text: The conclusions may also be sensitive to stochastic eﬀects and to the treatment of baryonic physics and so large samples, ideally drawn from simulations with diﬀerent plausible baryonic physics, are needed to reach a deﬁnitive conclusion regarding the relevance of dark discs. The implications of the dark disc for direct detection signals has been explored be- fore [29, 32, 55–57]. Depending on the density of the dark disc, the event rates in direct detection experiments could be enhanced, especially in the low recoil energy range. The annual modulation signal and its phase could also be altered. In particular, depending on the lag velocity of the DM particles in the disc compared to the stellar particles, the phase of the annual modulation could be shifted. However, most previous simulations have shown that the increase in the local DM density due to the dark disc is relatively small [29, 32, 56] and therefore that direct detection signals do not change signiﬁcantly due to the presence of a dark disc. To assess whether a dark disc component might exist for any of our MW analogues, we extract the azimuthal velocity distribution of the star particles in the torus for each halo. We then ﬁt the azimuthal components of both the DM and stellar velocity distributions with a double Gaussian. Our aim is to search for galaxies in which there is a DM component rotating as fast as the stars. Among the four haloes in our sample which show evidence of rotation with signiﬁcant positive mean azimuthal speed, only two haloes (E7 and E12) have a rotating DM component in the disc with mean velocity comparable (within 50 km/s) to that of the stars. The azimuthal component of the DM velocity distribution for the same two haloes in the DMO simulation is symmetric around zero. Notice that both the E7 and E12 galaxies don’t have a prominent stellar disc, as deﬁned in ref. [39]. The two panels of ﬁgure 6 show the azimuthal components of the local DM (shown as black data points and 1σ error bars) and stellar (orange data points and 1σ error bars) velocity distributions for two MW-like haloes (E12 and E7) in the eagle HR simulation with comparable mean azimuthal velocities for the DM and stellar distributions. The left panel of ﬁgure 6 corresponds to the same halo (E12) as shown in the bottom left panel of ﬁgure 4. The black and orange solid lines show the best ﬁt double Gaussian distribution to the DM and stellar azimuthal velocity distributions, respectively. We can see from the left panel that there are two kinematically distinct structures in the azimuthal component of the stellar velocity distribution in the torus centered at 8 kpc. The structure with a negative mean azimuthal speed contains most of the mass and is also present in a torus centered at 5 kpc as well as a torus centered at 10 kpc. While the structure with a positive mean azimuthal speed which contains less mass is not present at radii smaller than the Solar position, and may be due to the accretion of a massive satellite into the disc region. Hence, this structure is likely a “ring” of stars, the relic of a “ring” of gas that accretes late in the formation of the galaxy. Such features are not infrequent in simulations of galaxy formation (see e.g. [58]). In the left (right) panel of ﬁgure 6, when ﬁtting with a double Gaussian distribution, the mean azimuthal velocity for the second Gaussian (with higher µ) is 136 km/s (88 km/s) for the DM distribution, and 171 km/s (81 km/s) for the stellar distribution. Thus, the diﬀerences in the second peak of the double Gaussian between the DM and stellar distributions are 35 km/s and 7 km/s, in the left and right panels, respectively. The fraction of the rotating component of the double Gaussian DM distribution with the larger mean velocity is 32% and 43% in the left and right panels, respectively. In summary, from a total of 14 MW-like haloes in our sample, evidence for a signiﬁcant – 16 –

text: The

label: paragraph

text: conclusions

text: may

text: also

text: be

text: sensitive

text: to

text: stochastic

text: eﬀects

text: and

text: to

text: the

text: treatment

text: of

text: baryonic

text: physics

text: and

text: so

text: large

text: samples,

text: ideally

text: drawn

text: from

text: simulations

text: with

text: diﬀerent

text: plausible

text: baryonic

text: physics,

text: are

text: needed

text: to

text: reach

text: a

text: deﬁnitive

text: conclusion

text: regarding

text: the

text: relevance

text: of

text: dark

text: discs.

text: The

text: implications

text: of

text: the

text: dark

text: disc

text: for

text: direct

text: detection

text: signals

text: has

text: been

text: explored

text: be-

text: fore

text: Depending

text: on

text: the

text: density

text: of

text: the

text: dark

text: disc,

text: the

text: event

text: rates

text: in

text: direct

text: detection

text: experiments

text: could

text: be

text: enhanced,

text: especially

text: in

text: the

text: low

text: recoil

text: energy

text: range.

text: The

text: annual

text: modulation

text: signal

text: and

text: its

text: phase

text: could

text: also

text: be

text: altered.

text: In

text: particular,

text: depending

text: on

text: the

text: lag

text: velocity

text: of

text: the

text: DM

text: particles

text: in

text: the

text: disc

text: compared

text: to

text: the

text: stellar

text: particles,

text: the

text: phase

text: of

text: the

text: annual

text: modulation

text: could

text: be

text: shifted.

text: However,

text: most

text: previous

text: simulations

text: have

text: shown

text: that

text: the

text: increase

text: in

text: the

text: local

text: DM

text: density

text: due

text: to

text: the

text: dark

text: disc

text: is

text: relatively

text: small

text: and

text: therefore

text: that

text: direct

text: detection

text: signals

text: do

text: not

text: change

text: signiﬁcantly

text: due

text: to

text: the

text: presence

text: of

text: a

text: dark

text: disc.

text: To

text: assess

text: whether

text: a

text: dark

text: disc

text: component

text: might

text: exist

text: for

text: any

text: of

text: our

text: MW

text: analogues,

text: we

text: extract

text: the

text: azimuthal

text: velocity

text: distribution

text: of

text: the

text: star

text: particles

text: in

text: the

text: torus

text: for

text: each

text: halo.

text: We

text: then

text: ﬁt

text: the

text: azimuthal

text: components

text: of

text: both

text: the

text: DM

text: and

text: stellar

text: velocity

text: distributions

text: with

text: a

text: double

text: Gaussian.

text: Our

text: aim

text: is

text: to

text: search

text: for

text: galaxies

text: in

text: which

text: there

text: is

text: a

text: DM

text: component

text: rotating

text: as

text: fast

text: as

text: the

text: stars.

text: Among

text: the

text: four

text: haloes

text: in

text: our

text: sample

text: which

text: show

text: evidence

text: of

text: rotation

text: with

text: signiﬁcant

text: positive

text: mean

text: azimuthal

text: speed,

text: only

text: two

text: haloes

text: and

text: have

text: a

text: rotating

text: DM

text: component

text: in

text: the

text: disc

text: with

text: mean

text: velocity

text: comparable

text: (within

text: km/s)

text: to

text: that

text: of

text: the

text: stars.

text: The

text: azimuthal

text: component

text: of

text: the

text: DM

text: velocity

text: distribution

text: for

text: the

text: same

text: two

text: haloes

text: in

text: the

text: DMO

text: simulation

text: is

text: symmetric

text: around

text: zero.

text: Notice

text: that

text: both

text: the

text: and

text: galaxies

text: don’t

text: have

text: a

text: prominent

text: stellar

text: disc,

text: as

text: deﬁned

text: in

text: ref.

text: The

text: two

text: panels

text: of

text: ﬁgure

text: show

text: the

text: azimuthal

text: components

text: of

text: the

text: local

text: DM

text: (shown

text: as

text: black

text: data

text: points

text: and

text: 1σ

text: error

text: bars)

text: and

text: stellar

text: (orange

text: data

text: points

text: and

text: 1σ

text: error

text: bars)

text: velocity

text: distributions

text: for

text: two

text: MW-like

text: haloes

text: and

text: in

text: the

text: eagle

text: HR

text: simulation

text: with

text: comparable

text: mean

text: azimuthal

text: velocities

text: for

text: the

text: DM

text: and

text: stellar

text: distributions.

text: The

text: left

text: panel

text: of

text: ﬁgure

text: corresponds

text: to

text: the

text: same

text: halo

text: as

text: shown

text: in

text: the

text: bottom

text: left

text: panel

text: of

text: ﬁgure

text: The

text: black

text: and

text: orange

text: solid

text: lines

text: show

text: the

text: best

text: ﬁt

text: double

text: Gaussian

text: distribution

text: to

text: the

text: DM

text: and

text: stellar

text: azimuthal

text: velocity

text: distributions,

text: respectively.

text: We

text: can

text: see

text: from

text: the

text: left

text: panel

text: that

text: there

text: are

text: two

text: kinematically

text: distinct

text: structures

text: in

text: the

text: azimuthal

text: component

text: of

text: the

text: stellar

text: velocity

text: distribution

text: in

text: the

text: torus

text: centered

text: at

text: kpc.

