The cosmological redshift illustrated.

    Cosmological Redshift Topics:

    1. Cosmological Redshift Explicated:

      1. Image 1 Caption: The cosmological redshift illustrated.

      2. As electromagnetic radiation (EMR) propagates through intergalactic space, it progressively redshifts due to the expansion of the universe: i.e., it increases in wavelength as space expands under it.

        Note the increase in wavelength is NOT a forced stretching. Energy is NOT being put into the EMR. It's actually being taken out. For an explication, see topic Cosmological Redshift and Energy Conservation below.

        wave packet with equal phase and group velocities

      3. Image 2 Caption: An animation of the "propagation of a wave packet in a non-dispersive transmission medium. We can see there is NO difference between phase velocity and group velocity." (Slightly edited.) The animation is cartoon of a propagating wave packet or photon.

      4. Actually all lengths between gravitationally unbound astronomical objects scale up with the cosmic scale factor a(t).

        The cosmic scale factor for cosmic present t_0 (equal to the age of the observable universe = 13.797(23) Gyr (Planck 2018)) is conventionally set to 1 and given symbol a_0. Note subscript 0 indicates cosmic present by usual convention. Thus, a(t=t_0) = a_0 = 1.

        Cosmic time is measured from Big Bang which happened at lookback time the age of the observable universe = 13.797(23) Gyr (Planck 2018) (see Planck 2018: Age of the observable universe = 13.797(23) Gyr).

        Note from our understanding of general relativity (GR), the scaling up with cosmic scale factor a(t) is a literal growth of intergalactic space.

        From our understanding of the cosmic scale factor (which we further explicate below in section The Cosmic Scale Factor), the wavelength of EMR signal (or, in the quantum mechanical perspective, a photon) propagating through intergalactic space just scales with a(t): i.e.,

          λ_0 = [a_0/a(t)]λ , 
        where λ is the initial wavelength of emission, t is the cosmic time of emission, and λ_0 is the wavelength at cosmic present t_0 (equal to the age of the observable universe = 13.797(23) Gyr (Planck 2018)).

      5. The cosmological redshift is defined
              λ_0 - λ    a_0
          z = ------- = ---- - 1  .
                 λ      a(t) 
        We see that the cosmological redshift is the relative change in wavelength from emission to observation.

      6. A wonderfully simple fact is the relative cosmic scale factor a_0/a(t) (i.e., scaling-up factor of the observable universe) since the cosmic time t when the EMR signal was emitted---the signal we are observing right now at cosmic present t_0 = to the age of the observable universe = 13.797(23) Gyr (Planck 2018) is just
          a_0/a(t) = z + 1   or inverting to get  a(t) = a_0/(z+1)  , 
        where a_0 = a(t_0): i.e., the cosmic scale factor at cosmic present t_0 = to the age of the observable universe = 13.797(23) Gyr (Planck 2018) which conventionally we set to 1.

        Since cosmological redshift z is a direct observable, the relative cosmic scale factor a_0/a(t) is a direct observable. Just add 1 to z and we get it.

        But we do NOT get cosmic time t!!!!!

        The fact is that a(t)'s functional dependence on cosmic time t is NOT known observationally: it is a function dependent on the cosmological model. So the scaling-up formula does NOT give us t or lookback time which is t_0-t.

        If we did have cosmic time t, we'd easily be able to determine the cosmic scale factor a(t) throughout most of cosmic time and cosmology would be a lot easier.

        If only galaxies had clock faces on them that told cosmic time, but they do NOT.

        In fact, solving for cosmic scale factor a(t) from the Friedmann equation of cosmology (which is derived from general relativity along with certain assumptions) is one of the main goals of cosmology.

      7. How do we get direct observable cosmological redshift z from observations?

        1. If λ_0 is the wavelength of as single isolated spectral line (absorption line, (see examples in Image 1) or an emission line), then it CANNOT usually be identified. It is after all redshifted by some unknown amount from its intrinsic line wavelength (i.e., laboratory line wavelength).

        2. However, if we have a group of observed spectral lines from one atom (in some specific ionization stage) or molecule in some specific ionization stage), then the pattern of the spectral lines allows the atom or molecule to be identified.

        3. In fact, the group of observed spectral lines are usually (neutral) atomic hydrogen lines (see the figure below (local link / general link: line_spectrum_hydrogen.html) since hydrogen is what stars and galaxies (NOT counting dark matter) are mostly made of.

          With the atom or molecule identified, one does know all the intrinsic line wavelengths (i.e., the λ's) and then the cosmological redshift z can easily be calculated from the above formula z = (λ_0 - λ)/λ using the observed wavelengths (i.e., the λ_0's).


