Transcript Lecture 23 Models with Cosmological Constant
Lecture 23 Models with Cosmological Constant
ASTR 340 Fall 2006 Dennis Papadopoulos Problems 11.3-11.6 Due 12/5/06 Chapter 11
Review
• How does
R(t)
(and
H
) change in time? And what is the value of the curvature
k
?
• Need to solve Einstein’s equation!
G
8
G
T
c
4
• • •
STANDARD COSMOLOGICAL MODELS
In general Einstein’s equation relates geometry to dynamics That means curvature must relate to evolution Turns out that there are three possibilities… k=-1 k=0 k=+1
Important features of standard models…
• All models begin with in the past R=0 at a finite time – This time is known as the
BIG BANG
– Space and time come into existence at this moment… there is no time before the big bang!
– The big bang happens everywhere in space… not at a point!
• There is a connection between the geometry and the dynamics – Closed ( k=+1 ) solutions for universe expand to maximum size then re-collapse – Open ( k=-1 ) solutions for universe expand forever – Flat ( k=0 to a halt).
) solution for universe expands forever (but only just barely… almost grinds
•
Hubble time
We can relate this to observations… • Once the Hubble parameter has been determined accurately from observations, it gives very useful information about age and size of the expanding Universe… • • Recall Hubble parameter is ratio of rate of change of size of Universe to size of Universe:
H
1
R
R t
1
R dR dt
If
Universe
were
R/
t=constant
expanding at a constant rate, we would have and
R(t) =t
(
R/
t)
; then would have
H= (
R/
t)/R=1/t i.e.
t H =1/H would be age of Universe since Big Bang R(t) t
Hubble time for nonuniform expansion
Slope of R(t) curve is dR/dt
Big Bang NOW
–
Hubble time
is t H =1/H=R/(dR/dt) – Since rate of expansion varies, t H =1/H of the Universe gives an estimate of the age – This tends to overestimate the actual age of the Universe since the Big Bang
Terminology
– – – –
Hubble distance,
D=ct H
(distance that light travels in a Hubble time). This gives an approximate idea of the size of the observable Universe.
Age of the Universe
,
t age
(the amount of cosmic time since the big bang). In standard models, this is always less than the Hubble time.
Look-back time
,
t lb
(amount of cosmic time that passes between the emission of light by a certain galaxy and the observation of that light by us)
Particle horizon
(a sphere centered on the Earth with radius
ct age
; i.e., the sphere defined by the distance that light can travel since the big bang). This gives the edge of the actual observable Universe.
Friedmann Equation
•
Where do the three types of evolutionary solutions come from?
• Back to Einstein’s eq….
G
8
c
4
G
T
• When we put the RW metric in Einstein’s equation and go though the GR, we get the
Friedmann Equation
… this is what determines the dynamics of the Universe
R
2 8 3
R
2
kc
2 2
GM R
kc
2 • What are the terms involved?
– G is Newton’s universal constant of gravitation – – –
R
is the rate of change of the cosmic scale factor same as
R/
t
is the total matter and energy density
k
is the geometric curvature constant
• If we divide Friedmann equation by
R 2
, we get:
R
( )
R
2
H
2 8 3
G
kc
2
R
2 • Let’s examine this equation… • H 2 must be positive … so the RHS of this equation must also be positive.
• Suppose density is zero ( =0 ) – Then, we
must
have negative k (i.e., k=-1 ) – So, empty universes are open and expand forever – Flat and spherical Universes can only occur in presence of (enough) matter.
Critical density
• What are the observables for flat solution (
k=0
) – Friedmann equation then gives
H
2 8 3
G
– So, this case occurs if the density is exactly equal to the
critical density
…
crit
3
H
8
G
2 – “Critical” density means “flat” solution for a
given
value of
H
, which is the most easily observed parameter
H
2 8 3
G
kc
2
R
2 • In general, we can define the
density parameter
…
crit
8
G
3
H
2 • Can now rewrite Friedmann’s equation yet again using this… we get 1
kc
2
H
2
R
2
Omega in standard models
1
kc
2
H
2
R
2 • Thus, within context of the standard model: – <1 if k=-1 ; then universe is hyperbolic and will expand – – forever =1 if k=0 ; then universe is flat and will (just manage to) expand forever >1 if k=+1 ; then universe is spherical and will recollapse • Physical interpretation: If there is more than a certain amount of matter in the universe ( > critical ), the attractive nature of gravity will ensure that the Universe recollapses!
