Cosmology

Cosmology

B.F. Roukema

(c) CC BY-SA 4.0 2020-09-30

Introduction

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 2

basic cosmology model:

Introduction

basic cosmology model:

■ GR: curvature = matter-energy content

Introduction

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 2

basic cosmology model:

■ GR: curvature = matter-energy content

■ verbal averaging: homogeneous, isotropic spatial slices

Introduction

basic cosmology model:

■ GR: curvature = matter-energy content

■ verbal averaging: homogeneous, isotropic spatial slices

■ shape of space (curvature + topology)

Introduction

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 2

basic cosmology model:

■ GR: curvature = matter-energy content

■ verbal averaging: homogeneous, isotropic spatial slices

■ shape of space (curvature + topology)

+ observations of expansion ⇒ hot big bang ⇒ black body radiation, nucleosynthesis

Introduction

basic cosmology model:

■ GR: curvature = matter-energy content

■ verbal averaging: homogeneous, isotropic spatial slices

■ shape of space (curvature + topology)

+ observations of expansion ⇒ hot big bang ⇒ black body radiation, nucleosynthesis

Introduction

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 2

basic cosmology model:

■ GR: curvature = matter-energy content

■ verbal averaging: homogeneous, isotropic spatial slices

■ shape of space (curvature + topology)

+ observations of expansion ⇒ hot big bang ⇒ black body radiation, nucleosynthesis

■ but: ∃ galaxies ⇒ inhomogeneous, anisotropic spatial slices

Introduction

basic cosmology model:

■ GR: curvature = matter-energy content

■ verbal averaging: homogeneous, isotropic spatial slices

■ shape of space (curvature + topology)

+ observations of expansion ⇒ hot big bang ⇒ black body radiation, nucleosynthesis

■ but: ∃ galaxies ⇒ inhomogeneous, anisotropic spatial slices

■ standard model: density perturbations (anisotropy)

Introduction

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 2

basic cosmology model:

■ GR: curvature = matter-energy content

■ verbal averaging: homogeneous, isotropic spatial slices

■ shape of space (curvature + topology)

+ observations of expansion ⇒ hot big bang ⇒ black body radiation, nucleosynthesis

■ but: ∃ galaxies ⇒ inhomogeneous, anisotropic spatial slices

■ standard model: density perturbations (anisotropy)

Introduction

basic cosmology model:

■ GR: curvature = matter-energy content

■ verbal averaging: homogeneous, isotropic spatial slices

■ shape of space (curvature + topology)

+ observations of expansion ⇒ hot big bang ⇒ black body radiation, nucleosynthesis

■ but: ∃ galaxies ⇒ inhomogeneous, anisotropic spatial slices

■ standard model: density perturbations (anisotropy)

verbal averaging

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 3

verbal averaging

■ w:Cosmological principle

verbal averaging

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 3

■ w:Cosmological principle

■ practical meaning:

verbal averaging

■ w:Cosmological principle

■ practical meaning:

1. assume homogeneity and isotropy

2. find the (differential 4-pseudo-manifold, metric) pairs (M, g) that solve G = 8πT

verbal averaging

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 3

■ w:Cosmological principle

■ practical meaning:

1. assume homogeneity and isotropy

2. find the (differential 4-pseudo-manifold, metric) pairs (M, g) that solve G = 8πT

3. assume that (M, g) remains unchanged if we add density perturbations to an early time slice

verbal averaging

verbal averaging

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 4

verbal averaging

■ w:Comoving coordinates ■

verbal averaging

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 4

■ w:Comoving coordinates ■

∆x(t) = a(t)∆r

verbal averaging

verbal averaging

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 4

■ w:Comoving coordinates ■

Z (t,r₂,θ,φ)

(t,r1,θ,φ)

ds = a(t)∆r = a(t)|r2 − r1|

verbal averaging

where all expansion/contraction → w:scale factor a(t)

FLRW metric

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 5

FLRW metric

■ w:Friedmann–Lemaître–Robertson–Walker metric

■ w: w:

w:Howard Percy Robertson w:Arthur Geoffrey Walker

FLRW metric

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 6

FLRW metric

FLRW metric

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 6

FLRW metric

FLRW metric

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 6

FLRW metric

ds2 _{= −dt}2 + a2(t) [dr2 + r_⊥2 (dθ2 + cos2 θdφ2)]

