Cosmological Models - hu-berlin.defjeger/Cosmolect1-5.pdf · Cosmological Models (mainly...

Cosmology

Cosmological Models(mainly relativistic Cosmology)

[References]Objects in the universe are essentially electrically neutral, such that gravity is theforce driving the dynamics of the universe. Therefore the basic theory is generalrelativity theory, which in the first instance expresses our local experience.

One has to make other assumption: T µνuniverse, boundary conditions, etc.

There are a number of global properties we can assign to the universe: radius,curvature, mean radiation density and mean mass density as well as dynamicalproperties like the expansion or the fluctuations in the microwave radiation.

Basic Postulates:

One of the basic observations is that velocities of matter systems on astronomicalvicinities are small, which together with the validity of the equivalence principle (allfreely falling bodies move in the same way) suggest the validity of ahydrodynamic model (streaming fluid) and hence to the assumption:

c© 2009, F. Jegerlehner ≪x Lect. 5 x≫ 301

http://www-com.physik.hu-berlin.de/~fjeger/Cosmolect1-4.pdf


Cosmology

Ê Weyl’s Postulate:The world lines form a bundle of geodesic lines, which diverge from a point (finiteof infinite) in the past or in the future.

This means that geodesic lines meet in not more than one single singular point.

Remark: it should be stressed that this is a strong idealization and a specialassumption about the true distribution and movement of matter. At the same timeit is natural in the sense of the equivalence principle: matter moves independent ofindividual properties universal under the influence of gravitation.

Consequence: there exist a bundle of hypersurfaces orthogonal to the geodesics.

Which means: matter defines e natural coordinate system, the co-moving restsystem of matter:

l xi = constant along world lines

l t = constant on hypersurfaces orthogonal to the world lines




Cosmology

Consequently: a co-moving observer measures t as proper time

Therefore the line element in the co-moving system takes the form:

ds2 = (c dt)2 + gik dxi dxk ; i, k = 1, 2, 3

with t the cosmic time. The latter defines a universal simultaneity, which is actuallynecessary to have a meaningful application of GRT to the universe.

Note: the existence of a cosmic time is preconditioned by the way the universe ispopulated with matter (galaxies etc.).

Exercise: discuss whether the postulate of a universal time is in contradiction tothe special relativity principle?

Ë Cosmological Principle:A co-moving observer xi = constant is not able to distinguish his position nor anydirection.




Cosmology

Which means: the subspaces t = constant are homogeneous and isotropic(thereby the homogeneity is a consequence of isotropy, see below).

Notice: this assumption is very special as it implies that for t = constant densityρ(x) and pressure p(x) are constant in the entire universe.

Consequence: the subspaces t = constant are geometrically similar in the sensethat the spatial metric has the form:

gik = −S 2(t) hik ; S 2(t) > 0 ,

and hik is homogeneous, isotropic and time-independent.




Cosmology

In fact hik is the metric of a 3D space of constant curvature (any point is center):

hik dxi dxk = δik dxi dxk︸︷︷︸dr2+r2 dΩ2

+

(xidxi

)2

a2 − xixi

=dr2

1 − r2/a2 + r2(dθ2 + sin2 θ dϕ2

)in spherical coordinates. For a→ ∞ we have flat 3D-space. Note that(xidxi)2 = r2 (dr)2 and

r2

a2 − r2 dr2 + dr2 =a2 dr2

a2 − r2 =(dr)2

1 − r2/a2

with the above form as a result.




Cosmology

1/a2 is the space curvature =

> 0= 0< 0

For finite a, the transformation of variables:

k = a2/α2 ; α = |a| ; r/α→ r ; α S (t)→ S (t)

leads to the well known Robetson-Walker (RW) line element of a homogeneous,isotropic cosmological model:

ds2 = (c dt)2 − S 2(t)

dr2

1−kr2 + r2(dθ2 + sin2 θ dϕ2

)The case k = 0 corresponding to 1/a2 = 0, which remains unchanged.




Cosmology

with

k =

1 is the positive spherical closed0 space zero flat infinite−1 curvature negative hyperbolic open

whereby the transformation r → r/(1 + k r2/4) leads back to the standard form.

Key points:

r hypersurfaces t = constant are 3D spaces of constant curvature

r time development is a conformal mapping of he 3D geometrieswith scale factor S (t).

r distances between two points are proportional to S (t) everywhere.If S (t) is increasing we have a general expansion.

r the spatial structure and the motion of galaxies cause each other.




Cosmology

We mention here another interesting fact: Infeld & Schild have shown that everyhomogeneous isotropic model is conformally flat6 (Minkowskian):

ds2 = e2ζ(x) gµν dxµdxν

The transformation (t, r)→ (τ, ρ) which maps the RW line element to manifestlyconformal form reads:

6The mapping

ds2 → ds2 = α2(x) ds2 ; α2 > 0

is a conformal transformation (specifically a dilatation)




Cosmology

ct∫0

dzS (z)

=

1/2 (cτ+ρ)∫0

dx1 + k x2 +

1/2 (cτ−ρ)∫0

dx1 + k x2

r∫0

du√

1 − k u2=

1/2 (cτ+ρ)∫0

dx1 + k x2 −

1/2 (cτ−ρ)∫0

dx1 + k x2 .