Value of critical density
• For present best-observed value of the Hubble constant, H 0 =72 km/s/Mpc critical density is equal to critical =10 -26 kg/m 3 ; i.e. 6 H atoms/m 3 • Compare to: – – – water = 1000 kg/m 3 air =1.25 kg/m 3 (at sea level) interstellar gas =2 10 -21 kg/m 3
The deceleration parameter,
q • The
deceleration parameter
measures how quickly the universe is decelerating (or accelerating) • In standard models, deceleration occurs because the gravity of matter slows the rate of expansion • For those comfortable with calculus, actual definition of
q
is:
q
R RH
2
Matter-only standard model
• In standard model where density is from rest mass energy of matter only, it turns out that the value of the deceleration parameter is given by
q
2 • This gives a consistency check for the standard, matter dominated models… we can attempt to measure in two ways: i.e. measure mass density and compare to critical value – Use measurement of deceleration parameter – Measurement of
q
is analogous to measurement of Hubble parameter, by observing change in expansion rate as a function of time: need to look at how
H
changes with redshift for distant galaxies
Direct observation of
q • Deceleration shows up as a deviation from Hubble’s law… • A very subtle effect – have to detect deviations from Hubble’s law for objects with a large redshift
• Newtonian interpretation is therefore: – k=-1 is “positive energy” universe (which is why it expands forever) – k=+1 is “negative energy” universe (which is why it recollapses at finite time) – k=0 is “zero energy” universe (which is why it expands forever but slowly grinds to a halt at infinite time)
k=-1 k=0 k=+1
Expansion rates
• For flat ( k=0, =1 ), matter-dominated universe, it turns out there is a simple solution to how
R
varies with
t
:
R
(
t
)
R
(
t
0 )
t t
0 2 / 3 – This is known as the Einstein-de Sitter solution • In solutions with >1, expansion is slower (followed by re-collapse) <1, expansion is faster
Modified Einstein’s equation
• But Einstein’s equations most generally also can include an extra constant term; i.e. the
T
term in 8
G
G
T
c
4 • • has an additional term which just depends on space-time geometry times a constant factor, • This constant (Greek letter “Lambda”) is known as the “
cosmological constant
”; corresponds to a vacuum energy, i.e. an energy
not
associated with either matter or radiation could be positive or negative – Positive would act as a repulsive force which tends to make Universe expand faster – Negative would act as an attractive force which tends to make Universe expand slower • Energy terms in cosmology arising from positive often referred to as “
dark energy
” are now
Modified Friedmann Equation
• When Einstein equation is modified to include , the Friedmann equation governing evolution of R(t) changes to become:
R
2 2
H R
2 8 3
G R
2
R
2 3
kc
2 • Dividing by
(HR) 2
, we can consider the relative contributions of the various terms evaluated at the present time,
t 0
• The term from matter at
t 0
• Two additional “ ” has subscript “M”; density parameter terms at
t 0
are defined:
M
0
crit
(3
H
0 2 0 /8
G
) 3
H
0 2
k
kc
2
R
0 2
H
0 2 • Altogether, at the present time,
t 0 ,
we have 1
M
k
Generalized Friedmann Equation in terms evolution of
R(t)
of ’s • The generalized Friedmann equation governing is written in terms of the present ’s (density parameter terms) as:
R
2 2
H R
0 0 2
M
R
0
R
R R
0 2
k
• The only terms in this equation that vary with time are the scale factor
R
and its rate of change
R
• Once the constants H 0 , M , , k are measured empirically (using observations), then whole future of the Universe is determined by solving this equation!
• Solutions, however, are more complicated than when =0
Special solution
Static model (Einstein’s) – Solution with c =4 G – No expansion:
H=0
, , k= c R 2
R=constant
/c 2 – Closed (spherical) – Of historical interest only since Hubble’s discovery that Universe is expanding!
R
2 /
R
2 (8
R
0
kc
2 /
R
2 )
Effects of
• Deceleration parameter (observable) now depends on both matter content and (will discuss more later) • This changes the relation between evolution and geometry. Depending on value of , – closed ( k=+1 ) Universe could expand forever – flat ( k=0 ) or hyperbolic ( k=-1 ) Universe could recollapse
Consequences of positive
• Because term appears with
positive
power of Friedmann equation, effects of
R
in increase with time if
R
keeps increasing
R
2 2
H R
2 8 3
G R
2
R
2 3
kc
2 • Positive can create accelerating expansion!
THE DE SITTER UNIVERSE “de Sitter” model: Solution with k =0 =0 (no matter!), and (flat space!), M >0 Hubble parameter is constant Expansion is exponential
R
R
0
e Ht
/
t
0
• Steady solution: – Constant expansion rate – Matter constantly created – No Big Bang – Ruled out by existing observations: • Distant galaxies (seen as they were light travel time in the past) differ from modern galaxies • Cosmic microwave background implies earlier state with uniform hot conditions (big bang) • Observed deceleration parameter differs from what would be required for steady model