Modified Dark Matter: Does Dark Matter Know about the Cosmological Constant?

Douglas Edmonds Emory & Henry College

(moving to Penn State, Hazleton) Collaborators

Duncan Farrah Chiu Man Ho Djordje Minic

Y. Jack Ng Tatsu Takeuchi

Outline •  Evidence for CDM •  Problems with CDM •  Observations of a universal acceleration

scale •  Constructing MDM mass profiles: Heuristic

argument based on gravitational thermodynamics

•  Observations: – MDM in galaxies – MDM in galaxy clusters

•  Conclusions and future work

Evidence of Dark Matter

© M33 Image: NOAO, AURA, NSF, T.A.Rector.

Evidence of Dark Matter

Coma Cluster

Fritz Zwicky virial theorem 2K = −U

M ~r v2

G

Kinetic energy is equal to half the potential energy if the cluster is virialized

Then the mass can be estimated once we know velocities of galaxies and their distance to the center

In 1937, Zwicky finds ~10 x more matter than observed

Evidence of Dark Matter

Sanders 1999

virial discrepancy in galaxy clusters

Evidence of Dark Matter

X-ray: NASA/CXC/CfA/M.Markevitch et al.; Optical: NASA/STScI; Magellan/U.Arizona/D.Clowe et al.; Lensing Map: NASA/STScI; ESO WFI; Magellan/U.Arizona/D.Clowe et al.

Evidence of Dark Matter

Copyright: ESA and the Planck Collaboration

Evidence of Dark Matter

Dodelson 2011

Evidence it is Cold

© 2008 - 2016 Kavli Institute for Cosmology, Cambridge

Evidence it is Cold

Evidence it is Cold

CDM Tensions at Small Scales • Core-cusp problem

F568-3

Black dots with error bars are GRC dataGreen dotted line is the GRC predicted by baryonic matter onlyRed dashed line is the GRC predicted by NFW CDMBlue solid line is the GRC predicted by an isothermal dark matter halo with a constant density coreSimulated dark matter halos overpredict the rotation speed in the inner few kpc

CDM Tensions at Small Scales• Missing satellites

• Too big to fail

1012 M¤ CDM halo formed in N-body simulation (shown on left); Milky Way’s dwarf satellites (shown on right)Missing satellites: Simulations predict many more satellites than observations revealToo big to fail: On the left, the 9 most massive subhalos produced in the simulation are circled. Central regions of these simulated halos exceed the mass in observed dwarf satellites by a factor of ~5

D.H. Weinberg+4 2015

CDM Tensions at Small Scales

We will take the view that tensions between CDM predictions and

observations are telling us something about the

fundamental nature of dark matter

Modified Newtonian Dynamics In MOND, Newton’s equation of motion is modified to

with an interpolating function that satisfies

For a (baryonic) source mass, the above modification to the equation of motion implies

On the outskirts of galaxies, this implies that

leading to flat rotation curves, the baryonic Tully-Fisher relation

Modified Newtonian Dynamics

Brownstein and Moffat 2005

MOND is very successful at galactic scales

Red points with error bars are data: horizontal axis is distance in kpc; vertical axis is rotation speed in km/sGreen dashed line is the GRC predicted by gaseous diskMagenta dotted line is the GRC predicted by starsCyan dash-dotted line is the GRC predicted by MONDBlack solid line is the GRC predicted by MSTG (metric skew tensor gravity – J. Moffat

MOND Tensions at Large ScalesBut MOND fails at large scales Sanders 1999

DE et al. 2016

Top: Virial mass using MONDian dynamics versus Observed mass. While MOND resolves much of the mass discrepancy, a factor ~2 still remainsNote that this remaining mass discrepancy can be resolved by changing the interpolating function used (though, perhaps, difficult to justify physically) – A more severe problem for MOND is that it fails to reproduce the proper shape of the mass distributionBottom: Mass as a function of radius in a galaxy cluster: Solid black line is virial mass; Blue shaded region represents error; Green dot-dashed line is gas mass; Black dotted line is MOND effective mass; Black dashed line is CDM fit; Red solid line is MDM fit

Dodelson 2011

MOND Tensions at Large Scales

Milgrom Identified an Acceleration Scale

ac ≈1.2×10−10m/s2 ≈ cH0 / 2π

But the really interesting thing is the appearance of a universal acceleration scale that appears to be related

to the Hubble scale

McGaugh+2 2016

“The dark matter contribution is fully specified by that of the baryons.”

Observational Constraint on Dark Matter

“2693 data points for 153 galaxies with very different morphologies, masses, sizes, and gas fractions. The correlation persists even when dark matter dominates.”

Modified dark matter (MDM) is a(non-local) form of dark matter

that behaves like MOND at galactic scales and CDM at cluster and cosmic scales.

Our (phenomenological) model for MDM

makes explicit the appearance ofthe Hubble scale (equivalently, Λ)

and the observed correlation between dark matter and baryonic matter.

Ho,Minic&Ng,2010,Phys.Le<.B,693,567DE,Farrah,Ho,Minic,Ng,&Takeuchi,2014,ApJDE,Farrah,Ho,Minic,Ng,&Takeuchi,2016,arXiv

Modified Dark Matter •  InthecontextofGR,themissingmassproblemcanbesolvedintwo(disYnct)ways:WecanchangetheEinsteintensor(modifiedgravity)oraddanextraenergy-momentumtensor(e.g.,CDM).

