Analytical Model for Non-thermal Pressure in Galaxy Clusters - PowerPoint PPT Presentation

Analytical Model for Non-thermal Pressure in Galaxy Clusters Eiichiro Komatsu (MPA) Gravity Lunch Seminar, Princeton University February 13, 2015

References • Shi & EK, MNRAS, 442, 512 (2014) • Shi, EK, Nelson & Nagai, MNRAS, 448, 1020 (2015) Xun Shi (MPA) Kaylea Nelson (Yale)

Motivation • We wish to determine the mass of galaxy clusters accurately

Where is a galaxy cluster? Subaru image of RXJ1347-1145 (Medezinski et al. 2009) http://wise-obs.tau.ac.il/~elinor/clusters

Where is a galaxy cluster? Subaru image of RXJ1347-1145 (Medezinski et al. 2009) http://wise-obs.tau.ac.il/~elinor/clusters

Subaru image of RXJ1347-1145 (Medezinski et al. 2009) http://wise-obs.tau.ac.il/~elinor/clusters

Hubble image of RXJ1347-1145 (Bradac et al. 2008)

Chandra X-ray image of RXJ1347-1145 (Johnson et al. 2012)

Image of the Sunyaev-Zel’dovich effect at 150 GHz [Nobeyama Radio Observatory] (Komatsu et al. 2001) Chandra X-ray image of RXJ1347-1145 (Johnson et al. 2012)

Multi-wavelength Data Z σ T k B Z dl n 2 I X = e Λ ( T X ) I SZ = g ν dl n e T e m e c 2 Optical: X-ray: SZ [microwave]: •10 2–3 galaxies •hot gas (10 7–8 K) •hot gas (10 7-8 K) •velocity dispersion •spectroscopic T X •electron pressure •gravitational lensing •Intensity ~ n e2 L •Intensity ~ n e T e L

Galaxy Cluster Counts • We count galaxy clusters over a certain region in the sky [with the solid angle Ω obs ] • Our ability to detect clusters is limited by noise [limiting flux, F lim ] • For a comoving number density of clusters per unit mass, dn/dM, the observed number count is Z ∞ Z ∞ dz d 2 V dn dM N = Ω obs dF dzd Ω dM dF 0 F lim ( z )

Dark Energy vs Galaxy Clusters • Counting galaxy clusters provides information on dark energy by • Providing the comoving volume element which depends on the distance [d A (z)] and the expansion rate [H(z)] • Providing the amplitude of matter fluctuations as a function of redshifts, σ 8 (z)

6 ’redshift_volume_1000_w1.txt’u 1:($2*1e-9) ’redshift_volume_1000_w09.txt’u 1:($2*1e-9) ’redshift_volume_1000_w11.txt’u 1:($2*1e-9) Comoving Volume, V(<z), over 1000 deg 2 [Gpc 3 /h 3 ] Z z d 2 V 5 Z dz 0 V ( < z ) = 1000 deg 2 d Ω dz 0 d Ω 0 4 w =–1.1 3 w =–0.9 2 w=P DE / ρ DE 1 Ω m = 0 . 3 Ω de = 0 . 7 0 0 0.5 1 1.5 2 Redshift, z

Mass Function, dn/dM • The comoving number density per unit mass range, dn/dM, is exponentially sensitive to the amplitude of matter fluctuations, σ 8 , for high-mass, rare objects • By “high-mass objects”, we mean “high peaks,” satisfying 1.68/ σ (M) > 1

Mass Function, dn/dM • The comoving number density per unit mass range, dn/dM, is exponentially sensitive to the amplitude of matter fluctuations, σ 8 , for high-mass, rare objects • By “high-mass objects”, we mean “high peaks,” satisfying 1.68/ σ (M) > 1

0.01 Comoving Number Density of DM Halos [h 3 /Mpc 3 ] (Tinker et al. 2008) ’Mh_dndlnMh_z0_s807.txt’ ’Mh_dndlnMh_z05_s807.txt’ ’Mh_dndlnMh_z1_s807.txt’ ’Mh_dndlnMh_z0_s808.txt’ ’Mh_dndlnMh_z05_s808.txt’ 0.0001 ’Mh_dndlnMh_z1_s808.txt’ z=0 σ 8 =0.8 1e-06 z=0.5 σ 8 =0.7 Ω b = 0 . 05 , Ω cdm = 0 . 25 σ 8 =0.8 Ω de = 0 . 7 , w = − 1 z=1 H 0 = 70 km / s / Mpc σ 8 =0.7 1e-08 σ 8 =0.8 • dn/dM falls off exponentially in the cluster-mass range [M>10 14 Msun/h], 1e-10 σ 8 =0.7 and is very sensitive to the value of σ 8 and redshift 1e-12 • This can be understood by the exponential dependence on 1.68/ σ (M,z) 1e-14 1e+14 1e+15 Dark Matter Halo Mass [Msun/h]

Cumulative mass function from X-ray cluster samples Chandra Cosmology Project Vikhlinin et al. (2009)

Cumulative mass function from X-ray cluster samples Chandra Cosmology Project Vikhlinin et al. (2009)

The Challenge Z ∞ Z ∞ dz d 2 V dn dM N = Ω obs dF dzd Ω dM dF 0 F lim ( z ) • Cluster masses are not directly Mis-estimation of the masses observable from the observables severely compromises the statistical power of galaxy clusters as a DE probe • The observables “F” include • Number of cluster member • X-ray intensity [X-ray] galaxies [optical] • X-ray spectroscopic • Velocity dispersion [optical] temperature [X-ray] • Strong- and weak-lensing • SZ intensity [microwave] masses [optical]

