1. Dark Matter, Dark Energy Interaction: Cosmological Implications, Observational Signatures and Theoretical Challenges
Bin Wang
Shanghai Jiao Tong University
May 4, 2015, Beijing

2. Outline Cosmological Implications:
Why do we need the interaction between DE&DM?
Perturbation theory when DE&DM are in interaction
Observational Signatures:
Is the interaction between DE&DM allowed by observations?
CMB
ISW effect
Kinetic SZ
Galaxy Cluster scale tests
Growth factor and Structure formation
Number of Galaxy Cluster counting
Theoretical Challenges:
How to understand the interaction between DE&DM?
Relation to the modified gravity

3. 95% Unknown: 25% DM+70% DE DE-- ?
QFT value 123 orders larger than the observed
Coincidence problem:
Why the universe is accelerating just now?
In Einstein GR: Why are the densities of DM and DE of precisely the same order today?
is not the end story to account for the cosmic acceleration
Reason for proposing Quintessence, tachyon field, Chaplygin gas models etc.
No clear winner in sight
Suffer fine-tuning

4. Why do we need the interaction between DE&DM?
A phenomenological generalization of the LCDM model is
LCDM model,
Stationary ratio of energy densities
Coincidence problem less severe than LCDM
The period when energy densities of DE and DM are comparable is longer
The coincidence problem
is less acute
can be achieved by a suitable interaction between DE & DM

5. Do we need to live with Phantom?
Degeneracy in the data.
SNe alone however are consistent with w in the range, roughly
-1.5 ≤ weff ≤ -0.7
WMAP: w=-1.06{+0.13,-0.08}
One can try to model w<-1 with scalar fields like quintessence. But that requires GHOSTS: fields with negative kinetic energy, and so with a Hamiltonian not bounded from below:
3 M42 H2 = - (f’)2/2 + V(f)
`Phantom field’ , Caldwell, 2002
Phantoms and their ills: instabilities, negative energies…,
w<-1 from data is strong!
Theoretical prejudice against w<-1 is strong!

6. MAYBE NOT!!
Conspiracies are more convincing if they DO NOT rely on supernatural elements!
Ghostless explanations:
1) Modified gravity affects EVERYTHING, with the effect to make w<-1.
S. Yin, B. Wang, E.Abdalla, C.Y.Lin, arXiv: , PRD (2007)
A. Sheykhi, B. Wang, N. Riazi, Phys. Rev. D 75 (2007)
R.G. Cai, Y.G. Gong, B. Wang, JCAP 0603 (2006) 006
2) Another option: Interaction between DE and DM
Super-acceleration (w<-1) as signature of dark sector interaction
B. Wang, Y.G.Gong and E. Abdalla, Phys.Lett.B624(2005)141
B. Wang, C.Y.Lin and E. Abdalla, Phys.Lett.B637(2006)357.
S. Das, P. S. Corasaniti and J. Khoury, Phys.Rev. D73 (2006)

7. Is Einstein’s GR successful on large scale?
Modified gravity: f(R) model Extended GR by introducing
Non-minimal coupling between DE and DM
Motivations to introduce the interaction between DE and DM:
Alleviate the coincidence problem
Accommodate the DE with w<-1
Relate to the study of the modified gravity

8. Interaction
70% Dark Energy
25% Dark Matter

9. The Interaction Between DE & DM
Phenomenological interaction forms:
For Q > 0 the energy proceeds from DE to DM
Phenomenological forms of Q

10. Signature of the interaction in the CMB
Sachs-Wolfe effects:
non-integrated
photons’ initial conditions
Early ISW
integrated
has the unique ability to probe the
“size” of DE: EOS, the speed of sound
Late ISW
Signature of the interaction between DE and DM?

11. Perturbation theory when DE&DM are in interaction
Choose the perturbed spacetime
DE and DM, each with energy-momentum tensor
denotes the interaction between different components.
The perturbed energy-monentum tenser reads ,

12. Perturbation Theory The perturbed Einstein equations
The perturbed pressure of DE:

13. Perturbation Theory DM: DE: He, Wang, Jing, JCAP(09);
He, Wang, Abdalla, PRD(11)
We have not specified the form of the interaction between dark sectors.

