Review of Homogeneous Cosmology - Discussion of characteristics of GR derived models for our homogeneous and isotropic expanding universe followed ...
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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 1
Basic Ingredients
of the Big Bang Model
n 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
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 2Ingredients Follow Naturally from
Basic Concepts and Principles
n 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 isotropic
n General relativity
n Physics the same everywhere:
n Cosmological principle
n Structure formation:
n Gravitational collapse
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 3Big Bang Model Observationally
Supported
n Cosmology has rested on four observational
pillars for 5 decades now
n The Universe expands homogeneously
n The night sky is dark
n There exists a cosmic microwave background that is a pure black
body spectrum
n 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 Cosmo-LSS | Mohr | Lecture 1 4A Modern Measure of the Expansion
Riess, Press & Kirshner ApJ 1996 All studies provide consistent
results in the local universe:
Measurements to 19 Supernovae 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)
Blue points: 19 SNe v r = Hod
Red line: Hubble Law with
Ho=19.6 km/s/MLy vr
so H o =
d
The Hubble parameter has units
of velocity over distance.
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 5Why is the Night Sky Dark?
n Suppose that the universe is infinite and
homogeneous
n every line of sight intercepts a star
n sky should glow as brightly as the surface of an
average star
n but the night sky is dark…
n Olbers’ Paradox
n Heinrich Olbers in 1826
n Thomas Digges in 1576
Johannes Kepler in 1610
Observer
n
n Edmund Halley in 1721
n Therefore, Universe cannot be infinite and
homogeneous!
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 6COBE Spectrum: Blackbody Emission
TCBR = 2.735 K
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 7Cosmic Elemental Abundances
H
12 Stars
He
10 O Fe
C Ne Si
Big Mg S Cr Ni
8 Ar Ca
Bang Ti
Log N N
6
Na Al Cl
K Co Cu
P Mn
F
4 B V
Sc
2
Be
Li
Atomic Number
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 8Cosmic Densitometer
n 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
Deuterium
n Precise observational test
n 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
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 9Consider now dynamics in Spacetime Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 10
Dynamics in Spacetime – the Geodesic
n 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 Cosmo-LSS | Mohr | Lecture 1 11Geodesics
n Any path between two points that is an extremum
(i.e. longest or shortest) is a geodesic
n Examples:
n Straight line in Euclidean geometry
n Great circle in spherical geometry
n “Hiker’s path” in saddle geometry
Moscow
Chicago These three geometries (flat, spherical and
hyperbolic) are important because they are
homogeneous and isotropic {Cosmological
Principle}
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 12How Can Geometry seem like a Force?
Parable of the
two travelers
•Two travelers start out walking in parallel.
traveler traveler •Mutual gravitational attraction draws them
A B closer.
•This is similar to the behavior of parallel
lines in a closed or spherical geometry!
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 13The Metric Equation
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
2 2 2
Δs = f (u , v)Δu + g (u , v)ΔuΔv + h(u , v)Δv
Metric coefficients
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 14The Metric Equation:
Euclidean Geometry
Δs 2 = f (u , v)Δu 2 + g (u , v)ΔuΔv + h(u , v)Δv 2
n Pythagorean Theorem is
2 2 2
Δs = Δx + Δy the metric equation for
Euclidean geometry
∴
f ( x, y ) = 1
g ( x, y ) = 0 Homogeneous, orthogonal, isotropic
h ( x, y ) = 1
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 15Metric Equation:
Spherical Polar Geometry
2 2 2
Δs = f (u , v)Δu + g (u , v)ΔuΔv + h(u , v)Δv
Δs 2 = R 2 Δθ 2 + ( R cosθ ) 2 Δφ 2 q
∴ (q+Dq,f+Df)
f (θ , φ ) = R 2
g (θ , φ ) = 0 f
2 2
h(θ , φ ) = R cos θ
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 16Spacetime 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
2 2 2 2
Δs = αc Δt − βcΔtΔx − γΔx
n a, b, g are the metric coefficients
