text: 10-2 N-body inferred SHM 10-3 MB fit rotating MB fit 10-4 10-5 10-6 g1536DM g1536 g15784 bands: MB simulation 1-σ 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 relative to best-fit rotating MB -0.8 dashed line: simulation average 0 50 100 150 200 250 300 350 400 450 100 200 300 400 500 600 100 200 300 400 500 600 700 800 vmin [km/s] Figure 6. (Upper panels) The time-averaged (over one year) mean inverse speed η(v ) for the min three N-body simulations. (Lower panels) Diﬀerences in speed distributions relative to the best-ﬁt rotating MB. Curves and bands are as described in ﬁgure 3. sensitive to a larger range of the velocity distribution, diluting the eﬀect of the deviations in the tail. 4.2 Results for mean inverse speed The diﬀerential count rate of dark matter scatters in a direct detection experiment is directly proportional to the mean inverse speed η of the dark matter particles as given in eq. (4.6). In this subsection we study the results from the N-body simulations for this quantity and compare to the same models examined in previous sections. In ﬁgure 6 we plot the time-averaged mean inverse speed η(v ) for the three simulations using the same color min schemes as in the previous ﬁgures. As anticipated, the inferred SHM is again a very poor ﬁt to the N-body distribution in the dark matter only simulation (g1536DM). As expected the best-ﬁt rotating and non- rotating MB do provide very good ﬁts for this case as they are ﬁts to the actual particle velocities. Once baryonic physics is included in the simulation, the SHM ﬁt improves dra- matically but still shows large deviations (see relative diﬀerence plots) at high velocities. This is true for both g1536 (SHM is again behind the best ﬁt MB band) and g15784. It is again not clear whether this eﬀect is mostly caused by limitations in the simulation due to small numbers of particles with these higher speeds or an actual inability of the SHM to accurately describe the relevant physics. 4.3 Results for Amplitude of Annual Modulation The direct detection signal arising from WIMP dark matter is expected to exhibit an annual modulation due to the motion of the Earth around the Sun. Detection of such an annual modulation was previously described as an important test in establishing the dark matter origin of a signal in direct detection experiments [5]. In contrast many backgrounds are isotropic and would not exhibit an annual variation. 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text: Yet, in the community there has been the belief that this simple SHM cannot be trusted as it does not adequately capture the true velocity distribution of the Galactic WIMPs. Dark matter halos in cosmological simulations (with dark matter only) were known to be triaxial and have anisotropic velocity distributions owing to their complex merger history. The anisotropic velocity distributions have been shown to deviate signiﬁcantly from the SHM [6] and alternative analytic ﬁts that were thought to better match the true distribution, particularly at the high-velocity tail have been explored [7, 8]. These alternatives to the SHM were motivated in part by results of dark matter only N-body simulations of structure formation which are limited by the fact that they do not include baryonic physics. In this work we examine the results of simulations that do include baryons. It is the purpose of this paper to investigate whether or not the SHM is a good representation of the true WIMP distribution in these simulations for the analysis of direct detection data. The process of galaxy formation proceeds hierarchically via the clustering and aggre- gation of dark matter halos. After the ﬁrst dark matter halos form, gas begins to be grav- itationally attracted into them where it cools, condenses and forms the ﬁrst generations of stars. As a galaxy gains mass (by accreting smaller galaxies) the dark matter halos also grow. Consequently galaxies like the Milky Way are embedded in dark matter halos that formed via the repeated merger of protogalactic fragments consisting of dark matter, gas, and early generations of stars. The measurement of the shapes of dark matter halos is challenging and the shape of the Milky Way Halo is currently uncertain [e.g. 9–11]. Since dark matter constitutes about 85% of the total mass in the Universe, most early simulations of galaxy formation relied purely on collisionless simulations of dark matter particles. These early collisionless N-body simulations showed that hierarchical growth results in dark matter halos whose shapes are highly triaxial or prolate [12–16] with radially anisotropic velocity distributions [17, 18]. More recently, N-body simulations have been performed that take into consideration not only collisionless dark matter particles but also the eﬀects of the condensation of baryons. These simulations ﬁnd somewhat diﬀerent halo shapes that are more spherical or axisym- metric [12, 19–22]. When baryons are taken into account, analysis of the orbital properties of dark matter particles in both idealized simulations [23, 24] and cosmological hydrodynamical simulations [25] show that the orbits remain box-like or chaotic (characteristic of triaxial halos) at the very center of the halo even though the halo is nearly spherical, but at inter- mediate radii (similar to the location of the Sun in the Milky Way) the orbits have more angular momentum and are comprised of tube-like orbits. There has been much attention paid to the possibility of a ‘dark disk’ in the Milky Way either produced as a result of preferential accretion of satellites in the plane of the baryonic disk which results in an enhanced density of dark matter in this region [26, 27], or due to non-spherical adiabatic contraction of a more spherical dark matter halo in response to the growth of a baryonic disk [28]. A recent study of the metallicity distribution of stars in the Milky Way disk from the Gaia-ESO survey [29] ﬁnds no observational evidence for the populations of stars that are expected to be brought into the disk by satellites that would be required to create a dark disk, suggesting the Milky Way halo had few in-plane accretion events. A previous study of a disk galaxy from a cosmological hydrodynamical simulation found that the dark matter in the solar neighborhood had a signiﬁcant amount of rotation similar to a ‘dark disk’ [30]. We do not ﬁnd a signiﬁcant disk-like rotating dark matter component in either of the two galaxy halos with baryons that we examine, suggesting that dark disks – 2 –

text: Yet,

label: paragraph

text: in

text: the

text: community

text: there

text: has

text: been

text: the

text: belief

text: that

text: this

text: simple

text: SHM

text: cannot

text: be

text: trusted

text: as

text: it

text: does

text: not

text: adequately

text: capture

text: the

text: true

text: velocity

text: distribution

text: of

text: the

text: Galactic

text: WIMPs.

text: Dark

text: matter

text: halos

text: in

text: cosmological

text: simulations

text: (with

text: dark

text: matter

text: only)

text: were

text: known

text: to

text: be

text: triaxial

text: and

text: have

text: anisotropic

text: velocity

text: distributions

text: owing

text: to

text: their

text: complex

text: merger

text: history.

