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Cosmology
The Chapter Cosmological Principle : the universe is homogeneous
and isotropic on sufficiently large scales
The universe looks pretty much like this everywhere "walls" and "voids" are present but no larger structures are seen....
It follows that the Universe has no "edge" or center.
But is the Universe the same at all times?
Remember universal expansion (Hubble's Law)?
recession velocity = Ho x distance
Thus, the cosmological principle does not imply that the Universe is constant at all times (this was once thought to be the case - Steady State Universe). Universal expansion points to a beginning of the Universe and implies that the Universe is changing over time - more on this in Chapter 21
Expansion of the Universe - Olbers's Paradox
-is the Universe infinite??
If so, every line of sight eventually hits a star and the sky is always bright. Number of stars goes up by r2 for each shell, but brightness decreases for each shell by r2. Brightness per shell is a constant.
The fact that the sky is not uniformly bright indicates that either the Universe has a finite size and/or the Universe has a finite age
+ redshifting of light from distance sources reduces flux
Newtonian Gravity in Cosmology
What can we learn about the evolution of an expanding Universe by applying Newtonian gravity?
Isotropy implies that the Universe is spherically symmetric from any point A spherical volume evolves under its own influence.
Consider mass m moving on the surface of a sphere with mass M(r) at position r m(d2r/dt2) = -GM(r)m/r(t)2
(20.13)
As the Universe expands, any given mass occupies a larger volume (t) = o(R3o/R3(t))
Homogeneity = constant
qu E
R is the scale factor where R = r(t)/r(to)
ion at
fm o
ion ot
One cannot be zero without the other... A Universe with matter cannot be static!
To integrate the equation of motion, first multiply both sides by dR/dt
Case 1:
k=0
This must be a constant (call it k)
R is proportional to t2/3 The Universe expands at an ever decreasing rate!
k can be either 0, +, or -
Borderline Universe or Marginally Bound
Case 2:
k>0
Case 3:
k<0
This term becomes smaller when R increases, eventually reaching a point where Rdot = 0. Expansion stops at Rmax.
If k is negative, then -k is positive and the right-hand side of 20.19 is always positive. As R increases, first term goes to zero and Rdot2 -k or Rdot (-k)1/2 Expansion continues forever! Open or Unbound Universe
How does the presence of Gravity in an expanding Universe affect the relationship between the Hubble Time (1/Ho) and the true age of the Universe? The Universe was expanding faster in the past it took less time to reach its current size than would be est imated from its current expansions rate
What quantity can we measure to tell if we live in an open or closed Universe? Define a dimensionless deceleration parameter q
Use 20.11, 20.16, 20.17 to eliminate R
k=0
Use previous equations plus Rdot = H(t)R(t) to get
Then, put this into previous 20.24 to get q = 1/2
k>0
q approaches infinity as R approaches Rmax Any value of q > 1/2 produces a closed universe
k<0
q must be positive (since expansion is slowing, Rdoubledot is - and q is then +) Any value of 0 < q < 1/2 produces an open universe
The Fate of the Universe
Whether or not the universe continues to expand forever or eventually collapses depends on its density.
High density = lots of mass = enough matter to gravitationally halt expansion and cause gravitational collapse
Low density = little mass = not enough gravitational attraction to stop expansion...it goes on forever
The density of a marginally bound universe - where gravity is just sufficient to halt the expansion at t = infinity - is called the
critical density
Setting q=1/2 in eq. 20.24 crit = 3H2/(8G) Cosmic density parameter = /crit
If H = 70 km/s/Mpc, then crit 1 x 10-29 g/cm3
(about one H atom in 200 L volume of space)
The possibility of a closed Universe has led to the idea of an oscillating Universe....
Cosmology and General Relativity
Einstein's principle of equivalence found that space-time is curved in the presence of a gravitational field. The mass density of the Universe tells us about the geometry of space-time. Curvature of space (in 2-d) can be described in terms of angles/areas of a triangle. If the 3 angles on the triangles drawn below are , , then + + = + (A/rc2) where is the curvature constant and rc is the radius of curvature
= +1 or > 1, Bound/Closed Universe
= 0 or = 1, Flat/Marginally Bound/Critical
2-D analogy of the Universe
Consider the Universe confined to the surface of an expanding sphere. The horizon is due to the finite age of the Universe, t (we only see light emitted within distance ct). This horizon is growing. Curvature becomes more apparent as survey area (area within the horizon) grows. As the Universe expands, the radius of curvature changes. "Radius" of the Universe is meaningless (in 3-d our sphere has a radius, but in 2-d we can only see surface). Thus R(t) keeps better track of expansion. We can investigate the "radius of curvature" on our 2-d surface. Also, in 2-d, there is no "center" to the Universe. The center is better represented as a space-time event in the past.
Cosmological Redshift
As Universe expands, photon's expands proportionally to the scale factor R(t)
Since redshift arises due to expanding wavelength of all photons traveling through an expanding Universe, it is called a cosmological redshift. As a consequence, we don't normally convert a z to a distance, since we need to assume a particular model for how R(t) has evolved.
