1. MASAHIDE YAMAGUCHI (山口 昌英)
Are Redshift-Space Distortions Actually a Probe of Growth of Structure?
Discriminating couplings of baryon and dark matter through redshift space distortion
MASAHIDE YAMAGUCHI (山口　昌英)
(Tokyo Institute of Technology)
Rampei Kimura, Teruaki Suyama, MY, Daisuke Yamauchi, Shuichiro Yokoyama
ArXiv

2. Contents Introduction Couplings of baryon & DM to DE Basic equations
Baryon, dark matter, dark energy
Couplings of baryon & DM to DE
Basic equations
How to probe them observationally ?
Redshift space distortion
Discussion and conclusions

3. Introduction

4. The (dominant) contents of our Universe
(Big-bang nucleosynthesis)
(rotation curve of our galaxy)
(supernovae IA)
(gravitational lensing)
Perlmutter et al.
もちろん、photonやneutrino, electronもある。
Particle data group
HST
nucleus (baryon)
dark matter
dark energy 4

5. The (dominant) contents of our Universe II
PLANCK
The energy density of our Universe is mostly occupied
by nucleus (baryon), dark matter, and dark energy.
もちろん、photonやneutrino, electronもある。
(Unfortunately, we have yet neither elucidated the identifications
of DM and DE, nor the reason of baryon asymmetry.) 5

6. The presence of dark energy
The Universe is now accelerating !!
Unknown new Energy is introduced or
GR may be modified in the IR limit
PLANCK
In either case, the dark energy responsible for
present acceleration might be dynamical,
which we identify with scalar field φ here.
We do not know how these components couple to one another. 6

7. How do baryons and dark matter couple to dark energy (& gravity) ?
In the same way ???
(See, e.g. B. Wang, E. Abdalla, F. Atrio-Barandela, D. Pavon 2016
for a review of an interaction between DM & DE)

8. Usual assumptions : the same coupling
Gravity & DE
Total Matter
( for simplicity)
The same coupling of baryon and DM :
phi is not necessarily dark energy in general, though the case with phi being DE is the most interesting.
baryon DM
The same coupling through gμν.
But, is this true ? or confirmed by observations ? 8

9. Non-minimal (non-universal) couplings
(Gleyzes et al. 2016)
New metric (invertible to original metric) :
(Bekenstein 1993)
conformal factor
disformal factor
This setting can easily pass the test of Etvos experiments.
The merit of this formalism is to quantify the couplings without knowing the concrete form of the action of dark matter.
How to confirm these different couplings
observationally ???
N.B. All baryons freely fall in the same way. 9

10. Einstein equations and energy momentum tensors
Energy momentum conservations :
Even for DM, energy momentum tensor is defined by g not bar{g}.
The pressureless condition is the same in either way (at least at linear order).
Baryon is conserved individually. 10

11. Scalar (DE) equations of motion
coupling between DM & φ(DE)
Energy momentum conservations :
Even for DM, energy momentum tensor is defined by g not bar{g}.
The pressureless condition is the same in either way (at least at linear order). 11

12. Concrete form of Q coupling between DM & φ(DE)
Even for DM, energy momentum tensor is defined by g not bar{g}.
The pressureless condition is the same in either way (at least at linear order). 12

13. What kind of observations can probe the couplings ?
Answer
Observations of large scale structure
Before explaining what redshift distortion is and what growth rate is, we would like to investigate how background quantities and perturbations evolve.
Redshift space distortion
Growth rate of matter perturbations 13

14. Metric and matter (perturbations)
Metric (perturbations in Newtonian gauge) :
(gravitational potential)
(curvature perturbation)
Matter energy momenta (perturbations in Newtonian gauge) :
density perturbations
Even for DM, energy momentum tensor is defined by g not bar{g}.
The pressureless condition is the same in either way (at least at linear order).
velocity perturbations
I = c (DM) or b (baryon) or m (total matter) 14

15. Background equations Einstein equations :
EOMs for baryon (b), DM (c), and φ :
Background component of Q
Non-minimal coupling
between DM & φ
A, AX, B, and BX are evaluated at background fields. 15