One gets

ds2 = (c dt)2 − S 2(t)

dr2

1 − kr2 + r2(dθ2 + sin2 θ dϕ2

)= e2ζ

(c dτ)2 −

[dρ2 + ρ2

(dθ2 + sin2 θ dϕ2

)]c© 2009, F. Jegerlehner ≪x Lect. 5 x≫ 309



Cosmology

with conformal function

e2ζ =S 2(t)

[1 + k/4 (cτ + ρ)2][1 + k/4 (cτ − ρ)2].

This result is important because Electrodynamics is conformally invariant.Maxwell’s equations thus apply without modification to homogeneous, isotropicworld models.

Exercise: Prove the conformal invariance of Maxwell’s equations.




Cosmology

Observable Quantities

For the physical interpretation the influence of the RW-metric

ds2 = (c dt)2 − S 2(t)

dr2

1 − kr2 + r2(dθ2 + sin2 θ dϕ2

)on observable quantities must be discussed.

Exercise: Show that in the co-moving spatial coordinates (r, θ, ϕ)

d`2 =dr2

1 − kr2 + r2 dΩ2

is invariant under the following transformations:




Cosmology

a) Rigid rotations: xi′ = Rik xk ; RRT = 1

b) Quasi-translations: xi′ = xi + ai√

1 − k r2 −(1 −√

1 − k a2)~x·~aa2

where ai is an arbitrary vector with

√1 − k a2 real, and a2 = aiai. This

transformation maps the origin xi = 0 into the point ai.

For later use we note that for

gik = S 2(t) hik ; hik =

1

1−k r2 0 00 r2 00 0 r2 sin2 θ

the determinant of the spatial metric is given by

det gik = S 6(t) det hik ; det hik =1

1 − k r2 r4 sin2 θ .




Cosmology

Spectral Shift (Red Shift)

For radial light propagation ds = 0 (null geodesics) and hence

c dtS (t) = ± dr

√1−k r2

A light source Q1 which in its rest frame (t1, r1) has period ∆t1. An observer in itsrest system (t0, r0) has a reference source Q0 of period ∆t1 and he observes aperiod ∆t0 from Q1. Assuming ∆t0,∆t1 |t0 − t1| we denote by λ = c∆t the wavelength and by ν = 1/∆t the frequency.

One then defines the spectral ratio

ζ = 1 + z ν1ν0

= 1 + ∆λλ1

; ∆λ = c (∆t0 − ∆t1)




Cosmology

z is the red shift:

z = ∆λλ1

z > 0 red shiftz < 0 blue shift

For the RW-metric we obtain:

1c

r1∫r0=0

dr√

1−k r2 =

t0∫t1

dtS (t)

independent of t!

The r integral yields

r∫0

dr′√

1 − k r′2=

arcsin r ; k = 1

r ; k = 0arcsinh r ; k = −1

= r + k ·r3

6+ O(r5)




Cosmology

The time independence implies

tt0 +∆t0

t0

r0 = 0

t1 +∆t1t1

r1

r

t0∫t1

dtS (t)

=

t0+∆t0∫t1+∆t1

dtS (t)

=

t0+∆t0∫t1

dtS (t)−

t1+∆t1∫t1

dtS (t)

=

t0∫t1

dtS (t)

+

t0+∆t0∫t0

dtS (t)−

t1+∆t1∫t1

dtS (t)




Cosmology

Notice: t1 < t0, t1 emission time, t0 receiving time, as ∆t0,∆t1 |t0 − t1| (clockperiod small with respect to transmission time) we must have:

∆t0S (t0) =

∆t1S (t1) hence ζ = 1 + z =

∆t0∆t1

=S (t0)S (t1)

Result: the time dependent scale factor S (t) implies a red shift

z =S (t0)S (t1) − 1

Expansion: S (t0) > S (t1) growing z > 0 red shift

stationary S (t) = constant ./ z = 0 no shift

Contraction: S (t0) < S (t1) decreasing z < 0 blue shift

[Link to Lect.: 13]


http://www-com.physik.hu-berlin.de/~fjeger/Cosmolect1-13.pdf#lightraysCosLec13



Cosmology

Observational fact:

Ê for very distant galaxies (d ≥ 1 Gpc) one indeed observes a systematic red shift(expansion, Hubble’s law)

Ë for nearer objects (d <∼ 1 Gpc) z ' 0 which tells us that S (t) is very slowly varying.

One therefore can make a number of approximations, by expanding in τ = t0 − t1.

First we define:

H0 = H(t0) =S (t0)S (t0) Hubble constant

q0 = q(t0) = −S (t0) S (t0)

(S (t0))2 deceleration parameter




Cosmology

and then write

S (t1) = S (t0)

1 − τH0 −τ2

2H2

0 q0 − · · ·

S −1(t1) = S −1(t0)

1 + τH0 +

τ2

2H2

0 (2 + q0) + · · ·

.

Thereby τ is determined by the distance to the source. For the red shift we thenhave

z = τH0 + τ2

2 H20 (2 + q0) + · · ·

Exercise: show that

1z

dzdt0

=S (t0) − S (t1)S (t0) − S (t1)

' −q0 H0 .




