modified gravity vs. dark matter successes of dark matter why try anything else? modified gravity...
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Modified Gravity vs. Dark Matter
Modified Gravity vs. Dark Matter
Successes of Dark Matter Why try anything else? Modified Gravity
Successes of Dark Matter Why try anything else? Modified Gravity
Scott Dodelson w/ Michele LiguoriOctober 17, 2006
Four reasons to believe in dark matter
Four reasons to believe in dark matter
Galactic Gravitational Potentials
Cluster Gravitational Potentials
Cosmology
Theoretical Motivation
Potential Wells are much deeper than can be explained with visible
matter
Potential Wells are much deeper than can be explained with visible
matter
We have measured this for many years on galactic scales Kepler: v=[GM/R]1/2
Fit Rotation Curves with Dark Matter
Fit Rotation Curves with Dark Matter
Chris Mihos Applet
Bullet ClusterBullet Cluster
Gas clearly separated from potential peaks
Gravity is much stronger in clusters than it should be:Gravity is much stronger in clusters than it should be:
This is seen in X-Ray studies as well as with gravitational lensing
Sanders 1999
Tyso
n
CosmologyCosmologySuccesses of Standard Model of Cosmology (Light Elements, CMB, Expansion) now supplemented by understanding of perturbationsAt z=1000, the photon/baryon distribution was smooth to one part in 10,000.Perturbations have grown since then by a factor of 1000 (if GR is correct)!
Simplest Explanation is Dark Matter
Simplest Explanation is Dark Matter
Clumpiness
Without dark matter, potential wells would be much shallower, and the universe would be much less clumpy
Large Scales
Supersymmetry: Add partners to each particle in the Standard Model
Supersymmetry: Add partners to each particle in the Standard Model
Beautiful theoretical idea invented long before it was realized that neutral, stable, massive, weakly interacting particles are needed: Neutralinos
Paves the way for a multi-prong Experimental
Approach
Paves the way for a multi-prong Experimental
Approach
Why consider Modified Gravity?
Why consider Modified Gravity?
Dark Matter has not been discovered yet. The game is not over!
Recent Developments This is an age-old debate
…
Dark Matter has not been discovered yet. The game is not over!
Recent Developments This is an age-old debate
…
Remember how Neptune was discovered
Remember how Neptune was discovered
Formed a design in the beginning of this week, of investigating, as soon as possible after taking my degree, the
irregularities of the motion of Uranus, which are yet unaccounted for; in order
to find out whether they may be attributed to the action of an
undiscovered planet beyond it; and if possible thence to determine the
elements of its orbit, etc.. approximately, which would probably
lead to its discovery.
John Adams (not that one)
Undergraduate Notebook, July 1841
Not everyone believed a new planet was responsible
Not everyone believed a new planet was responsible
Astronomer Royal, George Airy, believed deviation from 1/r2 force responsible for irregularities
Adams informed Airy of his plans, but Airy did not grant observing time.
By June 1846, both Adams and French astronomer LeVerrier had calculated
positions
By June 1846, both Adams and French astronomer LeVerrier had calculated
positions
Competition is a good thing: Airy instructed Cambridge Observatory to begin a search in July, 1846, and Neptune was discovered shortly thereafter.
Anomalous precession of Mercury’s perihelion went the other way
Anomalous precession of Mercury’s perihelion went the other way
LeVerrier assumed it was due to a small planet near the Sun and searched (in vain) for such a planet (Vulcan).
We now know that this anomaly is due to a whole new theory of gravity.
How can gravity be modified to fit rotation curves?
How can gravity be modified to fit rotation curves?
Change Newton’s Law far from a
point mass
Change Newton’s Law far from a
point mass)1(
02 r
r
r
MGag
constantr
MG
r
MGv
r
0
2
Equate with centripetal acceleration, v2/r
Expect to see largest deviation from Newton in largest galaxies
So Inferred Mass/Light ratio should be largest for large
galaxies
So Inferred Mass/Light ratio should be largest for large
galaxies
It isn’t!