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 17Minkowski Spacetime Metric
n Spacetime interval
2 2 2 2
Δs = c Δt − Δx
n Minkowski spacetime called flat spacetime
2 2 2 2
Δs = αc Δt − βcΔtΔx − γΔx
α (x, t) = 1
β (x, t) = 0
γ (x, t) = 1
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 18Geodesic 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 Cosmo-LSS | Mohr | Lecture 1 19Illustration: Minkowski Spacetime
t 2 2 2
Δs AB + Δs BC = 2cΔt 1 − Δx / c Δt
C
(0, 2Dt)
2 2 2
Δs BC = c Δt − Δx
Δs AC = 2cΔt B (Dx, Dt)
Δs AB = c 2 Δt 2 − Δx 2
(0,0)
x
A Spacetime interval larger for geodesic motion
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 20Dynamics of Particles and Light
n 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 Cosmo-LSS | Mohr | Lecture 1 21Einstein’s Field Equations
n 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
µν 8πG µν
G = 4 T
c
€
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 22Einstein’s Field Equation(s)
µν 8π G µν
G =− 4 T
c
µ = (1, 2, 3, 4);ν = (1, 2, 3, 4)
" 11 12 13 14 % " 11 12 13 14 %
$ G G G G ' $ T T T T '
21 22 23 24
$ G G G G ' 8π G $ T 21 T 22 T 23 T 24 '
$ G 31 G 32 G 33 G 34 ' = c 4 $ T 31 T 32 T 33 T 34 '
$ ' $ '
$# G 41
G 42
G 43 44
G '& $# T 41 T 42 T 43 T 44 '&
Riemann curvature tensor Stress-energy tensor
These tensors are symmetric, so there are only 10 components
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 23Comments
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 Cosmo-LSS | Mohr | Lecture 1 24Comments: Action at a Distance?
n In GR the Newtonian problem of instantaneous action
at a distance is removed
n Gravity is reflected in local curvature of spacetime, which
responds to local density of matter and energy
n Matter and energy subject to propagation speed of light
n Gravitational radiation is natural implication of theory
n Orbiting masses will produce ripples in spacetime- sending
out waves of curvature called gravitational radiation
n These waves propagate outward from their source at the
speed of light
n Detection of gravitational wave effects is one of the great
triumphs of general relativity!
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 25Consider now the expanding Universe Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 26
Friedmann-Robertson-Walker metric
n Within Peacock there is a convenient form adopted
for the RW metric which allows the curvature
dependence to be abstracted
#sin r ( k = 1)
%
Sk ( r) = $ r (k = 0)
%
&sinh r ( k = −1)
n The metric for all three curvature families is then
€
c dτ = c dt − R ( t )[ dr + S ( r) dϕ
2 2 2 2 2 2 2
k
2
]
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 27Comoving Separation
n In this coordinate system we can calculate the
separation between two astronomical sources at rest
within this expanding model
n Choose to locate them along coordinate r at r and r+dr
n Separation d at time t is R(t)dr
n Note the time dependence R(t), so the distances will simply
scale up over time with the scale factor
n Effective recession velocity would be
Δd R( t 2 )δr − R( t1 )δr ( R(t 2 ) − R(t1 ))
Vr = = = δr
Δt t 2 − t1 t 2 − t1
n Allowing us to recover the Hubble law where at any time an
observer sees Vr is proportional to separation…
€
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 28Redshift
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
t obs
cdt
2 2 2 2
0 = c dt − R (t)dr so r = ∫ R(t)
t emit
n A subsequent crest in this light wave will be emitted
at temit€+dtemit, and observed at tobs+dtobs so we can
write t obs +dt obs
cdt
r= ∫
R(t)
t emit +dt emit
Apr 16, 2021
€ Cosmo-LSS | Mohr | Lecture 1 29Redshift (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
t obs +dt obs t obs
cdt cdt cdt obs cdt emit
0= ∫ − ∫ = −
R(t) temit R(t) R(t obs ) R(t emit )
t emit +dt emit
cdt obs R(t obs )
or = Using ! =
#
= % &'
cdt emit R(t emit ) $
λobs R(t obs )
so = ≡ 1+ z
λemit R(t emit )
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 30Dynamics of the expansion
n 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 Cosmo-LSS | Mohr | Lecture 1 31Expanding Shell in Homogeneous Universe
homogeneous shell of mass
m
mass density expanding with
r speed V=HR
R
+
mass interior to R
M
V
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 32Newtonian Motivation for
the Friedmann Equation
K.E.+ P.E. = T.E. Consider shell of matter
1 # GMm & moving outward.
