spectrum, we must numeri-cally integrate the Einstein equations and the energy conser-vation equation ~14! for an imperfect fluid. This isconsiderably simpler than integrating the coupled Einsteinand Boltzmann equations.Let us briefly outline here the equations and approxima-

tions used. We perform our calculations in the synchronousgauge using the metric ~3!. The equations governing the evo-lution of perturbations are more easily written in Fourierspace. If we only consider only scalar-type perturbations, themetric perturbations can be written in terms of two scalarfields h(k ,t) and d(k ,t) as @27#

hi j~x ,t !5E d3keikx@

kik jh~

k ,t!

1$

kik j213 d i j6d~

k ,t!

%

#

.~39!

As in the FRW case, the equations governing the evolutionof matter density perturbations can be derived from the con-servation equation ~14!. For a relativistic fluid, with viscositybut no bulk viscosity or heat conduction, Eqs. ~25! and ~26!become

d5243 S u2

12 h D , ~40!

u1aa u5p2S d

42h

r

~

u23 d! D . ~41!

This result differs from @7# or @27# because our matter doesnot behave as a perfect fluid. As the temperature drops pastthe neutrino mass, the neutrinos no longer behave as relativ-istic particles. The evolution of density perturbations in theE;m regime is complicated by the time-dependent relationbetween the neutrino momentum and energy @27#. This addsterms to Eqs. ~40! and ~41! and makes the numerical calcu-lation of the pressure and density more difficult @32#. Tosimplify the calculation, we assume the transition from arelativistic gas to presureless dust to be instantaneous. Wealso assume that neutrino interactions are frozen in the non-

relativistic regime due to the small mean velocity. As a re-sult, we treat the neutrino gas as CDM particles with zerovelocity in the synchronous gauge, whose evolution is givenby

d512 h . ~42!

As previously noted, perturbations in a relativistic fluidare damped by viscosity. The off-diagonal spatial elementsof the stress-energy tensor ~parametrized by the viscosity!are also gauge invariant, which makes them simple to calcu-late. We therefore include only the viscosity ~no heat con-duction coefficient and no bulk viscosity! in our numericalcalculation of the power spectrum. We take the physical in-teraction time t to scale as a5 ~comoving t;a4!, and as-sume an initial Harrison-Zel’dovich spectrum of perturba-tions from inflation.In Fig. 1 we give the numerical integration of the previous

set of differential equations coupled with the equations forthe evolution of metric perturbations, baryons, and photons.We used a modified version of the COSMIC package madepublicly available by Berstchinger and Ma, and described in@27#. The dashed line corresponds to standard CDM and isplotted for comparison. The three solid lines correspond, indecreasing amplitude, to zero viscosity ~zero-mean-free path!and comoving mean-free paths at T

g

510 eV of 1022 and 0.1Mpc. In the case of zero viscosity, the neutrino perturbationsoscillate as acoustic waves around the CDM power spec-trum. The behavior is similar to the baryon-photon plasmawhere the interactions cause the perturbations in the baryoncomponent to oscillate as sound waves. When the viscosityis introduced the oscillations are damped approximately atthe scale given by the diffusion length. The effect of viscos-ity is to damp the power spectrum at the scale given by Eq.~37!: 3 and 7 Mpc21, respectively, for comoving mean-free

FIG. 1. Power spectrum for three different ‘‘sticky neutrino’’models. The solid lines correspond ~in decreasing amplitude! tointeractions with no neutrino diffusion, and with comoving mean-free paths at T

g

510 eV of 0.01 and 0.1 Mpc. These give rise toestimated damping scales of 7 and 3 Mpc21, respectively. Thedashed line correspond to standard CDM and is plotted for com-parison. The y-axis scale is arbitrary. We took H0550 km s21Mpc21.

5892 55FERNANDO ATRIO-BARANDELA AND SACHA DAVIDSON

Atrio-Barandela & Davidson (1997)


for our two different cases. In both cases, the power spec-trum cut-off is pushed towards higher k if self-interaction isassumed. The HDM (m!10 eV) results are in agreementwith the results of Atrio-Barandela and Davidson !25", for ksmaller than the cut-off scale. At small scales, their resultsare somewhat different from ours, probably because of anerroneous term in their perturbation equations #as explainedin the Appendix$.For our choice of particle masses, the dividing line be-

tween the non-interacting and strongly interacting regimes isroughly at

%0!10"36 cm2. #32$

Note that this is much lower than the cross section whichis needed to explain structure on galactic scales in the self-interacting cold dark matter model. In that case, the dividingline is closer to 10"23 cm2. For the case where the darkmatter is hot, self-interactions are not able to improve theagreement with observations significantly because the powerspectrum cut-off is still at much too large a scale. As dis-cussed in Ref. !14", warm dark matter provides a good fit toobservations of dwarf galaxies if the power spectrum cut-offis at roughly 2h50Mpc"1, corresponding to a mass of 1 keV.However, explaining the core structure of dark matter halosrequires that m#300 eV !14", so that even though the un-certainties involved in determining the best cut-off scale areas large as a factor two !14", the collisionless warm darkmatter model is inconsistent with observations. Our resultsindicate that it might be possible to lower the warm darkmatter particle mass to this smaller value and compensate by

making the warm dark matter self-interacting, which de-creases the cut-off length scale by about a factor of 1.6 com-pared to the non-self-interacting case. Numerically we findthat the k where &2(k) takes its maximum value is wellapproximated by

&2#k $max

!" 1.1# m1 keV$ 3/4Mpc"1 collisionless,

1.7# m1 keV$ 3/4Mpc"1 strongly self-interacting.

#33$

For the collisionless case this corresponds to the free-streaming scale, whereas in the strongly interacting case itcorresponds to the Jeans scale for a given particle mass.From this result we conclude that self-interacting warm darkmatter is marginally consistent with the present observationalconstraints.For the cosmic microwave background #CMB$, the fluc-

tuations are usually expressed in terms of the Cl coefficients,Cl!'%alm%2(, where the alm coefficients are determined interms of the real angular temperature fluctuations asT() ,*)!+ lmalmY lm() ,*). Figure 2 shows the CMB spectrafor the same two particle masses. If the dark matter is hot,the CMB spectrum is changed relative to cold dark matter,because the DM particles are not completely non-relativisticat recombination. This gives rise to what is called the earlyintegrated Sachs-Wolfe #ISW$ effect. Self-interactions havevery little impact because they only affect scales within thedark matter sound horizon at recombination. Even for a darkmatter mass of 10 eV, this is at too small a scale to have asignificant impact. For a dark matter particle mass of 1 keV,

FIG. 1. Matter power spectra for two different dark matter par-ticle masses. The dashed line is for no self-interaction, the triple-dot-dashed is for %0!1.2$10"38 cm2, the dot-dashed for %0!8$10"36 cm2, and the dotted is assuming complete pressure equi-librium. For reference we have plotted the spectrum for standardcold dark matter #full line$.

FIG. 2. CMB power spectra for the same models as in Fig. 1.The curve labels are also identical to those in Fig. 1.

STEEN HANNESTAD AND ROBERT J. SCHERRER PHYSICAL REVIEW D 62 043522

043522-4

Hannestad and Scherrer (2000)

2

0.01

1

100

10000

1e-06 0.0001 0.01 1

|δin

t|

scale factor a

no couplingQ=10-38cm2MeV-1

0

1000

2000

3000

4000

5000

6000

7000

10 100 1000

l(l+1

)Cl/2π

(µK2 )

l

100

1000

10000

100000

0.01 0.1

P(k)

(h-3

Mpc

3 )

k (h Mpc-1)

FIG. 1: A perturbation of interacting dark matter (redline) of wavenumber k = 0.81 h Mpc!1 is plotted against aperturbation of non-interacting dark matter (black dottedline) in the upper panel. The e!ects of this interactionare clearly seen in the angular power spectum of the CMB(middle panel) and on the matter power spectum (lowerpanel). We have chosen a very large value of Q to magnifythe e!ect on the main cosmological observables.

likely that the relic abundance of these non standarddark matter candidates formed via the well knownfreezing phenomenon, as the pair annihilations wouldbe quite e!cient till recent times in depleting theirnumber density, unless they represent a too lightspecies. On the other hand, a di"erent scenario has

been also considered in the literature, where dark mat-ter relic density today is the remnant of an initial par-ticle – antiparticle asymmetry produced in the earlyuniverse, thus similar to the mechanism of baryogene-sis which leads to a baryon density today much largerthan what is expected by freezing of strong interac-tions alone [18, 19]. This possibility may also ac-count for the intriguing similarity between the valuesof #b and #dm we observe today, which di"er by a fac-tor five only, a feature which may call for a commonmechanism for their formation. In the following wetherefore, consider the case of a non self–conjugatedfermion or scalar particle ! with a conserved globalU(1) charge, at least in the low energy scale regime,say below their mass scale, which also corresponds tothe relevant stages for structure formation we are in-terested in.

