Halo Model. Vinicius Miranda. Cosmology Course - Astro 321, Department of Astronomy and Astrophysics University of Chicago

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1 Introduction program Thanks Department of Astronomy and Astrophysics University of Chicago Cosmology Course - Astro 321, 2011

2 Introduction program Thanks Outline 1 Introduction 2 Linear Theory Press - Schechter Theory Dark Matter Galaxies velocities 3 program Good Qualities Bad Qualities 4 Thanks

3 Introduction program Thanks Since 1998, type IA SN indicates that the universe started to accelerate at some point in the past.

4 Introduction program Thanks Background evolution is determined by equation of state.

5 Introduction program Thanks Knowing Ω m (CMB or BAO), EOS is mainly obtained by SN assuming w constant

6 Introduction program Thanks Limitations of Kinematic Tests Theoretical alternatives: Assuming the dynamics of Dark Energy is given by a scalar field, Two very distinct possibilities are subject of intense research. Modified gravity. R µν 1 2 g µν + L µν (g µν ) = κ 2 T (m) µν (1) New non-gravitational field. R µν 1 ( ) 2 g µν = κ 2 T µν (m) + T µν (φ) (2)

7 Introduction program Thanks Example of the former approach (JF slide) Scalar Field Dark Energy aka quintessence General features: m eff < 3H 0 ~ ev (w < 0) (Potential > Kinetic Energy) V(ϕ) V ~ m 2 ϕ 2 ~ ρ crit ~ ev 4 ϕ ~ ev ~ M Planck ev (10 3 ev) 4 Ultra-light particle: Dark Energy hardly clusters, nearly smooth Equation of state: usually, w > 1 and evolves in time Hierarchy problem: Why m/φ ~ 10 61? Weak coupling: Quartic self-coupling λ φ < ϕ

8 Introduction program Thanks Example of the second approach Our Work ( V. Miranda, S. Jorás, I. Waga and M. Quartin PRL v102, p ,2009) In our work we tried to figure out if it is possible to build a viable f(r) model, without high curvature corrections, that is compatible with the existence of relativistic star and which does not suffer from Frolov s singularity. We started from the following generalization of the HSS models R f ( R) = R α R* β R * β R f ( R) = R α R* ln 1+ R* 1 n β β =1 1 f ( R) = R α R* 1 n R Hu & Sawicki s model 1+ R * n = 2 β = m α = % α / β Starobinsky s model 1 f ( R) = R % α R* 1 2 m R 1 + R*

9 Introduction program Thanks Dinamical Tests Can we differentiate the two approaches by measuring only the geometry? NO! We need the growth of structures Cluster Counts Lensing

10 Introduction program Thanks

11 Introduction program Thanks (Mortonson, Hu,Huterer) "We have shown that any observation that purports to rule out ΛCDM from the existence of massive clusters at any redshift also rules out quintessence, since quintessence models can suppress but not enhance the abundance of rare clusters compared to ΛCDM. This conclusion still holds if dark energy is a non-negligible fraction of the total density at high redshift. Once normalized to the CMB, quintessence models can only reduce the number of clusters"

12 Introduction program Thanks Gravitational Lensing: "Light Propagate as if in a medium with spatially varying index of refraction" n = 1 2Φ( x, τ) c 2 (3) JF slide Mean 3D Cluster Mass Profile from Statistical Lensing Virial mass 2-halo term NFW fit Johnston, etal 33

13 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Linear Power Spectrum In linear theory, it is possible to analyze the Dark Matter Power Spectrum with the following pieces: Initial Conditions given by inflation. R = A s ( k k norm ) ns 1 The linear growth rate of the gravitational potential as a function of time. (4) G(a) = Ψ(k norm, a) Ψ(k norm, a init ) (5)

14 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie How to calculate the linear growth rate? δ t + 2H δ t where δ(k, a) = G(a)aδ(k, 0). 4piG ρ = 0 (6) d 2 [ G 5 d(ln a) ] dg 2 w(a)ω DE(a) d ln a [1 w(a)] Ω DE(a)G = 0 (7)

15 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Transfer function that defines subhorizon evolution (Eisenstein and Hu)

16 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Press - Schechter Theory Can we estimate qualitatively the non linear power spectrum without doing a N Body simulation? ANSATZ: Choose some particle in the initial density Field. Smooth the density filed around this particle with a top hat filter of scale R. Assume the collapse of dark matter particle is spherical. For R of the other if Hubble volume, the overdensity of this smoothed field should be very small. As R decreases, the overdensity value will vary, sometimes up, sometimes down. Eventually the ovedensity will cross some threshold value δ sc (z).

