Review of Homogeneous Cosmology - LMU

Apr 16, 2021 1

Review of Homogeneous Cosmology

Discussion of characteristics of GR derived models for our homogeneous and isotropic expanding universe followed by presentation of distances and time within an expanding universe context

Apr 16, 2021 2

Basic Ingredients of the Big Bang Modeln Universe began a finite time in the past t0 in a hot,

dense state

n Subsequent expansion R(t) obeys Friedman equation (GR) globally

n Matter and radiation interact according to known laws of physics

n Structures condense out of expanding and cooling primordial material

Cosmo-LSS | Mohr | Lecture 1

Apr 16, 2021 3

Ingredients Follow Naturally from Basic Concepts and Principlesn Finite time:

n Expansion observed. Implies higher density in past, and that at some point the density goes to infinity- a beginning

n Friedmann evolution:n Homogeneous and isotropicn General relativity

n Physics the same everywhere:n Cosmological principle

n Structure formation:n Gravitational collapse


Apr 16, 2021 4

Big Bang Model Observationally Supportedn Cosmology has rested on four observational

pillars for 5 decades nown The Universe expands homogeneouslyn The night sky is darkn There exists a cosmic microwave background that is a pure black

body spectrumn The abundance of the light elements requires primordial

nucleosynthesis

n Ever more sensitive observational studies now point to a Universe with a finite age that began in a phase of high density and temperature


Apr 16, 2021 5

A Modern Measure of the Expansion

Measurements to 19 Supernovae

Riess, Press & Kirshner ApJ 1996

Blue points: 19 SNeRed line: Hubble Law with

Ho=19.6 km/s/MLy

All studies provide consistent results in the local universe: other galaxies are receding from us, and their recession velocities are proportional to their distances.

The farther away the galaxy, the faster it travels away from us (or the more the universe has expanded during the time it took the light to reach us)

The Hubble parameter has units of velocity over distance.

€

vr = Hod

so Ho =vrd


Apr 16, 2021 6

Why is the Night Sky Dark?n Suppose that the universe is infinite and

homogeneousn every line of sight intercepts a starn sky should glow as brightly as the surface of an

average starn but the night sky is dark…

n Olbers’ Paradoxn Heinrich Olbers in 1826n Thomas Digges in 1576n Johannes Kepler in 1610n Edmund Halley in 1721

n Therefore, Universe cannot be infinite and homogeneous!

Observer


Apr 16, 2021 7

COBE Spectrum: Blackbody Emission

€

TCBR = 2.735 K


Apr 16, 2021 8

Cosmic Elemental AbundancesH

He

Be

B

C O NeMg

SiS Ar Ca

TiCr

Fe

Ni

N

FNa Al

P Cl K

ScV

Mn Co Cu

LiAtomic Number

Log N

2

4

6

8

10

12 Stars

BigBang


Apr 16, 2021 9

Cosmic Densitometern Primordial nucleosynthesis

n explains observed, light element abundances if the density of normal matter (baryons) in the universe lies around 3.5x10-31 g/cm3 or 0.21 hydrogen atoms per cubic meter

n Precise observational testn independent measurements of

abundances of four different light elements lead to consistent constraints on the density of normal matter

n provides confidence that primordial or Big Bang nucleosynthesis provides a correct explanation of the formation of the light elements.

Burles, Nollett & Turner

Deuterium


Consider now dynamics in Spacetime

Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 10

Apr 16, 2021 11

Dynamics in Spacetime – the Geodesicn Any observer traveling along a geodesic in spacetime

is an unaccelerated, inertial observer

n So dynamics in general relativity comes down to calculating the curvature of space for a given mass distribution and then defining a geodesic in that space


Apr 16, 2021 12

Geodesicsn Any path between two points that is an extremum

(i.e. longest or shortest) is a geodesic

n Examples:n Straight line in Euclidean geometryn Great circle in spherical geometryn “Hiker’s path” in saddle geometry

Chicago

Moscow

These three geometries (flat, spherical and hyperbolic) are important because they are homogeneous and isotropic {Cosmological Principle}


Apr 16, 2021 13

How Can Geometry seem like a Force?

travelerA

travelerB

Parable of thetwo travelers

•Two travelers start out walking in parallel.