In case of a spin zero species, the interaction la-grangian can be chosen as a Yukawa term

Lint = hFR"L! + h.c., (1)

with F a spinor field, or via coupling with an inter-mediate vector-boson field Uµ

Lint = ig(!!#µ! ! !#µ!!)Uµ + g2!!!UµUµ

+ g!"L$µ"LUµ. (2)

In both cases F and U fields will be assumed to havemass larger than !, to prevent fast ! decay at treelevel, see [13].

In case of spin 1/2 dark matter, one has

Lint = h!R"LF + h.c., (3)

with F a scalar field or finally,

Lint = g(cL!L$µ!L + cR!R$

µ!R)Uµ

+ g!"L$µ"LUµ. (4)

We notice that, as we will see in the next Section,bounds on dark matter – neutrino scattering cross sec-tion from CMB and large scale data, correspond to amass scale for ! as well as the other field involved inthe interaction lagrangian (F or U) larger than MeV.Thus, we will assume this lower bound in the follow-ing, mdm " MeV, mF,U " MeV.

It is quite easy to compute the thermal averagedscattering cross section corresponding to these inter-action terms in the non-relativistic limit for the darkmatter particle ! and F or U . For example in thecase of Eq. (1) or (3) and (2) one gets, respectively

#%dm"! |v|$ % |h|4T 2

!

(m2F ! m2

dm)2, (5)

#%dm"! |v|$ % g2g2!

T 2!

m4U

. (6)

The first result holds for mF &= mdm, otherwise crosssection takes a constant value, as for the well known

Cyr-Racine & Sigurdson (2012)

4

1000

10000

10 100 1000C

l l(l+

1)/2π

[µK

]2l

ΛCDM WMAP7ΔNeff=1, c2

eff=0, Q0=10-32 cm2 MeV-1

ΔNeff=1, c2eff=0.33, c2

vis=0.33, Q0=10-32 cm2 MeV-1

ΔNeff=1, c2vis=0, Q0=10-32 cm2 MeV-1

WMAP7 datasetsSPT datasets

0.1

1

10

100

1000

10000

100000

0.0001 0.001 0.01 0.1 1

P(k)

[(h-1

Mpc

)3 ]

k [h/Mpc]

LRG sample from Data Release 7ΛCDM WMAP7

ΔNeff=1, c2eff=0, Q0=10-32 cm2 MeV-1

ΔNeff=1, c2eff=0.33, c2

vis=0.33, Q0=10-32 cm2 MeV-1

ΔNeff=1, c2vis=0, Q0=10-32 cm2 MeV-1

FIG. 1: Upper panel: The magenta lines depict the CMB temperature power spectra CTT

l for the best fit parameters for a!CDM model from the WMAP seven year data set. The dotted curve shows the scenario with a constant interacting crosssection with Q0 = 10!32cm2 MeV!1 for "Ne! = 1 and assuming canonical values for c2e! = c2vis = 1/3. The dashed (dotdashed) curve illustrates the same interacting scenario but with c2e! = 0 and c2vis = 1/3 (c2e! = 1/3 and c2vis = 0). We depictas well the data from the WMAP and SPT experiments, see text for details. Lower panel: matter power spectrum for thedi#erent models described in the upper panel. The data correspond to the clustering measurements of luminous red galaxiesfrom SDSS II DR7 [31].

following set of parameters:

{!b, !c, !s, ", ns, log[1010As], "Ne!, c

2vis, c

2e! , Q0},

where !b ! #bh2 and !c ! #ch2 are the physical baryonand cold dark matter energy densities, !s is the ratio be-tween the sound horizon and the angular diameter dis-tance at decoupling, " is the optical depth, ns is thescalar spectral index, As is the amplitude of the primor-dial spectrum, "Ne! is the extra dark radiation compo-nent, c2vis is the viscosity parameter, c2e! is the e$ectivesound speed and Q0, in units of cm2 MeV!1, encodes thedark radiation-dark matter interaction. The flat priorsconsidered on the di$erent cosmological parameters are

specified in Tab. I.

For CMB data, we use the seven year WMAP data[6, 33] (temperature and polarization) by means of thelikelihood supplied by the WMAP collaboration. Weconsider as well CMB temperature anisotropies from theSPT experiment [10], which provides highly accuratemeasurements on scales ! 10 arcmin. We account as wellfor foreground contributions, adding the SZ amplitudeASZ , the amplitude of the clustered point source contri-bution, AC , and the amplitude of the Poisson distributedpoint source contribution, AP , as nuisance parameters inthe CMB data analysis.

Furthermore, we include the latest constraint from

Diamanti et al. (2012)

10-2

10-1

100

100 101 102 103 104 105 106

[P (k

) / P

cdm

(k)]1/

2

k [1/Mpc]

(tanθ, ξRH) = (0.8, 0.8)mX = 1 TeV

10 GeV100 MeV

10-2

10-1

100

100 101 102 103 104 105 106 107 108

[P (k

) / P

cdm

(k)]1/

2

k [1/Mpc]

(tanθ, ξRH) = (0.8, 0.1)mX = 1 TeV

10 GeV100 MeV

10-2

10-1

100

100 101 102 103 104 105 106 107 108

[P (k

) / P

cdm

(k)]1/

2

k [1/Mpc]

(tanθ, ξRH) = (10, 0.1)mX = 1 TeV

10 GeV100 MeV

FIG. 4: Transfer functions of the normalized dark matter density perturbation amplitude for(tan !h

W , "RH) = (!

3/5, 0.8) (top), (!

3/5, 0.1) (middle), and (10, 0.1) (bottom).

11

Hidden Charged DM

Feng et al. (2009)

Neutrino-DM Interaction

Serra et al. (2009)

Interacting Warm DM

DM-DR Interaction

⌅ ⇥ 0.37⇤D ⇥ 0.008BD ⇥ 1 keVmD ⇥ 1 GeV

10�5 0.001 0.1 1010�17

10�13

10�9

10�5

0.1

1000

k �h⇧Mpc⇥

P⇤k⌅�⇤h

�1Mpc⌅3 ⇥

Atomic Dark Matter

Standard CDM

Atomic DM


Generic Signature: Dark Acoustic Oscillation (DAO)


•  A telltale signature of interacting DM is the presence of damped acoustic oscillations in the DM fluid at early times.

x = 0.37aD = 0.008BD= 5 keVmD= 10 GeVHaL HbL HcL HdL HeL

10-8 10-7 10-6 10-5 10-4 0.001

10-4

0.001

0.01

0.1

1

10

100

1êHz+1L

»d D»

Atomic Dark MatterLCDM

But wait, what about WIMPs ?


FIG. 2. Transfer function of the CDM density pertur-bation amplitude (normalized by the primordial amplitudefrom inflation). We show two cases: (i) Td/M = 10!4 andTd/Teq = 107; (ii) Td/M = 10!5 and Td/Teq = 107. In eachcase the oscillatory curve is our result and the other curve isthe free-streaming only result that was derived previously inthe literature [4,7,8].

kpeak = (8, 15.7, 24.7, ..)!!1d !

Mpl

T0Td. (22)

This same scale determines the “oscillation” damping.The free streaming damping scale is,

!dcd(!d) ln(!eq/!d) !MplM1/2

T0T3/2d

ln(Td/Teq), (23)

where Teq is the temperature at matter radiation equal-ity, Teq " 1 eV. The free streaming scale is parametri-cally di!erent from the “oscillation” damping scale. How-ever for our fiducial choice of parameters for the CDMparticle they roughly coincide.

The vanishing of the sound speed during the QCDphase transition amplifies perturbations which have""!QCD > 1, where "!QCD is the duration of the tran-sition. In Figure 2 the a!ected modes are those withxd = k!d >

#3 (!QCD/"!QCD) (!d/!QCD). Typical

values of !d/!QCD $ 10%15 and !QCD/"!QCD $ 3%10relate this condition to modes with xd > 50% 260. Thusthe a!ected scales are severely damped by the e!ects con-sidered in this paper.

Finally we want to stress the fact that the dampingscale is significantly smaller than the scales observed di-rectly in the Cosmic Microwave Background or throughlarge scale structure surveys. For example, the ratio ofthe damping scale to the scale that entered the horizonat equality is !d/!eq $ Teq/Td $ 10!7 and to our presenthorizon !d/!0 $ (TeqT0)1/2/Td $ 10!9. In the contextof inflation, these scales were created 16 and 20 e–foldsapart. Given the large extrapolation, one could certainlyimagine that a change in the spectrum could alter theshape of the power spectrum around the damping scale.