•  InCDM,theextraenergy-momentumtensorisindependentofthebaryonictensor.

•  InMDM,wea<empttorecastEinstein’sequaYonsuchthattheenergy-momentumpartcontainsthecosmologicalconstantterminordertoseeifCDMmassprofilescouldknowaboutthecosmologicalconstant.

Modified Dark Matter from Gravitational Thermodynamics

following Jacobson 1995

Local observer with acceleration a in a spatially flat de Sitter space

In such a space, the thermodynamic relation has T as the Unruh temperature associated with the local accelerating (Rindler) observer

The acceleration a can be interpreted as surface gravity on the associated (Rindler) horizon.

The entropy is then associated with the area of this horizon.

The energy is the integral of the energy-momentum tensor

Modified Dark Matter from Gravitational Thermodynamics

following Jacobson 1995

The link between the thermodynamic relation and Einstein’s equations is the Raychaudhuri equation (where λ is the appropriate affine parameter)

Using the Raychaudhuri equation along with the Unruh temperature, the Bekenstein-Hawking entropy, and the thermodynamic relation between energy and entropy, it follows that

Applying local conservation of energy and momentum then yields Einstein’s equations

We preserve entropy (in order to remain consistent with Einstein’s theory) but change the temperature in such a way that the energy-momentum tensor knows about the inertial properties that temperature knows about.

Modified Dark Matter from Gravitational Thermodynamics

following Jacobson 1995

Our modification is to introduce a fundamental acceleration that is related to the cosmological constant .

a0 = c2 Λ / 3 and Λ = 3H 2

0 / c2

To change the temperature, we look to our observer with acceleration a in de Sitter space. The Unruh temperature experienced by this observer is

Our model is thus where dS remains unchanged and we define in analogy with the normalized temperature.

Modified Dark Matter from Gravitational Thermodynamics

following Jacobson 1995

Since the de Sitter space has a cosmological horizon, it has a horizon temperature. We define an effective temperature such that we get zero temperature for zero acceleration.

Therefore, energy is not changed in an arbitrary way, but instead in accordance with the change in temperature that should be fixed by the background.

MDM Mass Profiles

Note that entropy is unchanged so that the Einstein tensor is unchanged. However, due to the change in temperature, energy is changed.

If we rewrite the temperature as

we can also write

We now interpret the unprimed part as corresponding to baryonic matter. Then,

Thus, the energy-momentum tensor of the extra sources must be related to that of the baryons.

MDM Mass Profiles

Expanding the formula for the de Sitter temperature, we find a relation between the extra source (primed) and baryonic matter (unprimed).

The 00 component gives us a relation between the mass of the extra source (primed) and the baryonic mass (unprimed).

This extra mass, we call dark matter. Our dark matter mass profile knows about the baryonic matter as well as the cosmological background (and the inertial properties of masses moving in that background). In essence, we have vacuum origin for the observed fundamental acceleration and a quantum origin for the dark matter mass profiles.

MDM Mass Profiles To get the full expression for the gravitational force, we must consider our temperature modification. From we get and

Note that we recover Milgrom’s scaling (usually associated with MOND).

MDM Mass Profiles Given the heuristic nature of our argument, the MDM mass profile can, in principle, be modified due to some physical effects associated with scale. For example, the temperature could be changed using the Tolman-Ehrenfest formula

where and is determined by

boundary conditions. We are therefore led to the MDM mass profile:

M ' = f (r)(a0 / a)2M

For the following data fits, we adopt the following mass profile:

where rMDM is a scale factor and is a dimensionless constant that depends, in principle, on the ratio of dimensionful values of at different scales.

Galactic Rotation Curves

Data – black squares Stars – blue line Gas – green line [Sanders & Verheijen, 1998]

MDM – red line CDM (NFW) – black line

DE+5 2014

Galaxy Clusters

Galaxy ClustersM (r) = kT (r)r

µmpGd lnρgd ln r

+d lnT (r)d ln r

!

"#

$

%&

sphericalsymmetryandhydrostaYcequilibrium(Sarazin1988)

Vikhlininetal.(2006) ρg =1.2mp nenp

modificaYonoftradiYonalβ-model

T (r) = T0tcool (r)t(r)

Allenetal.(2001)

Galaxy Clusters

MMOND =Mbaryonic

1+ ac / a( )2

!M =Mbaryonic 1+α

1+ r / Rs

⎛

⎝⎜

⎞

⎠⎟a20a2

⎡

⎣⎢

⎤

⎦⎥

MOND effective mass:

Total mass with MDM:

black solid: virial mass blue shaded: 1-σ error red solid: MDM dashed: CDM dot-dashed: gas dotted: MOND

M (r) = kT (r)rµmpG

d lnρgd ln r

+d lnT (r)d ln r

!

"#

$

%&

Virial Mass: spherical symmetry and hydrostatic equilibrium (Sarazin 1988)

Galaxy Clusters

DE+5 2016

Conclusions We have constructed a dark matter mass profile that knows about the cosmological constant. The Modified Dark Matter mass profile depends on the baryonic mass profile, naturally accounting for the observed correlation between dark matter and baryonic matter. We have shown that the mass profile is consistent with galaxy and galaxy cluster data, and performs just as well as CDM and considerably better than MOND.