Cosmology

Measuring Distances

Distance measurements on cosmological scales are quite problematic andcorrespondingly there are quite a number of definitions of distances.

The coordinate distance between an event (t1, r1) and the observer (t0, r0 = 0) isr1.

The proper distance D1 can be defined by setting dt = 0:

D1 = S (t0)

r1∫0

dr√

1 − k r2= S (t0) ·

arcsin r1 ; k = 1

r1 ; k = 0arcsinh r1 ; k = −1

= S (t0)

r1 + kr3

1

6+ · · ·




Cosmology

D1 = c

t0∫t1

dtS (t0)S (t)

= c

0∫−τ=t0−t1

dτ′(1 − τ′H0 +

τ′2

2H2

0 (2 + q0) + · · ·

)

= c τ

1 +τ

2H0 +

τ2

6H2

0 (2 + q0) + · · ·

Inverting this relation, we obtain

τ =D1

c−

H0

2

(D1

c

)2

+H2

0

6(1 − q0)

(D1

c

)3

+ · · ·




Cosmology

and the red shift is the given by

z = H0D1c +

H20

2 (1 + q0)(

D1c

)2+ · · ·

Here, D1 is the present distance (observation time) of the galaxy.

Unfortunately D1 is not directly observable, however, it is related to observabledistances.

The form of the relation exhibits the two key parameters H0 and q0 and

z = H0D1c

is the Hubble law.

The latter would be valid as an exact relation if the expansion would benon-accelerated: q0 = 0 such that




Cosmology

S (t) = S (t0) (1 + (t − t0) H0) ; q0 = 0

Depending on the value of q0 we have:

v q0 > 0 the expansion is decelerated

v q0 < 0 the expansion is accelerated

From observations one presently extracts:

H0 = 74.2 ± 3.6 km/s Mpc (HST) 71 ± 4 km/s Mpc (WMAP) ; q0 = −0.60 ± 0.02




Cosmology

Cepheid Variable Stars in Spiral Galaxy NGC 3021 allow for more precisedetermination of the Hubble constant




Cosmology

Type Ia super nova reveal an accelerated expansion of our universe




Cosmology

From theory one expects that themaximum Luminosity in the lightburst of a SN Ia has a “uni-versal” value (standard candle).The figure shows raw data ver-sus corrected results, exhibitingan absolute universal shape ofthe light-curve




Cosmology

The result is more than remarkable as it implies that it is established nowthat we are living in a universe with accelerated expansion. In fact, the neg-ative value for q0 is a mind-boggling result! Observations of distant type Iasupernovas R≫ indeed indicate that q0 is negative. An accelerating expan-sion of the universe is in contrast to “naive” expectations: namely, that onthe cosmological scale the gravitational attraction of matter determines thescene. An accelerating universe requires the attraction of normal matter(including dark matter) to be more than counteracted by negative pressuredark energy. The latter could be in the form of either quintessence or a pos-itive cosmological constant. What is the meaning of that within the contextof cosmological models (GRT) will be discussed below.

For q0 = 0 the age of the universe is

l H−10 = (13.7 ± 0.2) × 109 years [Hubble age]

the time back a Big Bang must have been taking place.


http://hyperphysics.phy-astr.gsu.edu/hbase/astro/snovcn.html#c4



Cosmology

The corresponding horizon of the universe, where the escape velocity reaches thespeed of light c is

l Dmax = cH0' 3.34 Gpc

about 10.9 × 109 light years.

Remember that cosmology takes place at distances D ≥ 1 Gpc, which means thatonly a relatively small fraction of the universe is accessible to direct observation.In order to see more we have to wait longer.




Cosmology

Hubble Flow: more precisely

Cosmological Redshift⇔ The Hubble Flow cz = H0 d ⇔ due to expansion of theUniverse. To measure H0 requirel Distancel RedshiftMust correct for local motions / contaminations

1 + z = (1 + zobs)(1 + 30/c + 3G/c)30 = radial velocity of observer

- Measured from CMB Dipole ∼ 220 km/s3G = radial velocity of galaxy

- contributions include Virgocentric infall,- Great attractor etc

Decomposition of velocity field(Mould et al. 2000, Tonry et al. 2000)

[Link to Lect.: 11,13]


http://www-com.physik.hu-berlin.de/~fjeger/Cosmolect1-11.pdf#dipoleCosLec11

http://www-com.physik.hu-berlin.de/~fjeger/Cosmolect1-13.pdf#dipoleCosLec13



Cosmology

Observable Distances

Ê Parallax Distance (for nearby objects)




Cosmology

v Observe a star six months apart,(opposite sides of Sun)

v Nearby stars will shift against background star field

v Measure that shift. Define parallax angle as half this shift

Nearest star - Proxima Centauri is at 4.3 light years =1.3 pc à parallax 0.8”Smallest parallax angles currently measurable 0.001” à 1000 parsecsà parallax is a distance measure for the local solar neighborhood.