However, for smooth inflaton potentials with small de-partures from scale invariance this is not likely to be thecase. On scales much smaller than the horizon at matterradiation equality, the spectrum of perturbations densitybefore the e!ects of the damping are included is approx-imately,

"2(k) ! exp

!

(n % 1) ln(k!eq) +1

2#2 ln(k!eq)

2 + · · ·"

& ln2(k!eq/8) (24)

where the first term encodes the shape of the primordialspectrum and the second the transfer function. Primor-dial departures from scale invariance are encoded in theslope n and its running #. The e!ective slope at scale kis then,

$ ln "2

$ ln k= (n % 1) + # ln(k!eq) +

2

ln(k!eq/8). (25)

For typical values of (n % 1) $ 1/60 and # $ 1/602

the slope is still positive at k $ !!1d , so the cut-o! in the

power will come from the e!ects we calculate rather thanfrom the shape of the primordial spectrum. Howevergiven the large extrapolation in scale, one should keep inmind the possibility of significant e!ects resulting fromthe mechanisms that generates the density perturbations.

Implications We have found that acoustic oscilla-tions, a relic from the epoch when the dark matter cou-pled to the cosmic radiation fluid, truncate the CDMpower spectrum on a comoving scale larger than e!ectsconsidered before, such as free-streaming and viscosity[4,7,8]. For SUSY dark matter, the minimum mass ofdark matter clumps that form in the universe is there-fore increased by more than an order of magnitude to avalue of "

Mcut =4%

3

#

%

kcut

$3

#M&crit

' 10!4

#

Td

10 MeV

$!3

M#, (26)

where &crit = (H20/8%G) = 9 & 10!30 g cm!3 is the crit-

ical density today, and #M is the matter density for theconcordance cosmological model [2]. We define the cut-o! wavenumber kcut as the point where the transfer func-tion first drops to a fraction 1/e of its value at k ( 0.This corresponds to kcut " 3.3 !!1

d .Recent numerical simulations [15,16] of the earliest and

smallest objects to have formed in the universe [17], need

"Our definition of the cut-o! mass follows the convention ofthe Jeans mass, which is defined as the mass enclosed withina sphere of radius !J/2 where !J ! 2"/kJ is the Jeans wave-length [14].

5

Loeb and Zaldarriaga (2005); Bertschinger (2006).

particle. The relevant scattering partners are the stan-dard model leptons which have thermal abundances. Fordetailed expressions of the cross section in the case of su-persymmetric (SUSY) dark matter, see Refs. [10,7]. Forour purpose, it is su!cient to specify the typical size ofthe cross section and its scaling with cosmic time,

! !T 2

M4!

, (6)

where the coupling mass M! is of the order of the weak-interaction scale (" 100 GeV) for SUSY dark matter.This equation should be taken as the definition of M!,as it encodes all the uncertainties in the details of theparticle physics model into a single parameter. The tem-perature dependance of the averaged cross section is aresult of the available phase space. Our results are quiteinsensitive to the details other than through the decou-pling time. Equating "!1

c /a to the Hubble expansion rategives the temperature of kinematic decoupling:

Td =

!

M4!M

Mpl

"1/4

! 10 MeV

!

M!

100 GeV

" !

M

100 GeV

"1/4

.

(7)

The term k2c2s#c in Eq. (4) results from the pressure

gradient force and cs is the dark matter sound speed. Inthe tight coupling limit, "c # H!1 we find that c2

s !fcT/M and that the shear term is k2!c ! fvc2

s"c$c. Herefv and fc are constant factors of order unity. We willfind that both these terms make a small di"erence on thescales of interest, so their precise value is unimportant.

By combining both equations in (4) into a single equa-tion for #c we get

#""c +1

x[1 + Fv(x)] #"c + c2

s(x)#c

= S(x)$3Fv(x)#" +x4

d

x5(3$"0 $ #"c) , (8)

where xd = k%d and %d denotes the time of kinematicdecoupling which can be expressed in terms of the de-coupling temperature as,

%d = 2td(1 + zd) !Mpl

T0Td! 10 pc

!

Td

10 MeV

"!1

% M!1! M!1/4, (9)

with T0 = 2.7K being the present-day CMB temperatureand zd being the redshift at kinematic decoupling. Wehave also introduced the source function,

S(x) & $3#"" + # $3

x#". (10)

For x # xd, the dark matter sound speed is given by

c2s(x) = c2

s(xd)xd

x, (11)

where c2s(xd) is the dark matter sound speed at kinematic

decoupling (in units of the speed of light),

cs(xd) ! 10!2f1/2c

!

Td

10 MeV

"1/2 !

M

100 GeV

"!1/2

.

(12)

In writing (11) we have assumed that prior to decouplingthe temperature of the dark matter follows that of theplasma. For the viscosity term we have,

Fv(x) = fvc2s(xd)x2

d

#xd

x

$5

. (13)

Free streaming after kinematic decoupling In the limitof the collision rate being much slower than the Hub-ble expansion, the CDM is decoupled and the evolutionof its perturbations is obtained by solving a Boltzmanequation:

&f

&%+

dxi

d%

&f

&xi+

dqi

d%

&f

&qi= 0, (14)

where f('x, 'q, %) is the distribution function which de-pends on time, position and comoving momentum 'q. Thecomoving momentum 3-components are dxi/d% = qi/a.We use the Boltzman equation to find the evolution ofmodes that are well inside the horizon with x ' 1. Inthe radiation era, the gravitational potential decays afterhorizon crossing (see Eq. 2). In this limit the comovingmomentum remains constant, dqi/d% = 0 and the Boltz-man equation becomes,

&f

&%+

qi

a

&f

&xi= 0. (15)

We consider a single Fourier mode and write f as,

f('x, 'q, %) = f0(q)[1 + #F ('q, %)ei"k·"x], (16)

where f0(q) is the unperturbed distribution,

f0(q) = nCDM

!

M

2(TCDM

"3/2

exp

%

$1

2

Mq2

TCDM

&

(17)

where nCDM and TCDM are the present-day density andtemperature of the dark matter.

Our approach is to solve the Boltzman equation withinitial conditions given by the fluid solution at a time%# (which will depend on k). The simplified Boltzmanequation can be easily solved to give #F ('q, %) as a functionof the initial conditions #F ('q, %#),

#F ('q, %) = #F ('q, %#) exp[$i'q · 'k%#

a(%#)ln(%/%#)]. (18)

The CDM overdensity #c can then be expressed interms of the perturbation in the distribution function as,

3

Kinematic decoupling is too early…

Not all models are captured by our approach.



•  What do cosmological data actually say about dark matter interactions??

1.  Cosmic Microwave Background (and its lensing).

2.  Galaxy clustering.

3.  (Weak lensing).

•  A useful toy model to answer this question is Atomic Dark Matter.

•  This toy model naturally contains the two ingredients of a “cosmologically-interesting” interacting DM scenario.

Interacting Dark Matter: DAO and Small-scale Damping

Atomic Dark Matter

•  Postulate a new U(1) gauge force in the (hidden) Dark Sector.

•  The dark matter is made of two oppositely-charged fermions (dark ‘electron’ and dark ‘proton’).

•  The Dark Sector is neutral overall (no long-range force).

•  The Model is fully described by 4 parameters:

This model has a very rich phenomenology despite its minimal set of ingredients.


3

parameters are subject to the consistency constraint

mD

BD� 8

↵2D

� 1, (1)

which ensures that me,p (mD + BD)/2. This boundis saturated when the two fermions have equal masses.Here, me and mp stand for the dark-electron mass andthe dark-proton mass, respectively. We give their valuesin terms of mD, BD, and ↵D in appendix A.

On a more general level, the atomic DM scenario canbe considered as a toy model of a more complete the-ory involving a hidden dark plasma in the early Uni-verse. Indeed, atomic DM contains the all key ingredientsof a dark-plasma theory (dark radiation, multiple par-ticles, kinetic and thermal decoupling, modified growthof DM fluctuations, long-range and short-range interac-tions, etc) with only minimal physical inputs. As such,the results presented in this paper should be understoodin the broader context of a generalized dark-plasma the-ory ( see e.g. [61] for general cosmological constraints onthis type of model).

Interestingly, the atomic DM scenario naturally en-globes the hidden charged DM models discussed in[62, 63] as special cases. Moreover, in the limit of verylarge atomic binding energy and large dark fine-structureconstant, dark atoms become basically undistinguishablefrom standard CDM particles. Therefore, the atomic DMscenario is a rather general testbed for physics beyond thevanilla WIMP/CDM paradigm.