2
mV + % − ( = E = const.
2 $ R '
+1 G % 4 π 3 (. Use Hubble law:
2
m- (HR) − ' R ρ *0 = E = const. V=HR
€ ,2 R& 3 )/
1
÷ both sides by mR 2
2
2
8π G 2E −kc Define kc2=-2E/m:
H2 − ρ= 2 =
3 mR R2 Friedmann Equation
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 33The Friedmann Equation
ΔR ˙
V= ≡R definition of velocity
Δt
V = HR Hubble Law
∴
2
2
% ˙
R ( 8πG kc 2
H ≡' * = ρ− 2
& R) 3 R
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 34Density and Geometry
n Friedmann equation implies that balance of expansion energy
and gravitational potential energy determines geometry of
spacetime
2 28πG 2 2
H R − R ρ = −kc
3
n For zero curvature k=0 models, r=rcrit where
2
3H
ρcrit =
€ 8πG
n For r>rcrit k=+1 (closed space) and for rDensity, 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 Cosmo-LSS | Mohr | Lecture 1 36Scenarios for Evolution of the R(t)
R R
expanding,
expanding, constant speed
decelerating
t t
R expanding, R
accelerating contracting,
accelerating
t t
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 37The deceleration equation
n We can take the time derivative of the Friedmann equation, too
˙ 2 8πG 2
R − ρR = −kc 2
3
and d( ρc 2 R 3 ) = − pd(R 3 ),
which is dE = − pdV
n Giving us ˙R˙ = − 4 πGR ( ρc 2 + 3p)
3
n Acceleration: rc2Density parameter
n The density parameter
is denoted as the
density divided by the
critical density ρ 8πGρ
Ω= = 2
n For k=0, W=1 at all ρ crit 3H
times, but otherwise
W=W(t), and the present
epoch value is Wo
€
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 39What is the Critical Density?
2
3H 0
ρ crit (today) ≡
8π G
-18 2
3⋅ (2.27 ×10 )
= −11
8 π (6.67 ×10 )
−27 3
≈ 9 ×10 kg/m
About 10 hydrogen atoms per cubic meter
~ 1011 M 0 Mpc 3
Mass of ~1 galaxy per Mpc3
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 40Density parameters and evolution
2
kc 8πG
n Generically, the universe
2
H + 2 =
R 3
( ρm + ρr + ρΛ )
contains radiation, dark matter,
matter and dark energy. kc 2
1+ 2 2 = Ωm + Ωr + ΩΛ
H R
n Note that the variation of the Ωm + Ωr + ΩΛ + Ωk = 1
energy densities of these
components differs, but they or Ωk = 1 − Ωm − Ωr − ΩΛ
evolve in such a way as to Matter : ρ ( z ) = ρ0 (1+ z )
3
maintain a constant value when 4
an effective curvature density Radiation : ρ ( z ) = ρ 0 (1+ z )
parameter Wk is introduced Cosmological constant : ρ ( z ) = ρ0
€
3(1+w)
3
In general : ρ ( z ) = ρ0 (1+ z )
# R & # 1 &3
Volume ∝ % ( = % ( p
$ R0 ' $1+ z ' where w ≡
ρ
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 41Scale factor and Horizon scale
n The Friedmann equation also
specifies the scale factor Ro.