•Mutual gravitational attraction draws them closer.

•This is similar to the behavior of parallel lines in a closed or spherical geometry!


Apr 16, 2021 14

The Metric Equation

222 ),(),(),( vvuhvuvuguvufs Δ+ΔΔ+Δ=Δ

n Defining geodesics requires an ability to calculate distances-even in curved geometries

n Metric equation provides the relationship between coordinate distances and metric distances (real or physical distances)

n For two points in a two dimensional, curved space (u,v) and (u+Du,v+Dv), general form for metric equation is

Metric coefficients


Apr 16, 2021 15

The Metric Equation: Euclidean Geometry

1),(0),(1),(

222

=

=

=

∴

Δ+Δ=Δ

yxhyxgyxf

yxsn Pythagorean Theorem is

the metric equation for Euclidean geometry

Homogeneous, orthogonal, isotropic



Apr 16, 2021 16

Metric Equation: Spherical Polar Geometry

θφθ

φθ

φθ

φθθ

22

2

22222

cos),(

0),(),(

)cos(

Rhg

Rf

RRs

=

=

=

∴

Δ+Δ=Δ q

f


(q+Dq,f+Df)�


Apr 16, 2021 17

Spacetime Metric

n Consider two events in spacetime n Event 1 (x,t) and Event 2 (x+Dx, t+Dt)

n General expression for spacetime interval

n a, b, g are the metric coefficients

2222 xxtctcs Δ−ΔΔ−Δ=Δ γβα


Apr 16, 2021 18

Minkowski Spacetime Metricn Spacetime interval

2222 xxtctcs Δ−ΔΔ−Δ=Δ γβα

€

α(x, t) =1β(x, t) = 0γ(x, t) =1

2222 xtcs Δ−Δ=Δ

n Minkowski spacetime called flat spacetime


Apr 16, 2021 19

Geodesic Motion

n Freely falling frames are inertial frames in General Relativity

n Bodies in free-fall follow geodesics in spacetime

n A geodesic in spacetime maximizes spacetime interval

shortest distance between two points

largest spacetime interval between two events


Apr 16, 2021 20

Illustration: Minkowski Spacetime

222 xtcsAB Δ−Δ=Δ

x

t

(0, 2Dt)

(0,0)

(Dx, Dt)tcsAC Δ=Δ 2

A

C

B

222 xtcsBC Δ−Δ=Δ

222 /12 tcxtcss BCAB ΔΔ−Δ=Δ+Δ

Spacetime interval larger for geodesic motionCosmo-LSS | Mohr | Lecture 1

Apr 16, 2021 21

Dynamics of Particles and Lightn Because free-falling bodies follow geodesics in spacetime, we can

understand the dynamics of moving bodies if we can calculate geodesics

n Calculate the geometry of spacetime given some distribution of mass and energy

n Use the metric equation to define geodesics- paths which produce a maximum of the spacetime interval

n But what is the relationship between the geometry of spacetime and the mass-energy distribution?


Apr 16, 2021 22

Einstein’s Field Equationsn Einstein developed a set of equations that relate the curvature

of spacetime to the distribution of mass and energy

n In their most compact form, the Einstein field equations can be written as

€

Gµν =8πGc 4

T µν


Apr 16, 2021 23

Einstein’s Field Equation(s)

Gµν = −8πGc4

T µν

µ = (1, 2,3, 4);ν = (1, 2,3, 4)

G11 G12 G13 G14

G21 G22 G23 G24

G31 G32 G33 G34

G41 G42 G43 G44

"

#

$$$$$

%

&

'''''

=8πGc4

T11 T12 T13 T14

T 21 T 22 T 23 T 24

T 31 T 32 T 33 T 34

T 41 T 42 T 43 T 44

"

#

$$$$$

%

&

'''''

Riemann curvature tensor Stress-energy tensor

These tensors are symmetric, so there are only 10 components Cosmo-LSS | Mohr | Lecture 1

Apr 16, 2021 24

Comments

n Einstein equation suggests matter+energy are the source for the curvature of spacetime

n In the weak field limit this formulation reproduces the familiar Newtonian concept of gravity

n One important difference: components of stress-energy tensor include mass and energy.

n Thus, a gas contributes to curvature through its mass density r and through its pressure p!