17 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Press - Schechter Theory Can we estimate qualitatively the non linear power spectrum without doing a N Body simulation? ANSATZ: Choose some particle in the initial density Field. Smooth the density filed around this particle with a top hat filter of scale R. Assume the collapse of dark matter particle is spherical. For R of the other if Hubble volume, the overdensity of this smoothed field should be very small. As R decreases, the overdensity value will vary, sometimes up, sometimes down. Eventually the ovedensity will cross some threshold value δ sc (z).

18 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Press - Schechter Theory Can we estimate qualitatively the non linear power spectrum without doing a N Body simulation? ANSATZ: Choose some particle in the initial density Field. Smooth the density filed around this particle with a top hat filter of scale R. Assume the collapse of dark matter particle is spherical. For R of the other if Hubble volume, the overdensity of this smoothed field should be very small. As R decreases, the overdensity value will vary, sometimes up, sometimes down. Eventually the ovedensity will cross some threshold value δ sc (z).

19 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Press - Schechter Theory Can we estimate qualitatively the non linear power spectrum without doing a N Body simulation? ANSATZ: Choose some particle in the initial density Field. Smooth the density filed around this particle with a top hat filter of scale R. Assume the collapse of dark matter particle is spherical. For R of the other if Hubble volume, the overdensity of this smoothed field should be very small. As R decreases, the overdensity value will vary, sometimes up, sometimes down. Eventually the ovedensity will cross some threshold value δ sc (z).

20 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie First crossing of this value indicates the scale where the field is dense enough to form a virialized halo at redshift z.

21 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Spherical Collapse Equation of motion. d 2 R dt 2 = GM(R) R 2 + ΛR 3, (8) ( ) dr 2 = 2GM dt R + ΛR2 3 E i. (9) Assume Λ = 0. Make a rescale in time: dη = 2GM/R m dt/r. R = R m 2 (1 cos(η)), (10) t = R3/2 m (η sin(η)). (11) 2GM

22 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Spherical Collapse Equation of motion. d 2 R dt 2 = GM(R) R 2 + ΛR 3, (8) ( ) dr 2 = 2GM dt R + ΛR2 3 E i. (9) Assume Λ = 0. Make a rescale in time: dη = 2GM/R m dt/r. R = R m 2 (1 cos(η)), (10) t = R3/2 m (η sin(η)). (11) 2GM

23 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Spherical Collapse Equation of motion. d 2 R dt 2 = GM(R) R 2 + ΛR 3, (8) ( ) dr 2 = 2GM dt R + ΛR2 3 E i. (9) Assume Λ = 0. Make a rescale in time: dη = 2GM/R m dt/r. R = R m 2 (1 cos(η)), (10) t = R3/2 m (η sin(η)). (11) 2GM

24 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Overdensity: For δ << 1 and η << 1: 1 + δ(t) = 9 (η sin η) 2 (1 cos η) 2. (12) δ(t) = δ L (t) = 3 20 ( ) 6πt 2/3. (13) t m δ L (2t m ) = (14) Virializarion ρ(t v ) ρ(t v ) = 18π2 (15)

25 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Overdensity: For δ << 1 and η << 1: 1 + δ(t) = 9 (η sin η) 2 (1 cos η) 2. (12) δ(t) = δ L (t) = 3 20 ( ) 6πt 2/3. (13) t m δ L (2t m ) = (14) Virializarion ρ(t v ) ρ(t v ) = 18π2 (15)

26 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Overdensity: For δ << 1 and η << 1: 1 + δ(t) = 9 (η sin η) 2 (1 cos η) 2. (12) δ(t) = δ L (t) = 3 20 ( ) 6πt 2/3. (13) t m δ L (2t m ) = (14) Virializarion ρ(t v ) ρ(t v ) = 18π2 (15)

27 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Comoving number density dn = ρ νf (ν)dν M ν Spherical Collapse ν = ( ) δsc (z) 2 (16) σ(m) νf (ν) = ν 2π exp( ν/2) (17) Ellipsoidal Collapse ( ( νf (ν) = A(p) 1 + (qν) p qν ) 1/2 ) exp qν/2 2π (18) A(p) = [ p Γ(1/2 p)/ π ] 1 (19)

