On the gravitational redshift of cosmological objects

Transcription

1 On the gravitational redshift of cosmological objects Kazuhiro Yamamoto (Hiroshima University) Collaboration with D. Sakuma(Hirohima), A. Terukina(Hiroshima), C. Hikage(IPMU)

2 Outline 1 Introduction gravitational redshift 2 Theoretical formulation 3 Satellite galaxy virialized in a halo 4 Intracluster gas 5 galaxies of void 6 Summary and conclusions

3 1. Introduction gravitational redshift Negative value of δz Relativistic effect in large scale structures Taruya-san s talk Gravitational redshift in cluster of galaxies Blueward <--- Shift (km/s) ---> Redward Zhao, Peacock, Li (2013) 1.5(σ 2 obs -σ2 bcg )/c TD f R TD anisotropic TD f R GR+TD GR+TD Obs. shift - Φ - Φ bcg /c Blue shift relative to BCG Wojtak, et al, (2011) Zhao, Peacock, Li (2013) Kaiser, (2013) Jimeno, et al. (2015) Cai et al., (2016) redshift relative to BCG BCG Projected Radius of the Composite Cluster R (Mpc) Signal has been detected. Other possibility for detection? Useful for a test of general relativity and gravity theories?

4 2. Theoretical formulation Newtonian gauge ds 2 = a( ) 2 [ (1 + 2 )d 2 +(1+2 )d~x 2 ] Geodesic equation: 1 p dp d d (,x i ( )) d p is a physical energy for cosmological rest frame obs. i, ij ˆp i ˆp j =1, + i dx i d =ˆpi (1 + ) Solution up to the first order of the metric perturbation 1 1+z j = ISW term Potential term ( Z ) p( 0 ) p( j ) = a( 0 j) a( 0 ) exp ( 0 ( ) 0 ( )) d ( 0, ~x( 0 )) + ( j, ~x( j )) j j Time at the emission of photon from the j-th object 0 Present epoch time a( 0 )=1

5 a( 0 )=1 η 0 Observer Observer η 0 η ( Z ) ηj 1+z j = p( j) p( 0 ) = 1 0 a( j ) exp ( 0 0 ) d + ( 0, ~x( 0 )) ( j, ~x( j )) j ~v 1 1+~ ~v j q 1 ~v 2 j 1 a( j ) ' 1 a( 1 ) j = 1 + ' 1+~ ~v j ~v2 j Hubble terms 1+z 1 1 a( 1 ) 1+z j '(1 + z 1 ) 1 H( 1 ) j + J-th object η j ~x( j ) j H 2 ( 1 ) η j 1 H( 1 ) j + H 2 ( 1 ) + ( 0, ~x( 0 )) ( j, ~x( j )) + ~ ~v j ~v j r γ η v j Doppler term a 00 ( 1 ) a( 1 ) Line of sight direction a 00 ( 1 ) a( 1 ) 2 j + 2 j χ ISW term Z 0 j ( 0 0 ) d Doppler effect 1+~ ~v j q 1 ~v 2 j., 1 Gravitational potential term Doppler terms

6 We omit the ISW term because we consider objects of subhorizon scale and consider the relative gravitational redshift. Z 0 j ( 0 0 ) d ' Z 0 1 ( 0 0 ) d ( 0 ( 1, ~x( 1 )) 0 ( 1, ~x( 1 ))) j ( 0, ~x( 0 )) Gravitational redshift of a observer does not contribute to the relative gravitational redshift. Redshift of the j-th object 1+z j '(1 + z 1 ) 1 H( 1 ) j + Hubble terms H 2 ( 1 ) 1 2 a 00 ( 1 ) a( 1 ) ( j, ~x( j )) + ~ ~v j ~v j 2 2 j 1+z 1 1 a( 1 ) Gravitational potential term Doppler terms Cluster s gravitational redshift

7 Effect of the light-cone coordinate Kaiser (2013) Expectation value integrated over the velocity space Distribution function on the light-cone coordinate f(~x, ~v) =(1+(~ ~v))f RF (~x, ~v) z (~ ~v)+ 1 2 ~v 2 h zi = R d 3 x R d 3 v zf(~x, ~v) R d3 x R d 3 vf(~x, ~v) h(~ ~v) 2 i h ~v 2 i Statistical isotropy in velocity space Additional second order Doppler term h(~ ~v) 2 i = 1 3 h ~v 2 i h(~ ~v) 2 i h ~v 2 i = 5 2 h(~ ~v)2 i makes some contribution to the results