Define a parsec (pc) which is simply 1 pc = 206265 AU =3.26ly.A parsec is the distance to a star which has a parallax angle of 1”




Cosmology

First stars distances:

v α Centauri Thomas Henderson 1832 [1.35 pc]v 61 Cygni Friedrich Wilhelm Bessel 1838 [3.48 pc]r parallax angle the smaller the more distant the starr atmospheric perturbations limit ground based measurements to 0.03”∼30 pcr Hipparcos satellite: (launched by ESA in 1989 terminated 1993)

measured precision parallaxes to an accuracy of about 0.001-arcsec.

l Hipparcos measured parallaxes for about 100,000 starsl Got 10% accuracy distances out to about 100 pcl Good distances for bright stars out to 1000 pc.

See also: R≫, R≫


http://www.esa.int/esaSC/120366_index_0_m.html

http://hyperphysics.phy-astr.gsu.edu/hbase/solar/hipparcos.html#c1



Cosmology

Parallax determination in curved space: an academic exercise

Note 1000 pc 1 Gpc curvature negligible!

ϑ

~n

~nb~n1 ~n1

b

⋆⋆

S

O

The trajectory of a light ray in the co-moving CS ofthe source S is

~x′ = ~nσ ; ~x′ = (r′, θ′, ϕ′)

where ~n is the direction of the ray leaving the sourceat ~x′ = 0 and σ is the parameter of the correspondingworld line.In the co-moving system of the observer ~x = (r, θ, ϕ)the source is located at ~x1.




Cosmology

The coordinate transformation between the two systems reads:

~x = ~x′ + ~x1

√1 − k r′2 −(1 −

√1 − k r2

1

)~x′ · ~x1

r21

,

such that the light ray in the observer system looks like

~x = ~nσ + ~x1

√1 − kσ2 −

(1 −

√1 − k r2

1

)~n · ~x1σ

r21

Let ~n1 =

~x1r1

be the unit vector in direction of the source. In the flat space limit~n = −~n1 and

~x = ~nσ − ~n(r1

√1 − kσ2 + σ − σ

√1 − k r2

1

),




Cosmology

where the 1st and the 3rd term cancel. Hence, the light signal hits the observer Ofor σ = r1.

In weakly curved space, which is reality in our context, we are interested in signalswhich pass close to O, an d which we may parametrize by the direction

~n = cos ε~n1 + sin ε~n⊥

with ε 1 ; ~n⊥ · ~n1 = 0 ; ~n2⊥ = 1 .

In linear approximation

~n = −~n1 + ε~n⊥

such that

~x(σ, ε) ' ~n1

(r1

√1 − kσ2 − σ

√1 − k r2

1

)+ εσ~n⊥




Cosmology

where σ ' r1.

Since the proper length in the RW-metric space is ` = S (t0) |~x |, for theimpact parameter b we obtain

b ' S (t0) |~x(r1, ε)| = S (t0) r1 ε

The tangent to the trajectory at σ = r1 is given by

d~x(σ, ε)dσ

∣∣∣∣∣∣σ=r1

' ε~n⊥ −~n1√

1 − k r21

,

and therefore the direction of observation

~nb ' −

√1 − k r2

1d~x(σ, ε)

dσ

∣∣∣∣∣∣σ=r1

' ~n1 −

√1 − k r2

1 ε~n⊥ .




Cosmology

With this we can calculate the angle of observation ϑ

tanϑ =

√1 − k r2

1 ε '√

1 − k r21

bS (t0) r1

' ϑ (ϑ 1) .

In Euclidean space with d1 as Euclidean distance of the source we would haveϑ ' b

d1or d1 '

bϑ.

Correspondingly we therefore define the parallax distance as

dP ≡bϑ'

S (t0) r1√1−k r2

1

; (ϑ, b→ 0)

Ë Angular Diameter Distance (distance of apparent size)

In case one has knowledge about the true size of an object, the diameter D, e.g.from model calculations, one can determine the distance from the apparent size,the angle ϑ:




Cosmology

Angular diameters of a) Sun b) Cartwheel Galaxy




Cosmology

ϑ

D

O

Coordinates of the source (t1, r1) in the equatorialplane θ = π/2. The observer O is at (t0, r0 = 0).Since dr1 = 0 ; dt = 0 the proper length

d`2 = S 2(t) r21 dϕ2

and hence

D = S (t1) r1 · ϑ , (ϑ 1) .

In Euclidean space we would have d1 = D/ϑ correspondingly, one defines

DA ≡Dϑ' S (t1) r1

as the distance of apparent size.




Cosmology

Ì Distance of proper motion

If one knows the true transversal velocity 3⊥, e.g., for binary stars, one candetermine the distance from the observed angular velocity of the observationangle:

∆ϑ

v⊥(t1)∆t1 = ∆D1

O

First we have

∆D1 = 3⊥(t1) ∆t1 = 3⊥(t1) ∆t0S (t1)S (t0)

.

The observed angular change per unit time is ω =

ϑ = ∆ϑ∆t0

. In Euclidean space we would have d1 =

3⊥∆t0∆ϑ

, and correspondingly

we define the distance of proper motion as

DM ≡3⊥

ϑ' S (t0) r1




Cosmology

The most important of the observable distances for very distant objects is thefollowing:

Í Luminosity Distance

This method requires the knowledge of the luminous power L1 of the source,usually obtained form model calculations or known known physical laws orregularities.