text: The

text: anisotropic

text: velocity

text: distributions

text: have

text: been

text: shown

text: to

text: deviate

text: signiﬁcantly

text: from

text: the

text: SHM

text: and

text: alternative

text: analytic

text: ﬁts

text: that

text: were

text: thought

text: to

text: better

text: match

text: the

text: true

text: distribution,

text: particularly

text: at

text: the

text: high-velocity

text: tail

text: have

text: been

text: explored

text: These

text: alternatives

text: to

text: the

text: SHM

text: were

text: motivated

text: in

text: part

text: by

text: results

text: of

text: dark

text: matter

text: only

text: N-body

text: simulations

text: of

text: structure

text: formation

text: which

text: are

text: limited

text: by

text: the

text: fact

text: that

text: they

text: do

text: not

text: include

text: baryonic

text: physics.

text: In

text: this

text: work

text: we

text: examine

text: the

text: results

text: of

text: simulations

text: that

text: do

text: include

text: baryons.

text: It

text: is

text: the

text: purpose

text: of

text: this

text: paper

text: to

text: investigate

text: whether

text: or

text: not

text: the

text: SHM

text: is

text: a

text: good

text: representation

text: of

text: the

text: true

text: WIMP

text: distribution

text: in

text: these

text: simulations

text: for

text: the

text: analysis

text: of

text: direct

text: detection

text: data.

text: The

text: process

text: of

text: galaxy

text: formation

text: proceeds

text: hierarchically

text: via

text: the

text: clustering

text: and

text: aggre-

text: gation

text: of

text: dark

text: matter

text: halos.

text: After

text: the

text: ﬁrst

text: dark

text: matter

text: halos

text: form,

text: gas

text: begins

text: to

text: be

text: grav-

text: itationally

text: attracted

text: into

text: them

text: where

text: it

text: cools,

text: condenses

text: and

text: forms

text: the

text: ﬁrst

text: generations

text: of

text: stars.

text: As

text: a

text: galaxy

text: gains

text: mass

text: (by

text: accreting

text: smaller

text: galaxies)

text: the

text: dark

text: matter

text: halos

text: also

text: grow.

text: Consequently

text: galaxies

text: like

text: the

text: Milky

text: Way

text: are

text: embedded

text: in

text: dark

text: matter

text: halos

text: that

text: formed

text: via

text: the

text: repeated

text: merger

text: of

text: protogalactic

text: fragments

text: consisting

text: of

text: dark

text: matter,

text: gas,

text: and

text: early

text: generations

text: of

text: stars.

text: The

text: measurement

text: of

text: the

text: shapes

text: of

text: dark

text: matter

text: halos

text: is

text: challenging

text: and

text: the

text: shape

text: of

text: the

text: Milky

text: Way

text: Halo

text: is

text: currently

text: uncertain

text: [e.g.

text: Since

text: dark

text: matter

text: constitutes

text: about

text: of

text: the

text: total

text: mass

text: in

text: the

text: Universe,

text: most

text: early

text: simulations

text: of

text: galaxy

text: formation

text: relied

text: purely

text: on

text: collisionless

text: simulations

text: of

text: dark

text: matter

text: particles.

text: These

text: early

text: collisionless

text: N-body

text: simulations

text: showed

text: that

text: hierarchical

text: growth

text: results

text: in

text: dark

text: matter

text: halos

text: whose

text: shapes

text: are

text: highly

text: triaxial

text: or

text: prolate

text: with

text: radially

text: anisotropic

text: velocity

text: distributions

text: More

text: recently,

text: N-body

text: simulations

text: have

text: been

text: performed

text: that

text: take

text: into

text: consideration

text: not

text: only

text: collisionless

text: dark

text: matter

text: particles

text: but

text: also

text: the

text: eﬀects

text: of

text: the

text: condensation

text: of

text: baryons.

text: These

text: simulations

text: ﬁnd

text: somewhat

text: diﬀerent

text: halo

text: shapes

text: that

text: are

text: more

text: spherical

text: or

text: axisym-

text: metric

text: When

text: baryons

text: are

text: taken

text: into

text: account,

text: analysis

text: of

text: the

text: orbital

text: properties

text: of

text: dark

text: matter

text: particles

text: in

text: both

text: idealized

text: simulations

text: and

text: cosmological

text: hydrodynamical

text: simulations

text: show

text: that

text: the

text: orbits

text: remain

text: box-like

text: or

text: chaotic

text: (characteristic

text: of

text: triaxial

text: halos)

text: at

text: the

text: very

text: center

text: of

text: the

text: halo

text: even

text: though

text: the

text: halo

text: is

text: nearly

text: spherical,

text: but

text: at

text: inter-

text: mediate

text: radii

text: (similar

text: to

text: the

text: location

text: of

text: the

text: Sun

text: in

text: the

text: Milky

text: Way)

text: the

text: orbits

text: have

text: more

text: angular

text: momentum

text: and

text: are

text: comprised

text: of

text: tube-like

text: orbits.

text: There

text: has

text: been

text: much

text: attention

text: paid

text: to

text: the

text: possibility

text: of

text: a

text: ‘dark

text: disk’

text: in

text: the

text: Milky

text: Way

text: either

text: produced

text: as

text: a

text: result

text: of

text: preferential

text: accretion

text: of

text: satellites

text: in

text: the

text: plane

text: of

text: the

text: baryonic

text: disk

text: which

text: results

text: in

text: an

text: enhanced

text: density

text: of

text: dark

text: matter

text: in

text: this

text: region

text: or

text: due

text: to

text: non-spherical

text: adiabatic

text: contraction

text: of

text: a

text: more

text: spherical

text: dark

text: matter

text: halo

text: in

text: response

text: to

text: the

text: growth

text: of

text: a

text: baryonic

text: disk

text: A

text: recent

text: study

text: of

text: the

text: metallicity

text: distribution

text: of

text: stars

text: in

text: the

text: Milky

text: Way

text: disk

text: from

text: the

text: Gaia-ESO

text: survey

text: ﬁnds

text: no

text: observational

text: evidence

text: for

text: the

text: populations

text: of

text: stars

text: that

text: are

text: expected

text: to

text: be

text: brought

text: into

text: the

text: disk

text: by

text: satellites

text: that

text: would

text: be

text: required

text: to

text: create

text: a

text: dark

text: disk,

text: suggesting

text: the

text: Milky

text: Way

text: halo

text: had

text: few

text: in-plane

text: accretion

text: events.

