Dark matter annihilation feedback in cosmological

MNRAS 472, 1214–1225 (2017) doi:10.1093/mnras/stx1974Advance Access publication 2017 August 3

Dark matter annihilation feedback in cosmological simulations – I: Codeconvergence and idealized haloes

N. Iwanus,1‹ P. J. Elahi2 and G. F. Lewis1

1Sydney Institute for Astronomy, School of Physics, A28, The University of Sydney, NSW 2006, Australia2International Centre for Radio Astronomy Research, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

Accepted 2017 July 31. Received 2017 July 19; in original form 2017 April 24

ABSTRACTWe describe and test a novel dark matter annihilation feedback (DMAF) scheme that has beenimplemented into the well-known cosmological simulation code GADGET-2. In the models con-sidered here, dark matter can undergo self-annihilation/decay into radiation and baryons. Theseproducts deposit energy into the surrounding gas particles and then the dark matter/baryonfluid is self-consistently evolved under gravity and hydrodynamics. We present tests of thisnew feedback implementation in the case of idealized dark matter haloes with gas componentsfor a range of halo masses, concentrations and annihilation rates. For some dark matter models,DMAF’s ability to evacuate gas is enhanced in lower mass, concentrated haloes where theinjected energy is comparable to its gravitational binding energy. Therefore, we expect thestrongest signs of dark matter annihilation to imprint themselves on to the baryonic structure ofconcentrated dwarf galaxies through their baryonic fraction and star formation history. Finally,we present preliminary results of the first self-consistent DMAF cosmological box simulationsshowing that the small-scale substructure is washed out for large annihilation rates.

Key words: galaxies: formation – dark matter – large-scale structure of Universe.

1 IN T RO D U C T I O N

The Lambda cold dark matter (�CDM) model of the Universesuccessfully predicts large-scale observations, such as the cosmicmicrowave background (Planck Collaboration XIII 2016), baryonicacoustic oscillations (Eisenstein et al. 2005; Alam et al. 2017) andsupernovae (Riess et al. 1998; Perlmutter et al. 1999). However,there appears to be tension with observations when galactic scalesare examined. �CDM N-body simulations predict that DM densityprofiles of galactic haloes should appear sharply peaked at theircentres (Navarro, Frenk & White 1996b; de Blok, McGaugh &Rubin 2001; Oh et al. 2008; de Blok 2010) – contrary to observedrotation curves measured in nearby dwarf galaxies, which implythe cores are flattened out, also known as the ‘cuspy halo’ problem(Moore 1994). Milky Way galaxy analogues that are produced inthese simulations imply that our galaxy should be isotropically sur-rounded by many hundreds of dwarf Galaxies, whereas observationsshow only a few dozen (Klypin et al. 1999) that are aligned prefer-entially across a large thin disc (Pawlowski, Pflamm-Altenburg &Kroupa 2012; Ibata et al. 2013).

While DM and energy are invoked in �CDM, there has beenlittle insight into the fundamental nature of the ‘Dark Sector’. Darkmatter’s existence is deduced by its gravitational influence on oth-

� E-mail: [email protected]

erwise discrepant small-scale behaviour, such as the rotation curvesof galaxies (Rubin, Ford & Thonnard 1980) and lensing studies ofgalactic collisions (Markevitch et al. 2004) and leaving the identityof DM largely theoretical.

Extensions of the standard model of particle physics, for example,yield many potential DM candidates (Feng 2010), the most popu-lar class being the weakly interacting massive particles (WIMPs).WIMPs meet many of the properties needed for a DM candidate –interacting gravitationally, electromagnetically neutral and coupledonly to the standard model at or less than weak force coupling. Theirpopularity compared to other candidates is largely due to theoreticalconsiderations from the early Universe, known as the WIMP mira-cle. In the WIMP miracle, stable particles are thermally producedwhen the hot dense plasma from the big bang cools as the Universeexpands. The abundances of these stable particles are set when therate of expansion is such that the rate of interaction cannot keep theparticles in thermal equilibrium with the rest of the plasma causingthese relics to decouple. To reproduce the correct DM abundance�DM ≈ 0.258 (Planck Collaboration XIII 2016), the DM parti-cle candidate requires a velocity-averaged cross-section of 〈σv〉 ≈3 × 10−26 cm3 ˜s−1 (Steigman, Dasgupta & Beacom 2012). Thisresult is especially tantalizing as this cross-section is in the rightorder of magnitude expected from heavy particles in the GeV/TeVmass ranges that interact through the weak force. Many of theseparticle have already been postulated for independent theoreticalconcerns in high-energy particle physics (Martin 1998; Bertone,

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Dark matter annihilation simulations 1215

Hooper & Silk 2005; Peccei 2008; Gaskins 2016). As a result, mostDM searches are dominated by the WIMP paradigm.