III. THERMAL HISTORY

In the early Universe, the dark sector forms a tightly-coupled plasma much like the standard baryon-photonplasma. As the Universe cools down, three importanttransitions need to carefully be taken into account. First,once the dark sector temperature falls below BD, it be-comes energetically favorable for the dark fermions torecombine into neutral dark atoms. Second, once themomentum transfer rate between the DR and the darkfermions falls below the Hubble expansion rate (kineticdecoupling), the DM e↵ectively ceases to be draggedalong by the radiation and can start to clump and formstructures. Finally, once the energy transfer rate betweenthe DM and the DR falls below the Hubble rate (thermaldecoupling), the DM temperature TDM ceases to trackthat of DR and start cooling adiabatically. Accuratelycapturing these transitions and computing their impacton cosmological observables is a major goal of this paper.

We begin this section by determining the big-bang nu-cleosynthesis (BBN) bound on the dark-photon temper-ature. We then discuss the recombination of dark atomsand their thermal coupling to the DR, emphasizing thedi↵erences between dark atoms and regular atomic hy-drogen. We finally present the solutions to the joint evo-lution of the dark-atom ionized fraction and temperature.

FIG. 1. E↵ective number of dark sector relativistic degrees offreedom at the time of nucleosynthesis as a function of ↵D andBD for dark atoms with mass mD = 1 GeV. Here, we havefixed ⇠BBN = 0.5. We also display the consistency constraintgiven by Eq. 1 above which dark atoms do not exist.

A. BBN Limit on Dark-Sector Temperature

Observations of the relative abundance of light ele-ments put a bound on the possible number of relativisticdegrees of freedom at the time of BBN. This limit isusually quoted in terms of the e↵ective number of lightneutrino species in thermal equilibrium at BBN; here weshall use the conservative estimate N⌫ = 3.24± 1.2 (95%confidence) derived in Ref. [64]. More recent estimates[65–69] of N⌫ are statistically consistent with this value.Assuming that the dark sector contributes gBBN

⇤,D relativis-tic degrees of freedom during BBN and further assumingthree species of SM neutrinos, we obtain the bound

gBBN⇤,D ⇠4BBN 2.52, (2)

where ⇠BBN is the ratio of the dark sector and visibletemperatures at the time of nucleosynthesis. In the min-imal atomic DM scenario considered in this work, DMis totally decoupled from SM particles and therefore theentropy of the dark sector and the visible sector are sep-arately conserved

gBBN⇤S,D⇠3BBN

gtoday⇤S,D ⇠3=

gBBN⇤S,vis

gtoday⇤S,vis, (3)

where g⇤S,D is the present-day e↵ective number of de-grees of freedom contributing to the entropy of the darksector, sD / g⇤S,DT 3

D. For the simplest model of dark

atoms considered here, we have gtoday⇤S,D = 2 (i.e. onlydark photons contribute). Similarly, g⇤S,vis is the e↵ec-tive number of degrees of freedom contributing to the

FYCR, K. Sigurdson, arXiv:1209.5752


Phys. Lett. B174 (1986) 151

Atomic Dark matter: Origins

See also D. E. Kaplan et al. 2011, 2012.

Which parameters really affect the cosmological observables?


1.  A quantity determining the momentum transfer rate between the dark radiation and the DM:

2.  The amount of dark radiation in the Universe:

3

Eqs. (6) and (7) describe the evolution of the dark-photonover-densities (��) and of the dark-photon velocity, re-spectively. It is also necessary to solve for the hierarchyof dark-photon multipoles (Eqs. (8) and (9)) to properlyaccount for DR di↵usion and its impact on atomic DMperturbations.

We solve these equations numerically together withthose describing the evolution of CDM, baryon, photon,and neutrino fluctuations using a modified version of thecode CAMB [14]. We first precompute the evolution of thedark plasma opacity as described in Ref. [10]. We assumepurely adiabatic initial conditions

�D(zi) = �c(zi) = �b(zi) ��(zi) = ��(zi), (10)

✓D(zi) = ✓�(zi) = ✓�(zi), (11)

F�l = 0, l � 2. (12)

where zi is the initial redshift which is determined suchthat all modes are superhorizon at early times, k⌧(zi) ⌧1. Here, the subscripts “c”, “b”, and “�” refer to CDM,baryon and regular photon, respectively. At early timeswhen k⌧D ⌧ 1 and ⌧D/⌧ ⌧ 1, Eqs. (5) and (7) are verysti↵ and we use a second-order tight-coupling scheme sim-ilar to that used for the baryon-photon plasma at earlytimes [15, 16].

IV. COSMOLOGICAL OBSERVABLES

Since it modifies the growth of DM fluctuations on avariety of scales, PIDM can imprint its signature on cos-mological observables such as the CMB and the matterpower spectrum. These observables are sensitive to mo-mentum transfer rate between the interacting DM andthe dark radiation which determines the kinetic decou-pling epoch. This rate is largely determined by the fol-lowing combination of parameters

�DAO ⌘ ↵D

✓BD

eV

◆�1 ⇣ mD

GeV

⌘�1/6. (13)

This quantity has a simple interpretation: ↵6D/B2 /

↵2D/m2

e is directly proportional to the dark Thomsoncross section, (1/B4

D) provides the scaling of the darkradiation energy density with the typical energy scale ofthe dark atoms, and mD is the factor converting the en-ergy density in dark atoms to their number density. For arelatively strongly coupled (↵D > 0.01) dark sector, anychanges to ↵D, BD, and mD that leaves �DAO invariantlead to the same cosmological observables. For smallervalues of the dark fine-structure constant however, thedetails of the dark recombination process becomes im-portant and the observables become dependent on both�DAO and ↵D.

A. Matter Power Spectrum

B. Cosmic Microwave Background

C. CMB Lensing Power Spectrum

V. DATA

VI. RESULTS

VII. IMPACT ON GALAXY FORMATION

Acknowledgements— The work of FYCR is supportedby the W.M. Keck Institute for Space Studies. The re-search of KS is supported in part by a National Scienceand Engineering Research Council (NSERC) of CanadaDiscovery Grant. FYCR thanks the Aspen Center forPhysics, where part of this work was completed, fortheir hospitality. This research was supported in part bythe National Science Foundation under Grant No. NSFPHY11-25915. Part of the research described in this pa-per was carried out at the Jet Propulsion Laboratory,California Institute of Technology, under a contract withthe National Aeronautics and Space Administration.

[1] J. Fan, A. Katz, L. Randall, and M. Reece,Phys.Rev.Lett. 110, 211302 (2013), 1303.3271.

[2] J. Fan, A. Katz, L. Randall, and M. Reece (2013),1303.1521.

[3] M. McCullough and L. Randall (2013), 1307.4095.[4] H. Goldberg and L. J. Hall, Phys.Lett. B174, 151 (1986).[5] D. E. Kaplan, G. Z. Krnjaic, K. R. Rehermann, and C. M.

Wells, JCAP 1005, 021 (2010), 0909.0753.[6] D. E. Kaplan, G. Z. Krnjaic, K. R. Rehermann, and C. M.

Wells, JCAP 1110, 011 (2011), 1105.2073.[7] S. R. Behbahani, M. Jankowiak, T. Rube, and J. G.

Wacker, Adv.High Energy Phys. 2011, 709492 (2011),1009.3523.

[8] K. Petraki, M. Trodden, and R. R. Volkas, JCAP 1202,044 (2012), 1111.4786.

[9] J. M. Cline, Z. Liu, and W. Xue, Phys.Rev. D85, 101302(2012), 1201.4858.

[10] F.-Y. Cyr-Racine and K. Sigurdson, Phys.Rev. D87,103515 (2013), 1209.5752.

[11] L. Ackerman, M. R. Buckley, S. M. Carroll, andM. Kamionkowski, Phys.Rev. D79, 023519 (2009),0810.5126.

[12] J. L. Feng, M. Kaplinghat, H. Tu, and H.-B. Yu, JCAP0907, 004 (2009), 0905.3039.

[13] C.-P. Ma and E. Bertschinger, Astrophys.J. 455, 7(1995), astro-ph/9506072.

2

pect the details of the UV completion to a↵ect the low-energy interactions responsible for modifying the growthof DM fluctuations on small scales. We further assumethat the CDM and the interacting component only inter-act through gravity.

