Post on 29-Mar-2021
Dark Matter, Galactic Dynamics, and GaiaTesting Alternative Theories
Matthew MoschellaPrinceton University
APS DPF 2019
Image: ESA/Gaia/DPAC
arXiv:1812.08169 & work in progresswith M. Lisanti, N. Outmezguine, O. Slone
Evidence for Dark Matter on Many ScalesGalaxies
Galaxy Clusters CosmologyEilers et al. (2018); Planck Collaboration (2018); Chandra X-Ray Observatory (2006)
Moschella (Princeton) Galactic Dynamics APS DPF 2019 2 / 34
The Missing Mass Problem in Galaxies
• Flat Rotation Curves
• Local Velocity Dispersions
• “Small Scale Crisis”• Missing Satellites• Too Big to Fail• Core vs. Cusp
• Dynamical Correlations with Baryons• Baryonic Tully-Fisher Relation• Diversity of Rotation Curves• Radial Acceleration Relation
Eilers et al. (2018);
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The Diversity of Rotation Curves
• Diversity of inner rotation curves for galaxies with similar halo mass• Correlates with surface brightness (baryons)
Ren et al. (2018)
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The Radial Acceleration Relation
• SPARC: rotation curves for 175disk galaxies
• Observe a tight correlation:
a =
{aN aN � a0√a0aN aN � a0
• a0 ∼ 10−10 m s−2Lelli et al. (2017); McGaugh et al. (2016)
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Solutions to the Missing Mass Problem in Galaxies
Which is most consistent with observed Milky Way dynamics?
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MOND-like Forces
• Built to reproduce rotation curves:
a =
{aN aN � a0√a0aN aN � a0
a0 ≈ 10−10 m s−2
• Regardless of theoretical mechanism, the empirical target is:
a = ν
(aN
a0
)aN
• For simplicity, we assume this relation holds going forward.
Milgrom (1983); Famaey, McGaugh (2012)
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MOND-like Forces vs. Dark Matter
• MOND-like forces predict a in the same direction as aN
• Dark Matter predicts a = aN + aDM
• In the Milky Way, aN is disk-like, but aDM is spherical
• Need to look at vertical acceleration to distinguish kinematically
• If you could measure a and aN, this is all you need – sincemeasurements are imperfect, the real situation is morecomplicated.
Image: ESO/NASA/JPL-Caltech/M. Kornmesser/R. Hurt
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A Teaser: The Direction of the Gravitational Acceleration
Data requires more enhancement in aR than in az
slope ∼ azaR
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A Generalized Framework
Bayesian Approach: marginalize over model parameters
ρB = ρ∗,bulge + ρ∗,disk + ρg,disk
Bland-Hawthorn, Gerhard (2016); Image: Wikipedia
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Milky Way Observables
For self-consistency, we must use only measurements that come fromdirect photometric observations, rather than dynamical fitting.
• Baryonic Surface Density
• Disk Scale Radius/Height
• Stellar Bulge Mass
• Rotation Curve
• Vertical Velocity Dispersions
σz(z)2 = − 1
n(z)
∫ ∞z
n(z′) az(z′) dz′
• probes vertical acceleration
• requires assumption ofequilibrium (Jeans equation)
• requires modelling the tracernumber density n(z)
Zhang et al. (2013); Read (2014)
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A MOND-like Force Model
• adopt interpolation function that fits the radial acceleration relation
ν(aN/a0) =1
1− e−√aN/a0
a0 = 1.20± 0.24× 10−10 m
s2
McGaugh et al. (2016)
• Compare Models:
BIC = k log n− 2 log L̂
• k: num. of model parameters• n: num. of data points• L̂: maximum likelihood
Table: Camarena, Marra (2018)
• Comparing to a reference DM model, we find that ∆BIC = 10.4• very strong preference for dark matter
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A Generalized MOND-like Force Model
• avoid dependence on a particular parametrization of ν(aN/a0)• expand ν(aN/a0) about the local value: aN,ref . v2
0/R0 ∼ a0
ν
(aN
a0
)≈ ν0 + ν1 · aN
• Comparing to a reference DM model, ∆BIC = 4.1• positive, but not strong, preference for dark matter
• Prefers very small enhancements ν ≈ 1
• Does not fit the radial accelerationrelation
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Superfluid Dark Matter
• Dark matter condenses to a superfluid core inside galaxies• In superfluid phase, phonons mediate a long-range MOND-like force
a = aN + aDM + aphonon
• Work in Progress: Because ρDM is small, expect to be similar topure MOND-like force scenario
Berezhiani, Khoury (2015); Berezhiani et al. (2017);
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Solutions to the Missing Mass Problem in Galaxies
Which is most consistent with observed Milky Way dynamics?
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Conclusions
• Local galactic observables can be used to test models withMOND-like forces precisely where they are most successful
• MOND-like forces appear in tension with existing data and are unableto simultaneously reproduce the observed radial and verticalaccelerations
• Currently working on applying this framework to Superfluid DarkMatter – similar results expected
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Thank You
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Supplementary Material
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The (Baryonic) Tully-Fisher Relation
• Look at many galaxies
• Observe Mb ∝ v4f
• Mb: total baryonic mass (luminosity)• vf : asymptotic rotational velocity
• Mb ∼ aNR2, vf ∼√Ra
⇒ a ∝ √aN• proportionality constant: Ga0
• a0 ≈ 10−10 m s−2
Milgrom (1983); McGaugh et al. (2016)
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Milky Way Observables
For self-consistency, we must use only measurements that come fromdirect photometric observations, rather than dynamical fitting.