n GR provides a solution for geometry even when stress-energy tensor vanishes--

n This may be an indication that gravity itself is a form of energy, thereby creating curvature through interaction with itself


Apr 16, 2021 25

Comments: Action at a Distance?n In GR the Newtonian problem of instantaneous action

at a distance is removedn Gravity is reflected in local curvature of spacetime, which

responds to local density of matter and energyn Matter and energy subject to propagation speed of light

n Gravitational radiation is natural implication of theoryn Orbiting masses will produce ripples in spacetime- sending

out waves of curvature called gravitational radiationn These waves propagate outward from their source at the

speed of lightn Detection of gravitational wave effects is one of the great

triumphs of general relativity!


Consider now the expanding Universe


Friedmann-Robertson-Walker metricn Within Peacock there is a convenient form adopted

for the RW metric which allows the curvature dependence to be abstracted

n The metric for all three curvature families is then

Apr 16, 2021 27

€

Sk r( ) =

sinr k =1( )r k = 0( )sinhr k = −1( )

#

$ %

& %

€

c 2dτ 2 = c 2dt 2 − R2 t( ) dr2 + Sk2 r( )dϕ2[ ]


Comoving Separation

n In this coordinate system we can calculate the separation between two astronomical sources at rest within this expanding modeln Choose to locate them along coordinate r at r and r+drn Separation d at time t is R(t)drn Note the time dependence R(t), so the distances will simply

scale up over time with the scale factorn Effective recession velocity would be

n Allowing us to recover the Hubble law where at any time an observer sees Vr is proportional to separation…

Apr 16, 2021 28

€

Vr =ΔdΔt

=R t2( )δr − R t1( )δr

t2 − t1= δr

R(t2) − R(t1)( )t2 − t1


Redshift

n Consider observing light emission from distant galaxy at some coordinate position r where we have conveniently placed ourselves at the origin

n Light travels null geodesics, so we can write

n A subsequent crest in this light wave will be emitted at temit+dtemit, and observed at tobs+dtobs so we can write

Apr 16, 2021 29

€

0 = c 2dt 2 − R2(t)dr2 so r =cdtR(t)temit

tobs

∫

€

r =cdtR(t)temit +dtemit

tobs +dtobs

∫


Redshift (cont)

n Differencing these, noting that (1) the coordinate distance r has not changed (distant galaxy at rest in expanding universe) and (2) R(t) has not changed substantially in the time for one cycle of a light wave we obtain

Apr 16, 2021 30

€

0 =cdtR(t)temit +dtemit

tobs +dtobs

∫ −cdtR(t)temit

tobs

∫ =cdtobsR(tobs)

−cdtemitR(temit )

orcdtobscdtemit

=R(tobs)R(temit )

soλobsλemit

=R(tobs)R(temit )

≡1+ z


Using ! = #$ = % &'

Dynamics of the expansionn Evolution of homogeneous and isotropic universe

captured through expansion history R(t) [or H(t)]

n This evolution is determined by Einstein equations, and basically is affected by the gravitational interactions of all the components of the universe (dark matter, photons, baryons, dark energy, etc)

n Follows from simple consideration of energy conservation within Newtonian context

Apr 16, 2021 31Cosmo-LSS | Mohr | Lecture 1

Apr 16, 2021 32

Expanding Shell in Homogeneous Universe

homogeneousmass density

r

+

R

mass interior to RM

V

shell of massm

expanding withspeed V=HR


Apr 16, 2021 33

Newtonian Motivation for the Friedmann Equation

€

K.E .+ P.E . = T.E .12mV 2 + −

GMmR

#

$ %

&

' ( = E = const.

Consider shell of matter moving outward.

€

m 12

(HR)2 −GR

4π3R3ρ

%

& '

(

) *

+

, -

.

/ 0 = E = const.