A. Atomic Dark Matter

In the atomic DM scenario, two oppositely-chargedmassive fermions can interact through a new unbrokenU(1)D gauge force to form hydrogen-like bound states.In the early Universe, these charged fermions are tightly-coupled to a thermal bath of dark radiation and henceform a hot ionized plasma similar to that of standardbaryons and photons. For simplicity, we refer to thelightest fermion as “dark electron” (mass me) while theheaviest fermion is denoted as “dark proton” (mass mp).We assume that these two oppositely-charged compo-nents come in equal number such that the dark sectoris overall neutral. This model is characterized by five pa-rameters which are the mass of the dark atoms mD, thedark fine-structure constant ↵D, the binding energy ofthe dark atoms BD, the present-day ratio of the dark ra-diation temperature to the cosmic microwave backgroundtemperature ⇠ ⌘ (TD/TCMB)|z=0, and the fraction ofthe overall DM density contained in dark atoms, fADM.These parameters are subject to the consistency condi-tion mD/BD � 8/↵2

D � 1, which ensures that the rela-tionship me +mp �BD = mD is satisfied.

The evolution of the dark plasma is largely governedby the opacity ⌧�1

D of the medium to dark radiation. Themain contributions to this opacity are Compton scat-terings of dark radiation o↵ charged dark fermions andRayleigh scatterings o↵ neutral dark atoms, that is,

⌧�1D = ⌧�1

Compton + ⌧�1R , (1)

where

⌧�1Compton = anADMxD�T,D

"1 +

✓me

mp

◆2#, (2)

and

⌧�1R = anADM(1� xD)h�Ri

' 32⇡4anADM(1� xD)�T,D

✓TD

BD

◆4

. (3)

Here, �T,D ⌘ 8⇡↵2D/(3m2

e) is the dark Thomson crosssection, a is the scale factor, xD is the ionized fractionof the dark plasma, nADM is the number density of darkatoms, �R is the Rayleigh scattering cross section, andwhere the angular bracket denotes thermal averaging.We note that the second line of Eq. 3 is only valid ofTD < BD. It is out of the scope of this paper to dis-cuss in detail the evolution of the ionized fraction. Werefer the reader to Ref. [10] for a thorough investigationof dark atom recombination.

III. EVOLUTION OF COSMOLOGICALFLUCTUATIONS

Since a fraction of the DM forms a tightly-coupledplasma in the early Universe, the evolution of cosmo-logical fluctuations in the PIDM model departs signif-icantly from that of a standard ⇤CDM Universe. In-deed, as Fourier modes enter the causal horizon, the darkradiation pressure provides a restoring force opposingthe gravitational growth of over densities, leading to thepropagation of dark acoustic oscillations (DAO) in theplasma. These acoustic waves propagates until dark ra-diation kinematically decouples from the interacting DMcomponent. Similar to the baryon case, the scale cor-responding to the sound horizon of the dark plasma atkinetic decoupling remains imprinted on the matter fieldat late times. This so-called DAO scale is a key signa-ture of cosmologically interesting interacting DM mod-els. Indeed, much like the free-streaming length of warmDM models, the DAO scale divides the modes that arestrongly a↵ected by the DM interactions (through damp-ing and oscillations) from those that behave mostly likein the CDM paradigm. We note however that, in con-trast to warm DM models, the suppression of small-scalefluctuations in the atomic DM scenario is mostly due toacoustic damping [10].The equations describing the evolution of atomic DM

density and velocity fluctuations are

�D + ✓D � 3� = 0, (4)

✓D +a

a✓D � c2Dk2�D � k2 =

RD

⌧D(✓� � ✓D), (5)

where we closely followed the notation of Ref. [13] in con-formal Newtonian gauge. Here, �D is the atomic DM den-sity contrast, ✓D and ✓� are the divergence of the atomicDM and dark radiation velocity, respectively; � and arethe gravitational scalar potentials, RD ⌘ 4⇢�/3⇢ADM, cDis the sound speed of atomic DM, and k is the wavenum-ber of the mode. Here, the subscript � always refers tothe dark radiation. The right-hand side of Eq. (5) rep-resents the collision term between the atomic DM andthe dark radiation. At early times, we generally haveRD � 1 and ⌧D ⌧ H�1, implying that the DM is e↵ec-tively dragged along by the dark radiation. The latterevolves according to the following Boltzmann equations:

�� +4

3✓� � 4� = 0; (6)

✓� � k2(1

4�� F�2

2)� k2 =

1

⌧D(✓D � ✓�); (7)

F�2 =8

15✓� � 3

5kF�3 �

9

10⌧DF�2; (8)

F�l =k

2l + 1

⇥lF�(l�1) � (l + 1)F�(l+1)

⇤� 1

⌧DF�l. (9)

Evolution of Fluctuations

8/12/13 Francis-Yan Cyr-Racine, JPL/Caltech

21 Francis-Yan Cyr-Racine, Caltech/JPL 2/01/13

•  A new “DAO” scale corresponding to the size of the “dark” sound horizon at kinetic decoupling emerges in the dark-matter density field.

•  On smaller scales, the interaction of dark matter with the dark radiation suppresses the amplitude of fluctuations.

Evolution of Fluctuations



Correlation Function

1 2 5 10 20 50 100

0

10

20

30

40

50

r �h�1Mpc⇥

r2⌅ L⇤r⌅

BD ⇥ 1 keV, ⇤D ⇥ 0.008BD ⇥ 1 keV, ⇤D ⇥ 0.08BD ⇥ 100 eV, ⇤D ⇥ 0.08

CDM

20

⌅ ⇥ 0.37⇤D ⇥ 0.08BD ⇥ 100 eVmD ⇥ 1 GeV

10�5 0.001 0.1 1010�16

10�12

10�8

10�4

1

104

k �h⇧Mpc⇥

P⇤k⌅�⇤h

�1Mpc⌅3 ⇥

Atomic Dark Matter

Standard CDM

⌅ ⇥ 0.37⇤D ⇥ 0.08BD ⇥ 1 keVmD ⇥ 1 GeV

10�5 0.001 0.1 1010�18

10�14

10�10

10�6

0.01

100

k �h⇧Mpc⇥

P⇤k⌅�⇤h

�1Mpc⌅3 ⇥

Atomic Dark Matter

Standard CDM

⌅ ⇥ 0.37⇤D ⇥ 0.008BD ⇥ 1 keVmD ⇥ 1 GeV

10�5 0.001 0.1 1010�17

10�13

10�9

10�5

0.1

1000

k �h⇧Mpc⇥

P⇤k⌅�⇤h

�1Mpc⌅3 ⇥

Atomic Dark Matter

Standard CDM

FIG. 12. Total linear matter power spectrum at z = 0 forthree atomic DM models. For reference, we also display thelinear matter power spectrum for a vanilla ⇤CDM model.

fine-structure constant ↵D. We observe that as ↵D isdecreased, the smallest comoving wavenumber a↵ectedby the DAOs moves toward higher values. At first, thisseems counterintuitive as a higher value of the dark cou-pling constant generally leads to a lower residual ioniza-tion fraction which in turns allows the dark photons torapidly decouple from the DM. We must however remem-ber that dark photons also interact with neutral darkatoms through Rayleigh scattering. For a fixed bindingenergy and DM mass, the Compton-scattering contribu-

tion to the dark-plasma opacity is roughly independentof ↵D since xD(TD ⌧ BD) / ↵�6

D and �T,D / ↵6D.

On the other hand, the Rayleigh scattering contributionto ⌧D is a steep function of ↵D with ⌧�1

R / ↵6D after

the onset of dark recombination. Thus, an increase ofthe dark fine-structure constant can considerably boostthe Rayleigh-scattering contribution to the opacity of thedark plasma, hence significantly postponing its epoch ofkinetic decoupling.It is also instructive to consider the correlation func-

tion of matter fluctuations in configuration space. Thelinear correlation function is related to the matter powerspectrum via a 3D Fourier transform which, after simpli-fication, can be reduced to

⇠L(r) =1

2⇡2

Zdk k2 P (k) j0(kr), (77)

where j0(kr) is the Spherical Bessel function of order0. In Fig. 13, we display the linear correlation functioncomputed from the three matter power spectra shown inFig. 12 as well as the correlation expected from a stan-dard ⇤CDM model. In all cases, the usual BAO scaleis clearly visible around r ⇠ 104h�1Mpc. In a similarmanner, the novel DAO length scale appears as a localenhancement of the correlation function at the scale cor-responding to the sound horizon of the dark plasma atdark decoupling. On scales smaller than this sound hori-zon, the correlation function is significantly damped com-pared to the ⇤CDM case, a consequence of the dampingof small-scale fluctuations discussed above.At late times, the key signature of these new features

in the matter power spectrum and correlation function isa minimal DM halo mass. Indeed, since most of primor-dial fluctuations on scales smaller than the dark-plasmasound horizon are e↵ectively wiped out by the di↵usionof dark photons, no self-bound object can form at latetimes at these scales. The first regions that can collapseinto self-bound DM halos must then have a minimal ini-tial comoving size ⇠ rDAO. Therefore, the first DM halos

1 2 5 10 20 50 100

0

10

20

30

40

50

r �h�1Mpc⇥

r2⌅ L⇤r⌅

BD ⇥ 1 keV, ⇤D ⇥ 0.008BD ⇥ 1 keV, ⇤D ⇥ 0.08BD ⇥ 100 eV, ⇤D ⇥ 0.08

CDM

FIG. 13. Linear correlation function for the three atomic DMmodels plotted in Fig. 12.