The power L1 of a source at (t0, r1) spreads over a surface of a sphere

4π r22 S 2(t0) .

Light which gets emitted during a time interval δt1 is received by the observer at




Cosmology

(t0, r0 = 0) during ∆t0 where

∆t0 = ∆t1S (t0)S (t1)

.

For the energy E = h · ν a frequency change

ν0 = ν1S (t1)S (t0)

is resulting, such that in total the measured Luminosity `0 is given by

`0 = L11

4π r21

S 2(t1)S 4(t0)

.




Cosmology

In Euclidean space we would have `0 = L11

4π d21

as d1 = r1.

Correspondingly one defines the Luminosity Distance




Cosmology

DL ≡

√L1

4π`20

= r1S 2(t0)S (t1) = r1 S (t0) · (1 + z)

The different observable distances satisfy the following relations:

DAdL

=S 2(t1)S 2(t0) = (1 + z)−2

DMdL

=S (t1)S (t0) = (1 + z)−1

We stress once more that the determination of the distances DL, DA and DM

require the knowledge of the object properties L1, D and 3⊥ (models, systematicsetc.)

One important result of the above considerations is that the proper distance D1 upto terms of 3rd orders agrees with DM:

D1 ' DM




Cosmology

which is important for the interpretation of the red shift formula (Hubble law).




Cosmology

Comparison of Distance Measures

Proper Motion Distance:

DM =3⊥

ϑ= S (t0) r1

Angular Diameter Distance:

DA = Dϑ

= S (t1) r1

Luminosity Distance:

DL =(

L4πF

)1/2= S (t0) r1 (1 + z)

DL = (1 + z) DM = (1 + z)2 DA




Cosmology

v The Luminosity Distance (DL) shows why distant galaxies are so hard to see -a very young and distant galaxy at red shift 15 would appear to be about 560billion light years from us

v Even though the Angular Diameter Distance (DA) suggests that it was actuallyabout 2.2 billion light years from us when it emitted the light that we now see.

v The Hubble Distance (DLT) tells us that the light from this galaxy has traveledfor 13.6 billion years between the time that the light was emitted and today.

v The Comoving Distance (DcM) tells us this same galaxy if seen today, wouldbe about 35 billion light years from us.

More on distance measurements R≫ R≫


http://www.talkorigins.org/faqs/astronomy/distance.html

http://www.astro.ucla.edu/%7Ewright/distance.htm



Cosmology




Cosmology

History of the determination of the Hubble constant. Illustrates the problem withcosmic distance measurementsCredit: J.P. Hucha R≫c© 2009, F. Jegerlehner ≪x Lect. 5 x≫ 348

http://www.cfa.harvard.edu/~huchra/hubble/



Cosmology

Number Counts

Counting the herd. Olber’s Paradox revised. Melodie: ”Weißt du, wieviel Sternleinstehen an dem blauen Himmelszelt” (see this: R≫)

Of particular interest:

l LUMINOSITY EVOLUTION (Galaxies in the past were brighter than today)

l DENSITY EVOLUTION (Galaxies in the past were more numerous than today)


http://terpsichore.stsci.edu/~summers/viz/hgast/hgast_video_640x480_30_ss_ew.mov



Cosmology




Cosmology

In the co-moving CS the spatial volume element reads

dV ′ =1

√1 − k r2

r2 sin2 θ dr dθ dϕ

and the surface element is given by

dV ′S = 4 πr2 dr√

1 − k r2

According to the cosmological principle we assume the density of the objects(galaxies, clusters of galaxies) to be homogeneous, isotropic and timeindependent (local rest system). For the presently observable universe this maybe a reasonable assumption. What we observe coming up at the horizon areobjects like the ones we know from “nearby”. In fact not quite: systematicdeviations→ evolution seen!

The number of sources in the volume element dV ′ with luminosity in the interval




Cosmology

L, L + dL is

dN′ = n′(L) dL dV ′ .

The element of the proper volume is

dV = S 3(t) dV ′ ,

such that

n(t, L) = n′(L) S −3(t)

hence

n(t, L) =(

S (t0)S (t)

)3n(t0, L)




Cosmology

Herewith, the number of sources at fixed cosmic time (dt = 0) in the intervalL1, L1 + dL1 in a spherical shell r1, r2 + dr1 reads

dN = 4 π n(t1, L1) S 3(t1)1√

1 − k r21

r21 dr1 dL1 .

The coordinate r1 may be replaced with the light time t1

r1 = r(t1) :

t0∫t1

c dtS (t)

=

r1∫r0=0

dr√

1 − k r2rsp.

cdt1S (t1)

= −dr1√

1 − k r21

.

This yields the change in the number of observed sources in the time intervalt1, t1 + dt1 as

dN = 4 π n(t0, L1)(S (t0)S (t1)

)3

S 2(t1) r2(t1) c dt1 dL1 .




Cosmology

It makes sense to ask for the number of sources in a sphere, the radius of which isfixed on the one hand by a maximum red shift zmax

rz = r(tz) ; tz : S (tz) =S (t0)

1 + zmax

on the other hand by a minimal luminosity `min

r` = r(t`) ; t` :r2(t`)S 2(t`)

=L

4 π `min S 4(t0).