Direct detection experiments involve measuring the recoil of nu-cleons colliding with DM particles inside large detectors that arebuilt deep underground to minimize environmental radiation noise.Currently, the DAMA/LIBRA experiment has a possible detectionof DM scattering off by an annual modulation in the detector signal– ostensibly caused by the earth’s rotation about the sun (Bernabeiet al. 2008, 2010). Indirect experiments involve the search for DM inastrophysical sources by the detection of annihilation products, likegamma rays or electron–positron pairs (Bertone et al. 2005; Kuhlenet al. 2008; Adriani et al. 2014; Wechakama & Ascasibar 2014;Anderson et al. 2016; Gaskins 2016; Hooper & Witte 2016; Lianget al. 2017). Dwarf spheroidal galaxies, in particular, have receiveda lot of interest, as these are regions where the DM content isexpected to be very dense but with little obscuring dust and gascontent (Ackermann et al. 2014). The Galactic Centre is also an-other target of interest, serving as a relatively nearby and extremelybright source of gamma rays. Its study requires modelling of diffuseemission through the dust and gas in the centre, but when this back-ground modelling is accounted for in the signal, there is an excessgamma-ray emission around 2 GeV, which cannot be accounted for.This excess is consistent with a dark matter annihilation (DMA)through a bottom and antibottom quark pairs channel (Ackermannet al. 2015; Daylan et al. 2016; Winter et al. 2016), though analternative explanation that the excess is due to a population of un-resolved millisecond pulsars has recently gained support (Bartels,Krishnamurthy & Weniger 2016; Clark et al. 2016a; Lee et al. 2016;Macias et al. 2016).

Many searches are model-dependent, and simplifications haveto be made to many aspects of DMA, such as the choice of anni-hilation channels and uncertainty in the DM distribution. Becausethe annihilation rate depends on the square of the density, it isextremely sensitive to the unresolved ‘substructure’ – which canprovide boosts of many orders of magnitude – depending on howthe measured smooth density field is extrapolated below resolvablescales (Mack 2014). The presence of unresolvable ultracompactminihaloes, for example, can boost the annihilation rate by 105 ifthey account for even small fractions (f = 0.01) of the total density(Clark et al. 2016b).

Another approach in DM research is studying the role DM playsin galactic and large-scale structure formation. Analytical studieshave shown that dark matter annihilation feedback (DMAF), wherethe energy from DMA is deposited into the nearby baryonic com-ponent over cosmic time, can cause the evacuation of gas in smallerhaloes and inhibit the star formation of the first galaxies (Asca-sibar 2007; Natarajan, Croton & Bertone 2008; Natarajan, Tan& O’Shea 2009; Ripamonti et al. 2010; Wechakama & Ascasi-bar 2011; Schon et al. 2015). The growth of large-scale structureis complicated, and analytical studies typically rely on linearizedperturbations or semi-analytical models that do not capture the non-linear evolution as the density perturbations become large, or theinterplay between baryonic and DM structure components. Thismotivates the need to model DMA with simulation codes to capturethe full effect of non-linear growth and understand how it impactsthe low-redshift Universe – perhaps imprinting clues from the darksector. In this paper, we present the first attempt at self-consistentlyfollowing the evolution of annihilating DM (which was also used inClark et al. 2016b, with a boost factor due to ultracompact minihalosubstructure). In Section 2, we outline the equations of our modeland outline the modifications to the GADGET-2 code. In Section 3,we detail how we produced simple test haloes and test our code

for different halo sizes and annihilation rates and check for conver-gence. In Section 4, we give an overview of how we might expectDMAF to affect the Universe and present preliminary results fromcosmological box simulations.

2 A N N I H I L AT I O N F E E D BAC K

We modified GADGET-2 (Springel 2005), a parallel N-body simula-tion code, implementing a DMA/decay feedback scheme. GADGET-2 and its descendants are ubiquitously used in the cosmologicalcommunity for studies of structure formation. Gravitational forcesare calculated using the TreePM method by a set of N-body par-ticles. The hydrodynamical forces in the code are calculated uti-lizing the smoothed particle hydrodynamics (SPH; Gingold &Monaghan 1977) formulation of fluid dynamics, where fluid ele-ments are represented by discrete particles. In the traditional SPHformulation,1 the density of a particle i is approximated by

ρi =N∑

j=1

MjW (rij , hi), (1)

where the sum is over nearby gas particles labelled j and the massof the particle is Mj weighted by a kernel W(r, h) that is a decreasingfunction of the distance between particle pairs r and a smoothinglength h. Typically, h is chosen such that there are N nearest neigh-bours, or that a constant mass is held within a kernel that falls tozero at h. We use the standard B-spline kernel, though higher orderkernels could be used.