Unfortunately, this
−1
c $ (Ω −1) '
n 2
expression is undefined for R0 = & )
the models consistent with
H0 % k (
observations (k=0)
n The Hubble length c/Ho is€a c
characteristic distance or = 14.1GLyr
H0
scale of the horizon
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 42Expansion history H(z)
n The expansion history H(z) just reflects the variation
of the energy densities of all the components in the
universe 8πG
H ( z) = ρ(z)
3
n For a universe composed of radiation, matter and
generalized dark energy, we can write the
expansion history as follows
€
[ 4 3 2
H 2 ( z) = H o2 Ωr (1+ z) + Ωm (1+ z) + (1 − Ωr − Ωm − ΩE )(1+ z) + ΩE (1+ z)
3(1+w )
]
n Any observational constraints on the expansion
€ history (e.g. from distance measurements) then p = wρ
allow measurements of w-- the equation of state
parameter-- and the density parameters
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 43Evolution of the density parameter
n If the current density parameter W=1 then it will be
unchanged for all time (constancy of curvature)
Ω −1
Ω( z) −1 =
(1 − Ωr − Ωm − ΩE ) + Ωr (1+ z)2 + Ωm (1+ z) + +ΩE (1+ z)(1+3w)
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 Cosmo-LSS | Mohr | Lecture 1 44Epoch of matter-radiation equality
n Currently the matter density
parameter is measured to be Ωm = 0.28
n The radiation density parameter is Ωr = 8.6x10 −5
measured to be
€
n The dark energy density parameter ΩE = 0.72
is about €
n Matter-radiation equality occurred at ( )
ρ zeq = ρ mo (1+ zeq ) 3 = ρ E
z~3300, and matter-DE equality € # ρE &
1
3
occurred at z~0.37 1+ zeq = % (
ρ
$ mo '
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 45
€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 back
n 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
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 46Expanding Universe Models Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 47
The Static Universe
n Einstein considered a solution with no 2 8πG 2
2
expansion and a cosmological constant H R − R ρ = −kc 2
3
n No expansion requires a particular density, 3kc 2
but that density depends on scale factor R ρ static = 2
8πGR
n It must be a mix of matter and €
vacuum
energy (see effective equation of state).
Vacuum energy provides the negative ˙R˙ = − 4 πGR ( ρ + 3p)
pressure that balances the gravitational 3
attraction.
€
Problem with this model is that it is
1
n
unstable
p = − ρ (remember p = wρ )
€
3
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 488πGρV
Λ=
De Sitter Model c2
2
n A phase of vacuum energy " ˙
R % Λc 2
domination leads to exponential H2 = $ ' =
expansion- an inflationary phase € # R& 3
" R˙ % Λc 2
n Note that this provides a H =$ ' =
mechanism for starting the # R& 3
expansion of the Universe at early
times- an early vacuum energy ⇒ R(t) = Rie H ( t −t i )
dominated phase
n Note that such a phase is generic
R
at “late times” in an expanding €
universe with matter, radiation and
dark energy components.
t
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 49Steady State Model
n 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.
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 50Spatially flat models
n Today, observations and
theoretical prejudices point
toward spatially flat models
n To be so close to W=1 today −0.0179 ≤ Ωk ≤ 0.0081
requires that the Universe has 95% confidence
been even closer to W=1 WMAP + BAO + SNe
throughout (Komatsu et al 2009)
n Uniformity of CMB temperature
Ωk = −0.052 +0.025
−0.027
requires mechanism to enlarge Planck TT + lowP
particle horizon. Inflation, this
mechanism, also drives the
Ωk = −0.005+0.008
−0.008
Planck TT + lowP + lensing
Universe to spatial flatness.
(Planck Coll. 2016)
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 51Initial Singularity
n If Big Bang is thought of as explosive instant, then this explosion
happened everywhere at once
n Universe isn’t expanding _into_ anything
n If infinite now then infinite before
n What started the expansion?
n Inflation provides a mechanism
n We know that a universe where the energy density is a positive
cosmological constant will expand exponentially
n 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
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 52Origin 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 effects
n Common to talk about “Hubble flow” as the inferred velocity due to the cosmological
redshift, but it is of course no flow at all!
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 53Nature of the Expansion
n 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.
