Lecture 20Hubble 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=Hd 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

2 2 2

2 2 22

r x y

r f x g x y h y

f, g, h Metric coefficients

2 2 2 2 2 2cosr R R sphere

Metric is invariant

2 2 2 2s c t c t x x 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 space-times 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

CfA redshiftsurvey

Each point is a bright galaxy

Las CampanasRedshift 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):

• In terms of radius and angles instead of x,y,z, this is written:

• General solution for isotropic, homogeneous curved space is:

• And in fact, in general the scale factor may be a function of time, i.e. R(t)

s2 (ct)2 R2 r2

1 kr2 2 sin2()2

s2 (ct)2 R2 x 2 y 2 z2

s2 (ct)2 R2 r2 2 sin2()2

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=1 R=2R=0.5

• What about k=-1 (hyperbolic) universe?– Scale factor gives “radius of curvature”

• For k=0 universe, there is no curvature… shape is unchanged as universe changes its scale (stretching a flat rubber sheet)

R=1 R=2

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 co-moving coordinates.– Let’s define a “co-moving” distance D– Then, the real (proper) distance between the

galaxies is d=R(t) D

DtR )(

DttR )(

Galaxies and galaxy clusters gravitationally bound. Their meter length does not change with expansion

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):

• 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=(Rt)/R

• For nearby galaxies, v=dH0 ,where the present value of the Hubble parameter is approximately H0 =70 km/s/Mpc

s2 (ct)2 R(t)2 r2

1 kr2 2 sin2()2

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.