2.1 Annihilation implementation

We assume a constant velocity-averaged annihilation cross-section〈σv〉 for a two-body interaction rate per volume that is proportionalto the number density of DM squared. Writing the annihilation ratein terms of the physical DM density ρ2

χ , we get

dA

dt= 〈σv〉

2m2χ

ρ2χ , (2)

where mχ is the DM particle mass. To implement this rate into thecode, we use the SPH estimate of ρχ (equation 1), where now wesum over nearby DM particles instead of gas. From equation (2), wecan calculate the quantities of interest – the DM mass annihilatingaway as well as the power released into nearby gas. The mass-lossrate is given by

dMi

dt= −〈σv〉

mχρχMi, (3)

where in each annihilation 2mχ units of mass is lost, and we haveestimated that the volume occupied by a DM particle is V = Mi/ρχ .

Annihilation products, like electron–positron pairs, also coupleto the gas and deposit their energy, the so-called DMAF. Againwe use equation (2) to estimate the annihilation rate – now at theposition of each gas particle. The energy produced at each gasparticle is equal to 2mχ c2. For this work, we assume that the mean

1 There have been numerous advances since traditional SPH that improveits treatment of instabilities and shocks (Read, Hayfield & Agertz 2010;Springel 2010; Hopkins 2013, 2015; Sembolini et al. 2016). However, al-though we use classic SPH here, the scheme is not reliant on a particularSPH implementation or even limited to SPH, in principle. It could also beused in mesh and moving mesh codes, estimating the density field at cellcentres.

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1216 N. Iwanus, P. J. Elahi and G. F. Lewis

deposition length-scale of energy is within SPH smoothing lengths,so all heating is local and perfectly couples with the gas. With theseassumptions, the energy deposition rate is

dui

dt= ε〈σv〉c2

mχ

ρ2χ

ρgas. (4)

We stress that ρχ is an estimate of the local DM density usingequation (1), summing over the nearest DM particles, while ρgas isthe same equation but now summing over the N nearest gas neigh-bours and ε is the coupling efficiency, where we have assumedε = 1, though it could, in principle, be less due to a neutrino-decaychannel, for example. This method is computationally efficient, asit requires only one extra density calculation loop per particle. Inaddition, our method can be trivially adapted to the case of decay-ing DM by simply changing the code to calculate a rate linearlyproportional to the local DM density and adjusting the appropriateconstants.

A weakness of this scheme is that as the mass-loss equation (3)and energy injection rate equation (4) are sampled over the DMparticles and gas particles, respectively, they measure the DM den-sity fields at different positions (separations ≈ h); hence, the totalamount of energy transferred between the two components will onlybe approximately conserved.

However, a formally rigorous treatment requires breaking up an-nihilation products into short- and long-range energy deposition,tracing the products from their origin DM particles. Given the un-certainty of what their annihilation channels are and their couplingefficiencies, local energy deposition is a reasonable assumption.

2.2 Code implementation

The main changes to the code involve the creation of new membersto GADGET-2’s particle and Message Passing Interface (MPI) com-munication structures and modifications to the first density loop (seeFig. 1). In order to calculate equation (4), we modify the densityloop to search for the nearest DM neighbours of the gas particles.We then alter the subsequent MPI communication process to sendthe relevant variables for the DM density calculations in additionto the regular baryonic variables. The termination condition of theloop for each particle is changed such that the number of gas andalso its DM neighbours must be within the specified tolerance; oth-erwise, the DM and/or gas smoothing lengths are adjusted and thedensity loop repeats. To avoid unnecessary recalculation, we tageach particle with a gas and DM flag, signalling that we have thecorrect number of SPH neighbours for that particle type and skip thecorresponding density calculations. The other parameters are nowjust constants of the specific DM model, and the energy injectionitself occurs at the end of the second SPH loop after the entropygeneration rate by shock heating is calculated, to which equation (4)is added.

