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Cosmology - Redshift and RadiationASTR 2120

Sarazin

Final ExamThursday, May 7, start anytime before 8

pm4 hoursYou may not consult the text, your notes,

or any other materials or any personYou can use three of 3x5 cards (both

sides) or 6x5 paper (one side) with equations only

Have pencils, paper, calculator

Final Exam~2/3 Quantitative Problems (like homework

problems)~1/3 Qualitative Questions

Multiple Choice, Short Answer, Fill In the Blank

Test done with Collab Tests & Quizzes ToolQualitative Problems:

Do work on work sheetsType Answers in Answer Boxes in Collab

Scan/photo worksheets and equation sheets, either upload at Collab Assignments “Final Exam Work Sheets” tool or email to me [email protected]

http://virginia.edu

Final Exam (Cont.)

Material:Final exam will cover the entire semester

Chapters (5), (7), 13-24Stars, Sun ® Cosmology

Extra emphasis on material not on first two testsExtragalactic Distances, Clusters of Galaxies

(problems), AGNs, CosmologyChapters 21, 23, 24Homeworks 9-11

Know pc, AU, Msolar, Lsolar, Rsolar, H0, TCMB

Final Exam ReviewReading DayWednesday, May 610 am – noon

Cosmology - Redshift and RadiationASTR 2120

Sarazin

Redshift in Cosmology z º (lobs - lem)/lem 0 ≤ z < ∞1 + z = lobs / lem

Straightforward to measureAstronomer prefer to

Distance from usModel dependent0 < d < dH (size of observable Universe), all of

early Universe compressed near d = dH)Age or time into past (compressed near t0)

Model dependent

Redshift in Cosmology Interpretation of Redshift z?Low redshifts, z << 1, nearby Universe:

Straightforward: redshift is Doppler shiftz º (lobs - lem)/lem » vr/c » H0d (vr << c)

High redshifts, z ≳ 1, distant Universe and past time:Hubble formula breaks downRelativistic Doppler shift?Effect of change in expansion rate of Universe, H(t)?Gravitational redshift?Effects of curvature of photon paths?

Redshift in Cosmology

Oh no, my head is going to explode!!

Then, a miracle occurs . . .

Sounds difficult to calculate, confusing, and very model dependent → a big pain??

Redshift in Cosmology Divide photon path into lots of very small pieces, treat

each as series of emissions and observations

For each piece, time dt << tH, length dl << dH

Just like low redshifts, dz << 1, nearby Universe:Straightforward: redshift is first order Doppler shiftRelativistic Doppler shift?Effect of change in expansion rate of Universe, H(t)?Gravitational redshift?Effects of curvature of photon paths?


€

dl = cdt distance traveled by photondv = H (t)dl = H (t)cdt Hubble expansion law

H (t)=1rdrdt

r = radius of Universe

dv = c 1rdrdtdt = c dr

r<< c

z =λobs −λemλem

redshift

dz =dλλ

<<1 where dλ = λobs −λem , and λobs ≈ λem

dz =dλλ

=dvc

=1cc drr

=drr


€

dλλ

=drr

integrate both sides

dλλ

∫ =drr∫

lnλ = ln r + constantlnλ = ln r + lnλem − ln rem

ln λλem

%

& '

(

) * = ln r

rem

%

& '

(

) *

λobsλem

=robsrem


€

λobsλem

=robsrem


=λobsλem

−1

1+ z =robsrem

Big Bang : rem → 0⇒ z→∞

Wavelengths just expand with Universe!!

Really, it could not have turned out to be simpler or nicer!

Redshift in Cosmology Example:Recently, there was a claim that the Hubble Ultra Deep

Field contains a galaxy with a redshift of z = 10When this galaxy emitted the light we now see, the

Universe was (1+z) = 11 times smaller than it is today!!

Cosmic Microwave Background (CMB)

T = 2.725 K Awfully cold - who cares?a) Most of known heat and free energy in Universeb) Most of photons in Universe

N(photons)/N(protons) ~ 109 (Homework problem)Where did this big number come from?Why isn’t it bigger?

If matter and antimatter were symmetric

N(photons)/N(protons) ≳ 1018

Bright and shiny (but empty) Universe!!