28 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Comoving number density dn = ρ νf (ν)dν M ν Spherical Collapse ν = ( ) δsc (z) 2 (16) σ(m) νf (ν) = ν 2π exp( ν/2) (17) Ellipsoidal Collapse ( ( νf (ν) = A(p) 1 + (qν) p qν ) 1/2 ) exp qν/2 2π (18) A(p) = [ p Γ(1/2 p)/ π ] 1 (19)

29 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Comoving number density dn = ρ νf (ν)dν M ν Spherical Collapse ν = ( ) δsc (z) 2 (16) σ(m) νf (ν) = ν 2π exp( ν/2) (17) Ellipsoidal Collapse ( ( νf (ν) = A(p) 1 + (qν) p qν ) 1/2 ) exp qν/2 2π (18) A(p) = [ p Γ(1/2 p)/ π ] 1 (19)

30 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie n_lnm

31 bias Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie δ = δ b + δ p (20) δ cp = δ c δ b (21) b(m) = 1 + qν 1 2p + [ δ c δ c 1 + (qν) p ] (22)

32 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie

33 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie z=0 Bias

34 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie NFR Profile Definition: Normalization ρ(r m) = ρ s (r/r s ) (1 + r/r s ) 2. (23) m = 4πρ s r 3 s [ ln(1 + c) c ] 1 + c (24) p(c m, z)dc = ( ) d(ln c) ln 2 [c/ c(m, z)] exp 2πσln 2 2σln 2 c c (25) c = 9 [ ] 0.20, 1+z σln m m c 0.25 and m M

35 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie NFR Profile Definition: Normalization ρ(r m) = ρ s (r/r s ) (1 + r/r s ) 2. (23) m = 4πρ s r 3 s [ ln(1 + c) c ] 1 + c (24) p(c m, z)dc = ( ) d(ln c) ln 2 [c/ c(m, z)] exp 2πσln 2 2σln 2 c c (25) c = 9 [ ] 0.20, 1+z σln m m c 0.25 and m M

36 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie

37 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie NFR Profile Normalize Fourier Transform: u(k m) = d 3 xρ( x m)e i k x d 3 xρ( x m) (26) For NFW profile: u(k m) = 4πρ sr 3 s m [ sin(kr s ) [Si([1 + c]kr s ) Si(kr s )] sin(ckr ] s) (1 + c)kr s (27) + 4πρ sr 3 s m [cos(kr s) [Ci([1 + c]kr s ) Ci(kr s )]]

38 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie

39 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie z=0 1 F_NFW M 1e 10, 1e 12, 1e 14, 1e

40 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Dark Matter Power Spectrum P(k) = P 1h (k) + P 2h (k) (28) P 1h (k) = ( d ln M M ρ m (z = 0) ) 2 dn d ln(m) u2 (k, M) (29) ( P 2h (k) = ( d ln M M ρ m (z = 0) ) ) dn 2 b(m)u(k, M) P l (k) d ln(m) (30)

41 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie z = 0 2 k Power Spectra 100 Linear Power Spectra Halo Term 2 Halo Term k

42 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Galaxy Power Spectrum P gal (k) = Pgal 1h (k) + P2h gal (k) (31) P 1h (k) = d ln M N gal(n gal 1) M n 2 gal dn d ln(m) up (k, M) (32) ( P 2h (k) = d ln M N gal M n gal dn d ln(m) ) 2 b(m)u(k, M) P l (k) (33)

43 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Galaxy Power Spectrum If N gal M < 1, then p = 1. Else p = 2. Variance: Mean: N gal (N gal 1) M = α(m) N gal where (34) ( ) α = log M/(10 11 M h 1 ) N gal M = 0 if M M h 1 (35) ( ) 0.9 = M/( ) if M M h 1 ( ) 0.8 ( ) 0.9 = 0.7 M/( ) + M/( ) if M (M h 1 )

44 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie

45 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie z = 0 k Power Spectra 100 Linear Power Spectra 1 1 Halo Term 2 Halo Term k

46 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie velocities p(v m)dv = Probability of a particle in a halo of mass m, has velocity in the range dv about v. f (v) = dmn(m)p(v m) dmmn(m) (36) Shape of p(v m) : 2 components. Virial Motions: Maxwell Boltzmann Distribution. Gaussian Field: Halo motions also obey Maxwell Boltzmann