The extra CPU time cost of these routines is expected to comemostly from the extra nearest neighbour searches and MPI commu-nication. Naively, now that ρgas and ρχ are calculated for each gasparticle, we should expect that the CPU time within the hydrody-namics part of the code is increased by a factor of 2, assuming thenearest neighbour search is roughly equal for gas and DM neigh-bours. In practice, however, the SPH smoothing lengths are setindependently, requiring particles to loop until both the numbersof DM and gas neighbours are within the set tolerance. The num-ber of MPI communications that are expected to occur betweenthreads will increase as well, since there exists the possibility thata DM particle will need to be communicated even if the gas loop

Figure 1. The modified density loop flowchart. Other smaller differencesinclude altering GADGET-2 data structures so that SPH particles hold DMproperties (e.g. DM smoothing lengths and density), inclusion of a DMnearest neighbour search function and DMAF energy injection in the secondSPH loop.

is local or complete. So overall, we expect the CPU time increaseto be some constant factor slightly greater than 2 in the densityloop. Practical tests of adiabatic cosmological box simulations andDMAF (results of which we will present in a future paper) seem toindicate an overall time cost (see Fig. 2) of a factor of less than 2 formodest annihilation rates in large N particle simulations. In the runs100 keV, the performance is much worse due to the large amountof energy injected. The increased pressure and subsequent largehydro-forces activate GADGET-2’s time-adaptive integration scheme,forcing smaller steps to maintain the accuracy of the code and alarger amount of time spent searching for neighbours.

Despite good performance, the code will be further optimized inthe future. A reduction in MPI communication time can be achievedby reducing the transfer of unneeded data structures during proces-sor communication; that is, only sending either gas or DM properitesif one is needed instead of sending both would lead to a smallspeed-up. communication; that is, only sending either gas or DMproperites if one is needed instead of sending both would lead toa small speed-up. Currently, the nearest neighbour search is con-ducted by ‘walking’ GADGET-2’s octtree data structure (Barnes &Hut 1986) and picking up particles of the correct type within thesmoothing length. A larger speed-up is expected if we streamlinethe search – for each walk picking up both DM and gas particleswithin the smoothing length instead of two separate searches.

2.3 Code verification on simple boxes and galaxies

We have tested our code in simple uniform boxes, static and ex-panding, and compared to analytical solutions for the overall heat-ing. The tests clearly show an error in the energy deposition of



Figure 2. The wall-time taken to run a DMAF 100 Mpc box simulation withdifferent resolutions. ICs are identical and generated with N-GENIC. Thesesimulations were run on the ARTEMIS High Performance Supercomputer atthe University of Sydney. The runs used 120 cores with identical simulationparameters, varying only the mass of the DM particle mχ and softeninglengths set at 5 per cent of the initial interparticle spacing. The 100 keVsimulations are most adversely affected due to having the largest amount ofenergy injection.

<2 per cent even in extreme models. This error can be significantlyreduced to < 0.3 per cent by using smaller time-steps. We associatethe remaining errors with the deviation in the SPH reconstructionof the uniform density, and we find we can further improve thiswith a greater particle resolution (see Appendix A). The error asso-ciated with the larger time-steps implies the need for a limiter as inSaitoh & Makino (2009) to ensure stability in the the heating rate,

particularly for our cosmological runs. None the less, we havechecked over our simulations here to ensure no shocks associatedwith this time-step instability developed.

We also tested our code in the simple case of ideal, isolated haloesthat formed the backbone of our tests. They were modelled using theNavarro–Frenk–White (NFW) profile (Navarro et al. 1996b) givenby

ρ(r) = ρ0(ra

) (1 + r

a

)2 , (5)

where a is the scale radius and ρ0 is a normalizing factor that de-termines the mass. We used GALACTICS (Kuijken & Dubinski 1995;Widrow & Dubinski 2005; Widrow, Pym & Dubinski 2008), to pro-duce our test haloes. For dark-matter-only simulations, we evolvedthe halo for 20 Gyr to allow initial instabilities in the halo to re-lax. For DMAF simulations, �b/�CDM ≈ 20 per cent of the DMparticles were converted into gas with the same mass, and a ther-mal energy equal to their local velocity dispersions with the kineticenergy correspondingly reduced to account for this additional ther-mal energy. These simulations were allowed to relax over 20 Gyr,though a small amount of mass was initially shed due an instabilitycaused by the gravitational force softening, not accounted for inGALACTICS at the central cusp (typically only a few per cent of themass was lost). The end results are stable haloes (see Fig. 3), andthese would serve as our initial conditions (ICs) for testing the code.