€

p + p → 2γ


VERY hot in the past:Theorem: Redshifted and expanded blackbody = blackbody

at redshifted temperature (homework)

€

λ ×Tγ = constant (Tγ ≡ CMB temperature)

Tγ z( ) = 1+ z( )Tγo


Example:Recently, there was a claim that the Hubble Ultra Deep

Field contains a galaxy with a redshift of z = 10When this galaxy emitted the light we now see, the CMB

temperature was (1+z) = 11 times larger than it is today = 30 K

CMB at z=10 Galaxy

Wilsoniac

Belliac Labs

Pensiac

30 K


€

λ ×Tγ = constant (Tγ ≡ CMB temperature)

Tγ z( ) = 1+ z( )Tγo

(1 + z) = ro / re ® ¥ as Big Bang is approached, so T ® ¥ at Big Bang

Hot Big Bang

The Hot Big BangASTR 2120

Sarazin

Radiation-Dominated Era

Radiation dominates thermal history of early Universe

(1+ z) = λobsλem

=robsrem

redshift = expansion

TCMB (z) = TCMB (0) (1+ z) Universe very hot in pastCMB(z) = Bν [TCMB (z)] Redshifted, expanded BB = BB

Radiation-Dominated EraIn early Universe, CMB dominates dynamics and gravityCompare to matter

ρmr3 = constant (mass conservation)

ρm (z) = ρ0 (ro / r)3 = ρ0 (1+ z)3

ργ = uγ / c2 ~ ργ 0 (ro / r)4 ∝ (1+ z)4 ∝T 4


€

ργ (z) = ργo (ro / r)4 = ργo(1+ z)4

ργ (z) /ρm (z) = (ργo /ρmo )(1+ z)→∞

as Big Bang is approached

ρmo = ρo =ΩMρcrit ≈ (1 / 3)×10−29 gm/cm3

ργo = 4.6×10−34 gm/cm3(homework)

(ργo / ρmo ) ≈10−4

ργ > ρm for z > 3600 t < 50, 000 years


Radiation dominated dynamical history of early UniverseGravityExpansion

ργ > ρm for z > 3600 t < 50, 000 years

Radiation-Dominated DynamicsSolve Cosmic Expansion Equation for

€

ρ = ργ ∝ (ro / r)4 not (ro / r)3

12v2 =

GMr

−12Kro

2c2 Cons. of Energy

As r→ 0, 1st term on r.h.s. dominates asM →∞ and (1/ r)→∞

v2 =drdt(

) *

+

, -

2

=2GMr

=2G(4π / 3)ργr

3

r

2

Radiation-Dominated Dynamics

€

drdt"

# $

%

& ' 2

=8πGργr

2

3

d(r / ro )dt

+

, - .

/ 0

2

=8πG3

ργrro

"

# $

%

& '

2

ργ = ργo(r / ro )−4

d(r / ro )dt

+

, - .

/ 0

2

=8πG3

ργo (r / ro )−4 rro

"

# $

%

& '

2

=8πG3

ργorro

"

# $

%

& '

−2

d(r / ro )dt

=8πG3

ργorro

"

# $

%

& '

−1


€

rro

"

# $

%

& ' d(r / ro )dt

=12d(r / ro )2

dt=

8πG3

ργo

rro

"

# $

%

& '

2

= 2 8πG3

ργo t + const. (= 0)

rro

"

# $

%

& ' =

32πG3

ργo"

# $

%

& '

1/4

t1/2

TγTγo

=rro

"

# $

%

& '

−1

=32πG

3ργo

"

# $

%

& ' −1/4

t −1/2


Tγ ≈1010 K t−1/2 (t in sec)for Tγ >10, 000 K

Correct for neutrinos

Very Hot!

Hot Big Bang

Thermal History of UniverseASTR 2120

Sarazin

Alpher, Gamow (floating in Ylem), & Hermann

Thermal History of UniverseDo in two passes:

t > 10-6 seconds, physics pretty well understoodVery early history, more speculative and more exotic physics. Tie in to frontiers of physics

Thermal History of UniverseGeorge Gamow

Thermal History of Universe

t ≲ 10-6 sec, T ≳ 1013 KkT ≳ mpc2 ~ mparticlec2 for most particles

Tγ ≈1010 K t−1/2 (t in sec)for Tγ >10, 000 K

€

particle + antiparticle ↔ 2γExample : p + p ↔ 2γN particles ≈ Nantiparticles ≈ Nγ

Bubbling sea of particles and antiparticlesp, p ,n,n ,e− ,e+ ,γ,ν,ν ,π,Ω− , . . . etc.

Thermal History of Universe10-5 sec ≲ t ≲ 1 sec, 1013 K > T ≳ 1010 K

kT < mparticlec2 for most particlesMost particles (except e’s, n’s) can be destroyed but

not made

a) Unstable particles decayt1/2 ~ 10-23 to 10-10 sec << tException: neutron, t1/2 = 11 minutes€

particle+ antiparticle→ 2γ2γ → particle+ antiparticle

€

p, p ,n,n ,e− ,e+ ,γ,νe,ν e,νµ ,ν µ ,ντ ,ν τ ,(dark matter particles)


b) Antimatter annihilates

If matter/antimatter symmetric, this is very efficientNp/Ng ≲ 10-18, not 10-9 as observed (homework)

If pure matter, Np ~ Ng initially, would still be trueNeed small, but non-zero asymmetry

€

p + p → 2γ (no reverse)n + n → 2γ

€

N p − N p

N p

~10−9


Existence of matter today requires Universe had a small matter/antimatter asymmetry by 1 sec

€

N p − N p

N p

~10−9


c) Electrons and neutrinos still producedkT >> mec2

d) Protons and neutrons in equilibrium

View p+ & n as different states (isotopic spin) of same particle

€

e−,e+ ,νe,ν e

€

p+ + e− ↔ n+νep+ +ν e ↔ n+ e+ , etc.