text: The

text: structure

text: with

text: a

text: negative

text: mean

text: azimuthal

text: speed

text: contains

text: most

text: of

text: the

text: mass

text: and

text: is

text: also

text: present

text: in

text: a

text: torus

text: centered

text: at

text: kpc

text: as

text: well

text: as

text: a

text: torus

text: centered

text: at

text: kpc.

text: While

text: the

text: structure

text: with

text: a

text: positive

text: mean

text: azimuthal

text: speed

text: which

text: contains

text: less

text: mass

text: is

text: not

text: present

text: at

text: radii

text: smaller

text: than

text: the

text: Solar

text: position,

text: and

text: may

text: be

text: due

text: to

text: the

text: accretion

text: of

text: a

text: massive

text: satellite

text: into

text: the

text: disc

text: region.

text: Hence,

text: this

text: structure

text: is

text: likely

text: a

text: “ring”

text: of

text: stars,

text: the

text: relic

text: of

text: a

text: “ring”

text: of

text: gas

text: that

text: accretes

text: late

text: in

text: the

text: formation

text: of

text: the

text: galaxy.

text: Such

text: features

text: are

text: not

text: infrequent

text: in

text: simulations

text: of

text: galaxy

text: formation

text: (see

text: e.g.

text: In

text: the

text: left

text: (right)

text: panel

text: of

text: ﬁgure

text: when

text: ﬁtting

text: with

text: a

text: double

text: Gaussian

text: distribution,

text: the

text: mean

text: azimuthal

text: velocity

text: for

text: the

text: second

text: Gaussian

text: (with

text: higher

text: µ)

text: is

text: km/s

text: km/s)

text: for

text: the

text: DM

text: distribution,

text: and

text: km/s

text: km/s)

text: for

text: the

text: stellar

text: distribution.

text: Thus,

text: the

text: diﬀerences

text: in

text: the

text: second

text: peak

text: of

text: the

text: double

text: Gaussian

text: between

text: the

text: DM

text: and

text: stellar

text: distributions

text: are

text: km/s

text: and

text: km/s,

text: in

text: the

text: left

text: and

text: right

text: panels,

text: respectively.

text: The

text: fraction

text: of

text: the

text: rotating

text: component

text: of

text: the

text: double

text: Gaussian

text: DM

text: distribution

text: with

text: the

text: larger

text: mean

text: velocity

text: is

text: and

text: in

text: the

text: left

text: and

text: right

text: panels,

text: respectively.

text: In

text: summary,

text: from

text: a

text: total

text: of

text: MW-like

text: haloes

text: in

text: our

text: sample,

text: evidence

text: for

text: a

text: signiﬁcant


text: − is in the range 0.41 0.66 GeV/cm3 for the MW-like haloes in the eagle HR simulation, and is 0.42 GeV/cm3 for the two apostle IR MW analogues. From the 14 MW analogues in both simulations (12 in eagle HR + 2 in apostle IR), the local DM density is higher by 2 – 27% for 10 haloes compared to the shell averaged value, while the enhancement is greater than 10% for ﬁve haloes, and greater than 20% for only two haloes. To evaluate if baryonic physics is responsible for this enhancement, we compare the average local DM density in an identical torus and shell for the same haloes in the DMO simulations. We ﬁnd that two of the DMO counterparts (haloes E1 and E11) show a similar increase of the DM density in the torus compared to the shell, which can be due to the halo itself not being a perfect sphere. The increase in the local DM density in the disc for some haloes could be due to the DM halo contraction as a result of dissipational baryonic processes in the hydrodynamic simulations (See [71] for a measure of dark halo contraction in the eagle galaxy formation model). Note that for the two haloes discussed in section which have a DM component co- rotating with the stellar disc with a mean azimuthal speed comparable to that of the stars, the local DM density in the Galactic disc is 27% higher for halo E12 (shown in the left panel of ﬁgure 6) and 9% lower for halo E7 (shown in the right panel of ﬁgure 6) compared to the shell averaged value. To further understand the eﬀect of baryonic physics on the local DM density, we can study the shape of the inner (< 8 kpc) DM halo. We ﬁrst calculate the inertia tensor of the DM particles within two diﬀerent radii (5 kpc and 8 kpc). The resulting inertia tensor ≥ ≥ then describes an ellipsoid with three axes of length a b c: these lengths provide information about the shape of the DM distribution. We also repeat this process for the DMO counterparts to our eagle HR galaxy sample. We ﬁrst calculate the sphericity, s = c/a. If the DM distribution is a perfect sphere, all three axes will have the same length, so c = a and thus sphericity s = 1; a value of s < 1 describes an increasing deviation from sphericity in our sample. We ﬁnd for our galaxy sample that the sphericities at 5 kpc cluster in the range s = [0.85, 0.95] and are typically lower by less than 10% at 8 kpc, thus haloes have a larger deviation from a sphere at larger radii. In both cases, however, due to dissipational baryonic processes the DM sphericity is systematically higher in the hydrodynamic simulations than is the case for the DMO runs, in which the range in s is [0.75, 0.85]. This result is in agreement with earlier detailed simulations [72, 73] and reproduces what was found in higher resolution simulations of the apostle series [74]. Where there is a deviation from sphericity, this can be described using the triaxiality parameter, T , which is deﬁned as − a2 b2 T = − . (5.1) a2 c2 ≈ (cid:29) ≈ For very oblate systems, a b c and thus T 0, whereas for the opposite, prolate (cid:29) ≈ ≈ case a b c and T 1. We ﬁnd that there is little correlation in our galaxy sample between the values of T at 5 kpc and 8 kpc, with both varying in the range [0.02, 0.8]. However, it should be noted that because these inner haloes are very close to spherical (s = [0.81, 0.96]), the deviation towards either prolate or oblate distributions is very small. By contrast, the DMO counterparts almost all have a preference for inner haloes that are – 18 – ##LTLine##

text: is

text: in

text: the

text: range

text: for

text: the

text: MW-like

text: haloes

text: in

text: the

text: eagle

text: HR

text: simulation,

text: and

text: is

text: for

text: the

text: two

text: apostle

text: IR

text: MW

text: analogues.

text: From

text: the

text: MW

text: analogues

text: in

text: both

text: simulations

text: in

text: eagle

text: HR

text: in

text: apostle

text: IR),

text: the

text: local

text: DM

text: density

text: is

text: higher

text: by

text: for

text: haloes

text: compared

text: to

text: the

text: shell

text: averaged

text: value,

text: while

text: the

text: enhancement

text: is

text: greater

text: than

text: for

text: ﬁve

text: haloes,

text: and

text: greater

text: than

text: for

text: only

text: two

text: haloes.

text: To

text: evaluate

text: if

text: baryonic

text: physics

text: is

text: responsible

text: for

text: this

text: enhancement,

text: we

text: compare

text: the

text: average

text: local

text: DM

text: density

text: in

text: an

text: identical

text: torus

text: and

text: shell

text: for

text: the

text: same

text: haloes

text: in

text: the

text: DMO

text: simulations.

text: We

text: ﬁnd

text: that

text: two

text: of

text: the

text: DMO

text: counterparts

text: (haloes

text: and

text: show

text: a

text: similar

text: increase

text: of

text: the

text: DM

text: density

text: in

text: the

text: torus

text: compared

text: to

text: the

text: shell,

text: which

text: can

text: be

text: due

text: to

text: the

text: halo

text: itself

text: not

text: being

text: a

text: perfect

text: sphere.

text: The

text: increase

text: in

text: the

text: local

text: DM

text: density

text: in

text: the

text: disc

text: for

text: some

text: haloes

text: could

text: be

text: due

text: to

text: the

text: DM

text: halo

text: contraction

text: as

text: a

text: result

text: of

text: dissipational

text: baryonic

text: processes

text: in

text: the

text: hydrodynamic

text: simulations

text: (See

text: for

text: a

text: measure

text: of

text: dark

text: halo

text: contraction

text: in

text: the

text: eagle

text: galaxy

text: formation

text: model).

text: Note

text: that

text: for

text: the

text: two

text: haloes

text: discussed

text: in

text: section

text: which

text: have

text: a

text: DM

text: component

text: co-

text: rotating

text: with

text: the

text: stellar

text: disc

text: with

text: a

text: mean

text: azimuthal

text: speed

text: comparable

text: to

text: that

text: of

text: the

text: stars,

text: the

text: local

text: DM

text: density

text: in

text: the

text: Galactic

text: disc

text: is

text: higher

text: for

text: halo

text: (shown

text: in

text: the

text: left

text: panel

text: of

text: ﬁgure

text: and

text: lower

text: for

text: halo

text: (shown

text: in

text: the

text: right

text: panel

text: of

text: ﬁgure

text: compared

text: to

text: the

text: shell

text: averaged

text: value.