text: A

text: previous

text: study

text: of

text: a

text: disk

text: galaxy

text: from

text: a

text: cosmological

text: hydrodynamical

text: simulation

text: found

text: that

text: the

text: dark

text: matter

text: in

text: the

text: solar

text: neighborhood

text: had

text: a

text: signiﬁcant

text: amount

text: of

text: rotation

text: similar

text: to

text: a

text: ‘dark

text: disk’

text: We

text: do

text: not

text: ﬁnd

text: a

text: signiﬁcant

text: disk-like

text: rotating

text: dark

text: matter

text: component

text: in

text: either

text: of

text: the

text: two

text: galaxy

text: halos

text: with

text: baryons

text: that

text: we

text: examine,

text: suggesting

text: that

text: dark

text: disks


text: N-body and reference SHM regions for both the CoGeNT and high-mass DAMA/LIBRA regions. Even more striking is the fact that the low-mass region for DAMA/LIBRA has completely disappeared. In this case, the tail of the modulation spectrum for the inferred SHM decays too quickly to produce a good ﬁt to the DAMA/LIBRA data. This is not entirely unexpected when considering the somewhat poor ﬁt for C by the inferred SHM 1 shown in ﬁgure 7. This feature supports the commonly held idea that the SHM does not provide a good ﬁt for the dark matter’s velocity distribution in galactic halos in dark matter only N-body simulations. As described previously, one of the most interesting results of this study is that the SHM provides a perfectly adequate ﬁt to the simulations with baryons (see ﬁgure 6 and ﬁgure 7), defying the “common lore” of the ﬁeld that suggests that the SHM does not adequately describe the dark matter distribution. This translates to much more similar results for the diﬀerent regions displayed in ﬁgure 9 for the g1536 and g15784 simulations. In particular, there are only very minor diﬀerences between the CoGeNT regions for the simulation and the inferred SHM for both g1536 and g15784. The diﬀerences between the simulation and the inferred SHM are more noticeable for the DAMA/LIBRA regions, especially for g15784. These results are anticipated based on the C distributions shown in ﬁgure 7. This can be 1 understood in terms of the physical nature of the modulation signal. As long as there is not signiﬁcant velocity substructure in the vicinity of the Earth, the amplitude of the modulation will be approximately equal to half the diﬀerence between the largest yearly scattering rate (time of year when the dark matter-Earth relative velocity is largest) and the smallest yearly scattering rate. This subtraction ampliﬁes any diﬀerences between the two models. Turning this logic around, the modulation will thus be a sensitive probe of departures from the SHM, i.e. velocity substructures, if they exist within the Milky Way. Finally, it is also interesting to note that the LUX limits are quite robust to changes in the velocity distribution. The largest diﬀerence again appears in the dark matter only simulation mainly at light masses. As can be observed in ﬁgure 6, the high speed tail of the SHM decays much more quickly than the N-body simulation. For low mass dark matter, only the high velocity tail can produce nuclear recoils above the energy threshold of a detector. This results in fewer expected dark matter scatters, and thus a weaker limit for the SHM at low masses. At large masses, the two limits are essentially indistinguishable for all of the simulations. This is expected, as for large dark matter masses, the bulk of the distributions produce scatters above threshold and thus similar numbers of expected events. 5 Discussion A generally accepted tenet among researchers in the dark matter ﬁeld is that while the SHM provides the general features of the true Milky Way Halo, it does not provide a good quantitative description of the actual distribution of dark matter in our Halo. A major piece of evidence that supports this claim comes from the poor ﬁt to the distributions in N-body simulations that include dark matter only. In our work, we have arrived at the same conclusion for our dark matter only simulation, and ﬁnd that it is due to an underestimate in v due to the still-rising rotation curve near the Sun. Intriguingly, however, we ﬁnd that 0 the presence of baryons both increases the steepness of the potential, and causes adiabatic compassion of the dark matter. These eﬀects combine to bring the peak of the rotation curve near the Sun, and as a result the SHM provides a much better approximation to the simulation data. In fact, the SHM provides very reasonable ﬁts in both of the simulations that include baryons that we studied. – 21 –

text: N-body

text: and

text: reference

text: SHM

text: regions

text: for

text: both

text: the

text: CoGeNT

text: and

text: high-mass

text: DAMA/LIBRA

text: regions.

text: Even

text: more

text: striking

text: is

text: the

text: fact

text: that

text: the

text: low-mass

text: region

text: for

text: DAMA/LIBRA

text: has

text: completely

text: disappeared.

text: In

text: this

text: case,

text: the

text: tail

text: of

text: the

text: modulation

text: spectrum

text: for

text: the

text: inferred

text: SHM

text: decays

text: too

text: quickly

text: to

text: produce

text: a

text: good

text: ﬁt

text: to

text: the

text: DAMA/LIBRA

text: data.

text: This

text: is

text: not

text: entirely

text: unexpected

text: when

text: considering

text: the

text: somewhat

text: poor

text: ﬁt

text: for

text: C

text: by

text: the

text: inferred

text: SHM

text: shown

text: in

text: ﬁgure

text: This

text: feature

text: supports

text: the

text: commonly

text: held

text: idea

text: that

text: the

text: SHM

text: does

text: not

text: provide

text: a

text: good

text: ﬁt

text: for

text: the

text: dark

text: matter’s

text: velocity

text: distribution

text: in

text: galactic

text: halos

text: in

text: dark

text: matter

text: only

text: N-body

text: simulations.

text: As

text: described

text: previously,

text: one

text: of

text: the

text: most

text: interesting

text: results

text: of

text: this

text: study

text: is

text: that

text: the

text: SHM

text: provides

text: a

text: perfectly

text: adequate

text: ﬁt

text: to

text: the

text: simulations

text: with

text: baryons

text: (see

text: ﬁgure

text: and

text: ﬁgure

text: defying

text: the

text: “common

text: lore”

text: of

text: the

text: ﬁeld

text: that

text: suggests

text: that

text: the

text: SHM

text: does

text: not

text: adequately

text: describe

text: the

text: dark

text: matter

text: distribution.