47 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Virial Motions: σvirial 2 GM GMr 2 r r ( 3 ) 3m 2/3 ( vir ρ crit ) (37) 4π vir ρ crit ( ) ( ) H(z) 1/3 σvirial 2 m = M km/s (38) /h H 0

48 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie Halo Motions: σ halo = d log(ag) d log(a) σ 1 1 σ0 4/σ4 1 σ4 1 (39) σj 2 (R) = 1 2π 2 dkk 2+2j P lin (k)w 2 [kr] (40)

49 Introduction program Thanks Linear Theory Press - Schechter Theory Dark Matter Galaxie z = 0 k v

50 Introduction program Thanks Good Qualities Bad Qualities Good Qualities It is organized. (readable) Calculates matter power spectrum, galaxy power spectrum, and peculiar velocity distribution. Input file allows you to change many physical and optimization parameters.

51 Introduction program Thanks Good Qualities Bad Qualities Good Qualities It is organized. (readable) Calculates matter power spectrum, galaxy power spectrum, and peculiar velocity distribution. Input file allows you to change many physical and optimization parameters.

52 Introduction program Thanks Good Qualities Bad Qualities Good Qualities It is organized. (readable) Calculates matter power spectrum, galaxy power spectrum, and peculiar velocity distribution. Input file allows you to change many physical and optimization parameters.

53 Introduction program Thanks Good Qualities Bad Qualities Allows you to choose between 2 different dark energy parameterizations: w de = w 0 + w 1 z (41) z w de = w 0 + w z. Allows you to choose between 3 different concentration parameters: c = 10 (42) c = c c given by p(c).

54 Introduction program Thanks Good Qualities Bad Qualities Input.txt

55 Introduction program Thanks Good Qualities Bad Qualities

56 Introduction program Thanks Good Qualities Bad Qualities Ouput.txt

57 Introduction program Thanks Good Qualities Bad Qualities

58 Introduction program Thanks Good Qualities Bad Qualities

59 Introduction program Thanks Good Qualities Bad Qualities run: make -fmakefile.mak final_project

60 Introduction program Thanks Good Qualities Bad Qualities Optimization It is optimized. Run: seconds/point. For a entire plot: 2-10 minutes (30-45 seconds to calculate and fit sigma + 1s/point). Can be better? Use of inline functions - Integral code that double (not triple) the number of points on each call...

61 Introduction program Thanks Good Qualities Bad Qualities Bad Qualities Normalization bug (someone really should make an independent check). Not written in a C++ class style. NR copyrighted $$$ code. However, integrands are written in gsl function style. Thus, it should be simple to use GSL integration code.

62 Introduction program Thanks Good Qualities Bad Qualities Bad Qualities Normalization bug (someone really should make an independent check). Not written in a C++ class style. NR copyrighted $$$ code. However, integrands are written in gsl function style. Thus, it should be simple to use GSL integration code.

63 Introduction program Thanks Good Qualities Bad Qualities Needs to incorporate more physics: subhalos structure for k >> 1Mpc/h 1 ), weak lensing, momentum power spectra, dark matter - galaxy cross power spectrum.

64 Introduction program Thanks Thanks and sorry about the thousands of questions (some "out of phase" with the lecture subject!) Funny thing about my "impatience". First Time I attended a class on was during the 2008 BSCG. Ravi Sheth BSCG proceedings: " I thank (..) V. Miranda for raising so many interesting questions, M. Calvao for keeping him under control.." Thanks for Keeping me under control! :)

65 Introduction program Thanks Thanks and sorry about the thousands of questions (some "out of phase" with the lecture subject!) Funny thing about my "impatience". First Time I attended a class on was during the 2008 BSCG. Ravi Sheth BSCG proceedings: " I thank (..) V. Miranda for raising so many interesting questions, M. Calvao for keeping him under control.." Thanks for Keeping me under control! :)

66 Introduction program Thanks Thanks and sorry about the thousands of questions (some "out of phase" with the lecture subject!) Funny thing about my "impatience". First Time I attended a class on was during the 2008 BSCG. Ravi Sheth BSCG proceedings: " I thank (..) V. Miranda for raising so many interesting questions, M. Calvao for keeping him under control.." Thanks for Keeping me under control! :)

N-body Simulations. Initial conditions: What kind of Dark Matter? How much Dark Matter? Initial density fluctuations P(k) GRAVITY

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