HSE: the leading method • Currently, most of the mass cluster estimations rely on the X-ray data and the assumption of hydrostatic equilibrium [HSE] • The measured X-ray intensity is proportional to ∫ n e2 dl, which can be converted into a radial profile of electron density, n e (r) , assuming spherical symmetry • The spectroscopic data give a radial electron temperature profile, T e (r) These measurements give an estimate of the electron pressure profile , P e (r)=n e (r)k B T e (r)

HSE: the leading method • Recently, more SZ measurements, which are proportional to ∫ n e k B T e dl, are used to directly obtain an estimate of the electron pressure profile

HSE: the leading method • In the usual HSE assumption, the total gas pressure [including contributions from ions and electrons] gradient balances against gravity [X=0.74 is the hydrogen mass abundance] • n gas = n ion +n e = [(3+5X)/(2+2X)]n e = 1.93n e • Assuming T ion =T e [which is not always satisfied!] • P gas (r) = 1.93P e (r) 1 ∂ P gas ( r ) = − GM ( < r ) • Then, HSE ρ gas ( r ) r 2 ∂ r • gives an estimate of the total mass of a cluster, M

Limitation of HSE 1 ∂ P gas ( r ) = − GM ( < r ) • The HSE equation ρ gas ( r ) r 2 ∂ r • only includes thermal pressure; however, not all kinetic energy of in-falling gas is thermalised • There is evidence that there is significant non- thermal pressure support coming from bulk motion of gas (e.g., turbulence) • Therefore, the correct equation to use would be 1 ∂ [ P th ( r ) + P non − th ( r )] = − GM ( < r ) r 2 ρ gas ( r ) ∂ r Not including P non-th leads to underestimation of the cluster mass!

Planck Collaboration XX, arXiv:1303.5080v2 Planck SZ Cluster Count, N(z) Planck CMB prediction with M HSE /M true =0.8 Planck CMB+SZ best fit with M HSE /M true =0.6 40% HSE mass bias?!

Shaw, Nagai, Bhattacharya & Lau (2010) • Simulations by Shaw et al. show that the non-thermal pressure [by bulk motion of gas] divided by the total pressure increases toward large radii. But why?

Battaglia, Bond, Pfrommer & Sievers (2012) AGN feedback, z = 0 1.1 x 10 14 M O • < M 200 < 1.7 x 10 14 M O • 1.7 x 10 14 M O • < M 200 < 2.7 x 10 14 M O • 2.7 x 10 14 M O • < M 200 < 4.2 x 10 14 M O 1.0 • 4.2 x 10 14 M O • < M 200 < 6.5 x 10 14 M O • 6.5 x 10 14 M O • < M 200 < 1.01 x 10 15 M O • 1.01 x 10 15 M O • < M 200 < 1.57 x 10 15 M O • Shaw et al. 2010 P kin / P th Trac et al. 2010 0.1 R 500 R vir 0.1 1.0 r / R 200 • Battaglia et al.’s simulations show that the ratio increases for larger masses, and…

Battaglia, Bond, Pfrommer & Sievers (2012) AGN feedback, 1.7 x 10 14 M O • < M 200 < 2.7 x 10 14 M O • z = 0 z = 0.3 z = 0.5 1.0 z = 0.7 z = 1.0 z = 1.5 Shaw et al. 2010, z = 0 P kin / P th Shaw et al. 2010, z = 1 0.1 R 500 R vir 0.1 1.0 r / R 200 • …increases for larger redshifts. But why?

Part I: Analytical Model Shi & Komatsu (2014) Xun Shi (MPA)

Analytical Model for Non- Thermal Pressure • Basic idea 1 : non-thermal motion of gas in clusters is sourced by the mass growth of clusters [via mergers and mass accretion] with efficiency η • Basic idea 2 : induced non-thermal motion decays and thermalises in a dynamical time scale • Putting these ideas into a differential equation: [ σ 2 =P/ ρ gas ] Shi & Komatsu (2014)

Finding the decay time, t d • Think of non-thermal motion as turbulence • Turbulence consists of “eddies” with different sizes

Finding the decay time, t d • Largest eddies carry the largest energy • Large eddies are unstable. They break up into smaller eddies, and transfer energy from large-scales to small- scales

Finding the decay time, t d • Assumption: the size of the largest eddies at a radius r from the centre of a cluster is proportional to r • Typical peculiar velocity of turbulence is r GM ( < r ) v ( r ) = r Ω ( r ) = r • Breaking up of eddies occurs at the time scale of 2 π t d ≈ Ω ( r ) ≡ t dynamical • We thus write: t d ≡ β 2 t dynamical

Shi & Komatsu (2014) Dynamical Time • Dynamical time increases toward large radii. Non-thermal motion decays into heat faster in the inner region

Source term • Define the “growth time” as ◆ − 1 ✓ d σ 2 tot t growth ≡ σ 2 tot dt

Shi & Komatsu (2014) Growth Time • Growth time increases toward lower redshifts and smaller masses. Non-thermal motion is injected more efficiently at high redshifts and for large-mass halos

Shi & Komatsu (2014) Non-thermal Fraction, η = turbulence injection efficiency f nth =P nth /(P th +P nth ) o t t fi e s t a n m o i i t x a o l u r p m p i a s o r d β = [turbulence y h decay time] / 2t dyn Non-thermal fraction increases with radii because of slower turbulence decay in the outskirts