■ aside: conformal coordinates: may define

u := Z

dt a(t)

FLRW metric

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 6

ds2 _{= −dt}2 + a2(t) [dr2 + r_⊥2 (dθ2 + cos2 θdφ2)]

■ aside: conformal coordinates: may define

u := Z

dt a(t) ⇒

FLRW metric

ds2 _{= −dt}2 + a2(t) [dr2 + r_⊥2 (dθ2 + cos2 θdφ2)]

■ aside: conformal coordinates: may define

u := Z

dt a(t) ⇒

ds2 = a2(u) −du2 + dr2 + r_⊥2 (dθ2 + cos2 θdφ2)

FLRW metric

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 6

ds2 _{= −dt}2 + a2(t) [dr2 + r_⊥2 (dθ2 + cos2 θdφ2)] where r⊥ :=    RC sinh _Rr_C k < 0 r k = 0 RC sin _Rr C k > 0

curvature

■ on a spatial slice (fixed value of t):

x + y

x

y

curvature

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 7

■ on a spatial slice (fixed value of t):

y

x

< x + y

k > 0

curvature

■ on a spatial slice (fixed value of t):

x

y

> x + y

k < 0

2D curvature intuition: k > 0

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 8

2D curvature intuition: k > 0

ds2_|φ=const,a=1 = dr2 + r_⊥2 dθ2, where r⊥ := RC sin _Rr

2D curvature intuition: k > 0

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 8

ds2_|φ=const,a=1 = dr2 + r_⊥2 dθ2, where r⊥ := RC sin _Rr

C

2D curvature intuition: k > 0

ds2_|φ=const,a=1 = dr2 + r_⊥2 dθ2, where r⊥ := RC sin _Rr

C

w: (al-Biruni, c. 1000 CE)

2D topology intuition (k = 0)

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 9

2D topology intuition (k = 0)

2D topology intuition (k = 0)

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 9

■ intuition 1: embed in higher dim. space

A

A

2D topology intuition (k = 0)

■ intuition 1: embed in higher dim. space

2D topology intuition (k = 0)

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 9

■ intuition 1: embed in higher dim. space

2D topology intuition (k = 0)

■ intuition 1: embed in higher dim. space

■ intuition 2: fundamental domain

2D topology intuition (k = 0)

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 9

■ intuition 1: embed in higher dim. space

■ intuition 2: fundamental domain

expansion

■ Hubble law (Lemaître 1927; ADS:1927ASSB...47...49L):

expansion

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 10

■ Hubble law (Lemaître 1927; ADS:1927ASSB...47...49L):

expansion

■ Hubble law (Lemaître 1927; ADS:1927ASSB...47...49L):

˙a(t₀)/a(t_{0) ≈ 600 km/s/Mpc > 0 (also Hubble 1929)}

expansion

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 10

■ Hubble law (Lemaître 1927; ADS:1927ASSB...47...49L):

˙a(t₀)/a(t_{0) ≈ 600 km/s/Mpc > 0 (also Hubble 1929)}

■ so what is the “Big Bang”?

expansion

■ Hubble law (Lemaître 1927; ADS:1927ASSB...47...49L):

˙a(t₀)/a(t_{0) ≈ 600 km/s/Mpc > 0 (also Hubble 1929)}

■ so what is the “Big Bang”?

expansion

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 10

■ Hubble law (Lemaître 1927; ADS:1927ASSB...47...49L):

˙a(t₀)/a(t_{0) ≈ 600 km/s/Mpc > 0 (also Hubble 1929)}

■ so what is the “Big Bang”?

expansion

■ Hubble law (Lemaître 1927; ADS:1927ASSB...47...49L):

˙a(t₀)/a(t_{0) ≈ 600 km/s/Mpc > 0 (also Hubble 1929)}

■ so what is the “Big Bang”?