text: To

text: further

text: understand

text: the

text: eﬀect

text: of

text: baryonic

text: physics

text: on

text: the

text: local

text: DM

text: density,

text: we

text: can

text: study

text: the

text: shape

text: of

text: the

text: inner

text: kpc)

text: DM

text: halo.

text: We

text: ﬁrst

text: calculate

text: the

text: inertia

text: tensor

text: of

text: the

text: DM

text: particles

text: within

text: two

text: diﬀerent

text: radii

text: kpc

text: and

text: kpc).

text: The

text: resulting

text: inertia

text: tensor

text: then

text: describes

text: an

text: ellipsoid

text: with

text: three

text: axes

text: of

text: length

text: a

text: b

text: c:

text: these

text: lengths

text: provide

text: information

text: about

text: the

text: shape

text: of

text: the

text: DM

text: distribution.

text: We

text: also

text: repeat

text: this

text: process

text: for

text: the

text: DMO

text: counterparts

text: to

text: our

text: eagle

text: HR

text: galaxy

text: sample.

text: We

text: ﬁrst

text: calculate

text: the

text: sphericity,

text: s

text: c/a.

text: If

text: the

text: DM

text: distribution

text: is

text: a

text: perfect

text: sphere,

text: all

text: three

text: axes

text: will

text: have

text: the

text: same

text: length,

text: so

text: c

text: a

text: and

text: thus

text: sphericity

text: s

text: a

text: value

text: of

text: s

text: describes

text: an

text: increasing

text: deviation

text: from

text: sphericity

text: in

text: our

text: sample.

text: We

text: ﬁnd

text: for

text: our

text: galaxy

text: sample

text: that

text: the

text: sphericities

text: at

text: kpc

text: cluster

text: in

text: the

text: range

text: s

text: and

text: are

text: typically

text: lower

text: by

text: less

text: than

text: at

text: kpc,

text: thus

text: haloes

text: have

text: a

text: larger

text: deviation

text: from

text: a

text: sphere

text: at

text: larger

text: radii.

text: In

text: both

text: cases,

text: however,

text: due

text: to

text: dissipational

text: baryonic

text: processes

text: the

text: DM

text: sphericity

text: is

text: systematically

text: higher

text: in

text: the

text: hydrodynamic

text: simulations

text: than

text: is

text: the

text: case

text: for

text: the

text: DMO

text: runs,

text: in

text: which

text: the

text: range

text: in

text: s

text: is

text: This

text: result

text: is

text: in

text: agreement

text: with

text: earlier

text: detailed

text: simulations

text: and

text: reproduces

text: what

text: was

text: found

text: in

text: higher

text: resolution

text: simulations

text: of

text: the

text: apostle

text: series

text: Where

text: there

text: is

text: a

text: deviation

text: from

text: sphericity,

text: this

text: can

text: be

text: described

text: using

text: the

text: triaxiality

text: parameter,

text: T

text: which

text: is

text: deﬁned

text: as

text: T

text: For

text: very

text: oblate

text: systems,

text: a

text: b

text: c

text: and

text: thus

text: T

text: whereas

text: for

text: the

text: opposite,

text: prolate

text: case

text: a

text: b

text: c

text: and

text: T

text: We

text: ﬁnd

text: that

text: there

text: is

text: little

text: correlation

text: in

text: our

text: galaxy

text: sample

text: between

text: the

text: values

text: of

text: T

text: at

text: kpc

text: and

text: kpc,

text: with

text: both

text: varying

text: in

text: the

text: range

text: However,

text: it

text: should

text: be

text: noted

text: that

text: because

text: these

text: inner

text: haloes

text: are

text: very

text: close

text: to

text: spherical

text: (s

text: the

text: deviation

text: towards

text: either

text: prolate

text: or

text: oblate

text: distributions

text: is

text: very

text: small.

text: By

text: contrast,

text: the

text: DMO

text: counterparts

text: almost

text: all

text: have

text: a

text: preference

text: for

text: inner

text: haloes

text: that

text: are

text: ##LTLine##


text: prolate rather than oblate, with T = [0.4, 0.9] at both radii for all of our candidates. We therefore conclude that spherical symmetry is a very good approximation for our galaxy halo centres with little dependence on radius, and that this marks an important diﬀerence from DMO studies. For halo E12 (shown in the left panel of ﬁgure 6), which has a local DM density in the torus enhanced by 27% compared to the shell averaged value, the triaxiality parameter T = 0.02 at 8 kpc, which implies an oblate halo, as expected. 6 Direct detection signals 6.1 Event rates We consider a DM particle χ scattering elastically oﬀ a nucleus with atomic mass number A and atomic number Z, and depositing the nuclear recoil energy E in the detector. The (cid:90) R diﬀerential rate (events per unit energy, per unit detector mass, per unit time) is given by dR ρ 1 dσ = χ d3v A vf˜ (v, t). (6.1) det dE m m dE R χ A v>vmin R Here ρ is the local DM density, m and m are the DM and nucleus masses, σ is the χ χ A A (cid:82) DM–nucleus scattering cross section and v is the 3-vector relative velocity between DM and ≡ | | the nucleus, while v v . f˜ (v, t) is the DM velocity distribution in the detector rest det frame normalised to unity, such that d3vf˜ (v, t) = 1. For the case of elastic scattering, d(cid:115)et the minimum speed v required for a DM particle to deposit a recoil energy E in the min R detector is given by m E A R v = , (6.2) min 2µ2 χA where µ = m m /(m + m ) is the reduced mass of the DM and nucleus. χA χ A χ A The diﬀerential cross section is in general a sum of spin-independent and spin-dependent contributions. Here we only consider the case of spin-independent scattering and assume equal couplings of DM to protons and neutrons. In this case the diﬀerential cross section is dσ m A2 A = A σ F 2(E ) , (6.3) dE 2µ2 v2 SI R R χp where σ is the spin-independent DM–nucleon scattering cross section, µ is the re- SI χp duced mass of the DM–nucleon system, and F (E ) is a form factor for which we use the R expression from Helm [75]. The DM velocity distribution is usually given in the Galactic rest frame, and to calculate the event rate in Eq. (6.7), one has to transform the velocity distribution from the Galactic rest frame to the rest frame of the detector, by f˜ (v, t) = f˜ (v + v + v (t)), (6.4) det gal s e where v (t) is the velocity of the Earth with respect to the Sun, and v = v + v is e s c pec the velocity of the Sun in the Galactic rest frame. Here v is the Sun’s circular velocity ≈ c (or Local Standard of Rest (LSR) velocity), and v (11.10, 12.24, 7.25) km/s [76] is pec – 19 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: prolate

text: rather

text: than

text: oblate,

text: with

text: T

text: at

text: both

text: radii

text: for

text: all

text: of

text: our

text: candidates.

text: We

text: therefore

text: conclude

text: that

text: spherical

text: symmetry

text: is

text: a

text: very

text: good

text: approximation

text: for

text: our

text: galaxy

text: halo

text: centres

text: with

text: little

text: dependence

text: on

text: radius,

text: and

text: that

text: this

text: marks

text: an

text: important

text: diﬀerence

text: from

text: DMO

text: studies.

text: For

text: halo

text: (shown

text: in

text: the

text: left

text: panel

text: of

text: ﬁgure

text: which

text: has

text: a

text: local

text: DM

text: density

text: in

text: the

text: torus

text: enhanced

text: by

text: compared

text: to

text: the

text: shell

text: averaged

text: value,

text: the

text: triaxiality

text: parameter

text: T

text: at

text: kpc,

text: which

text: implies

text: an

text: oblate

text: halo,

text: as

text: expected.

label: section

text: Direct

text: detection

text: signals

text: Event

text: rates

text: We

text: consider

text: a

text: DM

text: particle

text: χ

text: scattering

text: elastically

text: oﬀ

text: a

text: nucleus

text: with

text: atomic

text: mass

text: number

text: A

text: and

text: atomic

text: number

text: Z,

text: and

text: depositing

text: the

text: nuclear

text: recoil

text: energy

text: E

text: in

text: the

text: detector.

