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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

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Lecture 20 Hubble Time – Scale Factor

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Lecture 20 hubble time scale factor l.jpg

Lecture 20Hubble Time – Scale Factor

ASTR 340

Fall 2006

Dennis Papadopoulos

Hubble s law l.jpg

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


    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!

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Metric is invariant

f, g, h Metric coefficients


2D space-time metric

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  • 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.

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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 l.jpg

    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

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    Each point is a bright galaxy

    CfA redshift


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    Las Campanas

    Redshift survey

    Friedmann robertson walker metric l.jpg

    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 curvedspace is:

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

    Curvature in the frw metric l.jpg

    Curvature in the FRW metric

    • This introduces the curvature constant, k

    • Three possible cases…

      Spherical spaces (closed; k=+1)

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    Flat spaces (open; k=0)

    Hyperbolic spaces (open; k=-1)

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    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




    Slide13 l.jpg

    • 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)



    Co moving coordinates l.jpg

    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

    Slide15 l.jpg

    • 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

    Slide16 l.jpg

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

    Hubble law l.jpg

    Hubble Law

    Expansion rate l.jpg

    Expansion Rate

    Hubble time l.jpg

    Hubble Time

    Hubble sphere


    Acceleration decceleration l.jpg


    Scale factor robertson walker metric l.jpg

    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=(D R/D 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 l.jpg

    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.

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