One finds,

N(< zmax, > `min) = 4 π c∞∫0

dL1

t0∫maxtz,t`(t0)

dt1 r2(t1) S 2(t1) n(t1, L1)




Cosmology

Again we may expand in t0 − t1:

S (t1) = S (t0) 1 − H0 (t0 − t1) + · · ·

r(t1) = S −1 (t0 − t1)1 +

H0

2(t0 − t1) + · · ·

n(t1, L1) = n(t0, L1) 1 − β0(L1) H0 (t0 − t1) + · · ·

Note that for the RW-metric with taking n′(L1) = constant (conservation of matter inthe co-moving local rest system) we obtain

n(t1, L1) = n(t0, L1)(S (t0)S (t1

)3

' n(t0, L1)

1 + 3 H0 τ + 3 H20 (4 + q0)

τ2

2+ · · ·

which yields

β0(L1) = −3




Cosmology

In a so called steady-state theory one would have β0(L1) = 0. Of course on musttry to determine β0(L1) by observation, which is not an easy task, however.

In expanded form we obtain

N(< zmax, > `min) '4 π3

∞∫0

dL1 n(t0, L1) (t0 − tmax(L1))3

× 1 −34

(β0(L1) + 1) H0 (t0 − tmax(L1)) + · · ·

where tmax = max tz, t`(L1).




Cosmology

In spite of the fact that new telescopes, in particular the Hubble space telescope,have dramatically increased the number of observed objects, it remains verydifficult to determine the parameters β0(L), specific for number counts, and H0 oreven q0 just from number counts. But any kind of information is needed. In case ofradio sources one does not measure the total luminosity but the spectraldistribution. The above formulas have to be modified to include the appropriatefrequency shifts as well (see S. Weinberg I).




Cosmology

v Study Galaxy source counts to

à investigate the contribution of different galaxy populations to the Universeà compare the evolution of galaxies today with those in the pastà constrain Geometry of the Universe

v Source counts depend on

à Cosmologyà Luminosity Functionà K-Correctionà Evolution

v Source counts are a function of observing wavelength

à Different wavelengths are dominated by different classes of sourcesà To understand the star formation and evolutionary history à multiwavelength

analysis




Cosmology

v Present and next generation surveys

à Larger areas to greater depthsà increase statistical samples by orders of magnitudes!

Left: Differential UBVIJHK galaxy counts as a function of AB magnitudes. Note




Cosmology

the decrease of the logarithmic slope d log N/dm at faint magnitudes. Theflattening is more pronounced at the shortest wavelengths. Right: Extra-galacticbackground light per magnitude bin, (i = 10-0.4(mAB+48.6) N(m), as a function ofU (filled circles), B (open circles), V (filled pentagons), I (open squares), J (filledtriangles), H (open triangles), and K (filled squares) magnitudes. For clarity, theBVIJHK measurements have been multiplied by a factor of 2, 6, 15, 50, 150, and600, respectively.




Cosmology

More here:R≫ R≫


http://www.youtube.com/watch?v=vPqFwMWbQQo

http://www.youtube.com/watch?v=X5zVlEywGZg&feature=player_embedded



Cosmology

The Hubble Space Telescope (HST) has been upgraded recently and continues tolook deeper into space and time.




Cosmology




Cosmology

An “empty spot” region ...




Cosmology

... looked at closer (using new IR cameras)




Cosmology

References

v S. Weinberg, Gravitation and Cosmology, Wiley 1972

v S. Weinberg, Cosmology, Oxford 2008

v M. Roos, Introduction to Cosmology, Wiley 2003

v D. Harland, The Big Bang, Springer 2003

v J. Rich, Fundamentals of Cosmology, Springer 2001

v L. Bergstrom, A. Goodbar, Cosmology and Particle Physics, Springer 2004

v E. Harrison, Cosmology, Camb. Univ. Press 2000

[back]




Cosmology

Appendix: spaces of constant curvature

Theorem 3. Constant Gauss curvature in a point P

K(P) =Rµν,ρσξ

µξρηνησ(gµσgνρ − gµρgνσ

)ξµξρηνησ

= constant

./

Rµν,ρσ = K(gµσgνρ − gµρgνσ

).

Proof: 1) We define the auxiliary tensor

Tµν,ρσ Rµν,ρσ − K(x)(gµσgνρ − gµρgνσ

),

where K = K(x) may depend on the point x. By construction Tµν,ρσ has the




Cosmology

index symmetries of Rµν,ρσ, which are

Rµν,ρσ = −Rµν,σρ ; Rµν,ρσ = −Rνµ,ρσ

Rµν,ρσ = Rρσ,µν ; Rµ[ν,ρσ] = 0 ; [. . .] :∑cycl.

.

In compliance with the requirements

Tµν,ρσξµξρηνησ ≡ 0

holds for arbitrary vectors in the tangent space at P. The curvature K(x) at P isdirection independent (isotropy). We may take special vectors, with twoindependent components:

ξµ = a δµα + b δµβηµ = c δµγ + d δµδ




Cosmology

and consider the term proportional to abcd:

Tαγ,βδ + Tβγ,αδ + Tαδ,βγ + Tβδ,αγ = 0 ,

where in the 2nd and 4th term by symmetry we may interchange the index pairs:Tβγ,αδ = Tαδ,βγ etc. Thus

Tαγ,βδ + Tαδ,βγ = 0 .