We often to need study events (objects) spread out in space and time so we must computer distances between 2 events in 4-d spacetime
Space-time interval in spherical coordinates (ds)2 = (cdt)2 R2(ct)[(dr)2/(1-r2) + r2[(d)2+sin2(d)2]] Robertson-Walker metric where R is the scale factor of the Universe and r, , are the usual spherical coordinates of objects (like galaxies). These are the comoving coordinates of a point in space. If the expansion of the universe is homogeneous and isotropic, comoving coordinates are constant with time. The photons from distant galaxies follow null geodesics (where ds=0).
Einstein used GR to improve upon Newtonian cosmology and provide a description of the Universe as a whole. He introduced the cosmological constant into his GR equations since he found (as in eq. 20.17) that if the density of the Universe is non-zero, the Universe must be expanding. As this was before Hubble's work and most believed in a Steady State Universe, this constant allowed for a nonzero density and a static Universe. Let = /8G and a non-zero density is possible with Rdot=0 After Hubble's discovery, Einstein called this his "greatest blunder". Several cosmological theories were developed upon GR principles de Sitter k=0 and positive Friedmann =0 and zero pressure (low density) Lemaitre non-zero density and Results similar to Newtonian gravity with substitution.
Models of the Universe
Einstein's equation (analogous to Newtonian case eq. 20.19) has 2 relevant parts in discussing cosmology -
P is gas pressure (zero, except if matter is hot and dense)
These equations are integrated to give R(t) by looking at limiting cases (i.e. zero cosmological constant and zero density). Each case must be evaluated at zero, + and - curvature Friedmann models have =0 For a flat Universe (k = 0), R(t) = (t/to)2/3 (same as flat Newtonian case) = +/- differ from Newtonian math but have similar characteristics ("+" goes to maximum R and then contracts, "-" produces expansion that goes on forever) Lemaitre models have 0 These behave like those with mass density = -/8G (positive behaves like negative mass density expansion accelerates)
Expressing Distances in an Expanding Universe
The geometry and expansion rate of the Universe effects angular sizes and distances measured. DH = c/Ho Hubble Distance DA = L(size)/(angular size) Angular Distance DL = sqrt (Luminosity/4Flux) Luminosity Distance DL = (1+z)2 DA Distance Modulus is then DM = m M = 5 log (DL) 5 where DL is in pc m M = 25 + 5 log (DL) where DL is in Mpc If = 0, then DL = 2c/Ho [z/(G+1)] {1+[z/(G+1)]} where G = (1 + 2qoz)1/2 If non-zero , use Ned Wright's Javascript Cosmology Calculator: http://www.astro.ucla.edu/~wright/CosmoCalc.html
Angular size vs z (plotting DA/DH where DA=L/)
Luminosity distance vs z (plotting DL/DH)
DH=c/Ho= 3000h-1Mpc
At high z, angular diameter distance is such that 1 arcsec is about 5 kpc. flat, =0 solid open, =0 dotted flat, non-zero - dashed
(from Hogg 2000 astro-ph 9905116)
How does the age of the Universe differ with model assumptions?
Friedman models ( = 0) give to = (2/3)Ho-1 0 < to < (2/3)Ho-1 (2/3)Ho-1 < to < Ho-1
-with qo=1/2
=0 = +1 = -1
What is the relationship between z and age of the Universe? te(Gyr) = 10.5 (Ho/65) (1+z)-3/2 (gives age when light left source at z) tL = 2/3Ho-1[1-(1+z)-3/2]
What is the "lookback" time for z=1? z=6?
Lookback times vs z plotting tL/tH and age vs z plotting t/ tH tL is difference between age of Universe now and age te when photons left emitting source
flat, =0 solid open, =0 dotted flat, non-zero - dashed
How to determine if the Universe is Open, Closed or Flat?
1) Add up all matter in the Universe to determine density Luminous matter: Dark matter? only ~1% of crit
Still a factor of ~5 short to close the Universe with "known" DM (where dynamical evidence exists)
2) look for gravitational effects of the mass of the Universe, such as the slowing down of the expansion rate Hdot (to)
The rate of cosmic expansion is best determined by probing the greatest distances in the Universe where the geometry of the Universe is best detected - Type 1 supernovae These standard candles can be seen in distant galaxies (several Gpc).
Models showing unbound, decelerating universe more distant objects should be moving faster than nearby objects if the universe is decelerating.
Objects were receding less rapidly in the past!
70 km/s/Mpc
In 1998, astronomers found that the universe may actually be accelerating!
What might cause this acceleration? A nonzero cosmological constant. This is the basis of what is known as Dark Energy.
More results from Type 1a Supernovae....
3) Measure the curvature of space-time by surveying the Universe on large scales cosmologists have determined (using source counts and geometrical arguments based on galaxy sizes) that if the Universe if curved (=+/-1), radius of curvature would have to be comparable to or greater than the Hubble distance (c/Ho = 4300 Mpc) essentially flat!
Why should we be so close to the boundary ( = 1)?
must have been very close to unity in the past for it to be so close at the current time. With a non-zero , it is still possible to have a flat universe and low mass density M = M/crit Let ,eff = /8G then = /8Gcrit Total density parameter for the Universe is then tot = M +
One final comment on the Age of the Universe as calculated from various models. The accelerating universe model (M=0.3, =0.7) yields an age close to a constant Ho (Hubble time or 1/Ho).
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Age of a critical density universe is 9 billion years Age of an accelerating universe is about 14 billion years