      8. Since the procedure just described for obtaining cosmological redshifts is so simple and usually so accurate/precise, we consider cosmological redshift a DIRECT OBSERVABLE.

      9. Note physical distances (i.e., separations at our instant in cosmic time) and lookback times are NOT DIRECT OBSERVABLES, except asymptotically as cosmological redshift z goes to zero.

        They can only be known in general from a cosmological model fitted to the observable universe.

        Since circa 1998 to circa 2020????, the standard model of cosmology (SMC) has been the Λ-CDM model which fits all observations of the observable universe pretty well. It may need revision or replacement in the near future, but even so its predictions for physical distance and lookback time will probably still be pretty accurate.

      10. Since cosmological redshift is a DIRECT OBSERVABLE and physical distance and lookback time are NOT (except asymptotically as cosmological redshift z goes to zero), astronomers generally use cosmological redshift as the standard cosmological distance measure and NOT physical distance or lookback time.

        Thus, astronomers would usually say galaxy X is at redshift z_X.

    2. Cosmological Redshift and Doppler Shift Contrasted:

      1. Unfortunately, in the opinion of yours truly, many people conflate cosmological redshift and Doppler shift.

        They are DIFFERENT, though closely related, phenomena and their general formulae are DIFFERENT.

        Doppler shift is the shift in wavelength due to relative velocity in one inertial frame or between two inertial frames in relative acceleration. The situation is actually a bit complex in general for EMR: see Wikipedia: relativistic Doppler effect.

        Cosmological redshift, on the other hand is a shift wavelength due to propagation through a continuum of inertial frames that are separating due to the expansion of the universe.

        Note the cosmological redshift can be derived from the Doppler shift (see Li-38), but in the opinion of yours truly that does NOT make it Doppler shift in any simple sense---to repeat, the general formulae are DIFFERENT.

        However, many people are less finicky and call the cosmological redshift a Doppler shift.

      2. To add to the complexity of the distinction between cosmological redshift and Doppler effect, all galaxies and other extragalactic objects have peculiar velocities relative to their local inertial frames that participate in the mean universal expansion (i.e., their comoving frames)

        The peculiar velocities give real Doppler shifts that have to be "subtracted off" the actually observed z to get the cosmological redshift z.

        For large cosmological redshifts, the Doppler shifts are negligible, but when you get very close to the Milky Way, the Doppler shifts dominate and a correction is needed to get the cosmological redshift z.

        Inside the Milky Way, it is usually pointless to consider the cosmological redshifts at all.

        In intro astronomy courses, we do NOT worry about the complication of Doppler shift correction, but it is important.

        Cosmological redshift and the Doppler effect compared

      3. Image 3 Caption: A comparison of the Doppler effect (shift) (upper two panels) and the cosmological redshift (lower panel) for electromagnetic radiation (EMR). The observer is the Sun. Note there is just relative velocity in the upper two panels and the expansion of the universe in the lower panel.

      4. Note the 1st-order formulae for the cosmological redshift and the Doppler shift are the same: i.e.,
          z = Δλ/λ = v/c  << 1 , 
        where Δ = (λ_0-λ), λ is either of λ_0 or λ since they are the same to 1st-order, and v is relative velocity along the line of sight) between emitter and observer.

      5. Also note for cosmological motion, v is NOT an ordinary velocity (i.e., a velocity relative to an inertial frame) since it is the rate at which extragalactic space is growing between comoving frames: i.e., inertial frames participating in the mean expansion of the universe whose location is well approximated by sufficiently remote galaxies.

        In astro jargon, the non-ordinary velocities between comoving frames are called recession velocities.

        Recession velocities do exceed the vacuum light speed c = 2.99792458*10**8 m/s (exact by definition) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns as you go to high cosmological redshift z (i.e., to cosmologicallly remote proper distances). But this is NOT a violation of special relativity since recession velocities are between inertial frames NOT relative to a local inertial frame.

      6. The agreement of the 2 formulae for the 2 effects to 1st-order allowed early investigators in the 1920s of the universal expansion to interpret their data correctly without being clear about the nature of the universal expansion as a growth of space. The conceptual confusion between the cosmological redshift and the Doppler effect was probably cleared up pretty soon.

        In fact, those who understood Willem de Sitter's (1872--1934) de Sitter universe (1917) did understand the difference between the cosmological redshift and the Doppler effect. Edwin Hubble (1889--1953) who made the observation discovery of the expansion of the universe in 1929 was probably one of those who did understand the difference.

    3. The Cosmic Scale Factor:

      1. Recall that cosmic scale factor a(t) is the parameter that controls the expansion of the universe.

        Solving for it from the Friedmann equation of cosmology (which is derived from general relativity along with certain assumptions) is one of the main goals of cosmology.