text: This

text: translates

text: to

text: much

text: more

text: similar

text: results

text: for

text: the

text: diﬀerent

text: regions

text: displayed

text: in

text: ﬁgure

text: for

text: the

text: and

text: simulations.

text: In

text: particular,

text: there

text: are

text: only

text: very

text: minor

text: diﬀerences

text: between

text: the

text: CoGeNT

text: regions

text: for

text: the

text: simulation

text: and

text: the

text: inferred

text: SHM

text: for

text: both

text: and

text: The

text: diﬀerences

text: between

text: the

text: simulation

text: and

text: the

text: inferred

text: SHM

text: are

text: more

text: noticeable

text: for

text: the

text: DAMA/LIBRA

text: regions,

text: especially

text: for

text: These

text: results

text: are

text: anticipated

text: based

text: on

text: the

text: C

text: distributions

text: shown

text: in

text: ﬁgure

text: This

text: can

text: be

text: understood

text: in

text: terms

text: of

text: the

text: physical

text: nature

text: of

text: the

text: modulation

text: signal.

text: As

text: long

text: as

text: there

text: is

text: not

text: signiﬁcant

text: velocity

text: substructure

text: in

text: the

text: vicinity

text: of

text: the

text: Earth,

text: the

text: amplitude

text: of

text: the

text: modulation

text: will

text: be

text: approximately

text: equal

text: to

text: half

text: the

text: diﬀerence

text: between

text: the

text: largest

text: yearly

text: scattering

text: rate

text: (time

text: of

text: year

text: when

text: the

text: dark

text: matter-Earth

text: relative

text: velocity

text: is

text: largest)

text: and

text: the

text: smallest

text: yearly

text: scattering

text: rate.

text: This

text: subtraction

text: ampliﬁes

text: any

text: diﬀerences

text: between

text: the

text: two

text: models.

text: Turning

text: this

text: logic

text: around,

text: the

text: modulation

text: will

text: thus

text: be

text: a

text: sensitive

text: probe

text: of

text: departures

text: from

text: the

text: SHM,

text: i.e.

text: velocity

text: substructures,

text: if

text: they

text: exist

text: within

text: the

text: Milky

text: Way.

text: Finally,

text: it

text: is

text: also

text: interesting

text: to

text: note

text: that

text: the

text: LUX

text: limits

text: are

text: quite

text: robust

text: to

text: changes

text: in

text: the

text: velocity

text: distribution.

text: The

text: largest

text: diﬀerence

text: again

text: appears

text: in

text: the

text: dark

text: matter

text: only

text: simulation

text: mainly

text: at

text: light

text: masses.

text: As

text: can

text: be

text: observed

text: in

text: ﬁgure

text: the

text: high

text: speed

text: tail

text: of

text: the

text: SHM

text: decays

text: much

text: more

text: quickly

text: than

text: the

text: N-body

text: simulation.

text: For

text: low

text: mass

text: dark

text: matter,

text: only

text: the

text: high

text: velocity

text: tail

text: can

text: produce

text: nuclear

text: recoils

text: above

text: the

text: energy

text: threshold

text: of

text: a

text: detector.

text: This

text: results

text: in

text: fewer

text: expected

text: dark

text: matter

text: scatters,

text: and

text: thus

text: a

text: weaker

text: limit

text: for

text: the

text: SHM

text: at

text: low

text: masses.

text: At

text: large

text: masses,

text: the

text: two

text: limits

text: are

text: essentially

text: indistinguishable

text: for

text: all

text: of

text: the

text: simulations.

text: This

text: is

text: expected,

text: as

text: for

text: large

text: dark

text: matter

text: masses,

text: the

text: bulk

text: of

text: the

text: distributions

text: produce

text: scatters

text: above

text: threshold

text: and

text: thus

text: similar

text: numbers

text: of

text: expected

text: events.

text: Discussion

text: A

text: generally

text: accepted

text: tenet

text: among

text: researchers

text: in

text: the

text: dark

text: matter

text: ﬁeld

text: is

text: that

text: while

text: the

text: SHM

text: provides

text: the

text: general

text: features

text: of

text: the

text: true

text: Milky

text: Way

text: Halo,

text: it

text: does

text: not

text: provide

text: a

text: good

text: quantitative

text: description

text: of

text: the

text: actual

text: distribution

text: of

text: dark

text: matter

text: in

text: our

text: Halo.

text: A

text: major

text: piece

text: of

text: evidence

text: that

text: supports

text: this

text: claim

text: comes

text: from

text: the

text: poor

text: ﬁt

text: to

text: the

text: distributions

text: in

text: N-body

text: simulations

text: that

text: include

text: dark

text: matter

text: only.

text: In

text: our

text: work,

text: we

text: have

text: arrived

text: at

text: the

text: same

text: conclusion

text: for

text: our

text: dark

text: matter

text: only

text: simulation,

text: and

text: ﬁnd

text: that

text: it

text: is

text: due

text: to

text: an

text: underestimate

text: in

text: v

text: due

text: to

text: the

text: still-rising

text: rotation

text: curve

text: near

text: the

text: Sun.

text: Intriguingly,

text: however,

text: we

text: ﬁnd

text: that

text: the

text: presence

text: of

text: baryons

text: both

text: increases

text: the

text: steepness

text: of

text: the

text: potential,

text: and

text: causes

text: adiabatic

text: compassion

text: of

text: the

text: dark

text: matter.

text: These

text: eﬀects

text: combine

text: to

text: bring

text: the

text: peak

text: of

text: the

text: rotation

text: curve

text: near

text: the

text: Sun,

text: and

text: as

text: a

text: result

text: the

text: SHM

text: provides

text: a

text: much

text: better

text: approximation

text: to

text: the

text: simulation

text: data.

text: In

text: fact,

text: the

text: SHM

text: provides

text: very

text: reasonable

text: ﬁts

text: in

text: both

text: of

text: the

text: simulations

text: that

text: include

text: baryons

text: that

text: we

text: studied.