14. Perturbation Equations
Phenomenological interaction forms:
Perturbation equations:

15. Stability in the Perturbations
Choosing interactions
w>-1
Maartens et al, JCAP(08)
How about the other forms of the interaction? Stable??
How about the case with w<-1?
Curvature perturbation is not stable！
Is this result general??

16. Stability in the Perturbations
Stable perturbation depends on the form of the interaction and the value of the DE EoS
couplings
ξ~DE
ξ~DM
ξ~DE+DM
W>-1
Stable
Unstable
W<-1
For constant DE EoS
J.He, B.Wang, E.Abdalla, PLB(09)
For time varying DE EoS, stability can be kept when DM density appears in the coupling term for w> PLB(10,11)

17. ISW imprint of the interaction
The analytical descriptions for such effect
ISW effect is not simply due to the change of the CDM perturbation. The interaction enters each part of gravitational potential.
J.H. He, B.Wang, P.J.Zhang, PRD(09)
J.H.He, B.Wang, E.Abdalla, PRD(11)

18. Imprint of the interaction in CMB
Interaction proportional to the energy density of DE
Interaction proportional to the energy density of DM & DE+DM
EISW+SW
EISW+SW
J.H.He, B.Wang, P.J.Zhang, PRD(09)

19. Degeneracy between the ξ and w in CMB
The small l suppression caused by changing ξ can also be produced by changing w
ξ can cause the change of acoustic peaks but w cannot
Suppression caused by ξ cannot be distinguished from that by w
Suppression caused by ξ is more than that by w
ξ ~DE
ξ ~DM, DE+DM
He, Wang, Abdalla, PRD(11)

20. Degeneracy between the ξ and
The change of the acoustics peaks caused by ξ can also be produced by the abundance of the cold DM
To break the degeneracy between ξ and , we can look at small l spectrum. ξ can bring clear suppression when ξ~DM or DM+DE, but not for ξ~DE

21. Likelihoods of ,w and ξ
ξ ~DE
ξ ~DM
ξ ~DE+DM

22. Fitting results He, Wang, Abdalla, PRD(11) WMAP7-Y WMAP+SN+BAO+H
PLANCK data: A Costa, X Xu, B Wang, E Ferreira, E Abdalla

23. Alleviate the coincidence problem
Interaction proportional to the energy density of DM
J.H. He, B.Wang, P.J.Zhang, PRD(09); J.H.He, B.Wang, E.Abdalla, PRD(11)

24. Sunyaev Zel'dovich effect
The thermal SZ effect
CMB photon
1% probability
distortion in the CMB spectrum
free energetic
electron in
thermal motion
Inverse Compton scattering
scattered CMB photon
Blue shifted,
boost CMB photon ≤1mK
218GHz
Features of TSZ effect:
CMB at frequency <218GHz, CMB at frequency>218GHz
TSZ depends on the depth of the cluster gas, distortion strong at the center, weak at the edge
Independent of the redshift
Intensity of TSZ depends on the cluster mass

25. TCMB=2.73K
The kinetic SZ effect
electron motion
with big peculiar
velocity vp
vp: peculiar velocity
scattering probability
Interesting to study KSZ effect
TCMB=2.73+ΔT
PLANCK result: the upper limit of the peculiar velocity can be three times of the LCDM prediction

26. The KSZ effect
n_e: the electron density,
σ_T: the Thomson cross section
k: the Thomson optical depth
v: the peculiar velocity of the electrons
n: the unit vector along the l.o.s.
Using the comoving distance x and neglecting any interaction of electrons with other particles,
the mean electron number density at present
the ionization fraction
peculiar momentum
p_E : gradient component
p_B: curl component
p_E: no contribution to KSZ, cancels out when integrating along the l.o.s.
p_B: contributes to KSZ

27. This formula is in the same form as the KSZ effect of the LCDM model
the rotational mode of v
the cross-talk between the density and the velocity
In the linear regime, only the irrotational component of the velocity fields couples to gravity
The KSZ effect is due to the cross-talk between the density gradient and the velocity.
The contribution of the kSZ effect to CMB:
the correlation between the density gradient and the velocity in Fourier space
Where,
This formula is in the same form as the KSZ effect of the LCDM model