By a different choice of the vectors similarly we obtain

Tαγ,δβ + Tαβ,δγ = 0 .

Now we consider the “Jacobi-Identity” (it holds for R... and hence for T...):

Tα[γ,βδ] ≡ Tαγ,βδ + Tαδ,γβ + Tαβ,δγ = 0 .




Cosmology

Using the two relations derived above together with the antisymmetry of R... andhence T... in the second index pair we find: the 2nd term is Tαδ,γβ = −Tαδ,βγ = Tαγ,βδand the 3rd Tαβ,δγ = −Tαγ,δβ = Tαγ,βδ and therefore

Tα[γ,βδ] ≡ 3 Tαγ,βδ = 0

for any combination of the indices, i.e. T... ≡ 0 q.e.d. q

Corollary 1. An isotropic space has a curvature tensor

Rµν,ρσ = K(x)(gµσgνρ − gµρgνσ

)Proof: 2) For the isotropic space the Ricci-tensor reads:




Cosmology

Rµν = Rλµ,νλ = gλω Rωµ,νλ

= K(x) · gωλ(gωλgµν − gωνgµλ

)= K(x) · (n − 1) gµν ,

and together with the scalar curvature

R = gµν Rµν = K(x) · n (n − 1) ,

the Einstein-tensor obtained is

Gµν = Rµν −12

R gµν = −K(x)(n − 2)(n − 1)

2gµν .

The Bianchi-Identity: DµGµν = 0 infers




Cosmology

K(x) is a scalar field

and hence Dµ K(x) = ∂µ K(x), also Dµ

(δµν

)≡ 0, such that

DµGµν = −

12

(n − 2)(n − 1)∂ν K(x) ≡ 0

i.e. K(x) = K = constant q.e.d. q

Corollary 2. For n ≥ 3 an isotropic space is also homogeneous

K(x) = K = constant (n ≥ 3)!

Theorem 4. (uniqueness) Two Riemann spaces Mn and Mn with the same con-stant curvature are locally (in neighborhoods of given points P and P) isometric.




Cosmology

Proof: By assertion there exists a mapping

Λµν(x)

∂xµ

∂xν

such that

gµν Λµρ Λν

σ = gρσ . (∗)

The latter “mapping” in fact is a system of differential equations which is to besolved. By linear transformations one can achieve that in P and P the initialconditions

gµν(0) = gµν(0) =gµν and Λ

µν(0) = δ

µν

hold. The integrability condition for (*) we obtain by differentiation of it. Denoting∂λ ≡

∂∂xλ and utilizing the transformation laws for the Christoffel symbols,




Cosmology

we obtain

∂λΛµν = Γανλ Λ

µα − Γ

µβγ Λ

βν Λ

γλ (∗∗)

For each solution of (**)

gρσ − gµν Λµρ Λν

σ = Cρσ = constant

holds, where by the choice of the initial conditions

Cρσ ≡ 0 .

Together with (**) we have

∂λ Λµν =

∂2xµ

∂xλ∂xν= ∂ν Λ

µλ




Cosmology

where the symmetry is the integrability condition for

Λµν =

∂xµ

∂xν.

Now, the integrability condition for (**) is precisely

Rαβ,γδ = Rµν,ρσ Λµα Λν

β Λργ Λσ

δ .

However,

Rαβ,γδ = K(gαδgβγ − gαγgβδ

)and

Rµν,ρσ = K(gµσgνρ − gµρgνσ

)c© 2009, F. Jegerlehner ≪x Lect. 5 x≫ 375



Cosmology

are the preconditions of the theorem, such that because of (*) the integrabilityconditions indeed are satisfied.

q.e.d. q

Corollary 3. In a Riemannian space of constant curvature to two given points Pand P′ there exists an isometric mapping which maps a neighborhood of P into aneighborhood of P′.




Cosmology

Explicit construction

Special forms of the metric. Starting point is

Mn+1: n+1-dimensional Euclidean or pseudo-Euclidean space

ds2 =gAB dxAdxB ; A, B = 1, · · · , n + 1

xn+1 z ;gAB= diag (1, · · · , 1, k) ; k = ±1, 0

ds2 =(dxi

)2+ k dz2 = hik dxidxk ; i, k = 1, · · · , n




Cosmology

Hypersurface: xixi = ρ2 − k z2 defines an isometric embedding

xixi = ρ2 − k z2 ;

k = 1 , ρ2 > 0 spherek = −1 , ρ2 < 0 two sheet hyperboloidk = −1 , ρ2 > 0 one sheet hyperboloid

Take k , 0: k z2 = ρ2 − x2 2 k zdz = −2 xidxi thus

dz2 =

(xidxi

)2

k2 z2 =1k

(xidxi

)2

ρ2 − x2 ; k2 = 1

and

ds2 =(dxi

)2+ k (dz)2 =

(dxi

)2+

(xidxi

)2

ρ2 − x2 = hikdxidxk




Cosmology

hik = δik +xixk

ρ2 − x2 ; hik = δik +xixk

ρ2

Check:

hikhkl = δil −

xixl

ρ2 +xixl

ρ2 − x2 −xixl x2

ρ2 (ρ2 − x2)= δi

l 3

The derivative of the metric is

∂l hik =1

ρ2 − x2

xi δkl + xk δil + 2

xixkxl

ρ2 − x2

=

1ρ2 − x2 (xihkl + xkhil)