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text: are not a generic prediction of cosmological hydrodynamical simulations. In this paper we study a new generation of high resolution cosmological hydrodynamical and N-body simulations designed to mimic the formation of disk galaxies similar in size and global properties to the Milky Way. This state-of-the-art suite of simulations includes baryonic physics that results in disk galaxies which have ﬂat rotation curves, exponential surface brightness proﬁles, and match a wide range of observed disk scaling relationships [31]. We study two diﬀerent galaxies from the MaGICC simulations [32]. For one of these simulations we also examine a dark-matter only simulation that was generated from the same set of dark matter initial conditions and with the same cosmology to enable us to quantify the eﬀects of baryons on the dark matter detection signal. The simulations are described in detail in Section . While the orbits of dark matter particles in simulations of dark matter halos with baryonic condensation are diﬀerent from orbits in idealized singular isothermal spheres, it is unclear if these diﬀerences are signiﬁcant enough in the solar neighborhood that they would impact dark matter direct detection experiments. It is the goal of this paper to address this question. 2 Simulations The cosmological hydrodynamical simulations we use in this paper are from the Making Galaxies in a Cosmological Context [MaGICC, 32]. The initial conditions assume a Lambda − − Cold Dark Matter (ΛCDM), WMAP3 cosmology with H = 73 km s 1Mpc 1, Ω = 0.24, 0 m Ω = 0.76, Ω = 0.04 and σ = 0.79 [33]. A sample of 16 galaxy halos with masses between ∼Λ × b∼ × 8 5 1011 to 2 1012M(cid:12) at redshift zero were chosen at random from a dark-matter only simulation [34]. The selection criteria required that for each selected halo (at z = 0) there ∼ × was no structure within 2.7 Mpc with a mass greater than 5 1011M(cid:12). Once the galaxies were chosen from the DM only runs, the simulations for those regions containing the galaxies were re-evolved from high redshift with high resolution, this time with baryonic physics included. The simulation volume was large enough to ensure a realistic angular momentum distribution and merger history. In order to achieve suﬃcient mass and spatial resolution, the simulations employ a common zooming technique that adds high resolution particles in the region of interest while following other regions with much lower resolution particles. In the highest resolution region of each simulation the dark matter, gas and star particles have × × × masses of 1.11 106M(cid:12), 2.2 105M(cid:12) and < 6.3 104M(cid:12) respectively, and a gravitational softening length of 310 pc. The simulation was carried out using the SPH code GASOLINE [35] and includes low-temperature metal cooling [36] a Schmidt-Kennicutt star formation law [37] and UV background radiation. A prior generation of simulations to MaGICC called McMaster Unbiased Galaxy Simulations [MUGS, 34] uses the same initial conditions and cosmology but has a diﬀerent implementation of the stellar feedback. Both MUGS and MaGICC employ the ‘blast wave’ model of supernova feedback, where gas cooling is locally suspended in order to mimic the thermal heating of gas from supernovae [38]. MaGICC diﬀers from MUGS in that it also includes early energy input into the ISM, from massive stars. This early feedback heats the surrounding gas from the moment a star forms, rather ∼ than waiting for the ﬁrst core collapse supernovae which are triggered 4 Myr after the star formed [32]. As a consequence of the early feedback implemented in MaGICC, the simulated galaxies we use in this paper suﬀer much less from overcooling and have small (more realistic) bulges. – 3 –

text: are

text: not

text: a

text: generic

text: prediction

text: of

text: cosmological

text: hydrodynamical

text: simulations.

text: In

text: this

text: paper

text: we

text: study

text: a

text: new

text: generation

text: of

text: high

text: resolution

text: cosmological

text: hydrodynamical

text: and

text: N-body

text: simulations

text: designed

text: to

text: mimic

text: the

text: formation

text: of

text: disk

text: galaxies

text: similar

text: in

text: size

text: and

text: global

text: properties

text: to

text: the

text: Milky

text: Way.

text: This

text: state-of-the-art

text: suite

text: of

text: simulations

text: includes

text: baryonic

text: physics

text: that

text: results

text: in

text: disk

text: galaxies

text: which

text: have

text: ﬂat

text: rotation

text: curves,

text: exponential

text: surface

text: brightness

text: proﬁles,

text: and

text: match

text: a

text: wide

text: range

text: of

text: observed

text: disk

text: scaling

text: relationships

text: We

text: study

text: two

text: diﬀerent

text: galaxies

text: from

text: the

text: MaGICC

text: simulations

text: For

text: one

text: of

text: these

text: simulations

text: we

text: also

text: examine

text: a

text: dark-matter

text: only

text: simulation

text: that

text: was

text: generated

text: from

text: the

text: same

text: set

text: of

text: dark

text: matter

text: initial

text: conditions

text: and

text: with

text: the

text: same

text: cosmology

text: to

text: enable

text: us

text: to

text: quantify

text: the

text: eﬀects

text: of

text: baryons

text: on

text: the

text: dark

text: matter

text: detection

text: signal.

text: The

text: simulations

text: are

text: described

text: in

text: detail

text: in

text: Section

text: While

text: the

text: orbits

text: of

text: dark

text: matter

text: particles

text: in

text: simulations

text: of

text: dark

text: matter

text: halos

text: with

text: baryonic

text: condensation

text: are

text: diﬀerent

text: from

text: orbits

text: in

text: idealized

text: singular

text: isothermal

text: spheres,

text: it

text: is

text: unclear

text: if

text: these

text: diﬀerences

text: are

text: signiﬁcant

text: enough

text: in

text: the

text: solar

text: neighborhood

text: that

text: they

text: would

text: impact

text: dark

text: matter

text: direct

text: detection

text: experiments.

text: It

text: is

text: the

text: goal

text: of

text: this

text: paper

text: to

text: address

text: this

text: question.

text: Simulations

text: The

text: cosmological

text: hydrodynamical

text: simulations

text: we

text: use

text: in

text: this

text: paper

text: are

text: from

text: the

text: Making

text: Galaxies

text: in

text: a

text: Cosmological

text: Context

text: [MaGICC,

text: The

text: initial

text: conditions

text: assume

text: a

text: Lambda

text: Cold

text: Dark

text: Matter

text: (ΛCDM),

text: cosmology

text: with

text: H

text: km

text: s

text: Ω

text: m

text: Ω

text: Ω

text: and

text: σ

text: A

text: sample

text: of

text: galaxy

text: halos

text: with

text: masses

text: between

text: ∼Λ

text: b∼

text: to

text: at

text: redshift

text: zero

text: were

text: chosen

text: at

text: random

text: from

text: a

text: dark-matter

text: only

text: simulation

text: The

text: selection

text: criteria

text: required

text: that

text: for

text: each

text: selected

text: halo

text: (at

text: z

text: there

text: was

text: no

text: structure

text: within

text: Mpc

text: with

text: a

text: mass

text: greater

text: than

text: Once

text: the

text: galaxies

text: were

text: chosen

text: from

text: the

text: DM

text: only

text: runs,

text: the

text: simulations

text: for

text: those

text: regions

text: containing

text: the

text: galaxies

text: were

text: re-evolved

text: from

text: high

text: redshift

text: with

text: high

text: resolution,

text: this

text: time

text: with

text: baryonic

text: physics

text: included.