But … the anomaly is most apparent at low accelerations
But … the anomaly is most apparent at low accelerations
Sanders & McGaugh 2002
So, modify Newton’s Law at low acceleration:
So, modify Newton’s Law at low acceleration:
20 )/(r
MGaaaa Ngg
Acceleration due to gravity
New,fundamental scale
For a point mass
1,
1,1)(
xx
xx
MOdified Newtonian Gravity (MOND, Milgrom 1983)
This leads to a simple prediction
This leads to a simple prediction
MGavr
MGa
r
v0
42
02
Expect stellar luminosity to be proportional to stellar mass
4vL
… which has been verified (Tully-Fisher Law)
… which has been verified (Tully-Fisher Law)
L~v4
Sanders & Veheijen 1998
You want pictures!You want pictures!
Fit Rotation Curves of many galaxies w/ only one free parameter (recall 3 used in CDM).
You want pictures!You want pictures!
Newtonian-inferred velocity from Stars
Newtonian-inferred velocity from Gas
MOND does not do as well on galaxy clusters
MOND does not do as well on galaxy clusters
Sanders 1999
On cosmology, MOND is silent
On cosmology, MOND is silent
Not a comprehensive theory of gravity so cannot be applied to an almost homogeneous universe. We don’t even know if the true theory – which reduces to MOND in some limit – is consistent with an expanding universe.Need a relativistic theory which reduces to MOND
Scalar-Tensor TheoryScalar-Tensor Theory
geg 2~
)~(~16
1 4 gRgxdG
SEH
dxdxgemdxdxgmSm
~
The metric appearing in the Einstein-Hilbert action
is distinct from the metric coupling to matter (e.g. point particle)
They are related by a conformal transformation
Equations of motion for a point particle in this theoryEquations of motion for a
point particle in this theory
)(
dt
vd
)21,21,21,21(~ diagg
In a weak gravitational field, the metric that appears in the Einstein-Hilbert action is
where Φ is the standard Newtonian potential, obeying the Poisson equation. Then the eqn of motion for a point particle is
Extra term, dominates when
Standard term0a
MOND limit obtained by choosing Lφ
MOND limit obtained by choosing Lφ
)/(8
20
,,
20 aFG
aL
Bekenstein & Milgrom 1984
eVHMpc
km
Mpckm
km
cr
v
c
a
gal
gal 3305
22
0 10sec
27)005.0sec)(/103(
sec)/200(
There is a new fundamental mass scale in the Lagrangian
That may sound nutty, but remember …
That may sound nutty, but remember …
We are in the market for new physics with a mass scale of
order H0
We are in the market for new physics with a mass scale of
order H0
Quintessence Beyond Einstein-Hilbert
Curvature of order a02
μ~a0
Scalar Tensor Theories face a huge hurdle
Scalar Tensor Theories face a huge hurdle
Light is deflected as it passes by distances far from visible matter in galaxies
SDSS: Fischer et al. 2000
All of these points are farther from Galactic centers than the visible matter.
Theorem: Conformal Metrics have same null curves
Theorem: Conformal Metrics have same null curves
0~22
dxdxgedxdxgds
Bottom line: No extra lensing in scalar-tensor theories
Bekenstein & Sanders 1994
Need to modify conformal relation between the 2
metrics
Need to modify conformal relation between the 2
metrics)~( ,
,2
BgAeg
with A,B functions of φ,μφ,μ also doesn’t work (Bekenstein & Sanders 1994).
)2sinh(2~2
AAgeg But, adding a new vector field Aμ so that
does produce a theory with extra light deflection (Sanders 1997).
TeVeS (Bekenstein 2004)TeVeS (Bekenstein 2004)
)()~(~16
1,,
4
VAAggxd
GS
Two metrics related via (scalar,vector) as in Sanders theory; one has standard Einstein-Hilbert action, other couples to matter in standard fashion.
Scalar action:
Vector action: )1(2~32
1 4
AAFKFgxd
GSA
Auxiliary scalar field added (χ) to make kinetic term standard; two parameters in potential V
F2 standard kinetic term for vector field; Lagrange multiplier, fixed by eqns of motion, enforces A2=-1; K is 3rd free parameter in model.
ScorecardScorecard
Dark Matter
Modified Gravity
Rotation Curves
GoodGood ExcellentExcellent
Clusters ExcellentExcellent PoorPoor
Cosmology ExcellentExcellent ??