÷ both sides by 12

mR 2

Use Hubble law: V=HR

€

H 2 −8πG3

ρ =2EmR2

=−kc 2

R2Define kc2=-2E/m:

Friedmann Equation


Apr 16, 2021 34

The Friedmann Equation

€

V =ΔRΔt

≡ ˙ R definition of velocity

V = HR Hubble Law∴

H 2 ≡˙ R R%

& '

(

) *

2

=8πG

3ρ −

kc 2

R2


Density and Geometryn Friedmann equation implies that balance of expansion energy

and gravitational potential energy determines geometry of spacetime

n For zero curvature k=0 models, r=rcrit where

n For r>rcrit k=+1 (closed space) and for r<rcrit k=-1 (open space)

Apr 16, 2021 35

€

H 2R2 −8πG3

R2ρ = −kc 2

€

ρcrit =3H 2

8πG


Density, Geometry and Fate

n Curvature is a quantity like total energy in an energy equation, and so intuitively we can think that open universes continue to expand forever and closed universes eventually turn around and recollapse

n This is true as long as there is no cosmological constant or dark energy term in the energy density


Apr 16, 2021 37

Scenarios for Evolution of the R(t)R

t

expanding,decelerating

R

t

expanding,constant speed

R

t

expanding,accelerating

R

t

contracting,accelerating


The deceleration equationn We can take the time derivative of the Friedmann equation, too

n Giving us

n Acceleration: rc2<-3p è p< -1/3 rc2

n Deceleration: DM, radiation, dustn Derivation from Einstein Equation yields both equations

n See notes provided along with lecture

Apr 16, 2021 38

€

˙ R 2 − 8πG3

ρR2 = −kc 2

and d(ρc 2R3) = −pd(R3),which is dE = −pdV

€

˙ ̇ R = − 4πGR3

ρc 2 + 3p( )


Density parameter

n The density parameter is denoted as the density divided by the critical density

n For k=0, W=1 at all times, but otherwise W=W(t), and the present epoch value is Wo

Apr 16, 2021 39

€

Ω =ρρcrit

=8πGρ3H 2


Apr 16, 2021 40

What is the Critical Density?

€

ρcrit (today) ≡3H0

2

8πG

=3 ⋅ (2.27 ×10-18 )2

8π(6.67×10−11)≈ 9 ×10−27 kg/m 3

About 10 hydrogen atoms per cubic meter

€

~ 1011M0 Mpc3

Mass of ~1 galaxy per Mpc3


Density parameters and evolutionn Generically, the universe

contains radiation, dark matter, matter and dark energy.

n Note that the variation of the energy densities of these components differs, but they evolve in such a way as to maintain a constant value when an effective curvature density parameter Wk is introduced

Apr 16, 2021 41

€

H 2 +kc 2

R2=8πG3

ρm + ρr + ρΛ( )

1+kc 2

H 2R2=Ωm +Ωr +ΩΛ

Ωm +Ωr +ΩΛ +Ωk =1or Ωk =1−Ωm −Ωr −ΩΛ

Matter : ρ z( ) = ρ0 1+ z( )3

Radiation : ρ z( ) = ρ0 1+ z( )4

Cosmological constant : ρ z( ) = ρ0In general : ρ z( ) = ρ0 1+ z( )3 1+w( )

where w ≡ pρ

€

Volume∝ RR0

#

$ %

&

' (

3

=11+ z#

$ %

&

' ( 3


Scale factor and Horizon scale

n The Friedmann equation also specifies the scale factor Ro.

n Unfortunately, this expression is undefined for the models consistent with observations (k=0)

n The Hubble length c/Ho is a characteristic distance or scale of the horizon

Apr 16, 2021 42

€

R0 =cH0

Ω−1( )k

$

% &

'

( )

−12

€

cH0

=14.1GLyr


Expansion history H(z)n The expansion history H(z) just reflects the variation

of the energy densities of all the components in the universe

n For a universe composed of radiation, matter and generalized dark energy, we can write the expansion history as follows

n Any observational constraints on the expansion history (e.g. from distance measurements) then allow measurements of w-- the equation of state parameter-- and the density parameters

Apr 16, 2021 43

€

H z( ) =8πG3

ρ z( )

€

H 2 z( ) = Ho2 Ωr 1+ z( )4 +Ωm 1+ z( )3 + 1−Ωr −Ωm −ΩE( ) 1+ z( )2 +ΩE 1+ z( )3 1+w( )[ ]


p = wρ

Evolution of the density parameter

n If the current density parameter W=1 then it will be unchanged for all time (constancy of curvature)

n Departures from W=1 in the past are vastly smaller than any departures today. At face value this raises a fine tuning problem unless we live in a spatially flat (zero curvature) universe