text: The

text: R

text: diﬀerential

text: rate

text: (events

text: per

text: unit

text: energy,

text: per

text: unit

text: detector

text: mass,

text: per

text: unit

text: time)

text: is

text: given

text: by

text: dR

text: ρ

text: dσ

text: χ

text: A

text: vf˜

text: (v,

text: t).

text: det

text: dE

text: m

text: m

text: dE

text: R

text: χ

text: A

text: v>vmin

text: R

text: Here

text: ρ

text: is

text: the

text: local

text: DM

text: density,

text: m

text: and

text: m

text: are

text: the

text: DM

text: and

text: nucleus

text: masses,

text: σ

text: is

text: the

text: χ

text: χ

text: A

text: A

text: DM–nucleus

text: scattering

text: cross

text: section

text: and

text: v

text: is

text: the

text: relative

text: velocity

text: between

text: DM

text: and

text: the

text: nucleus,

text: while

text: v

text: v

text: f˜

text: (v,

text: t)

text: is

text: the

text: DM

text: velocity

text: distribution

text: in

text: the

text: detector

text: rest

text: det

text: frame

text: normalised

text: to

text: unity,

text: such

text: that

text: (v,

text: t)

text: For

text: the

text: case

text: of

text: elastic

text: scattering,

text: the

text: minimum

text: speed

text: v

text: required

text: for

text: a

text: DM

text: particle

text: to

text: deposit

text: a

text: recoil

text: energy

text: E

text: in

text: the

text: min

text: R

text: detector

text: is

text: given

text: by

text: m

text: E

text: A

text: R

text: v

text: min

text: χA

text: where

text: µ

text: m

text: m

text: /(m

text: m

text: is

text: the

text: reduced

text: mass

text: of

text: the

text: DM

text: and

text: nucleus.

text: χA

text: χ

text: A

text: χ

text: A

text: The

text: diﬀerential

text: cross

text: section

text: is

text: in

text: general

text: a

text: sum

text: of

text: spin-independent

text: and

text: spin-dependent

text: contributions.

text: Here

text: we

text: only

text: consider

text: the

text: case

text: of

text: spin-independent

text: scattering

text: and

text: assume

text: equal

text: couplings

text: of

text: DM

text: to

text: protons

text: and

text: neutrons.

text: In

text: this

text: case

text: the

text: diﬀerential

text: cross

text: section

text: is

text: dσ

text: m

text: A

text: A

text: σ

text: F

text: dE

text: SI

text: R

text: R

text: χp

text: where

text: σ

text: is

text: the

text: spin-independent

text: DM–nucleon

text: scattering

text: cross

text: section,

text: µ

text: is

text: the

text: re-

text: SI

text: χp

text: duced

text: mass

text: of

text: the

text: DM–nucleon

text: system,

text: and

text: F

text: (E

text: is

text: a

text: form

text: factor

text: for

text: which

text: we

text: use

text: the

text: R

text: expression

text: from

text: Helm

text: The

text: DM

text: velocity

text: distribution

text: is

text: usually

text: given

text: in

text: the

text: Galactic

text: rest

text: frame,

text: and

text: to

text: calculate

text: the

text: event

text: rate

text: in

text: Eq.

text: one

text: has

text: to

text: transform

text: the

text: velocity

text: distribution

text: from

text: the

text: Galactic

text: rest

text: frame

text: to

text: the

text: rest

text: frame

text: of

text: the

text: detector,

text: by

text: f˜

text: (v,

text: t)

text: f˜

text: (v

text: v

text: v

text: (t)),

text: det

text: gal

text: s

text: e

text: where

text: v

text: (t)

text: is

text: the

text: velocity

text: of

text: the

text: Earth

text: with

text: respect

text: to

text: the

text: Sun,

text: and

text: v

text: v

text: v

text: is

text: e

text: s

text: c

text: pec

text: the

text: velocity

text: of

text: the

text: Sun

text: in

text: the

text: Galactic

text: rest

text: frame.

text: Here

text: v

text: is

text: the

text: Sun’s

text: circular

text: velocity

text: c

text: (or

text: Local

text: Standard

text: of

text: Rest

text: (LSR)

text: velocity),

text: and

text: v

text: km/s

text: is

text: pec

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: the DM halo properties, namely the local DM density and DM velocity distribution. The simplest and most commonly adopted DM halo model is the Standard Halo Model (SHM): it assumes an isothermal, spherical DM halo with an isotropic Maxwell-Botzmann DM velocity distribution. In this model, the local DM density is usually assumed to be 0.3 GeV/cm3; the (cid:112) local velocity distribution in the Galactic rest frame is assumed to be a Maxwellian with a most probable (peak) speed equal to the local circular speed v (usually assumed to be 220 c km/s), a three-dimensional velocity dispersion of 3/2 v , and truncated at the escape speed c of 544 km/s from the Galaxy. Throughout this paper, when referring to the SHM Maxwellian, we consider the Maxwellian distribution with a most probable speed of 230 km/s in order to be consistent with the local circular speed assumed in deriving the observed MW rotation curves [5] (see section for more details). The local DM density enters as a normalisation factor in the event rate in direct de- tection experiments, whilst the DM velocity distribution enters in the event rate in a more complicated way (see eq. (6.1)). In particular, diﬀerent direct detection experiments with diﬀerent energy thresholds and target nuclei probe diﬀerent DM speed ranges, and thus their dependence on the DM velocity distribution varies. To overcome the astrophysical uncer- tainties in the interpretation of direct detection results, methods to compare the results of diﬀerent experiments in a halo-independent way have been developed and applied to direct detection data [6–25]. Such methods are based on comparing diﬀerent experiments in the space of the minimum DM speed, v , required for a DM particle to deposit a recoil energy min E in the detector, and thus can only be applied to compare experiments which are sensitive R to the same range of v . min High resolution numerical simulations of galaxy formation can provide information on the properties of the DM halo. Velocity distributions extracted from high resolution DM-only (DMO) simulations deviate substantially from a Maxwellian distribution, with fewer particles at the peak and an excess of particles at large speeds [26, 27]. DMO simulations can achieve high resolution, but have signiﬁcant systematic uncertainties since they neglect the eﬀect of the baryonic components during the galaxy formation process. Additionally, velocity distributions drawn from isolated simulations of the MW and the accretion of the Sagittarius dwarf galaxy show signiﬁcant deviations from the best ﬁt Maxwellian distribution, and this has important eﬀects on direct detection rates [28]. Nevertheless, the local DM distribution is expected to be very smooth: high resolution N-body simulations predict that the density at the Solar radius can diﬀer from the mean over the best-ﬁt ellipsoidal equidensity contour by at most 15% at the 99.9% conﬁdence level [26]. Recently, realistic cosmological simulations including baryons have become possible thanks to advances in physical modeling, as well as better understanding and treatment of numerical issues. The hydrodynamic simulation performed by Ling et al. [29] predicts a DM speed distribution which deviates signiﬁcantly from the Maxwellian, and instead gives an excellent ﬁt to a Tsallis [30] distribution1. In this simulation, the local DM density is found to be 0.37 – 0.39 GeV/cm3. The Eris hydrodynamic simulation [32] ﬁnds a speed distribution that is not a perfect Maxwellian, with an excess of particles at speeds less than 350 km/s compared to the SHM Maxwellian and fewer particles at higher speeds. The speed distribution is instead well ﬁt by the ﬁtting function proposed by Mao et al. [33]. The NI- 1See also ref. [31] for a variation of the Tsallis distribution function which is anisotropic. – 2 – ##LTLine## ##LTLine##

text: the

text: DM

text: halo

text: properties,

text: namely

text: the

text: local

text: DM

text: density

text: and

text: DM

text: velocity

text: distribution.

text: The

text: simplest

text: and

text: most

text: commonly

text: adopted

text: DM

text: halo

text: model

text: is

text: the

text: Standard

text: Halo

text: Model

text: (SHM):

text: it

text: assumes

text: an

text: isothermal,

text: spherical

text: DM

text: halo

text: with

text: an

text: isotropic

text: Maxwell-Botzmann

text: DM

text: velocity

text: distribution.

text: In

text: this

text: model,

text: the

text: local

text: DM

text: density

text: is

text: usually

text: assumed

text: to

text: be

text: the

text: local

text: velocity

text: distribution

text: in

text: the

text: Galactic

text: rest

text: frame

text: is

text: assumed

text: to

text: be

text: a

text: Maxwellian

text: with

text: a

text: most

text: probable

text: (peak)

text: speed

text: equal

text: to

text: the

text: local

text: circular

text: speed

text: v

text: (usually

text: assumed

text: to

text: be

text: c

text: km/s),

text: a

text: three-dimensional

text: velocity

text: dispersion

text: of

text: v

text: and

text: truncated

text: at

text: the

text: escape

text: speed

text: c

text: of

text: km/s

text: from

text: the

text: Galaxy.