8 Phase space density on lightcone coordinate (Kaiser 2013) Low density High density Light cone coordinate ~v f(~x, ~v) =(1+(~ ~v))f RF (~x, ~v) ~v

9 3 Satellite galaxies virializedin halos Halo approach Central galaxy at the center of a halo, we assume the random velocity is negligible. Satellite galaxy moving in a halo with random velocity LOWZ sample (Parejko et al) Halo occupation distribution (HOD) galaxy number in a halo with mass M Satellite galaxy random velocity Virial random velocity variance 2 v,off = GM 2r vir Finger of God effect N cen N sat satellite N sat N cen central

10 SDSS III LOWZ sample / multipole power spectrum P 0 (k) kp 2 (k) kp 4 (k) kp 6 (k) Halo approach explains the galaxy distribution in redsihft space

11 Redshift of satellites relative to central galaxy h zi = h zi = (0) R d 3 x R d 3 v j z jr f(~x, ~v j ) R d3 x R d 3 v j f(~x, ~v j ) h s i h(~ ~v j) 2 i = h z(m)i Assuming central galaxy s random velocity is zero h(~ ~v j ) 2 i = 1 3 h~v2 j i NFW density profile (0) = 4 G s r 2 s = GM vir h s i =4 Z rvir 0 r vir drr 2 NFW(r) M vir c m(c) (r) = m(c) =ln(1+c) c/(1 + c) c c log(1 + c) GM vir m 2 (c) 1+c r vir h(~ ~v j ) 2 i = h z(m)i = GM vir r vir h zi = 2 v,o (M vir )= GM vir 2R vir c m(c) + c (c log(1 + c)) m 2 (c) 1+c Integration over halo mass distribution function multiplied by <N sat > R dm dn dm hn satih z(m)i R dm dn dm hn sati + 5 4

12 Result general relativity LRG (SDSS II) LOWZ CMASS h zi ( 4.6km/s) ( 3.5km/s) ( 2.5km/s) (0) h s i ( 9.7km/s) ( 7.0km/s) ( 5.4km/s) 5 2 h(~ ~v)2 i (+5.1km/s) (+3.5km/s) (+3.0km/s) r vir 1.0 h 1 Mpc 0.85 h 1 Mpc 0.79 h 1 Mpc Amplitude of signal is δz = 5 2 km/s, depending on the HOD Significant contribution from the 2 nd order Doppler term Modified gravity test? v 2 / G e Consistent with the previous results Zhao et al (13) Jimeno et a l(15) G e = 4 3 G LRG LOWZ CMASS (SDSS II) h zi ( 2.9km/s) ( 2.3km/s) ( 1.4km/s (0) h s i ( 9.7km/s) ( 7.0km/s) ( 5.4km/s) 5 2 h(~ ~v)2 i (+6.8km/s) (+4.7km/s) (+4.0km/s) r vir 1.0 h 1 Mpc 0.85 h 1 Mpc 0.79 h 1 Mpc potentially being an interesting test of modified gravity

13 4 Intracluster gas Perseus cluster gas motions observed by Hitomi X-ray satellite b a c Km/s km/s ! - The observation of the lines profile, the gas motions could be investigated with the accuracy of the order, km/s.

14 direction r χ Non-thermal pressure (unsolved problem in cluster physics) Cosmological hydrodynamical numerical simulations Intracluster gas motions can be generated in the structure formation process, non-thermal random motions (bulk motion or turbulence) contributes to the non-thermal pressure. 10 % 30 % of the total pressure This might cause discrepancy between Hydroequilibrium mass from X-ray, SZ effect v.s. Lensing mass Small scale random motions of gas non-thermal pressure

15 Formulation reference frame distribution function of gas ( 3/2 2 f RF (~x, ~v) =n(~x) exp mt (~x) 1+z j ' 1+z 1 +(1+z 1 ) m [~v ~ V (~x)] 2 2T (~x) Density Temperature Peculiar velocity (random) ) ( j, ~x( j )) + ~ ~v j ~v2 j 1+hz(x? )i = R R d d 3 vf(~x, ~v)(1 + z j ) R R d d3 vf(~x, ~v) γ line of sight direction r χ 1+hz(x? )i =1+z 1 +(1+z 1 ) R d n n(~x) ~ ~V (~x)+(~ ~V (~x)) ~ V (~x) R d n(~x)(1 + ~ V ~ ) o T (~x) m (~x) Statistically spherical symmetry Z d n(~x) ~V =0