Matter Power Spectrum

⌅ ⇥ 0.37⇤D ⇥ 0.008BD ⇥ 1 keVmD ⇥ 1 GeV

10�5 0.001 0.1 1010�17

10�13

10�9

10�5

0.1

1000

k �h⇧Mpc⇥

P⇤k⌅�⇤h

�1Mpc⌅3 ⇥

Atomic Dark Matter

Standard CDM

kcut

BAO

DAO



•  Advantages of CMB: 1.  “Simulations” do include baryons! 2.  Linear physics. 3.  Clean probe.

•  Drawback: 1.  Only explores a limited range of scales (l<3000).

Cosmic Microwave Background


•  There are 2 effects that can allow us to distinguish an “interacting dark matter scenario” to a ΛCDM Universe (assuming the same background cosmology!): u Since the dark radiation is tightly-coupled at early time, the

photon fluctuations do not obtain the usual phase shift and amplitude suppression associated with extra free-streaming neutrinos:

u Dark matter fluctuations are only allowed to grow after it decouples from the radiation => change to the gravitational potential => change to CMB lensing.


17

−1 −0.5 0 0.5 1

−2

−1

0

1

2

x / τ a)

d a/(−3ζ

in)

for

R ν−>

0

dν

dγ

−1 −0.5 0 0.5 10

0.5

1

Φ± ,

in u

nits

2ζ in

/(3τ)

x / τ b)

Φ+Φ−Φ(R

ν−> 0)

FIG. 3: a) Adiabatic Green’s functions for neutrino (solid) and photon (dashed) number density perturbations in the radiationera. The neutrino fraction, R! , of the radiation density is assumed infinitesimal. b) Adiabatic Green’s functions for thegravitational potentials !± ! (" ± !)/2 in the radiation era. The solid and dashed curves are the sums of the O(R0

!) andO(R!) terms for three neutrino species. The dotted line is !+ = ! for R! " 0.

appearing on its right hand side is the one providedby the photon density perturbation (112). As for theleft hand side, where ! = "+ + "!, the only delta-function comes from the double derivative of the term!

!2 ! 13

"

p! "#

1"3! |!|

$

in eq. (106). The equality of

these contributions requires

p! = !"

3(1 ! R!)p" . (114)

Substituting eq. (106) in (113) and eliminating p! withthe relation above, we obtain

p" =1

1 ! 2R!

%

3

2#in !

& 1

!1d!F!(!)

'

. (115)

Calculating p! from the last two equations is somewhateasier than from eq. (107).

Now we have all the analytic tools to analyze how neu-trinos a#ect CMB perturbations. The evolution of metricperturbations without neutrinos is given by eqs. (108–109). Then the photon density Green’s function followsfrom eqs. (112, 115) as

d(R!#0)" = !3#in

("3 "#

1"3! |!|

$

!

! 12 $D

#

|!|! 1"3

$)

.

(116)

Its Fourier transform (93) leads to the photon densityFourier modes in the radiation era:

d(R!#0)" (%, k) = !3#in

*

2 sin&s

&s! cos&s

+

, (117)

with &s = k%/"

3. In particular, without neutrinos thephoton density modes oscillate under the acoustic hori-zon (&s # 1) as a pure &s cosine.

The predictions for both the phase and the amplitudeof the photon mode oscillations di#er when the gravity

of neutrino perturbations is taken into account. The os-cillations of the Fourier modes on subhorizon scales aredescribed by the singular terms in the real space Green’sfunctions. For the photon density (112) these are the$-function and (!± 1"

3)!1 singularities at ! = ± 1"

3:

d"(!) = p" $D

*

|!|!1"3

+

+2r"

!2 ! 13

+ . . . , (118)

where

r" = "+(1/"

3) (119)

and the dots stand for more regular terms. The Fouriertransform of eq. (118) follows from the first and thirdlines of Table II, where n is set to 0 and 1, as

d"(%, k) = 2#

p" cos&s ! r"'"

3 sin&s

$

+ O(&!1s ) . (120)

A non-zero phase shift with respect to the cos&s oscil-lations is generated whenever r" $= 0. By eq. (119) thiscan happen for adiabatic perturbations if only some per-turbations propagate faster than the sound speed in thephoton fluid, and thus are able to generate metric pertur-bations beyond the acoustic horizon. This is the case forthe neutrino perturbations, propagating with the speedof light, Fig. 3 a).

The values of p" and r" in eq. (118) are calculatedin O(R! ) order in Appendix C. With its results (C6)and (C7), the mode (120) can be presented as

d"(%, k) = 3#in(1 + $") cos (&s + $&) + O(&!1s ) , (121)

where

$" % ! 0.2683R! + O(R2!) ,

$& % 0.1912 'R! + O(R2!) .

(122)

As demonstrated in Fig. 4 a), our theoretical predictionsare in excellent agreement with numerical calculations

Balshinsky & Seljak, 2004

z = 5 ¥ 107

Geff = 10-4 MeV-2

0.0010 0.00500.0020 0.00300.00150

50

100

150

r @h-1MpcD

d n,gHrL

Photons Hwê standard neutrinosLPhotons Hwê SI neutrinosLStandard NeutrinosSelf-Interacting HSIL Neutrinos


•  Key Signatures: 1.  Nonuniform amplitude suppression and phase shifts

across the CMB temperature and polarization spectra.

2.  Modified ratios of odd and even peaks in the TT spectrum.

3.  Modified CMB lensing


Cosmic Microwave Background: TT Spectrum


25

aD = 0.01BD = 5 keVmD = 5 GeV

0 500 1000 1500 2000 2500 3000

50

100

500

1000

5000

l

lHl+1LC

lTTêH2pL@m

K2 D x = 0.5

x = 0.8x = 1.0


0 500 1000 1500 2000 2500 30000

10

20

30

40

l

lHl+1LC

lEEêH2pL@m

K2 D

x = 0.5x = 0.8x = 1.0

FIG. 18. CMB angular power spectra in the atomic DM sce-nario for di↵erent values of ⇠. We fix all other dark param-eters to the values indicated on the plots. The upper paneldisplays the TT spectra while the lower panel shows the EEpolarization spectra. All other cosmological parameters areheld fixed. Here, the helium fraction is fixed to Yp = 0.24 toisolate the e↵ect from the changing sound horizon.

the energy density of the dark photons leads to an en-hanced damping of the CMB anisotropies [106]. To un-derstand the origin of this e↵ect, we need to rememberthat the photon di↵usion distance scales as rd / H�0.5

(see Eq. (63)) while the angular diameter distance scalesas H�1. Thus, the damping angular scale ✓d ⌘ rd/DA

e↵ectively increases if the Hubble rate is sped up dueto the presence of extra radiation. We therefore expectthat as the value of ⇠ is raised, the CMB spectrum willbe increasingly a↵ected by Silk damping. This e↵ect isshown in Fig. 19 where we clearly observe the decline inamplitude associated with the increasing DR density. Inaddition, if the primordial helium fraction was allowedto vary according to Eq. (79), this would further increasethe amount of Silk damping. Therefore, it is clear thatmeasurements of the CMB damping tail provide strongconstraint on ⇠.

In summary, beyond the impact of the atomic DM sce-nario on the background cosmology caused by the DR,we have identified four key cosmological signatures thatdistinguish the atomic DM scenario from a ⇤CDM model

aD = 0.05mD = 100 GeVBD = 1 MeV

0 500 1000 1500 2000 2500 30000

5

10

15

l

l3Hl+1LC lTTêH2pL@10

2mK2 D

x = 0.4x = 0.6x = 0.8

FIG. 19. CMB temperature power spectra in the atomic DMscenario for di↵erent values of ⇠. We fix all other dark pa-rameters to the values indicated on the plots. We keep fixedthroughout the redshift of matter-radiation equality and theangular size of the baryon-photon sound horizon at decou-pling. Here, the helium fraction is fixed to Yp = 0.24 toisolate the e↵ect from the changing damping scale.

containing extra relativistic neutrinos. First, the emer-gence of the new DAO length scale in the late-time den-sity field results in a minimal mass for the first DMprotohalos that is generically larger than in the stan-dard WIMP paradigm. Also, as the dark photons tran-sition from being tightly-coupled to the dark plasma toa free-streaming state, they impart varying phase shiftsand amplitude suppressions to the CMB multipoles en-tering the horizon. Importantly, these suppressions andphase shifts asymptote to constant values for l � ldecand l ⌧ ldec, a distinct feature of atomic DM that isnot easily reproduced in the ⇤CDM scenario. Further-more, we have shown that the odd CTT

l peaks are sup-pressed on scales that enter the causal horizon before DMkinematically decouples. It is therefore clear that precisemeasurements on the CMB damping tail could providemeaningful constraints on the parameter space of atomicDM. We should however keep in mind that the modi-fied evolution of DM and DR fluctuations can only a↵ectthe CMB if the dark sector kinetic decoupling happensclose enough to the epoch of last scattering. As such, anon-detection of these signatures e↵ectively puts a lowerbound on the redshift of kinetic decoupling which itselfdepends on a combination of ↵D, BD, mD, and ⇠.