• Baryonic Surface Density
• Disk Scale Radius/Height
• Stellar Bulge Mass
• Rotation Curve
• Vertical Velocity Dispersions
Σ1.1j = 2
∫ 1.1 kpc
0ρj(R0, z
′) dz′Σ1.1∗,obs = 31.2± 3.0 M� pc−2
Σ1.1g,obs = 12.6± 1.6 M� pc−2
McKee et. al. (2015); Flynn et. al. (2006)
Moschella (Princeton) Galactic Dynamics APS DPF 2019 20 / 34
Milky Way Observables
For self-consistency, we must use only measurements that come fromdirect photometric observations, rather than dynamical fitting.
• Baryonic Surface Density
• Disk Scale Radius/Height
• Stellar Bulge Mass
• Rotation Curve
• Vertical Velocity Dispersions
ρ∗,disk = ρ̃∗ e−R/h∗,Re−|z|/h∗,z
h∗,z,obs = 300± 50 pc
h∗,R,obs = 2.6± 0.5 kpc
Bland-Hawthorn, Gerhard (2016); Juric et. al. (2008)
Moschella (Princeton) Galactic Dynamics APS DPF 2019 21 / 34
Milky Way Observables
For self-consistency, we must use only measurements that come fromdirect photometric observations, rather than dynamical fitting.
• Baryonic Surface Density
• Disk Scale Radius/Height
• Stellar Bulge Mass
• Rotation Curve
• Vertical Velocity Dispersions
ρ∗,bulge(r) =M∗,b
2π
r∗,br
1
(r + r∗,b)3
• Existing measurements have alarge variance
• Conservative range:0 < M∗,b < 2× 1010 M�
M∗,b,obs = 1.50± 0.38× 1010 M�
Licquia, Newman (2015); Calchi Novat et. al. (2008)
Moschella (Princeton) Galactic Dynamics APS DPF 2019 22 / 34
Milky Way Observables
For self-consistency, we must use only measurements that come fromdirect photometric observations, rather than dynamical fitting.
• Baryonic Surface Density
• Disk Scale Radius/Height
• Stellar Bulge Mass
• Rotation Curve
• Vertical Velocity Dispersions
vc(R0) =√R0 · a(R0)
∣∣∣z=0
vc,obs = 229± 12km
s
(dvc/dR)obs = −1.7± 0.47km
s kpc
Eilers et. al. (2018)
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A Reference DM Model
• For our baseline DM model, wetake an NFW Profile:
ρDM(r) =ρ̃DM
(r/rs)α (1 + r/rs)
3−α
and the dynamics are:
a = aN + aDM ∇ · aDM = −4πGρDM
Navarro et. al. (1996);
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Superfluid DM
a = −∇Φ +αΛ
MPl∇φ
∇2Φ = 4πG(ρDM + ρB)
ρDM =2√
2m5/2Λ(
3 (β − 1) µ̂+ (3− β) (∇φ)2
2m
)3
√(β − 1) µ̂+ (∇φ)2
2m
(∇φ)2 + 2m(
2β3 − 1
)µ̂√
(∇φ)2 + 2m (β − 1) µ̂∇φ = αMPlaN
Berezhiani, Khoury (2015); Berezhiani et al. (2017);
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A Lagrangian Formulation of MOND
• Gravitational Lagrangian:
LN = − 18πG (∇ΦN )2 ⇒ L = − a20
8πGF(
(∇Φ)2
a20
)• µ(x) = ∂F
∂x2 : an arbitrary function up to asymptotes• a = −∇Φ
• Equation of Motion:
∇ · [µ (a/a0)a] = −4πGρB
µ(a/a0)a = aN + S
• ∇ · S = 0, but S 6= 0 in general• For spherical (one dimensional) symmetries, S = 0• For disk-like potentials, S ≈ 0 outside of the disk
Bekenstein, Milgrom (1984); Brada, Milgrom (1994)
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Quasilinear MOND
• Gravitational Lagrangian:
L = − 18πG
(2∇Φ ·∇ΦN − a2
0Q(
(∇ΦN )2
a20
))• ν(x) = ∂Q
∂x2 : an arbitrary function up to asypmtotes• ΦN : Newtonian potential (∇2ΦN = 4πGρb)• a = −∇Φ, aN = −∇ΦN
• Equation of Motion:
∇ · a = ∇ · [ν(aN/a0)aN]
a = ν (aN/a0)aN + S
Milgrom (2010)
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Is S Small?
0.04
0.04
0.04
0.04
0.04
0.04
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.09
0.09
0.09
0.09
2 4 6 8 10 12
-3
-2
-1
0
1
2
3
R [kpc]
z[kpc]
δ=|S |/|ν(aN) aN|
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Is the Interpolation Function Linear?
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Is the Interpolation Function Linear?
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Results: Dark Matter Parameters
ρDM(R0) = 0.29± 0.06 GeV cm−3
α < 1.1 at 90% confidence
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Results: MOND Parameters
• Red: Our analysis• Gray: ν(aN/a0) = 1
1−e−√
aN/a0, a0 = 1.20± 0.24× 10−10 m s−2
excluded at ∼ 95% confidenceMcGaugh et al. (2016)Moschella (Princeton) Galactic Dynamics APS DPF 2019 32 / 34
Model Parameters and Priors
• rs = 19 kpc
• r∗,bulge = 600 pc
• h∗,z = 300 pc
• hg,z = 130 pc
• hg,R = 2h∗,R• Also enforce that ν0 + ν1 · aN > 1 and that the baryon-only rotation
curve reaches its maximum before R = 5 kpc
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Systematic Checks
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