text: The

text: simulation

text: volume

text: was

text: large

text: enough

text: to

text: ensure

text: a

text: realistic

text: angular

text: momentum

text: distribution

text: and

text: merger

text: history.

text: In

text: order

text: to

text: achieve

text: suﬃcient

text: mass

text: and

text: spatial

text: resolution,

text: the

text: simulations

text: employ

text: a

text: common

text: zooming

text: technique

text: that

text: adds

text: high

text: resolution

text: particles

text: in

text: the

text: region

text: of

text: interest

text: while

text: following

text: other

text: regions

text: with

text: much

text: lower

text: resolution

text: particles.

text: In

text: the

text: highest

text: resolution

text: region

text: of

text: each

text: simulation

text: the

text: dark

text: matter,

text: gas

text: and

text: star

text: particles

text: have

text: masses

text: of

text: and

text: respectively,

text: and

text: a

text: gravitational

text: softening

text: length

text: of

text: pc.

text: The

text: simulation

text: was

text: carried

text: out

text: using

text: the

text: SPH

text: code

text: GASOLINE

text: and

text: includes

text: low-temperature

text: metal

text: cooling

text: a

text: Schmidt-Kennicutt

text: star

text: formation

text: law

text: and

text: UV

text: background

text: radiation.

text: A

text: prior

text: generation

text: of

text: simulations

text: to

text: MaGICC

text: called

text: McMaster

text: Unbiased

text: Galaxy

text: Simulations

text: [MUGS,

text: uses

text: the

text: same

text: initial

text: conditions

text: and

text: cosmology

text: but

text: has

text: a

text: diﬀerent

text: implementation

text: of

text: the

text: stellar

text: feedback.

text: Both

text: MUGS

text: and

text: MaGICC

text: employ

text: the

text: ‘blast

text: wave’

text: model

text: of

text: supernova

text: feedback,

text: where

text: gas

text: cooling

text: is

text: locally

text: suspended

text: in

text: order

text: to

text: mimic

text: the

text: thermal

text: heating

text: of

text: gas

text: from

text: supernovae

text: MaGICC

text: diﬀers

text: from

text: MUGS

text: in

text: that

text: it

text: also

text: includes

text: early

text: energy

text: input

text: into

text: the

text: ISM,

text: from

text: massive

text: stars.

text: This

text: early

text: feedback

text: heats

text: the

text: surrounding

text: gas

text: from

text: the

text: moment

text: a

text: star

text: forms,

text: rather

text: than

text: waiting

text: for

text: the

text: ﬁrst

text: core

text: collapse

text: supernovae

text: which

text: are

text: triggered

text: Myr

text: after

text: the

text: star

text: formed

text: As

text: a

text: consequence

text: of

text: the

text: early

text: feedback

text: implemented

text: in

text: MaGICC,

text: the

text: simulated

text: galaxies

text: we

text: use

text: in

text: this

text: paper

text: suﬀer

text: much

text: less

text: from

text: overcooling

text: and

text: have

text: small

text: (more

text: realistic)

text: bulges.


text: g1536DM g1536 g15784 (DM-only) (DM+baryons) (DM+baryons) Simulation × × × Virial Mass [M(cid:12)] 7.48 1011 5.84 1011 1.50 1012 Virial Radius [kpc] 260 143 328 × × × DM particle mass [M(cid:12)] 1.33 106 1.11 106 1.11 106 Circular velocity (at R = 8 kpc) [km/s] 108 187 273 Torus (r = 8 kpc, r = 2 kpc) 1 2 Number of DM particles 3085 4849 6541 Average DM density [GeV/cm3] 0.270 0.346 0.493 − Average velocity (U,V ,W ) [km/s] (0.0, 5.3, 0.5) (2.7, 21.6, 2.3) (0.9, 18.5, 3.4) √ √ √ Velocity s.d. (σ ,σ ,σ ) [km/s] (109, 95, 90) (144, 128, 121) (205, 166, 177) U V W RMS speed [km/s] 98 3 133 3 184 3 Maximum speed [km/s] 359 454 600. Table 1. Properties of largest dark matter halos in the three N-body simulation used in this work. We use the deﬁnition of the virial radius of Ref [39] which corresponds approximately to the radius ∼ within which the average density of the halo is 100 times the critical density. In this paper we focus on the dark matter halos of two disk galaxies from the MaGICC simulations with global properties similar to that of the Milky Way. These are called ‘g1536’ and ‘g15784’. The latter has a more massive bulge and a larger total mass and consequently a more rapidly rising rotation curve. g15784 has a more active accretion history with its last major merger at z = 2 while g1536 has its most recent merger at z = 2.9. The use of two disk galaxies both broadly resembling the Milky Way, evolved with the same baryonic physics, but arising from diﬀerent initial conditions and having diﬀerent accretion histories, allows us to assess how sensitive our results are to the details of the galaxy assembly process. In addition to these two fully hydrodynamical simulations of disk galaxies, we also analyze a dark matter only simulation which was generated with cosmological initial conditions identical to g1536, which we refer to as ‘g1536DM’. The baryonic physics (especially early feedback which unbinds a number of the lower mass subhalos and baryonic compression) alters the virial mass, virial radius, shape of the dark matter halo and its phase space distribution function. A direct comparison of the dark matter detection rates in ‘g1536’ and ‘g1536DM’ enables us to directly assess how these rates might be aﬀected in the Milky Way, since both simulations arise from the same cosmological initial conditions. Since all three simulations have slightly diﬀerent total matter distributions, their rota- tion curves diﬀer from each other and from the rotation curve of the Milky Way. In table 1 we provide additional details on the three diﬀerent simulations and the properties of the spatial and velocity distributions of DM particles therein. The DM halos with baryons have signiﬁcantly diﬀerent shapes from the halo in the dark matter-only simulation. Halo shapes are measured in terms of the axis ratios of the mass distribution stratiﬁed on concentric ellipsoids. The axis ratio b/a is the ratio of the intermediate to long axis scale lengths and c/a is the ratio of the short to long axis scale lengths. The shape of the dark matter distribution in ellipsoidal shells is computed using shells centered on the minimum of the potential. The shape is determined at each radius ∼ r by ﬁnding the ellipsoidal shell of width 1.2 kpc whose principal axes a, b, and c self- – 4 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: table