■ it is: ∃tb such that t → t+_b ⇒ a(t) → 0+

expansion

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 11

expansion

■ matter density: ρm ∝ a−3

■ wavelength: λ ∝ a [GR—transport four-velocity: Synge (1960);

expansion

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 11

■ matter density: ρm ∝ a−3

■ wavelength: λ ∝ a [GR—transport four-velocity: Synge (1960);

Narlikar (1994; ADS:1994AmJPh..62..903N)] ⇔ λ_obs λ_em = a_obs a_em

expansion

■ matter density: ρm ∝ a−3

■ wavelength: λ ∝ a [GR—transport four-velocity: Synge (1960);

Narlikar (1994; ADS:1994AmJPh..62..903N)] ■ λ_obs λ_em = 1 a_em (Defn: a0 := 1)

expansion

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 11

■ matter density: ρm ∝ a−3

■ wavelength: λ ∝ a [GR—transport four-velocity: Synge (1960);

Narlikar (1994; ADS:1994AmJPh..62..903N)]

■

1 + z = 1

aem

expansion

■ matter density: ρm ∝ a−3 = (1 + z)3

■ wavelength: λ ∝ a [GR—transport four-velocity: Synge (1960);

Narlikar (1994; ADS:1994AmJPh..62..903N)]

■

1 + z = a−1 (light-cone convention: a often means aem)

expansion

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 11

■ matter density: ρm ∝ a−3 = (1 + z)3

■ wavelength: λ ∝ a [GR—transport four-velocity: Synge (1960);

Narlikar (1994; ADS:1994AmJPh..62..903N)]

■

1 + z = a−1 (light-cone convention: a often means aem)

Black body

■ Planck’s Law: I(ν, T ) = 2hν_c23 1

Black body

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 12

■ Planck’s Law: I(ν, T ) = 2hν_c23 1

ekThν −1

Black body

■ Planck’s Law: I(ν, T ) = 2hν_c23 1

ekThν −1

■ wavelength: λ ∝ a ⇒ frequency: ν ∝ a−1 = (1 + z)

Black body

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 12

■ Planck’s Law: I(ν, T ) = 2hν_c23 1

ekThν −1

■ wavelength: λ ∝ a ⇒ frequency: ν ∝ a−1 = (1 + z)

■ ⇒ temperature: kT ∝ hν ∝ (1 + z)

■ z ≫ 1 ⇒ early Universe dominated by hot, dense plasma = protons,

Black body

■ Planck’s Law: I(ν, T ) = 2hν_c23 1

ekThν −1

■ wavelength: λ ∝ a ⇒ frequency: ν ∝ a−1 = (1 + z)

■ ⇒ temperature: kT ∝ hν ∝ (1 + z)

■ z ≫ 1 ⇒ early Universe dominated by hot, dense plasma = protons,

electrons, photons

Black body

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 12

■ Planck’s Law: I(ν, T ) = 2hν_c23 1

ekThν −1

■ wavelength: λ ∝ a ⇒ frequency: ν ∝ a−1 = (1 + z)

■ ⇒ temperature: kT ∝ hν ∝ (1 + z)

■ z ≫ 1 ⇒ early Universe dominated by hot, dense plasma = protons,

electrons, photons

CMB discovery: McKellar 1941

■ T ≈ 2.3 K — Andrew McKellar (1941; ADS:1941PDAO....7..251M)

from observations by Walter S. Adams (1941;

ADS:1941ApJ....93...11A)

Black body: COBE (∼ 1992)

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 14

Black body: COBE (∼ 1992)

Black body: COBE (∼ 1992)

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 14

BBN: Big bang nucleosynthesis

BBN: Big bang nucleosynthesis

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 15

■ Alpher, Bethe, & Gamow (1948; ADS:1948PhRv...73..803A)

main reactions of varying probability:

p + n _→ 2H + γ p + 2_{H →} 3He + γ 2_{H +} 2 H → 3He + n 2_{H +} 2_H → 3H + p 3_{He +} 2 H → 4He + p 3_{H +} 2 H → 4He + n

BBN: Big bang nucleosynthesis

■ Alpher, Bethe, & Gamow (1948; ADS:1948PhRv...73..803A)

main reactions of varying probability:

p + n _→ 2H + γ p + 2_{H →} 3He + γ 2_{H +} 2 H → 3He + n 2_{H +} 2_H → 3H + p 3_{He +} 2 H → 4He + p 3_{H +} 2 H → 4He + n

FLRW: a(t) =?