16 Isotropy of the random motion of gas statistically 3h(~ ~V ) 2 i = h ~ V 2 i = ~ 2 rnd 1+hz(x? )i =1+z 1 +(1+z 1 ) random motion of gas non-thermal pressure Shi et al (2014) gas ~ 2 rnd = P non R d gas (~x) thermal Variance of random velocity of gas, 5 6 ~ rnd 2 (~x)+ 5 2 R d gas (~x) T (~x) m (~x) n / gas Fraction of the non-thermal pressure Nelson et al 2014 P non thermal r χ P tot γ line of sight direction r

17 Coma cluster Pnon thermal/ptotal Tgas [kev] temperature mass density σ2 rnd r [Mpc] P non thermal P total r [Mpc] ρgas [10 27 g/cm] M(< r)[10 14 M ] r [Mpc] M(<r) r [Mpc] σrnd [km/s] ψ(r) r [Mpc] ψ(r) r [Mpc]

18 Coma cluster r γ line of sight direction χ hz(x? )i gravitational total potential non-thermal pressure thermal pressure h z(x? )i = hz(x? )i hz(0)i non-thermal pressure thermal pressure total gravitational potential Amplitude of the signal is 5 10 km/s

19 Gravitational redshift of intraclustergas The relative amplitude of signal is 5 10 km/s Gravitational potential tem makes a dominant contribution to the gravitational redshift. Random motion of gas of non thermal pressure makes a slight contribution to the gravitational redshift. Measurements of lines of out skirt region is necessary.

20 5.Void Hawken et al. (2016) Stacked voids in VIPERS (VIMOS Public Extragalactic Redshift Survey) galaxies 0.55< z< (h -1 Mpc) voids galaxies inside void possible signal of gravitational redshift?

21 Simple void profile Hawken et al 2016 (r) = M(< r) 4 r 3 /3 = 3 r 3 Z r 0 dr 0 r 02 (r 0 )= c e (r/r v) (r) = 1 r 2 d dr r 3 (r) = c r r v e (r/r v) (r) r v 0.8 c r r v (r) r v 0.2 c r r v

22 Gravitational potential (r) = 3 m 2a H2 0 Z 1 r 1.5 dr 0 r 0 (r 0 ) 3 = H2 0 r 2 v 2 m a c (2/, (r/r v ) ) r Radial velocity V (r) = Hr (r) f(a) r v r r v

23 Gravitational redshift of galaxy of void R R d d 3 v j (1 + z j )f(~x, ~v j ) 1+hz(x? )i = R R d d3 v j f(~x, ~v j ) R d ng (,x? ) 1+hz(x? )i =1+z 1 +(1+z 1 ) e z(,x? ) R d ng (,x? ) e z(,x? )= H 2 ( 1 ) 1 2 a 00 ( 1 ) a( 1 ) 2 nd order Hubble term 2 j ( j, ~x( j )) + (~ ~V ) ~ V 2 Gravitational potential Doppler term Integration over line of sight coordinate η x? η 1 γ line of sight direction r v Projected radius Line of sight direction χ η h z(x? )i = hz(x? )i hz(0)i : Relative redshift

24 Line of sight direction η η 1 γ line of sight direction r v χ h z(x? )i = hz(x? )i hz(0)i η 0.12 H r v 2 (1 + z 1 ) x r v total gravitational potential Hubble Doppler Gravitational potential term dominates result Projected radius x r v

25 h z(x? )i = hz(x? )i H r v 2 r v (1 + z 1 ) Line of sight direction hz(0)i x η η η 1 Gravitational potential γ line of sight direction r v Large contamination of the Hubble term χ 0.00 Doppler term Total 2 nd Hubble term x r v

26 Gravitational redshift of galaxies of voids The amplitude of signal is hz(x? )i hz(0)i O(10 6 ) 2 r v 10h 1 (1 + z 1 ) Mpc The 2 nd order Hubble term makes a large contribution, when the range of projection over the line of sight coordinate is wide. Feasibility of the detection should be checked more carefully.

27 6. Conclusions gravitational redshift, Simple theoretical model Satellite galaxy virialized in a halo with the HOD description Amplitude of signal depends on the HOD, Doppler term makes significant contribution, test of gravity Ingracluster gas Gravitational potential tem makes a dominant contribution to the gravitational redshift, non-thermal pressure may make a contribution slightly. Void Possible signal of the gravitational redshift hz(x? )i hz(0)i O(10 6 ) r v 10h 1 Mpc 2 (1 + z 1 ) The 2 nd order Hubble term can make a large contribution, when the range of projection over the line of sight direction is wide.

28 Thank you!