3 R ESULTS

3.1 Annihilation mass reduction

For the dark-matter-only simulations, we studied a halo of massM = 1015 M� and concentration 6.9 and varied the number ofparticles to show that in our smooth NFW profiled test halo, thesimulation results converge. In Fig. 4, we plot the DM density

Figure 3. An example of one of the haloes used in our tests showing the DM (left-hand panel) and gas (right-hand panel) density of a large NFW profiled halo,(M = 1015 M�, c = 6.9) sampled with 105 particles. Haloes like this were created using GALACTICS and would serve as our ICs for the purpose of testing ourcodes. The gaseous component was created by converting approximately �b/�CDM ≈ 20 per cent of the DM particles into gas and assigning a thermal energyequal to its local velocity dispersion. These ICs were allowed to relax and re-virialize over 20 Gyr. The intensity corresponds to the SPH density integrateddown the z-axis and was created with the package PYNBODY (Pontzen et al. 2013).



Figure 4. To test the convergence of our mass-loss implementation (equa-tion 3), we ran three simulations of an NFW halo (M = 1015 M�, c = 6.9)with different particle number resolutions and an extreme annihilationmodel with mass-loss due to a 0.1 eV thermal relic DM particle, run for20 Gyr. We have good agreement everywhere except towards the centresof the halo where the smaller sampling volume leads to noise in the innerregions. The vertical lines show the gravitational softening length used forthe corresponding simulation.

profiles of the halo after 20 Gyr with DMA from an extreme model,a particle with mass 0.1 eV, for the same halo but sampled with adifferent number of particles. The profiles nicely converge down to100 kpc for our lowest resolution simulation. Further in, the halois not well sampled and noise occurs at a radius that is near thegravitational smoothing lengths of the simulations.

In Fig. 5, we again take the same ICs sampled by 106 particlesand evolved for 20 Gyr but now vary the particle mass. We see thatthe loss of mass is the most prominent towards the centre of ourhaloes, as expected in the high-density regions. The models usedin these runs are quite extreme for the purposes of highlighting thetwo cases with and without DMA, the highest mass particle being10 eV annihilating with the thermal relic cross-section. Despite this,the total mass difference during this time period amounts only toa total mass change of 2 per cent for the highest rate. Unless theannihilation cross-section (or DM particle mass) is much higher(lower) than what observational data permit, it seems that the overalldirect change of mass due to DMA is negligible.

3.2 Gas with DMAF

We ran simulations for a halo with similar properties but now with20 per cent of the DM particles randomly converted into gas andstabilized after 20 Gyr. These haloes were then used as the ICs forsimulations with DMAF energy injection given by equation (4),which ran for 10 Gyr (see Fig. 6). DMAF preferentially deposits alarge amount of thermal energy into the gas particles located withinthe high-DM-density regions at the centre. The resultant pressure

Figure 5. A pure DM simulation of an NFW profiled halo (M = 1015 M�,c = 6.9) was evolved for 20 Gyr. The simulation with the lowest DM mass(highest rate) exhibits a suppression of the density at the centre of the halowhere the annihilation is strongest.

gradients evacuate large amounts of gas; in the inner regions ofthe halo, the radial density decreases by two to three orders ofmagnitude for our strongest DMAF simulation (10 keV).

Having shown the code depositing energy into the gas as ex-pected, we now turn to the question of convergence. We ran DMAFsimulations on a Milky Way sized halo (M = 1012 M�, c = 15)sampled at three different resolutions; the ICs are shown in the firstrow of Fig. 7. Although initially the thermal energy profiles differ,we find a good convergence of the results with increasing resolutionin both the DM and gas profiles over large simulation times. Forthe highest particle resolution runs, over the course of 25 Gyr, thedifferences in the thermal profiles wash away as the annihilationenergy is deposited. In the lowest resolution simulation, near thegravitationally softened region, the DM profile is relatively flatterthan its higher resolution counterparts. As a consequence, the energyinjected into the halo is correspondingly lower. In the higher resolu-tion simulations, however, the DM profiles more closely agree andso do the thermal energy profiles. This demonstrates the dependenceour scheme has on the resolution of the dark matter resolution; thethermal energy injected converges when the DM distribution is alsoconverged.

DMAF should influence galaxies most when the amount of en-ergy injected into the halo is similar to its gravitational binding en-ergy. Indeed, we see galaxies with less mass more readily eject theirgas (see Fig. 8). These haloes of different masses and concentrationsbut with identical DMAF parameters, 100 MeV and cross-sectionof 〈σv〉 = 3 × 10−26 cm3 s−1 all evolved for only 0.5 Gyr, and wesee that while the largest of the haloes is barely affected, the smallerbut concentrated ones are altered substantially. In the ‘dwarf-halo’case, the effect is drastic enough such that even the DM profile isaffected at radii greater than the simulation smoothing length.