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 16

■ second choice FLRW coord system: r := w:orthographic projection

FLRW: a(t) =?

■ second choice FLRW coord system: r := w:orthographic projection

of radial comoving distance (cf r := radial comoving distance) ⇒ coord singularity at equator (:= π/2 from centre) if k > 0

FLRW: a(t) =?

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 16

■ second choice FLRW coord system: r := w:orthographic projection

of radial comoving distance (cf r := radial comoving distance) ⇒ coord singularity at equator (:= π/2 from centre) if k > 0

FLRW: a(t) =?

■ second choice FLRW coord system: r := w:orthographic projection

of radial comoving distance (cf r := radial comoving distance) ⇒ coord singularity at equator (:= π/2 from centre) if k > 0

■ ds2 _{= −dt}2 + a2(t)  dr2 1 − kr2 + r 2_(dθ2 _{+ cos}2 _θdφ2₎ 

FLRW: a(t) =?

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 16

■ ds2 _{= −dt}2 + a2(t)  dr2 1 − kr2 + r 2_(dθ2 _{+ cos}2 _θdφ2₎ 

FLRW: a(t) =?

■ universe content: diag(T) = (−ρ, p, p, p)

■ maxima: calculate G and G = 8πT and simplify:

FLRW: a(t) =?

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 16

■ ds2 _{= −dt}2 + a2(t)  dr2 1 − kr2 + r 2_(dθ2 _{+ cos}2 _θdφ2₎ 

■ universe content: diag(T) = (−ρ, p, p, p)

■ maxima: calculate G and G = 8πT and simplify:

https://cosmo.torun.pl/Cosmo/FLRWEquationsGR ■ Friedmann Eqn: c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3

FLRW: a(t) =?

■ universe content: diag(T) = (−ρ, p, p, p)

■ maxima: calculate G and G = 8πT and simplify:

https://cosmo.torun.pl/Cosmo/FLRWEquationsGR ■ Friedmann Eqn: c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 ■

acceleration Eqn: ¨a

a = −

4 π G (ρ + 3 p/c2) 3

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: _a˙a22 =

8 π G ρ 3 −

c2 k a2

FLRW matter-dominated epoch

■ Friedmann Eqn: _a˙a22 =

8 π G ρ 3 −

c2 k a2

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: _a˙a22 =

8 π G ρ 3 −

c2 k a2

FLRW matter-dominated epoch

■ Friedmann Eqn: _a˙a22 =

8 π G ρm0

3 a3 − c 2 _k

a2

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

Defn: Hubble parameter H := ˙a/a

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

Defn: Hubble parameter H := ˙a/a

⇒ Hubble constant H0 := H(z = 0)

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

Defn: Hubble parameter H := ˙a/a

⇒ Hubble constant H0 := H(z = 0)

⇒ k = 0 case: _a˙a = _3t2 ⇒ H(t) = _3t2 ;

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

Defn: Hubble parameter H := ˙a/a

⇒ Hubble constant H0 := H(z = 0)