View p+ & n as different states (isotopic spin) of same particle

€

Nn

Np

= e−ΔE /kT = e−Δmc2 /kT = e−(mn −mp )c

2 /kT

n

p

ΔE=(mn−mp) c2


1 sec ≲ t ≲ 103 sec, 1010 K > T ≳ 3 x 108 K(mn-mp)c2/k ~ 1010 K

Nn/Np decreasesAfter a few seconds, T < mec2/k ~ 6 x 109 K, can’t

make electrons anymore

Reactions between n, p stop

€

Nn

Np


2 /kT

€

e+ + e− → 2γ (no reverse)

€

n+ e+ ↔ p+ +ν e, etc. stop


1 sec ≲ t ≲ 103 sec, 1010 K > T ≳ 3 x 108 KNeutron to proton ratio freezes out at

What happens to neutrons?

If nothing else happened, neutrons would decay away

€

Nn

Np


2 /kT

€

n→ p+ + e− +ν e, beta decay, t1/2 =11 minutes€

Nn ≈ Np / 8

Fusion During Big Bang1 sec ≲ t ≲ 103 sec, 1010 K > T ≳ 3 x 108 K

rbaryons ≲ 10-2 gm/cm3

Hotter, lower density than center of star, but not completely dissimilar

Differences from star:a) Very little time (minutes)

No weak reactionsNo pp reaction (1010 years in Sun)

b) Free neutronsNever true in stars except during SN, only

last 11 minutes

Fusion During Big Bang

Tritium

p+ n→ 2H+γ2H+ n→ 3H+γ2H+ p→ 3He+γ3He+ n→ 4He+γ3H+ p→ 4He+γ 3 H→ 3 He+ e− +νe

t1/2 =12 years

All tritium è Helium-3


and similar

No significant reactions beyond 4HeIn stars, requires Triple-Alpha reaction, very slow,

not enough time in Big BangLose energy between 4He and 12C, only reaction is

3 4He ® 12C

€

p+ n→ 2H+γ2H + p→ 3He+γ3He +n→ 4He+γ


3a reaction


and similar

No significant reactions beyond 4HeIn stars, requires Triple-Alpha reaction, very slow,

not enough time in Big BangLose energy between 4He and 12C, only reaction is

3 4He ® 12C

€


Fusion During Big BangFusion in Big Bang makes

H & 4HeTraces of 2H & 3HeTiny bits of 6Li, 7Li, 7Be

Fusion During Big BangHow much helium?

Fusion reactions up to helium very efficientAll neutrons ® heliumInitially, Nn ~ Np / 8Do arithmetic (homework problem), findY = 0.22 (mass fraction of helium)X = 0.78 (mass fraction of hydrogen)

Agree with values in oldest stars

Fusion During Big BangHow much 2H (deuterium), 3He (helium-3), Li?

Fusion reaction rate depends on density of baryonsrbaryons

High density = less 2H, 3He, more LiLow density = more 2H, 3He, less Li

€


Fusion During Big BangGives rbaryons at t = 1 sec, T = 1010 K

rbaryons (today) = rbaryons (1 sec) x (r / ro)3

= rbaryons (1 sec) x (1 + z)-3

T (1 sec) = T (today) x (1 + z)(1 + z ) = 1010 K / 2.725 Krbaryons (1 sec) gives rbaryons (today)!!

Fusion During Big BangGives rbaryons at t = 1 sec, T = 1010 K

rbaryons (today) = rbaryons (1 sec) x (r / ro)3

= rbaryons (1 sec) x (1 + z)-3

T (1 sec) = T (today) x (1 + z)(1 + z ) = 1010 K / 2.725 Krbaryons (1 sec) gives rbaryons (today)!!rbaryons (today) = 3.5 x 10-31 gm/cm3

W (baryons) = rbaryons / rcrit = Wb = 0.046

Fusion During Big BangWb = 0.046 << WM

Dark Matter not anything which was ordinary matter at t = 1 second

Not planets, brown dwarfs (MACHOs)Not black holes from stars or collapse of matter

Dark Matter = weakly interacting particles made in Big Bang!

astr 2120 sarazin - people.virginia.edupeople.virginia.edu/~cls7i/classes/astr2120/... · astr 2120...

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