text: Throughout

text: this

text: paper,

text: when

text: referring

text: to

text: the

text: SHM

text: Maxwellian,

text: we

text: consider

text: the

text: Maxwellian

text: distribution

text: with

text: a

text: most

text: probable

text: speed

text: of

text: km/s

text: in

text: order

text: to

text: be

text: consistent

text: with

text: the

text: local

text: circular

text: speed

text: assumed

text: in

text: deriving

text: the

text: observed

text: MW

text: rotation

text: curves

text: (see

text: section

text: for

text: more

text: details).

text: The

text: local

text: DM

text: density

text: enters

text: as

text: a

text: normalisation

text: factor

text: in

text: the

text: event

text: rate

text: in

text: direct

text: de-

text: tection

text: experiments,

text: whilst

text: the

text: DM

text: velocity

text: distribution

text: enters

text: in

text: the

text: event

text: rate

text: in

text: a

text: more

text: complicated

text: way

text: (see

text: eq.

text: In

text: particular,

text: diﬀerent

text: direct

text: detection

text: experiments

text: with

text: diﬀerent

text: energy

text: thresholds

text: and

text: target

text: nuclei

text: probe

text: diﬀerent

text: DM

text: speed

text: ranges,

text: and

text: thus

text: their

text: dependence

text: on

text: the

text: DM

text: velocity

text: distribution

text: varies.

text: To

text: overcome

text: the

text: astrophysical

text: uncer-

text: tainties

text: in

text: the

text: interpretation

text: of

text: direct

text: detection

text: results,

text: methods

text: to

text: compare

text: the

text: results

text: of

text: diﬀerent

text: experiments

text: in

text: a

text: halo-independent

text: way

text: have

text: been

text: developed

text: and

text: applied

text: to

text: direct

text: detection

text: data

text: Such

text: methods

text: are

text: based

text: on

text: comparing

text: diﬀerent

text: experiments

text: in

text: the

text: space

text: of

text: the

text: minimum

text: DM

text: speed,

text: v

text: required

text: for

text: a

text: DM

text: particle

text: to

text: deposit

text: a

text: recoil

text: energy

text: min

text: E

text: in

text: the

text: detector,

text: and

text: thus

text: can

text: only

text: be

text: applied

text: to

text: compare

text: experiments

text: which

text: are

text: sensitive

text: R

text: to

text: the

text: same

text: range

text: of

text: v

text: min

text: High

text: resolution

text: numerical

text: simulations

text: of

text: galaxy

text: formation

text: can

text: provide

text: information

text: on

text: the

text: properties

text: of

text: the

text: DM

text: halo.

text: Velocity

text: distributions

text: extracted

text: from

text: high

text: resolution

text: DM-only

text: (DMO)

text: simulations

text: deviate

text: substantially

text: from

text: a

text: Maxwellian

text: distribution,

text: with

text: fewer

text: particles

text: at

text: the

text: peak

text: and

text: an

text: excess

text: of

text: particles

text: at

text: large

text: speeds

text: DMO

text: simulations

text: can

text: achieve

text: high

text: resolution,

text: but

text: have

text: signiﬁcant

text: systematic

text: uncertainties

text: since

text: they

text: neglect

text: the

text: eﬀect

text: of

text: the

text: baryonic

text: components

text: during

text: the

text: galaxy

text: formation

text: process.

text: Additionally,

text: velocity

text: distributions

text: drawn

text: from

text: isolated

text: simulations

text: of

text: the

text: MW

text: and

text: the

text: accretion

text: of

text: the

text: Sagittarius

text: dwarf

text: galaxy

text: show

text: signiﬁcant

text: deviations

text: from

text: the

text: best

text: ﬁt

text: Maxwellian

text: distribution,

text: and

text: this

text: has

text: important

text: eﬀects

text: on

text: direct

text: detection

text: rates

text: Nevertheless,

text: the

text: local

text: DM

text: distribution

text: is

text: expected

text: to

text: be

text: very

text: smooth:

text: high

text: resolution

text: N-body

text: simulations

text: predict

text: that

text: the

text: density

text: at

text: the

text: Solar

text: radius

text: can

text: diﬀer

text: from

text: the

text: mean

text: over

text: the

text: best-ﬁt

text: ellipsoidal

text: equidensity

text: contour

text: by

text: at

text: most

text: at

text: the

text: conﬁdence

text: level

text: Recently,

text: realistic

text: cosmological

text: simulations

text: including

text: baryons

text: have

text: become

text: possible

text: thanks

text: to

text: advances

text: in

text: physical

text: modeling,

text: as

text: well

text: as

text: better

text: understanding

text: and

text: treatment

text: of

text: numerical

text: issues.

text: The

text: hydrodynamic

text: simulation

text: performed

text: by

text: Ling

text: et

text: al.

text: predicts

text: a

text: DM

text: speed

text: distribution

text: which

text: deviates

text: signiﬁcantly

text: from

text: the

text: Maxwellian,

text: and

text: instead

text: gives

text: an

text: excellent

text: ﬁt

text: to

text: a

text: Tsallis

text: In

text: this

text: simulation,

text: the

text: local

text: DM

text: density

text: is

text: found

text: to

text: be

text: The

text: Eris

text: hydrodynamic

text: simulation

text: ﬁnds

text: a

text: speed

text: distribution

text: that

text: is

text: not

text: a

text: perfect

text: Maxwellian,

text: with

text: an

text: excess

text: of

text: particles

text: at

text: speeds

text: less

text: than

text: km/s

text: compared

text: to

text: the

text: SHM

text: Maxwellian

text: and

text: fewer

text: particles

text: at

text: higher

text: speeds.

text: The

text: speed

text: distribution

text: is

text: instead

text: well

text: ﬁt

text: by

text: the

text: ﬁtting

text: function

text: proposed

text: by

text: Mao

text: et

text: al.

text: The

text: NI-

text: also

text: ref.

text: for

text: a

text: variation

text: of

text: the

text: Tsallis

text: distribution

text: function

text: which

text: is

text: anisotropic.