⇒ k = 0 case: _a˙a = _3t2

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

■ Lemaître (1927) ADS:1927ASSB...47...49L: H0 ≈ 600

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

■ Lemaître (1927) ADS:1927ASSB...47...49L: H0 ≈ 0.6 Gyr−1

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

■ Lemaître (1927) ADS:1927ASSB...47...49L: H0 ≈ 0.6 Gyr−1

■ Hubble (1929) ADS:1929PNAS...15..168H: H0 ≈ 500

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

■ Lemaître (1927) ADS:1927ASSB...47...49L: H0 ≈ 0.6 Gyr−1

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

■ Lemaître (1927) ADS:1927ASSB...47...49L: H0 ≈ 0.6 Gyr−1

■ Hubble (1929) ADS:1929PNAS...15..168H: H0 ≈ 0.5 Gyr−1

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

■ Lemaître (1927) ADS:1927ASSB...47...49L: H0 ≈ 0.6 Gyr−1

■ Hubble (1929) ADS:1929PNAS...15..168H: H0 ≈ 0.5 Gyr−1

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

■ Lemaître (1927) ADS:1927ASSB...47...49L: H0 ≈ 0.6 Gyr−1

■ Hubble (1929) ADS:1929PNAS...15..168H: H0 ≈ 0.5 Gyr−1

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

■ Lemaître (1927) ADS:1927ASSB...47...49L: H0 ≈ 0.6 Gyr−1

■ Hubble (1929) ADS:1929PNAS...15..168H: H0 ≈ 0.5 Gyr−1

⇒ EdS would give t0 = _3H2₀ ≈ 1.3 Gyr < tEarth ≈ 4.5 Gyr

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

■ Lemaître (1927) ADS:1927ASSB...47...49L: H0 ≈ 0.6 Gyr−1

■ Hubble (1929) ADS:1929PNAS...15..168H: H0 ≈ 0.5 Gyr−1

⇒ EdS would give t0 = _3H2₀ ≈ 1.3 Gyr < tEarth ≈ 4.5 Gyr

FLRW matter-dominated epoch

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

■ Lemaître (1927) ADS:1927ASSB...47...49L: H0 ≈ 0.6 Gyr−1

■ Hubble (1929) ADS:1929PNAS...15..168H: H0 ≈ 0.5 Gyr−1

⇒ EdS would give t0 = _3H2₀ ≈ 1.3 Gyr < tEarth ≈ 4.5 Gyr

FLRW matter-dominated epoch

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 17

■ Friedmann Eqn: ˙a2 = 8 π G ρ_{3 a} m0 − c2 k

■ matter-dominated epoch: ρ = ρm = ρm0 a−3 ■ k = 0 case: a =  t t0 2/3

Einstein–de Sitter model (EdS)

⇒ H(t) = _3t2 ; H₀ = H(t₀) = _3t2

0

■ Lemaître (1927) ADS:1927ASSB...47...49L: H0 ≈ 0.6 Gyr−1

■ Hubble (1929) ADS:1929PNAS...15..168H: H0 ≈ 0.5 Gyr−1

⇒ EdS would give t0 = _3H2₀ ≈ 1.3 Gyr < tEarth ≈ 4.5 Gyr

FLRW: ρ

■ Friedmann Eqn: _a˙a22 =

8 π G ρ 3 −

c2 k a2

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 18

FLRW: ρ

■ Friedmann Eqn: H2 = 8 π G ρ₃ − c_a22k

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 18

■ Friedmann Eqn: H2 = 8 π G ρ₃ − c_a22k

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ ρm0 = 3H

2 0

FLRW: ρ

■ Friedmann Eqn: H2 = 8 π G ρ₃ − c_a22k

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ ρm0 = 3H

2 0

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 18

■ Friedmann Eqn: H2 = 8 π G ρ₃ − c_a22k

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ ρm0 = 3H 2 0 8πG ⇔ k = 0 flat ◆ ρm0 > 3H 2 0 8πG ⇔ k > 0

FLRW: ρ

■ Friedmann Eqn: H2 = 8 π G ρ₃ − c_a22k

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ ρm0 = 3H 2 0 8πG ⇔ k = 0 flat ◆ ρm0 > 3H 2 0 8πG ⇔ k > 0 spherical

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 18

■ Friedmann Eqn: H2 = 8 π G ρ₃ − c_a22k

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ ρm0 = 3H 2 0 8πG ⇔ k = 0 flat ◆ ρm0 > 3H 2 0 8πG ⇔ k > 0 spherical ◆ ρm0 < 3H 2 0 8πG ⇔ k < 0

FLRW: ρ

■ Friedmann Eqn: H2 = 8 π G ρ₃ − c_a22k

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ ρm0 = 3H 2 0 8πG ⇔ k = 0 flat ◆ ρm0 > 3H 2 0 8πG ⇔ k > 0 spherical ◆ ρm0 < 3H 2 0 8πG ⇔ k < 0 hyperbolic

