Transcript Lecture 20 Hubble Time – Scale Factor

Lecture 20
Hubble Time – Scale Factor
ASTR 340
Fall 2006
Dennis Papadopoulos
Hubble’s Law
• Hubble interpreted redshift-distance relationship as a
linear increase of the recession velocity of external
galaxies with their distance
• Mathematically, the Hubble law is
v=Hd
where v=velocity and d=distance
• Modern measurement gives the Hubble constant as
H=72 km/s/Mpc
• In fact, Hubble’s interpretation is only “sort of” correct
• What really increases linearly with distance is simply
wavelength of light observed, and this redshift is due to
the cosmological expansion of space over the time since
the light left the distant galaxy and arrived at the Milky
Way!
SPACE TIME STRUCTURE – THE METRIC EQUATION
Metric is invariant
f, g, h Metric coefficients
r 2  x 2  y 2
r  f x  2 g xy  hy
2
2
2
sphere
r 2  R2 2  R2 cos2  2
s 2   c2t 2   ct x  x2
2D space-time metric
POSSIBLE GEOMETRIES FOR
THE UNIVERSE
• The Cosmological Principles constrain the
possible geometries for the space-time that
describes Universe on large scales.
• The problem at hand - to find curved 4-d spacetimes which are both homogeneous and
isotropic…
• Solution to this mathematical problem is the
Friedmann-Robertson-Walker (FRW) metric.
Cosmological Principle
– Universe is homogeneous – every place in the
universe has the same conditions as every other
place, on average.
– Universe is isotropic – there is no preferred direction
in the universe, on average.
• Ignoring details…
– All matter in universe is “smoothed” out
– ignore details like stars and galaxies, but deal with a
smooth distribution of matter
Observational evidence for
homogeneity and isotropy
• Let’s look into space…
see how matter is
distributed on large
scales.
• “Redshift surveys”…
– Make 3-d map of galaxy
positions
– Use redshift & Hubble’s
law to determine
distance
Each point
is a bright
galaxy
CfA redshift
survey
Las Campanas
Redshift survey
Friedmann-Robertson-Walker metric
• A “metric” describes how the space-time intervals relate to local
changes in the coordinates
• We are already familiar with the formula for the space-time
interval in flat space (generalized for arbitrary space coordinate
scale factor R):
s2  (ct)2  R2 x 2  y 2  z 2 
• In terms of radius and angles instead of x,y,z, this is written:
s2  (ct)2  R2 r 2   2  sin2 ()2 
• General solution for 
isotropic, homogeneous curved space is:

2


r
2
2
2
2
2
2
s  (ct)  R 
   sin  ( ) 
2
1 kr

• And in fact, in general the scale factor may be a function of time,
i.e. R(t)

Curvature in the FRW metric
•
•
This introduces the curvature constant, k
Three possible cases…
Spherical spaces (closed; k=+1)
Flat spaces (open; k=0)
Hyperbolic spaces (open; k=-1)
Meaning of the scale factor, R.
• Scale factor, R, is a central concept!
– R tells you how “big” the space is…
– Allows you to talk about changing the size of the space
(expansion and contraction of the Universe - even if the
Universe is infinite).
• Simplest example is k=+1 case (sphere)
– Scale factor is just the radius of the sphere
R=0.5
R=1
R=2
• What about k=-1 (hyperbolic) universe?
– Scale factor gives “radius of curvature”
R=1
R=2
• For k=0 universe, there is no curvature… shape is unchanged
as universe changes its scale (stretching a flat rubber sheet)
Co-moving coordinates.
• What do the coordinates x,y,z or r,, represent?
• They are positions of a body (e.g. a galaxy) in the space that
describes the Universe
• Thus, x can represent the separation between two galaxies
• But what if the size of the space itself changes?
• EG suppose space is sphere, and has a grid of coordinates on
surface, with two points at a given latitudes and longitudes 1,1
and 2,2
• If sphere expands, the two points would have the same latitudes
and longitudes as before, but distance between them would
increase
• Coordinates defined this way are called comoving coordinates
• If a galaxy remains at rest relative to the
overall space (i.e. with respect to the average
positions of everything else in space) then it
has fixed co-moving coordinates.
• Consider two galaxies that have fixed comoving coordinates.
– Let’s define a “co-moving” distance D
– Then, the real (proper) distance between the
galaxies is d=R(t) D
Galaxies and galaxy clusters gravitationally bound. Their meter
length does not change with expansion
R (t ) D
R(t  t ) D
Hubble Law
Expansion Rate
Hubble Time
Hubble
sphere
DH
Acceleration-Decceleration
Scale Factor –Robertson Walker Metric
•
•
•

According to GR, the possible space-time intervals in a homogeneous,
isotropic Universe are the FRW metric forms with k=0 (flat), k=1
(spherical), k=-1(hyperbolic):
2


r
2
2
2
2
2
2
s  (ct)  R(t) 
   sin  () 
2
1 kr

The scale factor R(t) describes the relative expansion of space as a
function of time.
Both physical distances between galaxies and wavelengths of radiation
vary proportional to R(t).
– d(t) =Dcomoving R(t)
– (t)=emitted R(t)/R(emitted)
•
•
•
Observed redshift of radiation from distant source is related to scale
factor at emission time (t) and present time (t0) by 1+z=R(t0)/R(t)
Hubble observed that Universe is currently expanding; expansion can
be characterized by H=(R/t)/R
For nearby galaxies, v=dH0 ,where the present value of the Hubble
parameter is approximately H0 =70 km/s/Mpc
Interpretation of Hubble law in
terms of relativity
• New way to look at redshifts observed by Hubble
• Redshift is not due to velocity of galaxies
– Galaxies are (approximately) stationary in space…
– Galaxies get further apart because the space between
them is physically expanding!
– The expansion of space, as R(t) in the metric equation, also
affects the wavelength of light… as space expands, the
wavelength expands and so there is a redshift.
• So, cosmological redshift is due to cosmological
expansion of wavelength of light, not the regular
Doppler shift from local motions.