News concerning MOND laws
Astro Seminar November 2016 MOND basic tenets
• A theory of dynamics (gravity/inertia) involving a new constant a0 (beside G, ...)
• Standard limit (a0 → 0): The Newtonian limit
• Scale invariance: (t, r) → λ(t, r)
MOND limit : a0 → ∞, A0 ≡ Ga0 fixed (G → 0):
a0 is analog to c in relativity or ~ in QM
1 Scale invariance
X ma = F,F = mMG/r2
2 2 V ma /a0 = F,F = mMG/r , 1/2 or ma = F,F ∝ m(MGa0) /r
2 MOND laws of galactic dynamics
• Essentially follow from only the basic tenets of MOND
• Are independent as phenomenological lawse.g., if interpreted as eects of DM (just as the BB spectrum, the photo electric eect, H spectrum, superconduc- tivity, etc. are independent in QM)
• Pertain separately to properties of the DM alone (e.g., asymptotic atness, universal Σ), of the baryons alone (e.g., M − σ, maximum Σ), relations between the two (e.g., M − V )
• Revolve around a0 in dierent roles
3 Phenomenological lows derived from basic tenets alone
In QM, many laws were derived before the Schr. equation: BB spectrum, photoelecic eect, H-like atomic spectra, speci c heat of solids, Induced and spontaneous emission/absorption.
All involve ~.
In pre GR: equivalence principle ⇒ gravitational redshift, light bending (c).
4 MOND: Baryonic density ⇒ dynamical one (not in DM)
r ⇒ g r ⇒ r ≡ − −1∇⃗ · g . ρb( ) M ( ) ρD( ) (4πG) M
5 Some of the MOND laws
• Asymptotic constancy of orbital velocity: V (r) → V∞ (H)
• Light-bending angle becomes asymptotically constant (H)
4 • The velocity mass relation: V∞ = MA0 (H-B)
4 • DML virial relation: σ ∼ MA0
2 • Discrepancy appears always at V /R = a0 (H-B)
• Isothermal spheres have surface densities Σ¯ . a0/G (B)
• The central surface density of dark halos is ≈ a0/2πG (H)
• Universal baryonic-dynamical central surface densities relation (H-B).
• Full rotation curves from baryon distribution alone (H-B)
6 A universal Halo central surface density?
Dashed line [best t log(Σ0) = 2.15 ± 0.2] From Salucci et al. 2012
−2 ΣM ≡ a0/2πG = 138M⊙pc [log(ΣM ) = 2.14]
7 Spiral galaxies (open red circles); dwarf irregulars (full green circles), spirals and ellipticals (weak lensing black squares); Milky Way dSphs (pink triangles); nearby spirals in THINGS (small blue triangles); early-type spirals (full red triangles).
8 A universal Halo central acceleration?
a. b.
c. d.
The parallel lines correspond to gmax(halo) =, 0.2a0 (dashed), 0.3a0 (solid), 0.4a0 (dotted).
gmax(halo)/a0 ≈ 0.46ρ0Rc/ΣM
9 From Milgrom and Sanders et al. 2006
10 Maximal Halo acceleration
Brada and Milgrom 1999: We have recently discovered that the modi ed Newtonian dynamics (MOND) implies some universal upper bound on the acceleration that can be produced by a dark halo, which, in a Newtonian analysis, is assumed to account for the eects of MOND. Not surprisingly, the limit is on the order of the acceleration constant of the theory.
gH/a0 = ν(y)y − y, y = gN /a0
.
ν(y ≫ 1) ≈ 1; ν(y ≪ 1) ≈ y−1/2
11 MOND prediction
ΣM is a universal accumulation value of ΣH, for high-surface density galaxies.
1/2 For low-surface-density systems ΣH ≈ 2.4(ΣbΣM ) , Σb = ρR. ∫ ∞ −1∇⃗ · g − g ΣH = 2 0 (4πG) ( N )dr = ∫ ∞ −2 ∂ { 2 − } ΣM 0 r ∂r r y[ν(y) 1] dr.
y(r) ≡ gN (r)/a0.
12 Baryonic and dynamical central Surface densities
Milgrom 2016, data: Lelli et al. 2016. `Scatter largely driven by obs. uncertainties'. `virtually no intrinsic scatter'.
13 Nonrelativistic theories
Nonlinear Poisson equation:
∇⃗ · [µ(|∇⃗ ϕ|/a0)∇⃗ ϕ] = 4πGρ The deep-MOND limit is conformally invariant
Quaslinear MOND (QUMOND):
∆ϕN = 4πGρ, ∆ϕ = ∇⃗ · [ν(|∇⃗ ϕN |/a0)∇⃗ ϕN ] Derivable from actions
Limits of relativistic theories (TeVeS, BIMOND, Einstein Aether)
14 central-surface-densities relation (CSDR) ∫ 0 0 , ≡ , y ′ ′ ΣD = ΣM S(ΣB/ΣM ) ΣM a0/2πG S(y) = 0 ν(y )dy Asymptotes:
0 0 for 0 ≫ , ΣD = ΣB ΣB ΣM 0 0 1/2 for 0 ≪ ΣD = (4ΣM ΣB) ΣB ΣM
Quite unrelated to the mass-asymptotic-speed relation.