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 19

FLRW: ρ

■ Friedmann Eqn: H2 = 8 π G ρ₃ − c_a22k

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 19

■ Friedmann Eqn: H2 = ρH_ρ 2

crit −

c2 k a2

FLRW: ρ

■ Friedmann Eqn: H2 = ρH_ρ 2

crit −

c2 k a2

Defn: ρ_crit := 3H_8πG2 critical density

Defn: Ω_m := _ρρ

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 19

■ Friedmann Eqn: H2 = ΩmH2 − c

2 _k

a2

Defn: ρ_crit := 3H_8πG2 critical density

Defn: Ωm := _ρρ

FLRW: ρ

■ Friedmann Eqn: H2 = ΩmH2 − c

2 _k

a2

Defn: ρ_crit := 3H_8πG2 critical density

Defn: Ωm := _ρρ

crit matter density parameter

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 19

■ Friedmann Eqn: H2 = ΩmH2 − c

2 _k

a2

Defn: ρ_crit := 3H_8πG2 critical density

Defn: Ωm := _ρρ

crit matter density parameter

FLRW: ρ

■ Friedmann Eqn: H2 = ΩmH2 + ΩkH2

Defn: ρ_crit := 3H_8πG2 critical density

Defn: Ωm := _ρρ

crit matter density parameter

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 19

■ Friedmann Eqn: 1 = Ωm + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

FLRW: ρ

■ Friedmann Eqn: 1 = Ωm + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

a2_H2 curvature density parameter (sign reversal!)

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 19

■ Friedmann Eqn: 1 = Ωm + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

a2_H2 curvature density parameter (sign reversal!)

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

FLRW: ρ

■ Friedmann Eqn: 1 = Ωm + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

a2_H2 curvature density parameter (sign reversal!)

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 19

■ Friedmann Eqn: 1 = Ωm + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

a2_H2 curvature density parameter (sign reversal!)

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ Ωm0 = 1 ⇔ k = 0 flat

FLRW: ρ

■ Friedmann Eqn: 1 = Ωm + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

a2_H2 curvature density parameter (sign reversal!)

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ Ωm0 = 1 ⇔ k = 0 flat

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 19

■ Friedmann Eqn: 1 = Ωm + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

a2_H2 curvature density parameter (sign reversal!)

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ Ωm0 = 1 ⇔ k = 0 flat

◆ Ωm0 > 1 ⇔ k > 0 spherical

FLRW: ρ

■ Friedmann Eqn: 1 = Ωm + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

a2_H2 curvature density parameter (sign reversal!)

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ Ωm0 = 1 ⇔ k = 0 flat

◆ Ωm0 > 1 ⇔ k > 0 spherical

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 19

■ Friedmann Eqn: 1 = Ωtot + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

a2_H2 curvature density parameter (sign reversal!)

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ Ωm0 = 1 ⇔ k = 0 flat

◆ Ωm0 > 1 ⇔ k > 0 spherical

◆ Ωm0 < 1 ⇔ k < 0 hyperbolic

FLRW: ρ

■ Friedmann Eqn: 1 = Ωtot + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

a2_H2 curvature density parameter (sign reversal!)

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ Ωm0 = 1 ⇔ k = 0 flat

◆ Ωm0 > 1 ⇔ k > 0 spherical

◆ Ωm0 < 1 ⇔ k < 0 hyperbolic

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 19

■ Friedmann Eqn: 1 = Ωtot + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

a2_H2 curvature density parameter (sign reversal!)

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ Ωm0 = 1 ⇔ k = 0 flat

◆ Ωm0 > 1 ⇔ k > 0 spherical

◆ Ωm0 < 1 ⇔ k < 0 hyperbolic

FLRW: ρ

■ Friedmann Eqn: 1 = Ωtot + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

a2_H2 curvature density parameter (sign reversal!)

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ Ωm0 = 1 ⇔ k = 0 flat

◆ Ωm0 > 1 ⇔ k > 0 spherical

◆ Ωm0 < 1 ⇔ k < 0 hyperbolic

FLRW: ρ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 19

■ Friedmann Eqn: 1 = Ωtot + Ωk

Defn: ρcrit := 3H

2

8πG critical density

Defn: Ωm := _ρcritρ matter density parameter

Defn: Ωk := − c

2_k

a2_H2 curvature density parameter (sign reversal!)