16. Perturbed equations in quasi static approximations
We pay attention to “within sound horizon”,
Einstein equations :
EOMs for baryon (b), DM (c), and φ :
baryon :
$R_1$ characterizes the strength of the modification of the CDM peculiar velocity field from the disformal coupling.
On the other hand, we have introduced $R_2$ to represent the contribution from the scalar field
perturbation in $\delta Q$. We should emphasize that the $R_2$ term gives the non-vanishing contributions only if the conformal and/or
disformal factors depend on the kinetic term.
DM :
φ : 16

17. Modified continuity equation for DM
minimal coupling case (A=1, B=0) : Q0 = R1 = R2 = 0
conformal factor, A = A(φ), disformal factor, B = B(φ) case:
Inserting concrete expression for delta Q to continuity eq of delta_c yields modifed continuity eq.
(Gleyzes et al. 2016)
Q0 ≠ 0, R1 ≠ 0, R2 = 0
The continuity equation is unchanged up to the overall factor.
But, our case generally includes R2 ≠ 0 case. 17

18. Modified continuity equation for total matter
Two sources of the violation of continuity eq. for total matter:
Inserting concrete expression for delta Q to continuity eq of delta_c yields modifed continuity eq.
violation of continuity eq. for CDM associated with R2
deviation of the background dynamics associated with Q0 18

19. How to probe the difference of couplings?
(See also Marcondes, Landim, Costa, Wang, Abdalla 2016
for a similar idea)
Before explaining what redshift distortion is and what growth rate is, we would like to investigate how background quantities and perturbations evolve. 19 19

20. Growth rate (I = c or b) growth rate : effective growth rate :
Combining all the perturbed equations to eliminate velocities as usual
two coupled second-order differential equations for δc & δb.
growing mode
(I = c or b)
the (normalized) k-independent
linear growth factors
initial density contrast
assuming δb caught up with δc
growth rate :
We have chosen initial time to be much after the time of CMB decoupling (z ~1100)
but much before the effect of the dark interaction becomes important (z ~ 1) and assumed that
the baryon density contrast has caught up with the CDM density contrast by the initial time.
effective growth rate : 20

21. Evolution equations for δc & δb
Inserting concrete expression for delta Q to continuity eq of delta_c yields modifed continuity eq. 21

22. Baryon case
Continuity eq :
effective growth rate : 22

23. DM case
Modified continuity eq :
effective growth rate : 23

24. Effective growth rate of total matter
Two sources of the deviation from fm :
What are implications of these equations ?
violation of continuity eq. for CDM associated with R2
deviation of the background dynamics associated with Q0 24

25. Redshift space distortion
In galaxy survey,
the distance is
measured by
redshift, z.
Peculiar velocity
must be taken
into account
in measuring
actual distance
and significantly
modifies shape.
(taken from Dodelson’s textbook) 25

26. Standard Kaiser formula
unit vector of line-of-sight
cosmic expansion
peculiar velocity
(x-component)
: redshift space position
(including angular position)
real space position
The observed numbers (of galaxies) are the same in both spaces.
Continuity eq :
Strictly speaking, we use “distant observed approximation (plane-parallel appro)”, in which line-of-sight direction is unchanged with good approximation.
On large scales, where linear theory applies, we assume v^g = v_m.
Kaiser’s arugument: x : order of the size of the survey, k^(-1) : order of the Fourier mode we can hope to measure in the survey
Perturbations on the largest scale probed by the survey k ~ x^{-1} are poorly determined since there are only a few such modes.
There are more smaller modes and we effectively average over all of such smaller modes to estimate power spectrum. Then,
we are really interested only in modes with kx >> 1.
If we take (b : bias) ,
β(fm) can be probed by comparing monopole & quadrupole components for example. 26

27. Modified Kaiser formula
Single RSD measurement determines f_m^eff (if b is fixed by other methods).
< Delta T delta_g>, < Phi(lensing) delta_g> propto b while <delta_g delta_g> propto b^2 => b
f_m(actual growth rate) can be determined by multiple redshift observations to directly observe time evolution of the structure.
(baryon zero limit, ωb = 0)
Single RSD measurement cannot determine fc !! (even if b is fixed) 27

28. Summary We have addressed the question how differently
baryons and dark matter can couple to dark energy.
We have found that, in the presence of X-dependence
of conformal and/or disformal couplings, continuity
and Euler equations are significantly modified.
With different couplings, the (effective) linear growth rate, which is measured by the peculiar velocities of the distributed galaxies, no longer corresponds to the
time derivative of the density perturbations and is
rather characterized by the couplings as well.

29. 谢谢
山口　昌英 29