Theoretical Motivation
SUSYSUSY Hubble ScaleHubble Scale
Zero Order Cosmology in TeVeS
Zero Order Cosmology in TeVeS
),,,( 2222 aaaadiagg
3
8/2
effG
a
dtda
Metric coupling to matter is standard FRW:
Scale factor a obeys a modified Friedmann equation
Bekenstein 2004Skordis, Mota, Ferreira, & Boehm 2006Dodelson & Liguori 2006
Zero Order Cosmology in TeVeS
Zero Order Cosmology in TeVeS
VVG
e '
16
2
2
4
)]ln(/1[ add
GeGeff
with effective Newton constant
and energy density of the scalar field
Zero Order Cosmology in TeVeS
Zero Order Cosmology in TeVeS
These corrections however are small so standard successes are retained
15/(4χ)
Note the logarithmic growth of φ in the matter era
Inhomogeneities in TeVeSInhomogeneities in TeVeSSkordis 2006Skordis, Mota, Ferreira, & Boehm 2006Dodelson & Liguori 2006
Perturb all fields: (metric, matter, radiation) + (scalar field, vector field)
E.g., the perturbed metric is
)]21(),21(),21(),21([ 2222 aaaadiagg
where a depends on time only and the two potentials depend on space and time.
Inhomogeneities in TeVeSInhomogeneities in TeVeS
,1 aeA
Other fields are perturbed in the standard way; only the vector perturbation is subtle.
Constraint leaves only 3 DOF’s. Two of these decouple from scalar perturbations, so we need track only the longitudinal component defined via:
Inhomogeneities in TeVeSInhomogeneities in TeVeS
,21 Sbb
22
/2412
K
b
Vector field satisfies second order differential eqn:
The coefficients are complicated functions of the zero order time-dependent a and φ.
In the matter era,
Conformal time
4
1 b
Inhomogeneities in TeVeSInhomogeneities in TeVeSConsider the homogeneous part of this equation:
0)/241(24
2
K
This has solutions: α~ηp with
Kp /19212
1
2
3
α decays until φ becomes large enough (recall log-growth). Then vector field starts growing.
Inhomogeneities in TeVeSInhomogeneities in TeVeS
For large K, no growing mode: vector follows particular solution.For small K, growing mode comes to dominate.
Particular solnLarge K
Small K
Inhomogeneities in TeVeSInhomogeneities in TeVeS
This drives difference in the two gravitational potentials …
Small K
Large K
Inhomogeneities in TeVeSInhomogeneities in TeVeS
… which leads to enhanced growth in matter perturbations! Small K
Large K
Standard Growth
ScorecardScorecard
Dark Matter
Modified Gravity
Rotation Curves
GoodGood ExcellentExcellent
Clusters ExcellentExcellent PoorPoor
Cosmology ExcellentExcellent ? ? ++
Theoretical Motivation
SUSYSUSY Hubble ScaleHubble Scale++
Enhanced Enhanced GrowthGrowth
ConclusionsConclusions Dark Matter explains a wide variety of phenomena, extremely well on largest scales and good enough on smallest scales.
Modified Gravity is intriguing: it does well on small scales, poorly on intermediate scales, but there is no one theory that can be tested on cosmological scales.
We are uncovering some hints: Theorists and Experimenters all have work to do!
In June 1845, the French also began the relevant
calculations
In June 1845, the French also began the relevant
calculations
Urbain Le Verrier: I do not know whether M. Le Verrier is actually the most detestable man in France, but I am quite certain that he is the most
detested.
This first search (by Challis) was unsuccessful
This first search (by Challis) was unsuccessful
In September 1846, Dawes’ friend William Lassell, an amateur astronomer and a brewer by trade, had just completed building a large telescope that would be able to record the disk of the planet. He wrote to Lassell giving him Adams's predicted position. However Lassell had sprained his ankle and was confined to bed. He read the letter which he gave to his maid who then promptly lost it. His ankle was sufficiently recovered on the next night and he looked in vain for the letter with the predicted position.
In September 1846, Dawes’ friend William Lassell, an amateur astronomer and a brewer by trade, had just completed building a large telescope that would be able to record the disk of the planet. He wrote to Lassell giving him Adams's predicted position. However Lassell had sprained his ankle and was confined to bed. He read the letter which he gave to his maid who then promptly lost it. His ankle was sufficiently recovered on the next night and he looked in vain for the letter with the predicted position.
Both Adams and LeVerrier refined their predictions…
LeVerrier wrote to German astronomer Galle on September 18,
1846
LeVerrier wrote to German astronomer Galle on September 18,
1846
Galle discovered it in 30 minutes on September 23.