VI. ASTROPHYSICAL CONSTRAINTS ONATOMIC DARK MATTER

As the Universe expands and cools down, non-linearstructures begin to emerge and eventually form present-day astrophysical objects such as galaxies and clusters ofgalaxies. The internal dynamics of these objects is deeplyinfluenced by the microphysics governing DM because thelatter contributes the vast majority of the mass inside


Cosmic Microwave Background: TT Spectrum

aD = 0.2mD = 10 GeVx = 0.8

0 500 1000 1500 2000 2500 3000-0.05

0.00

0.05

0.10

l

DC lT

T êC lTT

BD = 10 keVBD = 5 keVBD = 2 keVBD = 1 keV


Cosmic Microwave Background: EE Spectrum

25


0 500 1000 1500 2000 2500 3000

50

100

500

1000

5000

l

lHl+1LC

lTTêH2pL@m

K2 D x = 0.5

x = 0.8x = 1.0


0 500 1000 1500 2000 2500 30000

10

20

30

40

l

lHl+1LC

lEEêH2pL@m

K2 D

x = 0.5x = 0.8x = 1.0






aD = 0.05mD = 100 GeVBD = 1 MeV

0 500 1000 1500 2000 2500 30000

5

10

15

l


2mK2 D

x = 0.4x = 0.6x = 0.8







Cosmic Microwave Background: Lensing BB Spectrum

25


0 500 1000 1500 2000 2500 3000

50

100

500

1000

5000

l

lHl+1LC

lTTêH2pL@m

K2 D x = 0.5

x = 0.8x = 1.0


0 500 1000 1500 2000 2500 30000

10

20

30

40

l

lHl+1LC

lEEêH2pL@m

K2 D

x = 0.5x = 0.8x = 1.0






aD = 0.05mD = 100 GeVBD = 1 MeV

0 500 1000 1500 2000 2500 30000

5

10

15

l


2mK2 D

x = 0.4x = 0.6x = 0.8






2

E-mode anisotropies. Because of this, B modes are ofgreat interest as a clean probe of two more subtle sig-nals: (1) primordial tensor perturbations in the earlyUniverse [4, 5], the measurement of which would providea unique probe of the energy scale of inflation; and (2)gravitational lensing, which generates a distinctive non-Gaussian B-mode signal [6] that can be used to measurethe projected mass distribution and constrain cosmolog-ical parameters such as the sum of neutrino masses (fora review, see [7]).

Previous experiments have placed upper limits on theB-mode polarization anisotropy [8–11]. In this letter wepresent the first detection of B modes sourced by grav-itational lensing, using first-season data from SPTpol,the polarization-sensitive receiver on the South Pole Tele-scope.

Gravitational lensing remaps the observed position ofCMB anisotropies as n ! n+r�(n), where � is the CMBlensing potential [12]. This remapping mixes some of the(relatively) large E-mode signal into B. The induced B

mode at Fourier wavevector ~lB

is given to first order in� as [13]

B

lens(~lB

) =

Zd

2

~

l

E

Zd

2

~

l

�

W

�(~lE

,

~

l

B

,

~

l

�

)E(~lE

)�(~l�

),

(1)where the weight function W

� specifies the mixing. Inthis letter, we use measurements of E and � to synthe-size an estimate for the lensing contribution, which wecross-correlate with measured B modes. Using maps ofthe cosmic infrared background measured by Herschel -SPIRE to estimate �, and measurements of the E- andB-mode polarization from SPTpol, we detect the lensingsignal at 7.7� significance.

CMB Data: The South Pole Telescope (SPT) [14]is a 10-meter telescope located at the geographic SouthPole. Here we use data from SPTpol, a polarization-sensitive receiver installed on the telescope in January2012. SPTpol consists of two arrays of polarization-sensitive bolometers (PSBs): 1176PSBs that observe at150GHz, and 360PSBs that observe at 95GHz. Theinstrument and its performance are described in [15–18].The observation strategy, calibration, and data reductionfor SPTpol data are similar to those used for the SPT-SZsurvey, described in [19]. Here we briefly summarize theimportant points.

We calibrate the PSB polarization sensitivities withobservations of a ground-based thermal source behind apolarizing grid. This allows us to measure the polariza-tion angle of individual PSBs with < 2o statistical un-certainty and the average angle of all PSBs with < 0.1o

statistical uncertainty. We estimate systematic uncer-tainty on the average angle to be <1o (1.5o) at 150GHz(95GHz).

Between March and November 2012, we used SPTpolto observe a 100 deg2 region of low-foreground sky, be-

tween 23h and 24h in right ascension and �50 and�60 degrees in declination. We process the SPTpoldata by “observations”, which are half-hour periods inwhich the telescope scans half of the field. Each observa-tion is recorded as time-ordered data (TOD) from eachPSB, in azimuthal scans separated by steps in elevation.For each scan, we apply a low-pass anti-aliasing filter aswell as a high-pass 4th-order polynomial subtraction toremove large-scale atmospheric fluctuations. This sup-presses modes along the scan direction, which we accountfor with a two-dimensional transfer function measuredfrom simulations of the filtering process.In each observation, we drop PSBs with cuts based on

noise level during the observation, response to elevation-dependent atmospheric power, and response to an in-ternal thermal calibration source. Typical observationsinclude ⇠800 PSBs (⇠230 PSBs) at 150GHz (95GHz).We cut scans for PSBs with glitches (caused, for exam-ple, by cosmic ray hits). In typical 150GHz (95GHz)observations, we lose ⇠ 1% (⇠ 4%) of the data due toglitch removal.Data from each PSB are accumulated into maps of

the I, Q, and U Stokes parameters using measured po-larization angles and polarization e�ciencies. We weightthe TOD for each PSB in a scan by the inverse of thevariance along the scan direction between 1Hz and 3Hz(1300 . l

x

. 3900 for the telescope scan speed of 0.28degrees per second). We make maps using the obliqueLambert azimuthal equal-area projection [20] with square20⇥20 pixels. This projection preserves area on the skybut introduces small distortions in angle; we account forthese distortions by rotating the Q and U components tomaintain a consistent angular coordinate system acrossthe map. For each observation we form a noise mapfrom the di↵erence of left- and right-going scans, cut-ting observations which are outliers in metrics such asoverall variance. This cut removes ⇠ 8% (⇠ 9%) of the150GHz (95GHz) data. Finally, we add the individualobservations together to produce full-season maps, withpolarization noise levels of approximately 10µK-arcminat 150GHz and 25µK-arcmin at 95GHz.Inaccuracy in PSB gain measurements can cause direct

leakage of the CMB temperature into polarization, whichwe fit for using the cross-spectra of I with Q and U. Wefind < 2% leakage at both 150GHz and 95GHz, whichwe correct for by subtracting appropriate fractions of Ifrom Q and U. We show below that this correction isunimportant for our final results.We calibrate the overall amplitude of the SPTpol maps

to better than 1% in temperature by cross-correlatingwith SPT-SZ temperature maps over the same regionof sky. The SPT-SZ maps are calibrated by comparingto the Planck surveyor 143 GHz maps [21] over the full2500 deg2 SPT-SZ survey region.CIB Data: We use maps of the cosmic infrared back-

ground (CIB) [22] obtained from the SPIRE instrument


0 100 200 300 400

2

4

6

8

10

l

l4 Clff@¥

107 D Planck Lensing

LCDM+nGDAO=10-5GDAO=10-4GDAO=10-3GDAO=10-2

Cosmic Microwave Background: Lensing Spectrum

25


0 500 1000 1500 2000 2500 3000

50

100

500

1000

5000

l

lHl+1LC

lTTêH2pL@m

K2 D x = 0.5

x = 0.8x = 1.0


0 500 1000 1500 2000 2500 30000

10

20

30

40

l

lHl+1LC

lEEêH2pL@m

K2 D

x = 0.5x = 0.8x = 1.0






aD = 0.05mD = 100 GeVBD = 1 MeV

0 500 1000 1500 2000 2500 30000

5

10

15

l


2mK2 D

x = 0.4x = 0.6x = 0.8






Galaxy Clustering


Linear

0.02 0.05 0.10 0.20 0.50

1000

2000

5000

1¥104

2¥104

5¥104

k @hêMpcD

P mHkL@HMp

cêhL3 D BOSS-DR9


Cosmological Constraints


�6 �5 �4 �3 �2 �1 0 1

log10(�DAO)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

⇠ D

Planck+WP+High-l+DR9+Lens

Partially-Interacting Dark Matter


•  Assume that only a fraction fADM of DM is interacting.