15 How is 0 measured? ΣD Toomre (1963):
∫ 2 0 1 ∞ V (r) ΣT = 2πG 0 r2 dr
16 Discrepancy-acceleration correlation for rotationally-supported systems
Scale invariance ⇒ V (r → ∞) → V∞.
4 MOND acceleration ⇒ V∞ = MGa0.
2 2 1/2 1/2 On the asymptote g(r) = V (r)/r = (a0MG/r ) = (a0gN ) . Elsewhere on the rotation curves of any galaxy?
Modi ed inertia ⇒ g = gN ν(gN /a0) at all radii. ν(y) is universal. ν(y ≫ 1) ≈ 1. ν(y ≪ 1) ≈ y−1/2 (scale invariance). Also true in modi ed gravity for spherical systems.
Predicted by MOND, Approximately, in general.
17 Discrepancy-acceleration correlation for rotationally-supported systems
η ≡ g/gN vs. gN for 73 disc galaxies (points with δV/V ≤ 0.05) McGaugh (2015).
−2 η → λη, gN → λ gN
18 Discrepancy-acceleration correlation for rotationally-supported systems
19 Discrepancy-acceleration correlation for pressure-supported systems
g vs. gN , Scarpa (2006)
20 DM? The unexpected diversity of dwarf galaxy rotation curves
Oman, Kyle A.; Navarro, Julio F.; Fattahi, Azadeh; Frenk, Carlos S.; Sawala, Till; White, Simon D. M.; Bower, Richard; Crain, Robert A.; Furlong, Michelle; Schaller, Matthieu; Schaye, Joop; Theuns, Tom: (2015)
"...The shape of the circular velocity pro les of simulated galaxies varies systematically as a function of galaxy mass, but shows remarkably little variation at xed maximum circular velocity. This is especially true for low-mass dark-matter-dominated systems, ... This is at odds with observed dwarf galaxies, which show a large diversity of rotation curve shapes, even at xed maximum rotation speed..."
"...We conclude that one or more of the following statements must be true: (i) the dark matter is more complex than envisaged by any current model; (ii) current simulations fail to reproduce the diversity in the eects of baryons on the inner regions of dwarf galaxies; and/or (iii) ... data are incorrect..."
Moti: "And, despite the diversity, they all sit on very tight "baryon-DM" relations."
21 Mass-asymptotic-speed relation
McGaugh (2011)
V (M) from scale invariance. Power 4 from acceleration
22 23 Galaxy-galaxy lensing
Data from Brimioulle et al. 2013, analysis from Milgrom 2013.
24 Crater II: Discovery
The feeble giant. Discovery of a large and diuse Milky Way dwarf galaxy in the constellation of Crater
G. Torrealba, S. E. Koposov, V. Belokurov, M. Irwin (Submitted on 26 Jan 2016 (v1), last revised 1 Aug 2016 (this version, v4))
We announce the discovery of the Crater 2 dwarf galaxy, identi ed in imaging data of the VST ATLAS survey. Given its half-light radius of 1100 pc, Crater 2 is the fourth largest dwarf in the Milky Way, surpassed only by the LMC, SMC and the Sgr dwarf. With a total luminosity of MV -8, this satellite galaxy is also one of the lowest surface brightness dwarfs. Falling under the nominal detection boundary of 30 mag arcsec−2, it compares in nebulosity to the recently discovered Tuc 2 and Tuc IV and UMa II. Crater 2 is located ∼ 120 kpc from the Sun and appears to be aligned in 3-D with the enigmatic globular cluster Crater, the pair of ultra-faint dwarfs Leo IV and Leo V and the classical dwarf Leo II. We argue that such arrangement is probably not accidental and, in fact, can be viewed as the evidence for the accretion of the Crater-Leo group.
25 The velocity dispersion of Crater II: MOND prediction
MOND Prediction for the Velocity Dispersion of the `Feeble Giant' Crater II (McGAugh arXiv:1610.06189)
Abstract: Crater II is an unusual object among the dwarf satellite galaxies of the Local Group in that it has a very large size for its small luminosity. This provides a strong test of MOND, as Crater II should be in the deep MOND regime 2 −2 −1 2 −2 −1 (gin ≈ 34 km s kpc ≪ a0 = 3700 km s kpc ). Despite its great distance (≈ 120 kpc) from the Milky Way, the external eld of the host 2 −2 −1 (gex ≈ 282 km s kpc ) comfortably exceeds the internal eld. Consequently, Crater II should be subject to the external eld eect, a feature unique to MOND. This leads to the prediction of a very low velocity dispersion: +0.9 −1. σefe = 2.1−0.6 km s
26 The velocity dispersion of Crater II: recent measurement
Nordita workshop on Dark Matter Distribution in the Era of Gaia
From Twitter:
Walker: Very diuse MW satellite Crater 2 has very, very cold velocity dispersion of 2.5 km/s. Implies very low dynamical density
Walker: Crater 2's low velocity dispersion is also right where MOND predicts it.
Boylan-Kolchin: Crater 2's velocity dispersion is pretty much exactly what DudeDarkmatter (McGaugh) predicted recently using MOND
McGaugh: The MOND prediction was 2 km/s. Low because of EFE. Very hard to understand why this would happen in LCDM.
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