Figure 6. DMAF simulation results under different particle mass models. The cross-section was set at 〈σv〉 = 3 × 10−26 cm3 s−1 and DMAF energy wasdeposited into an NFW profiled halo (M = 1015 M�, c = 6.9), for 10 Gyr. The top row of panels show the change in the DM and gas profiles due to DMAF(solid lines) compared with non-DMAF simulations (dotted lines), and the vertical dashed lines are the gravitational smoothing lengths. The middle row ofpanels likewise compare the thermal energy profile evolved with and without DMAF. The bottom row of panels show the density–thermal energy distributionof the DMAF simulation, showing the high-density gas is preferentially heated due to the halo cusp.

4 A N N I H I L AT I O N F E E D BAC K I N AC O S M O L O G I C A L C O N T E X T

DMA simulations have been performed by Smith et al. (2012) wherethey applied a model of DMAF in the context of protostars form-ing around DM minihaloes. They found that for modest annihila-tion rates (thermal relic cross-sections with 100 GeV mass), DMAFprevents fragmentation or at least preferential formation of wide bi-nary stars subsequent to collapse. However, their treatment did notsimulate a ‘live’ DM halo and instead used a spherically symmet-ric background halo, and accreting gas was then simulated on top

of this analytical halo on very small scales (∼0.01 pc). Natarajanet al. (2008) studied DMAF in a large-scale cosmological con-text. Their treatment involved semi-analytical models of baryoncollapse (Croton et al. 2006), again with analytical haloes, albeitwith NFW parameters fitted from the Millennium DM simulationruns (Springel et al. 2005). None the less, Natarajan et al. (2008)found that in very sharply sloped haloes, the annihilation rate couldbe large enough to offset the galactic cooling rate and have an ap-preciable effect on the gas distribution and luminosity function ofgalaxies.



Figure 7. ICs for a convergence run (top row) and the DMAF simulations after 10 Gyr (middle row) and 25 Gyr (bottom row) of a Milky Way sized halo(M = 1012 M�, c = 15) sampled at three different resolutions and undergoing DMAF from a thermal relic 100 MeV DM particle. The DM density profiles(left-hand panels) and the gas density profiles (centre panels) are initially near identical except for the inner regions near the smoothing length. The thermalenergy (right-hand panels) was set by each particle’s local velocity dispersion and adjusting the kinetic energy appropriately (see Section 2.3). Though thethermal energy profile differs initially in the inner regions and far outer edges, we find that with evolution under DMAF, the differences shrink and convergeas the injected energy becomes the dominant component in the gas.

A main advantage of using analytical haloes instead of ‘live’simulations is that it sidesteps issues of resolving the high-densitypeaks towards the centres of the haloes – which would other-wise underestimate the total annihilation emanating from withinan unresolved cusp. This comes at the cost of not having a‘true’ dynamic halo that is interacting with gas. It has beenshown that in full N-body simulations, baryonic feedback eventslike supernovae and star formation can cause rapid gravitationalinstabilities that facilitate the formation of flat cored profiles

(Navarro, Eke & Frenk 1996a; Lackner & Ostriker 2010; Pontzen& Governato 2012). It is therefore important to not only considerDMAF on the gas, but also to study it in the presence of ‘live’haloes with complicated formation histories, substructure and merg-ers. To this end, we have the first cosmological box simulations tocreate a sample of galaxies and study DMAF on a more realisticpopulation.

Though our models have ‘unresolved cusps’ smoothed over bygravitational softening, their size is simply tied to the scale of how



Figure 8. A comparison of the effect of DMAF on three different types of haloes. From the left- to right-hand side, we show a dwarf-sized halo (M = 1010

M�, c = 25), a Milky Way sized halo (M = 1012 M�, c = 15) and a large halo (M = 1015 M�, c = 6.9). These panels show the evolution after only 0.5 Gyrof evolution under DMAF from a 100 MeV DM particle (dotted lines show simulations without DMAF).

well the DM density is resolved. In the case of our haloes, this isgiven by the power criterion (Power et al. 2003), i.e. resolved downto near the gravitational softening length. From this, it followsthat the correct amount of DMAF energy will only be depositedwhen the size of a halo’s physical core is larger than the smoothinglength. But even in the case of an infinitely ‘cuspy halo’ with nocore, while the energy density injection (4), in principle, diverges,the total energy injected into the halo does not. DMAF injectionscales with ρ2

χ ∝ r−2 in our haloes, while the volumes at whichthese energies are deposited scale with V ∝ r3. Hence, we find thatthe relative uncertainty in energy injected will be at most the ratio

of the unresolved core size to the scale radius of our haloes (seeAppendix B).