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 54Distances, Volumes and Time in
Expanding Universes
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 55Comoving or Proper Distance
n Proper distance dp or comoving distance is
coordinate distance r times present epoch
scale factor Ro
d p = R0 r
n For a photon, we have a null geodesic, and
so we can use the metric to calculate the
coordinate distance traveled
€ 0 = c dt − R (t)dr so r = ∫
2 2 2 2 cdt
t obs
t emit R(t)
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 56Comoving 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
Rdr = cdt R0 dR
= (1+ z) R0 dr = c (1+ z)
R RH
R˙ dR
H ≡ so dt = R0 −R0 dz
R RH R= R0 dr = c (1+ z)
(1+ z) (1+ z) 2 RH
dR
Rdr = c dR R0 c
RH =− R0 dr = − dz
dz (1+ z) 2 H(z)
€
€ €
Cosmo-LSS | Mohr | Lecture 1
Apr 16, 2021 57Comoving 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
[ 4 3 2
H 2 ( z) = H o2 Ωr (1+ z) + Ωm (1+ z) + (1 − Ωr − Ωm − ΩE )(1+ z) + ΩE (1+ z)
3(1+w )
]
z
c
d p = R0 r = ∫ H(z) dz
0
€
n Note that this is derived from the metric and the
Friedmann equation,
€ so it is valid for all constant w
models
z
c dz
dp =
H0
∫ 12
0
[Ω (1+ z)
r
4 3 2
+ Ωm (1+ z) + (1 − Ωr − Ωm − ΩE )(1+ z) + ΩE (1+ z)
3(1+w)
]
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 58Distance- Redshift Relation
n 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.
z
dz
dp = c ∫
0 H(z)
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
z
c dz
dp =
H0
∫ 12
0
[Ω (1+ z)
r
4 3 2
+ Ωm (1+ z) + (1 − Ωr − Ωm − ΩE )(1+ z) + ΩE (1+ z)
3(1+w)
]
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 59
€d d ⎡ Ro ⎤
[1+ z ] = ⎢ ⎥
Change in Redshift dto dto ⎣ R ⎦
dz dRo 1 dR dte Ro
= − 2
n Because the universe continues to dto dto R dte dto R
expand, the redshift is expected to dz dRo 1 Ro dR 1 ⎡ dte Ro ⎤
change over time = − ⎢ ⎥
dto dto Ro R dte R ⎣ dto R ⎦
n Note that it is the impact of a change in Ro
that we are exploring for dz/dt
dz
= H o (1+ z) − H (z)
dto
dz 3
= H o (1+ z) − H o Ωm (1+ z ) + ΩE
dto
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/
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 60Volume Element
n Given the proper distance or equivalently the comoving distance we
can define the comoving volume element
dVc = 4 πdc2 R0 dr
4 πc dc2 dz
dVc = 12
H 0 Ω (1+ z) 4 + Ω (1+ z) 3 + 1 − Ω − Ω − Ω (1+ z) 2 + Ω (1+ z) 3(1+w)
r[ m ( r m E) E ]
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 Cosmo-LSS | Mohr | Lecture 1 61Angular 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
l
dA =
θ
n It is like the comoving distance, but uses the scale factor R(z)
at the time of emission rather than observation
€ dc
dA =
(1+ z)
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 62Luminosity Distance
n Luminosity distance is the relationship between the observed flux of an
object and its intrinsic luminosity
L
dL =
4 πf
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)
€
L
f= 2 2
so dL = dc (1+ z)
4π d (1+ z )
c
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 Cosmo-LSS | Mohr | Lecture 1 63Surface 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
Iemit
Iobs = 4
(1+ z)
n This has a dramatic impact on our ability to map the light
distributions of distant galaxies and clusters
€
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 64Age, Lookback time and Elapsed time
n Calculating time or age in an expanding universe is similar to calculating distance
Rdr = cdt
R 1
≡
R0 1+ z
R dr c
so cdt = 0 Previously R0 dr = dz
(1+ z) H(z)
n So we have an expression for the time elapsed between redshift z and the
present (lookback time):
cdz
€ cdt =
H(z)(1+ z)
z
€
1 dz
t lb =
H0
∫ 12
0
[ 4 3 2
(1+ z) Ωr (1+ z) + Ωm (1+ z) + (1 − Ωr − Ωm − ΩE )(1+ z) + ΩE (1+ z)
3(1+w )
]
n Age of the universe to is limit of tlb where z goes to infinity
n Time available for an object to form is telapse=to-tlb
€
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 65Anthropic Principle
n 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
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 66Anthropic 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
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 67References
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.
Apr 16, 2021 Cosmo-LSS | Mohr | Lecture 1 68You can also read
