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) 10/18/17@BRICS-AGAC Rampei Kimura, Teruaki Suyama, MY, Daisuke Yamauchi, Shuichiro Yokoyama ArXiv 1709.09371

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

Introduction

The (dominant) contents of our Universe (Big-bang nucleosynthesis) (rotation curve of our galaxy) (supernovae IA) http://www2.astro.psu.edu/~mce/A001/lect19.html (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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Baryon case Continuity eq : effective growth rate : 22

DM case Modified continuity eq : effective growth rate : 23

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

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

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

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

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.

谢谢 山口　昌英 29