While our idealized halo simulations lack more complicatedphysics, such as realistic cooling curves, they show that annihi-lation increases in importance for less massive but concentratedhaloes, due to weaker gravitational binding and highly concentrateddensity peaks. Incorporation of DMAF into a more complete codewith the advanced physics models (cooling, stars, etc.) as well asa more realistic model for annihilation products coupling to gas isunder research. Furthermore, our assumption of 100 per cent heat-ing efficiency is only approximately true for light DM models at



Figure 9. SPH images of box simulations (DM is shown in red, gas in blue) of a �CDM universe (left-hand panels) and the corresponding simulations for auniverse with DMAF (right-hand panels) from a 100 MeV DM particle annihilating with the thermal relic cross-section. Due to annihilation from this model,the gas is blown out from most galaxies except for the largest. Because of the loss of the gravitational pull by the ejected gas, many of the smaller DM structureshave also been washed away. The intensity corresponds to the SPH density integrated down the z-axis and was created with the package pynbody (Pontzenet al. 2013).

high redshifts (Ripamonti, Mapelli & Ferrara 2007; Slatyer, Pad-manabhan & Finkbeiner 2009; Evoli et al. 2012). DMAF couplingon galactic scales is an ongoing area of research with large un-certainties. None the less, our interpretations are consistent withSchon et al. (2015), who found that DMAF in the form of electron–positron annihilation pairs couples efficiently to the gas – enoughsuch that DMAF is likely to have an effect on primordial haloesof mass 105–106 M� at high redshifts z > 20, even for heavy DMmodels (100 GeV) that are far less constrained by observations. Inthe context of the Milky Way cosmic-ray electrons, Delahaye et al.(2010) find energy deposition lengths of the order of ≈1kpc.

Haloes with present-day masses of about 1010 M� are espe-cially interesting, as below this mass scale the baryon fraction ofthese dwarf haloes is significantly reduced compared to cosmo-

logical abundances (Papastergis et al. 2012). This is largely dueto ultraviolet-suppression feedback (Hoeft et al. 2006) balancingthe gas cooling rate and preventing its condensation. Additionalheating sources, like unaccounted DMAF, which precisely affectsthese smaller, concentrated haloes more prominently, as we see inFig. 8, could therefore tip the balance in this regime pushing upthe baryon fraction drop-off scale. Therefore, dwarf galaxies, inparticular, will be a focus of our cosmological runs. Indeed, pre-liminary results from our cosmological runs back this assertion up.Fig. 9 shows gas (blue) and DM (red) integrated density images of asubsection of our box in the case of plain �CDM (left-hand panels)and with DMAF from a 100 MeV DM particle (right-hand panels).The haloes appear much less dense, and much of the small galacticstructure has been washed away. More so, while the effect is not as



striking as in the gas image, many of the smaller DM haloes havebeen washed away as well. A fuller analysis of these simulationswill be made available in the next paper of this series.

5 C O N C L U S I O N S

We present the first hydrodynamical simulations to self-consistentlymodel DMAF. New routines are implemented into the simulationcode GADGET-2 that take into account the effect of DMA productsdepositing energy into the nearby baryonic structure of galaxies. Wetested this code on uniform box simulations and the case of simplegalaxies modelled as NFW haloes with a gas component (�b/�CDM

≈ 20 per cent). As expected, DMAF preferentially heats the gas atthe centre of the halo near the DM density peak. Subsequently, theheated gas rises out of the haloes, altering their density and ther-mal profiles. We repeated our simulations at different resolutionsto confirm convergence of our results and found good agreementbetween realizations of similar haloes sampled with 104, 105 and106 particles. Testing on haloes with different masses and concen-trations shows that DMAF is most likely to affect the smaller scalestructures of galaxies such as subhaloes and the dwarfs throughtheir star formation histories, X-ray profiles and baryonic content.Preliminary results of DM and gas cosmological box simulations(100 Mpc h−1, 5123 particles), with DMAF from 100 MeV DMparticles, show that much of the baryonic small-scale structure hasbeen washed away – leaving only the largest gaseous structuresand even influencing small DM structure. Much effort has beeninvested by simulators into generating hydrosimulations and feed-back prescriptions that reproduce small galaxies (Chen, Bryan &Salem 2016; Fitts et al. 2017; Read, Agertz & Collins 2016; Sawalaet al. 2016; Tollet et al. 2016; Vogelsberger et al. 2016; Garrison-Kimmel et al. 2017). Our results indicate that DMAF must be con-sidered when modelling this regime, and simulations with sufficientannihilation rates could have important implications on the ‘cuspyhalo’ and ‘missing-dwarf’ problems in cosmology.