text: ##LTLine##

text: ##LTLine##


text: � EAGLE HR � EAGLE HR, DMO ] � ] � � � - - ) ) � � /� � /� � � � ( ( � � - ��� - ��� � � � � [ [ ) ) � � �� ��� �� ��� � � ( ( η η [ / ] [ / ] ��� ��� � ��� ��� ��� ��� ��� ��� � ��� ��� ��� ��� ��� ��� � �� � � �� � ��� ��� APOSTLE IR APOSTLE IR, DMO � � ] ] � � - � - � ) ) � � / / � � � � � � ( ( � � - - � ��� � ��� � � [ [ ) ) � � �� ��� �� ��� � � ( ( η η [ / ] [ / ] ��� ��� � ��� ��� ��� ��� ��� ��� � ��� ��� ��� ��� ��� ��� � �� � � �� � ��� ��� Figure 7. Time averaged halo integrals (solid coloured lines) for the the two haloes closest to (halo E12 shown in green) and farthest from (halo E3 shown in magenta) the SHM Maxwellian in the eagle HR simulation (top left), and the two haloes (blue and red lines) in the apostle IR simulation (bottom left). The solid coloured lines and the shaded bands correspond to the halo integrals obtained ± from the mean velocity distribution and the velocity distribution at 1σ from the mean, respectively. The solid black line speciﬁes the halo integral for the SHM Maxwellian (with peak speed of 230 km/s), and the dashed coloured lines specify the halo integral obtained from the best ﬁt Maxwellian velocity modulus distribution for each halo (with matching colours). The right panels show the halo integrals for the same haloes in the corresponding left panel but in a DMO simulation. velocity distribution strongly at R = 8 kpc, typically resulting in a shift of the tails of the 0 halo integral to higher velocities with respect to the DMO case. The diﬀerence between the halo integral of the simulated haloes in the hydrodynamic case and the SHM Maxwellian, which is more pronounced at relatively large v , is mainly min due to the diﬀerent peak speed of the DM velocity distribution of the simulated haloes compared to the 230 km/s assumed for the SHM Maxwellian. Indeed, the halo integral for the best ﬁt Maxwellian velocity distribution falls within the 1σ uncertainty band of the halo integrals for all but two MW-like haloes in the hydrodynamic simulation. For those two haloes, there is only a slight deviation between the halo integral for the best ﬁt Maxwellian and the simulated halo at large v . The range of the best ﬁt v parameter for min 0 the Maxwellian distribution is 223 – 289 km/s in the hydrodynamic case, as can be seen in table 2 in Appendix . On the other hand, the shape of the DM velocity modulus distributions for the haloes – 21 – ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: DM mass and spin-independent cross section for these four eagle HR benchmark haloes. In the left panel, the results are shown for the two haloes with smallest (E6) and largest (E4) ρ , χ whereas in the right panel the results are shown for the two haloes with speed distributions closest to (E12) and farthest from (E3) the SHM Maxwellian. The bottom panel shows the results for the same two haloes in the top right panel (E3 and E12), but setting the local DM density to 0.3 GeV/cm3 for both haloes, in order to isolate the eﬀect induced by the deviation of the velocity distribution from the SHM Maxwellian distribution. For comparison, the allowed regions and exclusion limits assuming the SHM Maxwellian (with peak speed of 230 km/s) velocity distribution and a local DM density of 0.3 GeV/cm3 are shown in black. The shaded band in the exclusion limits and the two allowed regions of the same colour are obtained from the upper and lower 1σ limits of the halo integrals. One can see from ﬁgure 8 that the general vertical shift of the preferred regions and exclusion limits with respect to the SHM at all DM masses is due to the variation in the local DM density, whereas the shift at low DM masses is due to the high velocity tail of the DM distribution. For the halo with speed distribution farthest from the SHM Maxwellian (E3) and assuming a local DM density of 0.3 GeV/cm3, the allowed regions and exclusion ≤ ∼ limits at low DM masses (m 10 GeV) shift by a maximum of 2 GeV towards smaller χ DM masses (bottom panel of ﬁgure 8). When considering the local DM density extracted from simulations for each halo, all the regions and exclusion limits shift vertically at most ∼ − by a factor of 2 3 in the scattering cross section compared to the SHM (see left panel of ﬁgure 8). This vertical shift is due to the diﬀerent local DM densities of the simulated haloes compared to the SHM, which enters as a normalisation factor in the predicted recoil rate in direct detection experiments. It is also clear from ﬁgure 8 that the halo-to-halo scatter is larger than the 1σ uncertainty for each halo. Notice also that in general the preferred regions and exclusion limits all shift in the same direction, and thus the compatibility between the results of diﬀerent experiments is not improved for any of the simulated haloes. The left panel of ﬁgure 9 is the same as the top left panel of ﬁgure 8, but for the two apostle IR haloes (A1 and A2). The right panel shows the results for the same two haloes in the left panel but assuming a ﬁxed local DM density of 0.3 GeV/cm3 for both haloes. The same features discussed for the eagle HR haloes are also visible for the apostle IR haloes in ﬁgure 9. 8 Conclusions In this work we have studied the predictions of the set of cosmological, hydrodynamic simulations of the eagle [35, 36] and apostle [37, 38] projects for DM direct detection. O − Considering ﬁrst all simulated haloes in the mass range (1012 1013) M(cid:12), we applied two selection criteria for choosing galaxies that are MW-like: (i) the rotation curve of the simu- lated galaxy provides a good ﬁt to the recent compilation of observed MW rotation curves from ref. [5], and (ii) the stellar mass of the simulated galaxy is within the 3σ observed stellar × × mass range of the MW, 4.5 1010 < M∗/M(cid:12) < 8.3 1010 [49]. We have shown that these two selection criteria are especially relevant for accurate predictions of the local DM density and velocity distribution, which are important quantities for investigating the implications for DM direct detection searches. For our selected MW analogues (12 eagle HR and 2 apostle IR), we extracted the – 24 –

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text: for

text: the

text: two

text: haloes

text: with

text: speed

text: distributions

text: closest

text: to

text: and

text: farthest

text: from

text: the

text: SHM

text: Maxwellian.

text: The

text: bottom

text: panel

text: shows

text: the

text: results

text: for

text: the

text: same

text: two

text: haloes

text: in

text: the

text: top

text: right

text: panel

text: and

text: but

text: setting

text: the

text: local

text: DM

text: density

text: to

text: for

text: both

text: haloes,

text: in

text: order

text: to

text: isolate

text: the

text: eﬀect

text: induced

text: by

text: the

text: deviation

text: of

text: the

text: velocity

text: distribution

text: from

text: the

text: SHM

text: Maxwellian

text: distribution.

text: For

text: comparison,

text: the

text: allowed

text: regions

text: and

text: exclusion

text: limits

text: assuming

text: the

text: SHM

text: Maxwellian

text: (with

text: peak

text: speed

text: of

text: km/s)

text: velocity

text: distribution

text: and

text: a

text: local

text: DM

text: density

text: of

text: are

text: shown

text: in

text: black.

text: The

text: shaded

text: band

text: in

text: the

text: exclusion

text: limits

text: and

text: the

text: two

text: allowed

text: regions

text: of

text: the

text: same

text: colour

text: are

text: obtained

text: from

text: the

text: upper

text: and

text: lower

text: 1σ

text: limits

text: of

text: the

text: halo

text: integrals.

text: One

text: can

text: see

text: from

text: ﬁgure

text: that

text: the

text: general

text: vertical

text: shift

text: of

text: the

text: preferred

text: regions

text: and

text: exclusion

text: limits

text: with

text: respect

text: to

text: the

text: SHM

text: at

text: all

text: DM

text: masses

text: is

text: due

text: to

text: the

text: variation

text: in

text: the

text: local

text: DM

text: density,

text: whereas

text: the

text: shift

text: at

text: low

text: DM

text: masses

text: is

text: due

text: to

text: the

text: high

text: velocity

text: tail

text: of

text: the

text: DM

text: distribution.

text: For

text: the

text: halo

text: with

text: speed

text: distribution

text: farthest

text: from

text: the

text: SHM

text: Maxwellian

text: and

text: assuming

text: a

text: local

text: DM

text: density

text: of

text: the

text: allowed

text: regions

text: and

text: exclusion

text: limits

text: at

text: low

text: DM

text: masses

text: (m

text: GeV)

text: shift

text: by

text: a

text: maximum

text: of

text: GeV

text: towards

text: smaller

text: χ

text: DM

text: masses

text: (bottom

text: panel

text: of

text: ﬁgure

text: When

text: considering

text: the

text: local

text: DM

text: density

text: extracted

text: from

text: simulations

text: for

text: each

text: halo,

text: all

text: the

text: regions

text: and

text: exclusion

text: limits

text: shift

text: vertically

text: at

text: most

text: by

text: a

text: factor

text: of

text: in

text: the

text: scattering

text: cross

text: section

text: compared

text: to

text: the

text: SHM

text: (see

text: left

text: panel

text: of

text: ﬁgure

text: This

text: vertical

text: shift

text: is

text: due

text: to

text: the

text: diﬀerent

text: local

text: DM

text: densities

text: of

text: the

text: simulated

text: haloes

text: compared

text: to

text: the

text: SHM,

text: which

text: enters

text: as

text: a

text: normalisation

text: factor

text: in

text: the

text: predicted

text: recoil

text: rate

text: in

text: direct

text: detection

text: experiments.

text: It

text: is

text: also

text: clear

text: from

text: ﬁgure

text: that

text: the

text: halo-to-halo

text: scatter

text: is

text: larger

text: than

text: the

text: 1σ

text: uncertainty

text: for

text: each

text: halo.

text: Notice

text: also

text: that

text: in

text: general

text: the

text: preferred

text: regions

text: and

text: exclusion

text: limits

text: all

text: shift

text: in

text: the

text: same

text: direction,

text: and

text: thus

text: the

text: compatibility

text: between

text: the

text: results

text: of

text: diﬀerent

text: experiments

text: is

text: not

text: improved

text: for

text: any

text: of

text: the

text: simulated

text: haloes.

text: The

text: left

text: panel

text: of

text: ﬁgure

text: is

text: the

text: same

text: as

text: the

text: top

text: left

text: panel

text: of

text: ﬁgure

text: but

text: for

text: the

text: two

text: apostle

text: IR

text: haloes

text: and

text: The

text: right

text: panel

text: shows

text: the

text: results

text: for

text: the

text: same

text: two

text: haloes

text: in

text: the

text: left

text: panel

text: but

text: assuming

text: a

text: ﬁxed

text: local

text: DM

text: density

text: of

text: for

text: both

text: haloes.