Apr 16, 2021 44

€

Ω z( ) −1 =Ω−1

1−Ωr −Ωm −ΩE( ) +Ωr 1+ z( )2 +Ωm 1+ z( ) + +ΩE 1+ z( ) 1+3w( )


Epoch of matter-radiation equalityn Currently the matter density

parameter is measured to be

n The radiation density parameter is measured to be

n The dark energy density parameter is about

n Matter-radiation equality occurred at z~3300, and matter-DE equality occurred at z~0.37

Apr 16, 2021 45

€

Ωm = 0.28

€

Ωr = 8.6x10−5

€

ΩE = 0.72

€

ρ zeq( ) = ρmo(1+ zeq )3 = ρE

1+ zeq =ρEρmo

#

$ %

&

' (

13


What epochs of the universe are observable?

n Directly observed objects?

n After first objects formed and reionized the universe (to z~10)

n Directly observable by light?

n From recombination onward (to z~103)

n Observable through light element abundances?

n to z~104

n Also observations that probe much farther backn Character of the initial density perturbations?

n Matter-antimatter asymmetry?

n Gravitational wave background from Inflation?

n Any physical remnant from an early universe process


Expanding Universe Models


The Static Universen Einstein considered a solution with no

expansion and a cosmological constant

n No expansion requires a particular density, but that density depends on scale factor R

n It must be a mix of matter and vacuum energy (see effective equation of state). Vacuum energy provides the negative pressure that balances the gravitational attraction.

n Problem with this model is that it is unstable

Apr 16, 2021 48

€

ρstatic =3kc 2

8πGR2

€

˙ ̇ R = − 4πGR3

ρ + 3p( )

€

H 2R2 −8πG3

R2ρ = −kc 2

p = −13ρ (remember p = wρ)


De Sitter Modeln A phase of vacuum energy

domination leads to exponential expansion- an inflationary phase

n Note that this provides a mechanism for starting the expansion of the Universe at early times- an early vacuum energy dominated phase

n Note that such a phase is generic at “late times” in an expanding universe with matter, radiation and dark energy components.

Apr 16, 2021 49

€

H 2 =˙ R R"

# $

%

& '

2

=Λc 2

3

H =˙ R R"

# $

%

& ' =

Λc 2

3

⇒ R(t) = RieH t− ti( )

R

t

€

Λ =8πGρVc 2


Steady State Modeln Adherents of a perfect cosmological principle would have the universe be

unchanging/homogeneous in both time and space.

n Model has constant Hubble parameter H and therefore exponential expansion. Because we observe matter in our universe some tricks have to be played to keep the matter density constant in an exponentially expanding universe. It is postulated that matter is continuously created.

n Back in the 60’s this model was taken seriously.

n Discovery of the microwave background basically ended this line of thinking, because it implies a hot, dense early equilibrium phase. As we have discussed, the light element abundances are also naturally explained by the Big Bang model

n Interesting case of theoretical prejudice being confronted with data.


Spatially flat modelsn Today, observations and

theoretical prejudices point toward spatially flat models

n To be so close to W=1 today requires that the Universe has been even closer to W=1 throughout

n Uniformity of CMB temperature requires mechanism to enlarge particle horizon. Inflation, this mechanism, also drives the Universe to spatial flatness.

Apr 16, 2021 51

Ωk = −0.052−0.027+0.025

WMAP + BAO + SNe(Komatsu et al 2009)


Planck TT + lowP + lensing (Planck Coll. 2016)

Ωk = −0.005−0.008+0.008

Planck TT + lowP

−0.0179 ≤Ωk ≤ 0.008195% confidence

Initial Singularityn If Big Bang is thought of as explosive instant, then this explosion

happened everywhere at oncen Universe isn’t expanding _into_ anythingn If infinite now then infinite before

n What started the expansion?n Inflation provides a mechanismn We know that a universe where the energy density is a positive

cosmological constant will expand exponentiallyn If an early phase transition led to a positive cosmological constant

of sufficiently dominant energy density, then no matter what is happening with the expansion at the time (static, contracting, expanding), the universe transitions to exponential expansion


Origin of the Redshift

n We have already shown that there is a “cosmological redshift”, where the change in the scale factor between emission and observation is reflected in the stretching of the wavelength of the light

n An astrophysical object can also have a “peculiar velocity” (i.e. velocity with respect to the Universal reference frame, which is presumably tracked by the Cosmic Microwave Background reference frame).

n An astrophysical object can also have a redshift due to differences in the gravitational potential well depth of the emitter and the observer (i.e. general relativistic “gravitational redshift”)

n Thus, a source’s redshift is the combination of all these effectsn Common to talk about “Hubble flow” as the inferred velocity due to the cosmological

redshift, but it is of course no flow at all!