AC K N OW L E D G E M E N T S

The authors thank Chris Power for his helpful feedback. The au-thors also acknowledge the University of Sydney High PerformanceComputing (HPC) service for providing computational resourcesfrom ARTEMIS, which has contributed to the research results re-ported within this paper, as well as the assistance of resourcesand services from the National Computational Infrastructure (NCI),which is supported by the Australian Government. NI is supportedby the Australian Government through an Australian PostgraduateAward (APA).

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A P P E N D I X A : D M A F IN U N I F O R M B OXS O L U T I O N S

In a homogeneous and isotropic universe, the densities are constantwith respect to position and so equation (4) can be written as

du

dt= 〈σv〉c2

mχ

ρcrit

a(t)3

�2χ

�b, (A1)

where �b and �χ are the usual baryon and DM cosmological densityparameters, ρcrit is the critical density and a(t) is the scalefactor asdetermined by the Friedman equations. In the case of a static boxa(t) = 1, the solution simply integrates a constant and so the thermalenergy increases linearly. For an expanding box, we transform fromt and integrate in a(t) space. For the case of a flat-matter-dominateduniverse (�M = 1, �b = 0.1573, �χ = �M − �b), the expandingFriedman equation reads

da

dt= H0a

− 12 , (A2)

where H0 is the Hubble constant. Dividing equations (A1) by (A2)and adding a term − u

ato take into account cosmological adiabatic

expansion yields the heating rate

du

da= 〈σv〉c2

mχ

ρcrit

H0a52

�2χ

�b− u

a. (A3)

Collecting the constants of the first term on the right-hand side intoκ and using an integrating factor yields the solution

u(z) = 1 + z

1 + z0

⎡⎣u0 + 2κ(1 + z0)

32

(1 −

(1 + z

1 + z0

) 12)⎤

⎦, (A4)

where z is the redshift, and we have used a = 11+z

, and u0 is anyinitial thermal energy at z0.

Fig. A1 shows the results of an expanding simulation with variousDMAF models, and the sub-panel shows the errors. For the mostextreme model, we see an error no greater than 2.0 per cent (staticbox simulations yield similar results), which fluctuates about theexpected energy. These errors can be minimized by decreasing themaximum allowed time-step of the simulation and increasingthe resolution of the simulation, as shown in Fig. A2, demonstratingthe robustness of our method.

Figure A1. The time evolution of a flat-matter-dominated, homogeneousand isotropic universe with an initial gas temperature set to T = 106 K andN = 163 particles. Shown in the crosses are the average specific thermalenergy from DMAF simulations with zero perturbations in their ICs. Thecurves are the theoretical expectations as given by equation (A4). The sub-panel shows the percentage deviation of the simulation results from theexpected theoretical curves.

Figure A2. Focussing on the 100 MeV model, we show how the uniformsimulations behave as the time-steps are decreased from a starting temper-ature of 10 K, to show the error due to the injected energy. We find theerror diminishes as the maximal time-steps of the simulation are decreased,converging to an error of < 0.3 per cent for dtmax 0.002. This residual errorcan then be further decreased by increasing particle resolution.



A P P E N D I X B: D M A F U N C E RTA I N T Y FO RC U S P Y H A L O E S

The energy density injected at a point is given similar to equation(4):

de

dt= κρ2

χ , (B1)

where we have multiplied out ρgas from (4) and κ are the appropriatepre-factors. Focusing on the energy injected within the scale radiusof a generic cuspy halo, the density profile is approximately

ρχ ∝ r−γ , (B2)

where for γ = 1, we retrieve the cusp of an NFW profile. The energybeing injected within a volume of radius r in the halo is

dE(r)

dt= 4πκ

∫ r

0ρ2

χ r ′2dr ′. (B3)

For sufficiently shallow cusps with γ < 1.5, this integral convergesand is proportional to

dE(r)

dt∝ r (3−2γ ). (B4)

Neglecting DMAF from within a gravitationally unresolved cuspyregion, it is therefore at most

� =(

rcore

rs

)(3−2γ )

, (B5)

where rs is the scale radius and rcore is the radius up to which thegravitational force and, hence, the halo is said to be well resolved,i.e. near the gravitational smoothing length, as argued by Poweret al. (2003). For γ = 1, we retrieve the approximate uncertaintydue to an NFW profile, �nfw = rcore

rs.

This paper has been typeset from a TEX/LATEX file prepared by the author.


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