text: The

text: same

text: features

text: discussed

text: for

text: the

text: eagle

text: HR

text: haloes

text: are

text: also

text: visible

text: for

text: the

text: apostle

text: IR

text: haloes

text: in

text: ﬁgure

text: Conclusions

text: In

text: this

text: work

text: we

text: have

text: studied

text: the

text: predictions

text: of

text: the

text: set

text: of

text: cosmological,

text: hydrodynamic

text: simulations

text: of

text: the

text: eagle

text: and

text: apostle

text: projects

text: for

text: DM

text: direct

text: detection.

text: O

text: Considering

text: ﬁrst

text: all

text: simulated

text: haloes

text: in

text: the

text: mass

text: range

text: we

text: applied

text: two

text: selection

text: criteria

text: for

text: choosing

text: galaxies

text: that

text: are

text: MW-like:

text: (i)

text: the

text: rotation

text: curve

text: of

text: the

text: simu-

text: lated

text: galaxy

text: provides

text: a

text: good

text: ﬁt

text: to

text: the

text: recent

text: compilation

text: of

text: observed

text: MW

text: rotation

text: curves

text: from

text: ref.

text: and

text: (ii)

text: the

text: stellar

text: mass

text: of

text: the

text: simulated

text: galaxy

text: is

text: within

text: the

text: 3σ

text: observed

text: stellar

text: mass

text: range

text: of

text: the

text: MW,

text: We

text: have

text: shown

text: that

text: these

text: two

text: selection

text: criteria

text: are

text: especially

text: relevant

text: for

text: accurate

text: predictions

text: of

text: the

text: local

text: DM

text: density

text: and

text: velocity

text: distribution,

text: which

text: are

text: important

text: quantities

text: for

text: investigating

text: the

text: implications

text: for

text: DM

text: direct

text: detection

text: searches.

text: For

text: our

text: selected

text: MW

text: analogues

text: eagle

text: HR

text: and

text: apostle

text: IR),

text: we

text: extracted

text: the


text: velocity distribution with a free peak speed are similar to those obtained directly from the local DM velocity distributions of the simulated MW analogues. Thus a Maxwellian ve- locity distribution with a peak speed constrained by hydrodynamic simulations as well as observations could be used by the community when analysing the results of direct detection experiments. Acknowledgements We thank Christopher McCabe and Christoph Weniger for useful discussions on the results of this work. We especially thank Fabio Iocco and Miguel Pato for providing the extensive compilation of rotation curve measurements used in this paper. G.B. (P.I.) and N.B. acknowledge support from the European Research Council through the ERC start- ing grant WIMPs Kairos. The research of F.C. is part of the VIDI research programme “Probing the Genesis of Dark Matter”, which is ﬁnanced by the Netherlands Organisation for Scientiﬁc Research (NWO). This work is part of the D-ITP consortium, a program of the NWO that is funded by the Dutch Ministry of Education, Culture and Science (OCW). This work was supported by the Science and Technology Facilities Council (grant number ST/F001166/1); European Research Council (grant numbers GA 267291 “Cosmiway” and GA 278594 “GasAroundGalaxies”) and by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Oﬃce (AP P7/08 CHARM). R.A.C. is a Royal Society Research Fellow. This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www. dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. We acknowledge PRACE for awarding us access to the Curie machine based in France at TGCC, CEA, Bruy`eres-le-Chˆatel. This work makes use of PyMinuit2. 2http://code.google.com/p/pyminuit – 28 – ##LTLine##

text: velocity

text: distribution

text: with

text: a

text: free

text: peak

text: speed

text: are

text: similar

text: to

text: those

text: obtained

text: directly

text: from

text: the

text: local

text: DM

text: velocity

text: distributions

text: of

text: the

text: simulated

text: MW

text: analogues.

text: Thus

text: a

text: Maxwellian

text: ve-

text: locity

text: distribution

text: with

text: a

text: peak

text: speed

text: constrained

text: by

text: hydrodynamic

text: simulations

text: as

text: well

text: as

text: observations

text: could

text: be

text: used

text: by

text: the

text: community

text: when

text: analysing

text: the

text: results

text: of

text: direct

text: detection

text: experiments.

text: Acknowledgements

text: We

text: thank

text: Christopher

text: McCabe

text: and

text: Christoph

text: Weniger

text: for

text: useful

text: discussions

text: on

text: the

text: results

text: of

text: this

text: work.

text: We

text: especially

text: thank

text: Fabio

text: Iocco

text: and

text: Miguel

text: Pato

text: for

text: providing

text: the

text: extensive

text: compilation

text: of

text: rotation

text: curve

text: measurements

text: used

text: in

text: this

text: paper.

text: G.B.

text: (P.I.)

text: and

text: N.B.

text: acknowledge

text: support

text: from

text: the

text: European

text: Research

text: Council

text: through

text: the

text: ERC

text: start-

text: ing

text: grant

text: WIMPs

text: Kairos.

text: The

text: research

text: of

text: F.C.

text: is

text: part

text: of

text: the

text: VIDI

text: research

text: programme

text: “Probing

text: the

text: Genesis

text: of

text: Dark

text: Matter”,

text: which

text: is

text: ﬁnanced

text: by

text: the

text: Netherlands

text: Organisation

text: for

text: Scientiﬁc

text: Research

text: (NWO).

text: This

text: work

text: is

text: part

text: of

text: the

text: D-ITP

text: consortium,

text: a

text: program

text: of

text: the

text: NWO

text: that

text: is

text: funded

text: by

text: the

text: Dutch

text: Ministry

text: of

text: Education,

text: Culture

text: and

text: Science

text: (OCW).

text: This

text: work

text: was

text: supported

text: by

text: the

text: Science

text: and

text: Technology

text: Facilities

text: Council

text: (grant

text: number

text: European

text: Research

text: Council

text: (grant

text: numbers

text: GA

text: “Cosmiway”

text: and

text: GA

text: “GasAroundGalaxies”)

text: and

text: by

text: the

text: Interuniversity

text: Attraction

text: Poles

text: Programme

text: initiated

text: by

text: the

text: Belgian

text: Science

text: Policy

text: Oﬃce

text: (AP

text: CHARM).

text: R.A.C.

text: is

text: a

text: Royal

text: Society

text: Research

text: Fellow.

text: This

text: work

text: used

text: the

text: DiRAC

text: Data

text: Centric

text: system

text: at

text: Durham

text: University,

text: operated

text: by

text: the

text: Institute

text: for

text: Computational

text: Cosmology

text: on

text: behalf

text: of

text: the

text: STFC

text: DiRAC

text: HPC

text: Facility

text: (www.

text: dirac.ac.uk).

text: This

text: equipment

text: was

text: funded

text: by

text: BIS

text: National

text: E-infrastructure

text: capital

text: grant

text: STFC

text: capital

text: grant

text: and

text: STFC

text: DiRAC

text: Operations

text: grant

text: and

text: Durham

text: University.

text: DiRAC

text: is

text: part

text: of

text: the

text: National

text: E-Infrastructure.

text: We

text: acknowledge

text: PRACE

text: for

text: awarding

text: us

text: access

text: to

text: the

text: Curie

text: machine

text: based

text: in

text: France

text: at

text: TGCC,

text: CEA,

text: Bruy`eres-le-Chˆatel.