•  Intuitively, a broader range of parameters should be allowed.

•  Can lead to interesting galactic-scale phenomenon (e.g. DDDM, Fan et al. (2013).)

•  Useful complementarity between the “small” astrophysical scales and the “large” cosmological scales

CMB TT Spectrum, fADM = 5%



CMB EE Spectrum, fADM = 5%


CMB Lensing BB Spectrum, fADM = 5%


CMB Lensing Spectrum, fADM = 5%

0 100 200 300 400

2

4

6

8

10

l

l4 Clff@¥

107 D Planck Lensing


Galaxy Clustering, fADM = 5%


Linear

0.02 0.05 0.10 0.20 0.50

1000

2000

5000

1¥104

2¥104

5¥104

k @hêMpcD

P gHkL@HMp

cêhL3 D BOSS-DR9




�6 �5 �4 �3 �2 �1 0 1

log10(�DAO)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7⇠ D

Planck+WP+High-l+BAO+Lens


Marginalized over fADM



�6 �5 �4 �3 �2 �1 0 1

log10(�DAO)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7⇠ D


fADM = 5%

Interacting DM: Allowed Fraction


0.00 0.01 0.02 0.03 0.04 0.05 0.06fADM

0.0

0.2

0.4

0.6

0.8

1.0P

/Pm

axPlanck+WP+High-l+BAO+Lens


25


0 500 1000 1500 2000 2500 3000

50

100

500

1000

5000

llHl+

1LClTTêH2pL@m

K2 D x = 0.5

x = 0.8x = 1.0


0 500 1000 1500 2000 2500 30000

10

20

30

40

l

lHl+1LC

lEEêH2pL@m

K2 D

x = 0.5x = 0.8x = 1.0






aD = 0.05mD = 100 GeVBD = 1 MeV

0 500 1000 1500 2000 2500 30000

5

10

15

l


2mK2 D

x = 0.4x = 0.6x = 0.8






ΓDAO=10-2.5

Aside: Constraints on DDDM


t coolbt Univ

t eqbt cool

Not Ionized

WDDDMthermal b 0.05 WDM

10-6 10-5 10-4 10-3 10-210-4

10-3

10-2

0.100

mC@GeVD

aD

e = 0.05, mX = 100 GeV, nX = nC = 3.3 ¥ 10-6 cm-3

t coolb tun

ivt eqbt cool

Not Ionized


10-6 10-5 10-4 10-3 10-210-4

10-3

10-2

0.100

mC@GeVD

aD

e = 0.05, mX = 1 GeV, nX = nC = 3.3 ¥ 10-4 cm-3

t eqbt cool

t coolbt univ

Not Ionized


10-6 10-5 10-4 10-3 10-210-4

10-3

10-2

0.100

mC@GeVD

aD

e = 0.05, mX = 100 GeV, nX = nC = 7.3 ¥ 10-5 cm-3

t coolb tun

iv

t eqbt cool

Not Ionized


10-6 10-5 10-4 10-3 10-210-4

10-3

10-2

0.100

mC@GeVD

aD

e = 0.05, mX = 1 GeV, nX = nC = 7.3 ¥ 10-3 cm-3

Figure 5: Cooling in the (mC ,↵D) plane. The purple shaded region is the allowed region that coolsadiabatically within the age of the universe. The light blue region cools, but with heavy and light particlesout of equilibrium. We take redshift z = 2 and TD = T

CMB

/2. The two plots on the left are for mX = 100

GeV; on the right, mX = 1 GeV. The upper plots are for a 110 kpc radius virial cluster; the lower plots,a 20 kpc NFW virial cluster. The solid purple curves show where the cooling time equals the age of theuniverse; they have a kink where Compton-dominated cooling (lower left) transitions to bremsstrahlung-dominated cooling (upper right). The dashed blue curve delineates fast equipartition of heavy and lightparticles. Below the dashed black curve, small ↵D leads to a thermal relic X, ¯X density in excess of the Oortlimit. To the upper right of the dashed green curve, BXC is high enough that dark atoms are not ionizedand bremsstrahlung and Compton cooling do not apply (but atomic processes might lead to cooling).

18

x = 0.15x = 0.2

x = 0.3x = 0.4

x = 0.6

Conclusions


•  This is a unique time for “cosmological” interacting dark matter.

•  Current data already put strong constraints on interacting dark matter.

•  CMB lensing B-mode and lensing reconstruction will provide even stronger constraints/hints of interacting DM in the near future (Planck, ACTpol, STPpol, PolarBear. etc.)

•  The complementarity between the cosmological and astrophysical properties of DM might help us pinpoint its nature.

Challenges and Future Directions


•  Non-linearity

•  CMB lensing

•  Weak lensing

•  Small-astrophysical scales? Wang & Zentner (2012)

11

FIG. 5: Fractional di!erence of auto convergence lensingpower spectrum between standard "CDM model and decay-ing dark matter model from first tomographic redshift bin(lensing source galaxies between 0 < zp < 0.6, where zp isphotometric, and not necessarily true redshift). Solid linesare calculated using halo model with NFW profiles. Theselines include the alteration of the linear power spectrum onlarge-scales and the reduction in the abundance of dark mat-ter halos due to free-streaming. However, halos are assumedto have the same profiles as they would in standard "CDM.The Dash-dotted lines include the nonlinear corrections tohalo density profiles.

evident that the DDM power spectra evolve significantlymore than the spectra in massive neutrino models. Thereason is that the decay process continuously deposits ki-netic energy into the daughter dark matter distribution,in contrast to the neutrinos which have purely redshiftingkinetic energy distributions.

C. Forecast Constraints on DDM ModelParameters

To estimate of the power of weak lensing to constrainDDM, we adopt a variety of possible strategies. First,we consider constraints from data on scales where linearevolution of density fluctuations should be valid. Thevalue of this approach is that exploiting linear scales toconstrain DDM does not require a simulation programto confirm or refine nonlinear models of structure for-mation in these models. This can be done with con-temporary theoretical knowledge of the phenomenologyof these models. Moreover, relatively large-scale con-straints are less observationally challenging because theyexploit data on scales where cosmic variance, rather thangalaxy shape measurements, are the dominant error [26].In both cases, these constraints are conservative so weshould expect that forthcoming lensing surveys designed

FIG. 6: Comparison of the redshift evolution of decayingdark matter and massive neutrino lensing power spectra. Weplot fractional di!erence of auto convergence lensing powerspectra between standard "CDM model and decaying darkmatter (or massive neutrino) models in three tomographicredshift bins (labeled at the top). For simplicity, we show onlythe linear power spectra in this plot, though spectra computedwith our nonlinear model lead to a similar conclusion.

to address dark energy should do at least as well as ourlinear forecasts. To limit ourselves to linear scales, wetake data on multipoles ! < 300. All of the constraintsthat we show in this section have been marginalized overthe remaining cosmological parameters, including neu-trino mass.To show the maximum potential of lensing surveys,

we consider measurements that extend into the mildlynonlinear regime, as is commonly done for dark energyforecasts. The primary value of this extension is not thatparticular features in the power spectra induced by DDMare added to the data set. Rather the primary improve-ment in constraints comes from an increase in the signal-to-noise with which the power suppression can be de-tected [26]. In this case, we include information on multi-poles up to our quoted maximum multipole !max = 3000(see § V). Constraints on these scales will rely on reli-able modeling of clustering on mildly nonlinear scales,so a comprehensive simulation program will be necessaryto ensure the robustness of such constraints. A com-prehensive program is computationally-intensive and be-yond the scope of our present paper, as part of our goal isto emphasize that such a large-scale numerical programmay be interesting and useful.In Figure 7 we display our forecast 1" exclusion con-

tours alongside a variety of other contemporary con-straints. The most relevant contemporary constraintscome from modifications to the structures of dark matterhalos with virial velocities similar to the SDM kick veloc-ities [15] (orange region). Additional constraints may beplaced on unstable dark matter by examining the proper-

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users.physics.harvard.eduusers.physics.harvard.edu/...workshop_cyr-racine.pdfSelf-interacting DM and Astrophysics • In the last decade, self-interacting DM has been mostly discussed