28. X.D.Xu, B.Wang, P.J.Zhang, F.Atrio-Barandela, JCAP(13)
These equations are formally identical to the KSZ contribution in the concordance ΛCDM model, BUT…
In the small scale approximation k>>aH, neglect the time variation of the potential
Evolution of DE, DM perturbation;
background density
Density perturbation of electrons
Velocity field of electrons
Interaction
KSZ effect
X.D.Xu, B.Wang, P.J.Zhang, F.Atrio-Barandela, JCAP(13)

29. Tension with negative coupling.
ξ ~DE, w>-1
The amplitude increases with
decreasing of coupling
ξ >0, smaller than the LCDM
model
ξ <0, larger than the LCDM
X: upper limit from ACT 8.6uk^2 at l=3000
+: upper limit from SPT
2.8uk^2 at l=3000
Upper limits depend on:
reionization history
the modeling of CIB
TSZ contribution
We just consider: homogeneous, linear
patchy reionization, nonlinear
ξ ~DE, w<-1
ξ~DM, w<-1
Tension with negative coupling.
KSZ observations favor a positive coupling between DM-DE interaction !

30. The non-linear evolution of matter density perturbations enhances the kSZ effect.
Employ the halofit model
to estimate the non-linear
corrections.
Consider the nonlinear effect, more tension with negative coupling.
KSZ observations favor a positive coupling between DM-DE interaction !

31. Summary: complementary KSZ effect: potential, peculiar velocity,
Large at big l,
from the moment of reionization z ~10.
ISW effect:
time evolution of the potential,
at large angular scales,
during the period of acceleration
complementary
X.D.Xu, B.Wang, P.J.Zhang, F.Atrio-Barandela, JCAP(13)

32. observables in addition to CMB should be added.
To reduce the uncertainty and put tighter constraints on the value of the coupling between DE and DM, new
observables in addition to CMB should be added.
Galaxy cluster scale test
E. Abdalla, L.Abramo, L.Sodre, B.Wang, PLB(09)
E. Abdalla, L.Abramo, J.Souza, PRD(09)
Growth factor of the structure formation
J.He, B.Wang, Y.P.Jing, JCAP(09)
Number of cluster counting
J.H.He, B.Wang, E.Abdalla, D.Pavon, JCAP(10) ……

33. Growth of structures
In the subhorizon approximation k>>aH, DM perturbation
DE perturbation
Introducing the interaction
In the subhorizon approximation k>>aH

34. Growth of structures: The Influence of the DE perturbation on the structure formation
When is not so tiny and
w close to -1, in subhorizon approximation,
DE perturbation is suppressed
He, Wang, Jing, JCAP(09)

35. Growth of structures: The influence of the interaction between dark sectors
He, Wang, Jing, JCAP(09)

36. Growth Index: The influence of the interaction between dark sectors
The interaction influence on the growth index overwhelms the DE perturbation effect.
This opens the possibility to reveal the interaction between DE&DM through measurement of growth factor in the future.
He, Wang, Jing, JCAP09

37. The dynamics in the process of the structure formation.
How does the interaction influence the structure formation?
The dynamics in the process of the structure formation.
Layzer-Irvine equation
Virial condition of the galaxy cluster
E. Abdalla, L.Abramo, L.Sodre, B.Wang, PLB(09)
E. Abdalla, L.Abramo, J.Souza, PRD(09)
2. Spherical collapse model.
Critical density to collapse
Number of galaxy clusters at different redshift
J.H.He, B.Wang, E.Abdalla, D.Pavon, JCAP(10)

38. Layzer-Irvine equation for DM
Layzer-Irvine equation describes how a collapsing system reaches dynamical equilibrium in an expanding universe
For DM: the rate of change of the peculiar velocity is
Neglecting the influence of DE and the couplings, Newtonian mechanics
Multiplying both sides of this equation by
integrating over the volume
and using continuity equation,
describes how DM reaches dynamical equilibrium in the collapsing system in the expanding universe.
E. Abdalla, L.Abramo, L.Sodre, B.Wang, PLB(09)
J.H.He, B.Wang, E.Abdalla, D.Pavon, JCAP(10)