■ consider a fixed observation, e.g. H0 = 100 km/s/Mpc

◆ Ωm0 = 1 ⇔ k = 0 flat

◆ Ωm0 > 1 ⇔ k > 0 spherical

◆ Ωm0 < 1 ⇔ k < 0 hyperbolic

FLRW curvature constant

■ metric in

FLRW curvature constant

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 20

■ metric in

◆ azimuthal equidistant coords: RC

FLRW curvature constant

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

FLRW curvature constant

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 20

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

FLRW curvature constant

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

FLRW curvature constant

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 20

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2

■ Defn: Ωk := − c

2_k

FLRW curvature constant

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ Ωk0 = − c2k H₀2

FLRW curvature constant

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 20

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ k = − Ω_{k 0}H₀2 c2

FLRW curvature constant

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ R 2 C = − c 2 Ω_{k 0}H₀2

FLRW curvature constant

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 20

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ R 2 C = − c 2 H₀2 1 Ω_{k 0}

FLRW curvature constant

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2_H2 ⇒ R_C2 = − c 2 H₀2 1 1−Ω_tot0

FLRW curvature constant

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 20

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ RC = c H0 1 √ Ω_{tot0 − 1}

FLRW curvature constant

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ RC = c H0 1 √ Ω_{tot0 − 1} ■ Ωtot0 > 1

FLRW curvature constant

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 20

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ RC = c H0 1 √ Ω_{tot0 − 1} ■ Ωtot0 > 1 spherical

FLRW curvature constant

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ RC = c H0 1 √ Ω_{tot0 − 1}

FLRW curvature constant

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 20

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ RC = c H0 1 √ Ω_{tot0 − 1}

■ Ωtot0 > 1 spherical RC real

FLRW curvature constant

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ RC = c H0 1 √ Ω_{tot0 − 1}

■ Ωtot0 > 1 spherical RC real

FLRW curvature constant

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 20

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ RC = c H0 1 √ Ω_{tot0 − 1}

■ Ωtot0 > 1 spherical RC real

FLRW curvature constant

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ RC = c H0 1 √ Ω_{tot0 − 1}

■ Ωtot0 > 1 spherical RC real

■ Ωtot0 = 1 flat RC undefined

FLRW curvature constant

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 20

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ RC = c H0 1 √ Ω_{tot0 − 1}

■ Ωtot0 > 1 spherical RC real

■ Ωtot0 = 1 flat RC undefined

FLRW curvature constant

■ metric in

◆ azimuthal equidistant coords: RC

◆ orthographic coords: k

■ orthographic: 1 − kr2 = 0 coord singularity at equator

■ ⇒ kR_C2 = 1 ⇒ k = 1/R_C2 ■ Defn: Ωk := − c 2_k a2H2 ⇒ RC = c H0 1 √ Ω_{tot0 − 1}

■ Ωtot0 > 1 spherical RC real

■ Ωtot0 = 1 flat RC undefined

Einstein’s free parameter: Λ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 21

■ Einstein: prevent expansion/contraction via Λ

Einstein’s free parameter: Λ

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E

■ maxima: calculate G and G + gΛ = 8πT and simplify:

Einstein’s free parameter: Λ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 21

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E

■ maxima: calculate G and G + gΛ = 8πT and simplify:

https://cosmo.torun.pl/Cosmo/FLRWEquationsGR

Einstein’s free parameter: Λ

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3

Einstein’s free parameter: Λ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 21

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 acceleration Eqn (Λ 6= 0): ¨a a = − 4 π G (ρ + 3 p/c2) 3 + c2 Λ 3

Einstein’s free parameter: Λ

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 acceleration Eqn (Λ 6= 0): ¨a a = − 4 π G (ρ + 3 p/c2) 3 + c2 Λ 3 Defn: “dust solution”: p(t) = 0

Einstein’s free parameter: Λ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 21

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 acceleration Eqn (Λ 6= 0): ¨a a = − 4 π G (ρ + 3 p/c2) 3 + c2 Λ 3 Defn: “dust solution”: p(t) = 0

Einstein’s free parameter: Λ

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 Defn: “dust solution”: p(t) = 0

Defn: Ω_Λ := _{3 H}c2Λ2

acceleration Eqn (Λ 6= 0): ¨a

a = −

4 π G (ρ + 3 p/c2)

3 + ΩΛ H

Einstein’s free parameter: Λ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 21

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 Defn: “dust solution”: p(t) = 0