text: This

text: work

text: makes

text: use

text: of

text: ##LTLine##


text: HAO (Numerical Investigation of a Hundred Astrophysical Objects) simulations [34] ﬁnd a DM speed distribution which is very diﬀerent from the Maxwellian, being more symmetric around the peak velocity and falling rapidly at large velocities, thus agreeing with the results of the Eris simulation. All these hydrodynamic simulations help build an understanding of the processes that can shape and modify the local DM density and velocity distribution. However, to make more precise quantitative predictions it is necessary that all these eﬀects are modeled in such a way that the main galaxy population properties are broadly repro- duced. This ensures, for instance, that appropriate amounts of feedback are applied and that the corresponding impact of feedback on DM is reasonably realistic. In this work, we study the implications of simulated MW-like galaxies within the ea- gle (“Evolution and Assembly of GaLaxies and their Environments”) [35, 36] and apos- tle [37, 38] projects for DM direct detection. The eagle project is a suite of hydrodynamic simulations that are calibrated to reproduce the observed distribution of stellar masses and sizes of low-redshift galaxies as well as the relation between the stellar mass and black hole mass of galaxies; in this respect they are unique amongst simulations of galaxy formation. In a companion paper, we have studied the implications of the eagle and the related apos- tle simulations for DM indirect detection [39]. We ﬁrst consider a selection of MW-like O − galaxies within haloes with virial mass in the range (1012 1013) M(cid:12). We then impose a set of selection criteria crucial for accurately predicting the local DM density and velocity distribution. Namely, we require that the haloes satisfy observational constraints on the MW rotation curve [5], and have stellar masses within the observed MW stellar mass range. We then extract the local DM density and velocity distribution for the selected haloes, and use them to analyse current data from direct detection experiments. The paper is outlined as follows. In section we present the set of eagle and apostle cosmological hydrodynamic simulations used in this work. In section we discuss in detail the selection criteria used for choosing MW analogues, and how each criterion is relevant for determining the local DM distribution. In sections and we present the local DM velocity distribution and DM density extracted from the simulations. In section we introduce the formalism for computing the predicted signals in DM direct detection experiments. In section we perform an analysis of current direct detection data and discuss how the allowed regions and exclusion limits set by diﬀerent experiments in the DM mass and scattering cross section plane changes when the DM distribution of simulated haloes is used. Finally, in section we summarize the conclusions of this work. In Appendices , , and , we provide additional material relevant for this work. 2 Simulations The simulations used in this study comprise parts of the eagle [35, 36] and apostle [37, 38] projects, both of which we describe below. The eagle project combines a state-of-the-art hydrodynamical SPH implementation nicknamed anarchy [40, 41] – built on top of an optimized version of the gadget SPH code, last described by [42] – and a detailed subgrid model of galaxy formation that features metal-line cooling, photo-reionization, star formation, and feedback from star formation and active galactic nuclei [43–47]. The subgrid model is regulated by a number of parameters, which are calibrated to give a good ﬁt to the z = 0.1 galaxy stellar mass function and to the – 3 –

text: HAO

text: (Numerical

text: Investigation

text: of

text: a

text: Hundred

text: Astrophysical

text: Objects)

text: simulations

text: ﬁnd

text: a

text: DM

text: speed

text: distribution

text: which

text: is

text: very

text: diﬀerent

text: from

text: the

text: Maxwellian,

text: being

text: more

text: symmetric

text: around

text: the

text: peak

text: velocity

text: and

text: falling

text: rapidly

text: at

text: large

text: velocities,

text: thus

text: agreeing

text: with

text: the

text: results

text: of

text: the

text: Eris

text: simulation.

text: All

text: these

text: hydrodynamic

text: simulations

text: help

text: build

text: an

text: understanding

text: of

text: the

text: processes

text: that

text: can

text: shape

text: and

text: modify

text: the

text: local

text: DM

text: density

text: and

text: velocity

text: distribution.

text: However,

text: to

text: make

text: more

text: precise

text: quantitative

text: predictions

text: it

text: is

text: necessary

text: that

text: all

text: these

text: eﬀects

text: are

text: modeled

text: in

text: such

text: a

text: way

text: that

text: the

text: main

text: galaxy

text: population

text: properties

text: are

text: broadly

text: repro-

text: duced.

text: This

text: ensures,

text: for

text: instance,

text: that

text: appropriate

text: amounts

text: of

text: feedback

text: are

text: applied

text: and

text: that

text: the

text: corresponding

text: impact

text: of

text: feedback

text: on

text: DM

text: is

text: reasonably

text: realistic.

text: In

text: this

text: work,

text: we

text: study

text: the

text: implications

text: of

text: simulated

text: MW-like

text: galaxies

text: within

text: the

text: ea-

text: gle

text: (“Evolution

text: and

text: Assembly

text: of

text: GaLaxies

text: and

text: their

text: Environments”)

text: and

text: apos-

text: tle

text: projects

text: for

text: DM

text: direct

text: detection.

text: The

text: eagle

text: project

text: is

text: a

text: suite

text: of

text: hydrodynamic

text: simulations

text: that

text: are

text: calibrated

text: to

text: reproduce

text: the

text: observed

text: distribution

text: of

text: stellar

text: masses

text: and

text: sizes

text: of

text: low-redshift

text: galaxies

text: as

text: well

text: as

text: the

text: relation

text: between

text: the

text: stellar

text: mass

text: and

text: black

text: hole

text: mass

text: of

text: galaxies;

text: in

text: this

text: respect

text: they

text: are

text: unique

text: amongst

text: simulations

text: of

text: galaxy

text: formation.

text: In

text: a

text: companion

text: paper,

text: we

text: have

text: studied

text: the

text: implications

text: of

text: the

text: eagle

text: and

text: the

text: related

text: apos-

text: tle

text: simulations

text: for

text: DM

text: indirect

text: detection

text: We

text: ﬁrst

text: consider

text: a

text: selection

text: of

text: MW-like

text: O

text: galaxies

text: within

text: haloes

text: with

text: virial

text: mass

text: in

text: the

text: range

text: We

text: then

text: impose

text: a

text: set

text: of

text: selection

text: criteria

text: crucial

text: for

text: accurately

text: predicting

text: the

text: local

text: DM

text: density

text: and

text: velocity

text: distribution.

text: Namely,

text: we

text: require

text: that

text: the

text: haloes

text: satisfy

text: observational

text: constraints

text: on

text: the

text: MW

text: rotation

text: curve

text: and

text: have

text: stellar

text: masses

text: within

text: the

text: observed

text: MW

text: stellar

text: mass

text: range.

text: We

text: then

text: extract

text: the

text: local

text: DM

text: density

text: and

text: velocity

text: distribution

text: for

text: the

text: selected

text: haloes,

text: and

text: use

text: them

text: to

text: analyse

text: current

text: data

text: from

text: direct

text: detection

text: experiments.

text: The

text: paper

text: is

text: outlined

text: as

text: follows.

text: In

text: section

text: we

text: present

text: the

text: set

text: of

text: eagle

text: and

text: apostle

text: cosmological

text: hydrodynamic

text: simulations

text: used

text: in

text: this

text: work.

text: In

text: section

text: we

text: discuss

text: in

text: detail

text: the

text: selection

text: criteria

text: used

text: for

text: choosing

text: MW

text: analogues,

text: and

text: how

text: each

text: criterion

text: is

text: relevant

text: for

text: determining

text: the

text: local

text: DM

text: distribution.

text: In

text: sections

text: and

text: we

text: present

text: the

text: local

text: DM

text: velocity

text: distribution

text: and

text: DM

text: density

text: extracted

text: from

text: the

text: simulations.

text: In

text: section

text: we

text: introduce

text: the

text: formalism

text: for

text: computing

text: the

text: predicted

text: signals

text: in

text: DM

text: direct

text: detection

text: experiments.

text: In

text: section

text: we

text: perform

text: an

text: analysis

text: of

text: current

text: direct

text: detection

text: data

text: and

text: discuss

text: how

text: the

text: allowed

text: regions

text: and

text: exclusion

text: limits

text: set

text: by

text: diﬀerent

text: experiments

text: in

text: the

text: DM

text: mass

text: and

text: scattering

text: cross

text: section

text: plane

text: changes

text: when

text: the

text: DM

text: distribution

text: of

text: simulated

text: haloes

text: is

text: used.

text: Finally,

text: in

text: section

text: we

text: summarize

text: the

text: conclusions

text: of

text: this

text: work.

text: In

text: Appendices

text: and

text: we

text: provide

text: additional

text: material

text: relevant

text: for

text: this

text: work.

text: Simulations

text: The

text: simulations

text: used

text: in

text: this

text: study

text: comprise

text: parts

text: of

text: the

text: eagle

text: and

text: apostle

text: projects,

text: both

text: of

text: which

text: we

text: describe

text: below.

text: The

text: eagle

text: project

text: combines

text: a

text: state-of-the-art

text: hydrodynamical

text: SPH

text: implementation

text: nicknamed

text: anarchy

text: built

text: on

text: top

text: of

text: an

text: optimized

text: version

text: of

text: the

text: gadget

text: SPH

text: code,

text: last

text: described

text: by

text: and

text: a

text: detailed

text: subgrid

text: model

text: of

text: galaxy

text: formation

text: that

text: features

text: metal-line

text: cooling,

text: photo-reionization,

text: star

text: formation,

text: and

text: feedback

text: from

text: star

text: formation

text: and

text: active

text: galactic

text: nuclei

text: The

text: subgrid

text: model

text: is

text: regulated

text: by

text: a

text: number

text: of

text: parameters,

text: which

text: are

text: calibrated

text: to

text: give

text: a

text: good

text: ﬁt

text: to

text: the

text: z

text: galaxy

text: stellar

text: mass

text: function

text: and

text: to

text: the