Week 4 - University of Utah

ASTR/PHYS 4080: Introduction to Cosmology Spring 2021: Week 03

Week 4

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Read thru Chapter 5 Also read the Key Concepts for those chapters

Today: Eqn of States & Model Universes

HW 3 due Thursday HW 4 available

Office Hours: Tuesdays 2-3pm

Wednesdays 11am-12pm

Only available first 20min on Tuesday

http://www.astro.utah.edu/~wik/courses/astr4080spring2021/

Spring 2021: Week 04ASTR/PHYS 4080: Introduction to Cosmology

Equations of the Universe

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(·aa )

2

=8πG3c2

ε −κc2

R20a2

+Λ3

εΛ ≡c2

8πGΛ

·ε + 3·aa

(ε + P) = 0

P = wε

Friedmann Eqn: gravity

Fluid Eqn: conserve energy

Eqn of State: relations b/t thermodynamic properties

PV = NRT

Ideal Gas Law

P = nkT P =ρμ

kT

ε = ρc2

P ≈kTμc2

ε


Acceleration Equation

3

··aa

=4πG3c2

(ε + 3P) +Λ3

(·aa )

2

=8πG3c2

ε −κc2

R20a2

+Λ3

·ε + 3·aa

(ε + P) = 0+ +

=

P < −13

ε··a > 0

Λ = 0

If const, also const b/cΛ ε εΛ ≡c2

8πGΛ

so P = − εΛ = −c2

8πGΛ


Evolution of Components

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CMB dominates energy density of photons in universe today

Don’t know the mass of neutrinos, but energy density dominated by momentum and

decoupling in early universe. Calculated to be:

True for each neutrino flavor, so total is 3x above.

Relativistic Component: “Radiation” = Photons + Neutrinos

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Relativistic, Non-rel., & Components

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Radiation Matter:

Dark Energy:

Using evolution of each component with a, can compute when matter-dark energy and matter-radiation components were comparable

Ωr,0 =1.681εCMB,0

εcrit,0= 9 × 10−5

(neutrinos have ~68% of the energy density of CMB photons)


Model Universes

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Can now solve for a(t) generically, if not necessarily analytically:

• Empty • Matter only

• classic case: open, closed, flat • Radiation only (+curvature) • Lambda only (+curvature) • Various Combinations!


Empty

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Only 1 Constituent in a Flat Spacetime

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Only 1 Constituent in a Flat Spacetime

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mempty

r

m

empty

r


Classical Cosmology: Matter + Curvature

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Classical Cosmology: Matter + Curvature

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Density —> Destiny


Flat: Matter + Cosmological Constant

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Matter + Lambda + Curvature

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Benchmark Model

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6.5. BENCHMARK MODEL 119

−10 −8 −6 −4 −2 0−6

−4

−2

0

2

log(H0t)

log(

a)

trm

tmΛ

t0

a∝t1/2

a∝t2/3

a∝eKt

Figure 6.5: The scale factor a as a function of time t (measured in unitsof the Hubble time), computed for the Benchmark Model. The dotted linesindicate the time of radiation-matter equality, arm = 2.8 × 10−4, the time ofmatter-lambda equality, amΛ = 0.75, and the present moment, a0 = 1.

is roughly six times greater: Ωdm,0 ≈ 0.26. The bulk of the energy density inthe Benchmark Model, however, is not provided by radiation or matter, butby a cosmological constant, with ΩΛ,0 = 1 − Ωm,0 − Ωr,0 ≈ 0.70.

The Benchmark Model was first radiation-dominated, then matter-dominated,and is now entering into its lambda-dominated phase. As we’ve seen, radi-ation gave way to matter at a scale factor arm = Ωr,0/Ωm,0 = 2.8 × 10−4,corresponding to a time trm = 4.7 × 104 yr. Matter, in turn, gave way tothe cosmological constant at amΛ = (Ωm,0/ΩΛ,0)1/3 = 0.75, corresponding totmΛ = 9.8 Gyr. The current age of the universe, in the Benchmark Model, ist0 = 13.5 Gyr.

With Ωr,0, Ωm,0, and ΩΛ,0 known, the scale factor a(t) can be computednumerically using the Friedmann equation, in the form of equation (6.6).Figure 6.5 shows the scale factor, thus computed, for the Benchmark Model.Note that the transition from the a ∝ t1/2 radiation-dominated phase tothe a ∝ t2/3 matter-dominated phase is not an abrupt one; neither is thelater transition from the matter-dominated phase to the exponentially grow-ing lambda-dominated phase. One curious feature of the Benchmark Model


Benchmark Model

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ASTR/PHYS 4080: Introduction to Cosmology Spring 2021: Week 04

https://tritonstation.wordpress.com/2019/01/28/a-personal-recollection-of-how-we-learned-to-stop-worrying-and-

love-the-lambda/

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Week 4 - University of Utah

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