Nature of the Expansionn As the universe expands, the distances _between_ objects, which are

well separated, increase

n Dynamics within gravitationally bound objects is dominated by the local mass distribution and unaffected by universal expansion

n Picture is more like raisin bread– raisins remain same size and loaf rises and increases the distances between all pairs of raisins

n In structure formation, the gravitationally bound objects are typically growing through accretion, providing a sort of intermediate case-dynamics of central “virial region” is independent of expansion, but the flow of additional material accreting onto the bound object is affected by the universal expansion. Thus, studies of inflow regions and the overall growth of structure can provide constraints on the global expansion rate.


Distances, Volumes and Time in Expanding Universes


Comoving or Proper Distance

€

dp = R0r

n Proper distance dp or comoving distance is coordinate distance r times present epoch scale factor Ro

n For a photon, we have a null geodesic, and so we can use the metric to calculate the coordinate distance traveled

Apr 16, 2021 56

€

0 = c 2dt 2 − R2(t)dr2 so r =cdtR(t)temit

tobs

∫


Comoving or Proper Distance (2)

n But observationally what we start with is a redshift, and so we typically wish to change variables from dt to dz

Apr 16, 2021 57

€

Rdr = cdt

H ≡˙ R R

so dt =dRRH

Rdr = cdRRH

€

R0R

= 1+ z( )

R =R0(1+ z)

dRdz

= −R0

(1+ z)2

€

R0dr = c 1+ z( ) dRRH

R0dr = c 1+ z( ) −R0(1+ z)2

dzRH

R0dr = −c

H(z)dz


Comoving or Proper Distance (3)n With this expression we can use the Friedmann

equation for H(z) to provide a general expression for the proper distance

n Note that this is derived from the metric and the Friedmann equation, so it is valid for all constant w models

Apr 16, 2021 58

€

dp = R0r =c

H(z)dz

0

z

∫€

H 2 z( ) = Ho2 Ωr 1+ z( )4 +Ωm 1+ z( )3 + 1−Ωr −Ωm −ΩE( ) 1+ z( )2 +ΩE 1+ z( )3 1+w( )[ ]

€

dp =cH0

dz

Ωr 1+ z( )4 +Ωm 1+ z( )3 + 1−Ωr −Ωm −ΩE( ) 1+ z( )2 +ΩE 1+ z( )3 1+w( )[ ]1 2

0

z

∫


Distance- Redshift Relationn Note that the distance-redshift relation is perhaps the most

direct handle we have on the evolution of the Hubble parameter, the expansion history of the Universe.

n Given the dependences of the expansion history on the contents of the universe and equation of state of the dark energy, these distance measurements form one of the two major handles we have on the nature of the cosmic acceleration

Apr 16, 2021 59

€

dp = cdzH(z)0

z

∫

€

dp =cH0

dz


0

z

∫


Change in Redshiftn Because the universe continues to

expand, the redshift is expected to change over time

n Note that it is the impact of a change in Rothat we are exploring for dz/dt

Apr 16, 2021 60

ddto

1+ z[ ] = ddto

RoR

⎡

⎣⎢⎤

⎦⎥

dzdto

=dRodto

1R−dRdte

dtedto

RoR2

dzdto

=dRodto

1Ro

RoR−dRdte

1R

dtedto

RoR

⎡

⎣⎢

⎤

⎦⎥

dzdto

= Ho(1+ z)−H (z)

dzdto

= Ho(1+ z)−Ho Ωm 1+ z( )3 +ΩE

n The redshift will change by 1 part in 108 over a human lifetime, so it is measureable if it can be separated from changes in peculiar velocity and drift of wavelength calibration of similar scale (i.e. 3 m/s)

n The CODEX spectrograph, a key project on the ELT, is designed to carry out this measurement of the change in redshift over cosmic time

n http://www.iac.es/proyecto/codex/


Volume Elementn Given the proper distance or equivalently the comoving distance we

can define the comoving volume element

n This is a critical element in the study of abundance evolution (of, say, galaxy clusters) to learn about the cosmic acceleration, and we will come back to it.