Defn: Ω_Λ := _{3 H}c2Λ2 acceleration Eqn (Λ 6= 0): ¨a a = − H2 2 ρ ρcrit + Ω_Λ H2

Einstein’s free parameter: Λ

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 Defn: “dust solution”: p(t) = 0

Defn: Ω_Λ := _{3 H}c2Λ2

acceleration Eqn (Λ 6= 0): ¨a

a = −

H2 Ωm

2 + ΩΛ H

Einstein’s free parameter: Λ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 21

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 Defn: “dust solution”: p(t) = 0

Defn: Ω_Λ := _{3 H}c2Λ2 acceleration Eqn (Λ 6= 0): a¨ a a2 ˙a2 = − Ωm 2 + ΩΛ

Einstein’s free parameter: Λ

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 Defn: “dust solution”: p(t) = 0

Defn: Ω_Λ := _{3 H}c2Λ2 acceleration Eqn (Λ 6= 0): a¨ a a2 ˙a2 = − Ωm 2 + ΩΛ Defn: q := −¨aa_˙a2

Einstein’s free parameter: Λ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 21

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 Defn: “dust solution”: p(t) = 0

Defn: Ω_Λ := _{3 H}c2Λ2

Defn: q := −¨aa_˙a2 “deceleration parameter”

acceleration Eqn (Λ 6= 0): q = Ωm

Einstein’s free parameter: Λ

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 Defn: “dust solution”: p(t) = 0

Defn: Ω_Λ := _{3 H}c2Λ2

Defn: q := −¨aa_˙a2 “deceleration parameter”

Einstein’s free parameter: Λ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 21

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 Defn: “dust solution”: p(t) = 0

Defn: Ω_Λ := _{3 H}c2Λ2

Defn: q := −¨aa_˙a2 “deceleration parameter”

■ q = Ω₂m − ΩΛ acceleration equation

Einstein’s free parameter: Λ

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 Defn: “dust solution”: p(t) = 0

Defn: Ω_Λ := _{3 H}c2Λ2

Defn: q := −¨aa_˙a2 “deceleration parameter”

■ q = Ω₂m − ΩΛ acceleration equation

Einstein’s free parameter: Λ

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 21

■ Einstein: prevent expansion/contraction via Λ

ADS:1917SPAW...142E Friedmann Eqn (Λ 6= 0): c 2 _k a2 + ˙a2 a2 = 8 π G ρ 3 + c2 Λ 3 Defn: “dust solution”: p(t) = 0

Defn: Ω_Λ := _{3 H}c2Λ2

Defn: q := −¨aa_˙a2 “deceleration parameter”

■ q = Ω₂m − ΩΛ acceleration equation

distances in FLRW cosmology

distances in FLRW cosmology

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 22

■ azimuthal equidistant r: proper distance at t0 ≡ comoving radial

distances in FLRW cosmology

■ azimuthal equidistant r: proper distance at t0 ≡ comoving radial

distance r = R_tt0 _a(tc dt′′₎

distances in FLRW cosmology

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 22

■ azimuthal equidistant r: proper distance at t0 ≡ comoving radial

distance r = R_tt0 _a(tc dt′′₎

■ Friedmann Eq: 1 = Ωm + Ωk + ΩΛ

distances in FLRW cosmology

■ azimuthal equidistant r: proper distance at t0 ≡ comoving radial

distance r = R_tt0 _a(tc dt′′₎ ■ Friedmann Eq: 1 = Ωm + Ωk + ΩΛ ■ Ωm = _ρcritρ = ρ0 a −3 ρ_crit0 (H2_/H2 0)

distances in FLRW cosmology

Cosmology (C) CC BY-SA 4.0 0 | FLRW | k | top | 3obs | a(t): EdS Ω RC q r H0r dA d_L d(x, y) | ξ P (k) 2020-09-30 – 22

■ azimuthal equidistant r: proper distance at t0 ≡ comoving radial

distance r = R_tt0 _a(tc dt′′₎ ■ Friedmann Eq: 1 = Ωm + Ωk + ΩΛ ■ Ωm = _ρcritρ = ρ0 a −3 ρ_crit0 (H2_/H2 0) = Ωm0 H 2 0a−1 ˙a−2