Cosmology Cosmological Models (mainly relativistic Cosmology) [References] Objects in the universe are essentially electrically neutral, such that gravity is the force driving the dynamics of the universe. Therefore the basic theory is general relativity theory, which in the first instance expresses our local experience. One has to make other assumption: T μν universe , 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 dynamical properties like the expansion or the fluctuations in the microwave radiation. Basic Postulates: One of the basic observations is that velocities of matter systems on astronomical vicinities are small, which together with the validity of the equivalence principle (all freely falling bodies move in the same way) suggest the validity of a hydrodynamic model (streaming fluid) and hence to the assumption: c 2009, F. Jegerlehner ≪ ❘ Lect. 5 ❘ ≫ 301
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[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:
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
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.
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.
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.
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
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.
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.
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)
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”
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.
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.
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
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:
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:
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
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.
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
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 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).
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
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.
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:
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:
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.
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,
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′.
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