39. Virial condition For DM: equilibrium configuration of the system
If the DE is distributed homogeneously,
For DM:
For a system in equilibrium
Virial Condition:
presence of the coupling between DE and DM changes the
equilibrium configuration of the system
Galaxy clusters are the largest virialized structures in the universe
Comparing the mass estimated through naïve virial hypothesis with that from WL and X-ray
E. Abdalla, L.Abramo, L.Sodre, B.Wang, PLB(09)
E. Abdalla, L.Abramo, J.Souza, PRD(09)
galaxy clusters optical, X-ray and weak lensing data

40. Layzer-Irvine equation for DE
For DE: starting from
Multiplying both sides of this equation by
integrating over the volume
and using continuity equation, we have:
For DE:
For DM:
The time and dynamics required by DE and DM to reach equilibrium are different in the collapsing system.
DE does not fully cluster along with DM.
J.H.He, B.Wang, E.Abdalla, D.Pavon, JCAP(10)

41. Spherical collapse model
Homogenous DE :
In the background:
The energy balance equation
Friedmann equation
In the spherical region:
Raychaudhuri’s equation – dynamical motion of the spherical region
Local expansion
The energy balance equation in the spherical region:
DM perturbation equation

42. Spherical collapse model
Inhomogenous DE : DE does not trace DM
DE and DM have different four velocities
Rest on DM frame, we obtain the energy momentum tensor for DE,
The non-comoving perfect fluids
energy flux of DE observed in DM frame
The timelike part of the conservation law, DM DE
Additional expansion due to peculiar velocity of DE relative to DM

43. Spherical collapse model
The spacelike part of the conservation law
Only DE has non-zero component
is the DE flux observed in DM rest frame
Only keep linear, we obtain:
Raychaudhuri’s equation
We can describe the spherical collapse model when DE does not trace DM
J.H.He, B.Wang, E.Abdalla, D.Pavon, JCAP(10)

44. Press-Schechter Formalism
J.H.He, B.Wang, E.Abdalla, D.Pavon, JCAP(10)

45. More tests are needed CMB ……
A lot of effort is required to disclose the signature on the interaction between DE and DM
CMB
Cluster M
Cluster N
kSZ
Redshift Distortion ……
More tests are needed

46. Understanding the interaction between DE and DM from Field Theory
S. Micheletti, E. Abdalla, B. Wang, PRD(09)
E. Abdalla, L. Graef, B. Wang, PLB(13)
Two fields describing each of the dark components:
a fermionic field for DM, a bosonic field for the Dark Energy,

48. The scalar is directly coupled to matter
Relating the interaction between DE and DM to the modified gravity
J.H.He, B. Wang, E.Abdalla PRD(11,12)
Acceleration of our universe may originate from some geometric
modification from Einstein gravity.
The simplest model:
f(R) gravity
R: Ricci scalar
model:
The action
conformal transformation
the action in the Einstein frame
The scalar is directly coupled to matter

49. Jordan frame Vin the S Einstein Iframe
Background Dynamics
f(R) gravity
Einstein frame
Second order equation
conformal transformation
Two first order equations
GR: F=df/dR= =0

50. mass dilation rate due to the conformal transformation.
Physical meaning of
J.H.He, B.Wang, E.Abdalla, PRD(11)
The equation of motion under the transformation
where we have used D.20
in Wald’s book
for perfect fluid and drop pressure
the equation of motion of particles with varying mass
We have
introduce a scalar field Γ which satisfies
mass dilation rate due to the
conformal transformation.

51. Perturbation: (Newtonian gauge)
Jordan frame
Einstein frame
f(R) gravity
Conformal transformation

52. Einstein frame Interaction between DE and DM
Equivalence
The equation of motion for a free particle
Jordan frame
f(R) gravity
Action
Field equations
background
dynamics
Cosmological perturbation
Conformal
transformation
Conformal
transformation
Conformal
transformation
Action
Field equations
background
dynamics
Cosmological perturbation
Einstein frame Interaction between DE and DM
mass dilation

53. Summary Cosmological Implications: Observational Signatures:
Motivation to introduce the interaction between DE & DM
Observational Signatures:
CMB+SNIa+BAO+kSZ
Galaxy cluster scale tests
Alleviate the coincidence problem
Theoretical Challenges:
Understanding the interaction from field theory
Relating the interaction model with modified gravity

54. Report on Progress in Physics invited review 2015
Thanks!!!