Cosmology

and the Christoffel symbol reads

Γi,kl =12

(∂lhik + ∂khil − ∂ihkl)

=1

ρ2 − x2 xihkl

Γjkl = h jiΓi,kl =

1ρ2 x jhkl

∂r Γjkl =

1ρ2

(δ

jr hkl + x j∂rhkl

)=

1ρ2 (ρ2 − x2)

((ρ2 − x2

)δ

jr hkl + x jxkhlr + x jxlhkr

)c© 2009, F. Jegerlehner ≪x Lect. 5 x≫ 380



Cosmology

Rik,lm = ∂m Γi

kl − ∂l Γikm + Γi

λm Γλkl − Γiλl Γ

λkm

=1ρ2

(δi

mhkl − δilhkm

),

Rik,lm =1ρ2 (himhkl − hilhkm)

As a result space of constant curvature

K = 1ρ2




Cosmology

|x|

k = 1z

z2 + x2 = ρ2

|ρ|ρ2 > 0

|x|

k = −1z

−z2 + x2 = ρ2

i|ρ|

ρ2 < 0

|ρ|

ρ2 > 0

Corollary 4. Riemann spaces of constant curvature exist for arbitrary K ∈ R. Ac-cording to the previous theorem these are locally unique.

Result: Isotropic spaces are homogeneous and of constant curvature andcoordinates may chosen locally such that the metric is




Cosmology

ds2 =(dxi

)2+

(xidxi)2

ρ2−x2

Symmetries of spaces of constant curvature

The metric derived above per construction possesses an invariance group which isinduced by the orthogonal or pseudo-orthogonal group on Mn+1. The line element

ds2 =gAB dxAdxB

as well as the embedding condition

xAxA = ρ2

are invariant under

x′A = ΛAB xB ; ΛA

B real ,




Cosmology

where

gAC ΛA

B ΛCD =

gBD .

If we write

Λ =

(R ab c

); Λ

g ΛT =

g ,

the requirement of the invariance of the metric tensor can be split into theconditions: (

R RT)

ik+ k aiak = δik ,

Rik bTk + k c ai = 0 ,

bibi + k c2 = k .

On the induced space of constant curvature, we can distinguish two types of




Cosmology

transformations:

Ê Rotations:

ai = 0 R RT = 1 bi = 0 ; c = 1Evidently,

ds2 =(dxi

)2+

(xidxi

)2

ρ2 − x2 and k z2 + x2 = ρ2

both are invariant under x′i = Rik xk. These rotations leave the origin xi = 0

invariant.

Ë Quasi-translations:

ai , 0 bi , 0 ; c =√

1 − bibi/k

We are looking for a matrix

Rik δik − rik




Cosmology

which for ai = 0 goes to Rik = δik. This requires rik to be of the form

rik = λ k aiak .

Then(R RT

)ik

+ k aiak = δik ; k2 = 1 implies

−2λk aiak + λ2a2aiak + k aiak = 0

and hence

λ2 a2 − 2 λ k + k = 0 ./ λ2 k a2 − 2 λ + 1 = 0 .

Thus

λ± =1

k a2

(1 ±√

1 − k a2)

; a2 aiai




Cosmology

where only λ− is regular at a2 = 0 and hence

λ ≡ λ− = 1k a2

(1 −√

1 − k a2).

Furthermore, from R ki bT

k + k c ai = 0 we obtain

bTi − λ k aiakbT

k + k c ai = 0 (R)

with solutionbT

i = −k ai .

Here, we have used c =√

1 − bibi/k =√

1 − k a2 and (R) yields−1 − λ k a2 + c = 0 λ = 1−c

k a2 .

Altogether:Rik = δik − λ k aiak

bTi = −k ai

c =√

1 − k a2 ; λ = 1−ck a2




Cosmology

where ai is an arbitrary n-dimensional vector with the constraint that c =√

1 − k a2

is real! Therefore, for

x′i = Rikxk + ai z where k z2 = ρ2 − x2

we have

x′i = xi + ai(√|ρ2 − x2| − λ k x · a

)

This transformation in particular transforms the origin xi = 0 into the vectorx′i(0) = ai |ρ|, that’s why we call it quasi-translation. If we set ai

|ρ|= ai we finally

obtain:x′i = xi + ai

(√1 − K x2 − λK x · a

)where K = k/ρ2 is the Gauss curvature and ai is such that c = c =

√1 − K a2 is

real and λ = λ = 1−cK a2 . That’s it!




Cosmology

Previous ≪x , next x≫ lecture.






Cosmological Models - hu-berlin.defjeger/Cosmolect1-5.pdf · Cosmological Models (mainly...

Documents

Transcript of Cosmological Models - hu-berlin.defjeger/Cosmolect1-5.pdf · Cosmological Models (mainly...