text: (DM-only)

text: (DM+baryons)

text: (DM+baryons)

text: Simulation

text: Virial

text: Mass

text: Virial

text: Radius

text: [kpc]

text: DM

text: particle

text: mass

text: Circular

text: velocity

text: (at

text: R

text: kpc)

text: [km/s]

text: Torus

text: (r

text: kpc,

text: r

text: kpc)

text: Number

text: of

text: DM

text: particles

text: Average

text: DM

text: density

text: Average

text: velocity

text: (U,V

text: ,W

text: [km/s]

text: Velocity

text: s.d.

text: (σ

text: ,σ

text: ,σ

text: [km/s]

text: U

text: V

text: W

text: RMS

text: speed

text: [km/s]

text: Maximum

text: speed

text: [km/s]

text: Table

text: Properties

text: of

text: largest

text: dark

text: matter

text: halos

text: in

text: the

text: three

text: N-body

text: simulation

text: used

text: in

text: this

text: work.

text: We

text: use

text: the

text: deﬁnition

text: of

text: the

text: virial

text: radius

text: of

text: Ref

text: which

text: corresponds

text: approximately

text: to

text: the

text: radius

text: within

text: which

text: the

text: average

text: density

text: of

text: the

text: halo

text: is

text: times

text: the

text: critical

text: density.

text: In

text: this

text: paper

text: we

text: focus

text: on

text: the

text: dark

text: matter

text: halos

text: of

text: two

text: disk

text: galaxies

text: from

text: the

text: MaGICC

text: simulations

text: with

text: global

text: properties

text: similar

text: to

text: that

text: of

text: the

text: Milky

text: Way.

text: These

text: are

text: called

text: and

text: The

text: latter

text: has

text: a

text: more

text: massive

text: bulge

text: and

text: a

text: larger

text: total

text: mass

text: and

text: consequently

text: a

text: more

text: rapidly

text: rising

text: rotation

text: curve.

text: has

text: a

text: more

text: active

text: accretion

text: history

text: with

text: its

text: last

text: major

text: merger

text: at

text: z

text: while

text: has

text: its

text: most

text: recent

text: merger

text: at

text: z

text: The

text: use

text: of

text: two

text: disk

text: galaxies

text: both

text: broadly

text: resembling

text: the

text: Milky

text: Way,

text: evolved

text: with

text: the

text: same

text: baryonic

text: physics,

text: but

text: arising

text: from

text: diﬀerent

text: initial

text: conditions

text: and

text: having

text: diﬀerent

text: accretion

text: histories,

text: allows

text: us

text: to

text: assess

text: how

text: sensitive

text: our

text: results

text: are

text: to

text: the

text: details

text: of

text: the

text: galaxy

text: assembly

text: process.

text: In

text: addition

text: to

text: these

text: two

text: fully

text: hydrodynamical

text: simulations

text: of

text: disk

text: galaxies,

text: we

text: also

text: analyze

text: a

text: dark

text: matter

text: only

text: simulation

text: which

text: was

text: generated

text: with

text: cosmological

text: initial

text: conditions

text: identical

text: to

text: which

text: we

text: refer

text: to

text: as

text: The

text: baryonic

text: physics

text: (especially

text: early

text: feedback

text: which

text: unbinds

text: a

text: number

text: of

text: the

text: lower

text: mass

text: subhalos

text: and

text: baryonic

text: compression)

text: alters

text: the

text: virial

text: mass,

text: virial

text: radius,

text: shape

text: of

text: the

text: dark

text: matter

text: halo

text: and

text: its

text: phase

text: space

text: distribution

text: function.

text: A

text: direct

text: comparison

text: of

text: the

text: dark

text: matter

text: detection

text: rates

text: in

text: and

text: enables

text: us

text: to

text: directly

text: assess

text: how

text: these

text: rates

text: might

text: be

text: aﬀected

text: in

text: the

text: Milky

text: Way,

text: since

text: both

text: simulations

text: arise

text: from

text: the

text: same

text: cosmological

text: initial

text: conditions.

text: Since

text: all

text: three

text: simulations

text: have

text: slightly

text: diﬀerent

text: total

text: matter

text: distributions,

text: their

text: rota-

text: tion

text: curves

text: diﬀer

text: from

text: each

text: other

text: and

text: from

text: the

text: rotation

text: curve

text: of

text: the

text: Milky

text: Way.

text: In

text: table

text: we

text: provide

text: additional

text: details

text: on

text: the

text: three

text: diﬀerent

text: simulations

text: and

text: the

text: properties

text: of

text: the

text: spatial

text: and

text: velocity

text: distributions

text: of

text: DM

text: particles

text: therein.

text: The

text: DM

text: halos

text: with

text: baryons

text: have

text: signiﬁcantly

text: diﬀerent

text: shapes

text: from

text: the

text: halo

text: in

text: the

text: dark

text: matter-only

text: simulation.

text: Halo

text: shapes

text: are

text: measured

text: in

text: terms

text: of

text: the

text: axis

text: ratios

text: of

text: the

text: mass

text: distribution

text: stratiﬁed

text: on

text: concentric

text: ellipsoids.

text: The

text: axis

text: ratio

text: b/a

text: is

text: the

text: ratio

text: of

text: the

text: intermediate

text: to

text: long

text: axis

text: scale

text: lengths

text: and

text: c/a

text: is

text: the

text: ratio

text: of

text: the

text: short

text: to

text: long

text: axis

text: scale

text: lengths.

text: The

text: shape

text: of

text: the

text: dark

text: matter

text: distribution

text: in

text: ellipsoidal

text: shells

text: is

text: computed

text: using

text: shells

text: centered

text: on

text: the

text: minimum

text: of

text: the

text: potential.