n Note that the comoving volume Vc is the volume associated with a particular coordinate region at the present epoch. We can define a physical volume Vp, which would be the volume contained by this same coordinate region at a redshift z (i.e. when light was emitted). Vp=Vc/(1+z)3

Apr 16, 2021 61

€

dVc = 4πdc2R0dr

dVc =4πcH0

dc2dz



Angular Diameter Distance

n The distance that transforms the angular size q of an object into its physical size l at time of emission (which could be independent of expansion for bound object) is

n It is like the comoving distance, but uses the scale factor R(z) at the time of emission rather than observation

Apr 16, 2021 62

€

dA =lθ

€

dA =dc

(1+ z)


Luminosity Distancen Luminosity distance is the relationship between the observed flux of an

object and its intrinsic luminosity

n There are changes to the emitted photons in an expanding universe as they travel to the observer.

n The photon energies are reduced -- on factor of (1+z)n The time separations between arriving photons are delayed – another (1+z)

n Note that typically observations also are within a fixed band, and so the specific transformation and shape of the spectrum have to be accounted (k correction) for in relating observed flux and intrinsic luminosity

Apr 16, 2021 63

€

dL =L4πf

f = L4πdc

2 1+ z( )2so dL = dc (1+ z)


Surface Brightness Dimming or “Cosmological Dimming”

n In expanding universe, then, the surface brightness is not independent of distance because:

n Photon energies fall – one factor of (1+z)n Photon arrival times are delayed – another (1+z)n Scale factor increases, spreading brightness over larger physical surface

area – (1+z)2

n Thus, the surface brightness dims with redshift

n This has a dramatic impact on our ability to map the light distributions of distant galaxies and clusters

Apr 16, 2021 64€

Iobs =Iemit1+ z( )4


Age, Lookback time and Elapsed timen Calculating time or age in an expanding universe is similar to calculating distance

n So we have an expression for the time elapsed between redshift z and the present (lookback time):

n Age of the universe to is limit of tlb where z goes to infinityn Time available for an object to form is telapse=to-tlb

Apr 16, 2021 65

€

Rdr = cdtRR0

≡11+ z

so cdt =R0dr(1+ z)

€

Previously R0dr =c

H(z)dz

€

cdt =cdz

H(z)(1+ z)

tlb =1H0

dz

(1+ z) Ωr 1+ z( )4 +Ωm 1+ z( )3 + 1−Ωr −Ωm −ΩE( ) 1+ z( )2 +ΩE 1+ z( )3 1+w( )[ ]1 2

0

z

∫


Anthropic Principlen With the cosmological principle we postulate that we

do not live at a preferred place in the universe (and the physics of the universe is the same at all locations). Remember that observations led us to discard the perfect cosmological principle, which would state that no time in the universe is different from any other

n There is an anthropic principle as well, and at its core it is an acknowledgement that our presence here in the Universe likely places some constraints on the possible models that describe our Universe


Anthropic Principle

n Our existence is a valid observation like any other.

n Therefore, Universe must be old enough to allow for the development of life like ours

n More broadly, the Universe must allow for conditions for the development of life like ours

n Carbon based life form requires carbon- the triple alpha process is resonant, otherwise forming higher mass elements would be much more difficult


References

n Cosmological Physics, John Peacock, Cambridge University Press, 1999

n Foundations of Modern Cosmology, John Hawley and Katherine Holcomb, 2nd Edition, Oxford

University Press, 2005

n Gravitation and Cosmology, Steven Weinberg, John Wiley and Sons, 1972

n “Five-year Wilkinson Microwave Anisotropy Probe (WMAP1) Observations: Cosmological Interpretation,” 2009, Komatsu et al. The Astrophysical Journal Supplements, 180, 330.


Review of Homogeneous Cosmology - LMU

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