text: The

text: shape

text: is

text: determined

text: at

text: each

text: radius

text: r

text: by

text: ﬁnding

text: the

text: ellipsoidal

text: shell

text: of

text: width

text: kpc

text: whose

text: principal

text: axes

text: a,

text: b,

text: and

text: c

text: self-

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: WILSON LOOP AREA LAW FOR 2D YANG-MILLS IN GENERALIZED AXIAL GAUGE TIMOTHY NGUYEN Abstract. We prove that Wilson loop expectation values for arbitrary simple closed contours obey an area law up to second order in perturbative two-dimensional Yang-Mills theory. Our analysis occurs within a general family of axial-like gauges, which include and interpolate between holomorphic gauge and the Wu-Mandelstam-Liebrandt light cone gauge. Our methods make use of the homotopy-invariance properties of iterated integrals of closed one-forms, which allows us to evaluate the nontrivial integrals occurring at second order. We close with a discussion on complex gauge-ﬁxing and deformation of integration cycles for holomorphic path integrals to shed light on some of the quantum ﬁeld-theoretic underpinnings of our results. Contents 1. Introduction 1 2. The Setup 4 3. Analytic Preliminaries 12 4. Proof of Main Theorems 20 Appendix A. Remarks on Path Integrals Methods and Gauge-Fixing 24 References 28 1. Introduction Two-dimensional Yang-Mills theory provides a rich arena for understanding quantum ﬁeld theories due to the fact that the theory is soluble: exact formulas can be given for expectation values of general Wilson loop observables [17, 24, 15]. These formulas are given by integrals of products of heat kernels on the gauge group evaluated at times equal to the areas enclosed by the Wilson loops. This suggests that Wilson loops evaluated by the methods of perturbative quantum ﬁeld theory should yield a corresponding area law (heuristically, both the perturbative and exact area laws stem from the Yang-Mills path integral being formally invariant under area-preserving diﬀeomorphisms). Over the last twenty years, numerous checks for specialized contours to several orders [2, 3, 22] and for circular loops to all orders [11, 12] in perturbation theory have been made, both analytically and numerically using a variety of gauges, but no result for general contours beyond ﬁrst order in perturbation theory has been established for the expected perturbative Date: May 28, 2020. 1 ##LTLine##

text: WILSON

text: LOOP

text: AREA

text: LAW

text: FOR

text: YANG-MILLS

text: IN

text: GENERALIZED

text: AXIAL

text: GAUGE

text: TIMOTHY

text: NGUYEN

text: Abstract.

text: We

label: abstract

text: prove

text: that

text: Wilson

text: loop

text: expectation

text: values

text: for

text: arbitrary

text: simple

text: closed

text: contours

text: obey

text: an

text: area

text: law

text: up

text: to

text: second

text: order

text: in

text: perturbative

text: two-dimensional

text: Yang-Mills

text: theory.

text: Our

text: analysis

text: occurs

text: within

text: a

text: general

text: family

text: of

text: axial-like

text: gauges,

text: which

text: include

text: and

text: interpolate

text: between

text: holomorphic

text: gauge

text: and

text: the

text: Wu-Mandelstam-Liebrandt

text: light

text: cone

text: gauge.

text: Our

text: methods

text: make

text: use

text: of

text: the

text: homotopy-invariance

text: properties

text: of

text: iterated

text: integrals

text: of

text: closed

text: one-forms,

text: which

text: allows

text: us

text: to

text: evaluate

text: the

text: nontrivial

text: integrals

text: occurring

text: at

text: second

text: order.

text: We

text: close

text: with

text: a

text: discussion

text: on

text: complex

text: gauge-ﬁxing

text: and

text: deformation

text: of

text: integration

text: cycles

text: for

text: holomorphic

text: path

text: integrals

text: to

text: shed

text: light

text: on

text: some

text: of

text: the

text: quantum

text: ﬁeld-theoretic

text: underpinnings

text: of

text: our

text: results.

text: Contents

text: Introduction

text: The

text: Setup

text: Analytic

text: Preliminaries

text: Proof

text: of

text: Main

text: Theorems

text: Appendix

text: A.

text: Remarks

text: on

text: Path

text: Integrals

text: Methods

text: and

text: Gauge-Fixing

text: References

text: Introduction

text: Two-dimensional

text: Yang-Mills

text: theory

text: provides

text: a

text: rich

text: arena

text: for

text: understanding

text: quantum

text: ﬁeld

text: theories

text: due

text: to

text: the

text: fact

text: that

text: the

text: theory

text: is

text: soluble:

text: exact

text: formulas

text: can

text: be

text: given

text: for

text: expectation

text: values

text: of

text: general

text: Wilson

text: loop

text: observables

text: These

text: formulas

text: are

text: given

text: by

text: integrals

text: of

text: products

text: of

text: heat

text: kernels

text: on

text: the

text: gauge

text: group

text: evaluated

text: at

text: times

text: equal

text: to

text: the

text: areas

text: enclosed

text: by

text: the

text: Wilson

text: loops.

text: This

text: suggests

text: that

text: Wilson

text: loops

text: evaluated

text: by

text: the

text: methods

text: of

text: perturbative

text: quantum

text: ﬁeld

text: theory

text: should

text: yield

text: a

text: corresponding

text: area

text: law

text: (heuristically,

text: both

text: the

text: perturbative

text: and

text: exact

text: area

text: laws

text: stem

text: from

text: the

text: Yang-Mills

text: path

text: integral

text: being

text: formally

text: invariant

text: under

text: area-preserving

text: diﬀeomorphisms).

text: Over

text: the

text: last

text: twenty

text: years,

text: numerous

text: checks

text: for

text: specialized

text: contours

text: to

text: several

text: orders

text: and

text: for

text: circular

text: loops

text: to

text: all

text: orders

text: in

text: perturbation

text: theory

text: have

text: been

text: made,

text: both

text: analytically

text: and

text: numerically

text: using

text: a

text: variety

text: of

text: gauges,

text: but

text: no

text: result

text: for

text: general

text: contours

text: beyond

text: ﬁrst

text: order

text: in

text: perturbation

text: theory

text: has

text: been

text: established

text: for

text: the

text: expected

text: perturbative

text: Date:

text: May

text: ##LTLine##