        Image 4 below presents the cosmic scale factor a(t) for fiducial cosmological models.

        cosmic scale factor for fiducial cosmologies

      2. Image 4 Caption: The cosmic scale factor a(t) for fiducial cosmological models: 1) de Sitter universe (but also for the steady state universe), 2) Λ-CDM model (since circa 1998, the standard model of cosmology (SMC) with fiducial Λ-CDM model parameters), 3) Milne universe (AKA empty universe), 4) Einstein-de Sitter universe (1932, standard model of cosmology c.1960s--c.1990s), 5) A positive curvature universe (AKA closed universe).

    4. Cosmological Redshift and Energy Conservation:

      1. Recall photon energy E = hf = hc/λ = 1.2398419739(75) eV-μ/λ_μ. This implies that photons propagating through intergalactic space progressively lose energy due to the cosmological redshift and they do.

        But where does that lost energy go. We believe in energy conservation (as exemplified by an ideal Newton's cradle), right? Animation of Newton's cradle,

      2. Image 5 Caption: "An animation of Newton's cradle resting on a copy of Isaac Newton's (1643--1727) famous book the Principia (1687)---NOT an exact copy, though---Newton's portrait added on top is artistic liberty." (Slightly edited.)

        This ideal Newton's cradle exhibits exact conservation of mechanical energy since NO mechanical energy is dissipated to waste heat by friction, air drag, or any other process. Mechanical energy is the sum of kinetic energy and gravitational potential energy in this case.

      3. So where does the photon energy go from the photons that undergo the cosmological redshift?

        It just vanishes (see Car-120). It's somewhat distressing that energy conservation in the ordinary sense does NOT hold necessarily according to general relativity. But all is NOT lost. General relativity gives us a generalization of ordinary energy conservation: the general-relativity energy-momentum conservation equation (see also Car-120). Which equation we will NOT go into here.

        We just have to note that in certain contexts (which do NOT occur in everyday life NOR in most of astrophysics) ordinary energy conservation does NOT hold.

        The cosmological redshift is one of those cases.

        Another case is gravitational waves. Gravitational waves do convey energy and momentum across spacetime, but there is NO general proof that they conserve these quantities as they propagate across spacetime though there is a special case proof for energy conservation and momentum conservation which gives what is called the Bondi-Sachs energy-momentum conservation law (see Roger Penrose, The Road to Reality, 2004, p. 467--468).

        But what about gravitational waves in an expanding universe. Surely, their wavelength increases as they propagate, but do they lose energy thereby? We need a great expert to tell us.

      4. Non-Local Encoding of Gravitational Field Energy in General Relativity:

        Another interesting point about gravitational waves is that they make NO contribution to the energy-momentum tensor T_ij of the Einstein field equations. So though they convey energy and momentum across spacetime, there is NO way to assign so much energy density to any point in the gravitational waves. Somehow the energy is coded into the gravitational waves in a non-local way (see Roger Penrose, The Road to Reality, 2004, p. 467--468).

      5. Actually, the last point generalizes: the energy of the gravitational field in general is NOT localizable as energy density with so much energy here and so much energy there. Somehow the energy of the gravitational field is coded in a non-local way (see Roger Penrose, The Road to Reality, 2004, p. 464--468). Among other things, this means that the energy (more explicitly mass-energy) of the gravitational field does NOT itself cause a gravitational field in the way determined by Newton's law of universal gravitation. But that does NOT worry us, since the Einstein field equations give the correct gravitation.

        For more on the non-local gravitational field energy, see Relativity file: non_local_grav_field_energy.html. The description here should be conflated with the description that file someday.

    Images:
    1. Credit/Permission: © David Jeffery, 2004 / Own work.
      Image link: Itself.
    2. Credit/Permission: Frederic Perez (AKA User:Fffred), 2006 / Public domain.
      Image link: Wikipedia: File:Wave packet (no dispersion).gif.
    3. Credit/Permission: © User:Brews ohare, 2009 / CC BY-SA 3.0.
      Image link: Wikipedia: File:Two redshifts.JPG.
    4. Credit/Permission: © User:Greek3, 2017 / CC BY-SA 4.0.
      Image link: Wikimedia Commons: File:Mplwp universe scale evolution.svg.
    5. Credit/Permission: © Dominique Toussaint (AKA User:DemonDeLuxe), 2006 (uploaded to Wikimedia Commons by User:Scetoaux, 2008) / CC BY-SA 3.0.
      Image link: Wikimedia Commons: File:Newtons cradle animation book 2.gif.
    Local file: local link: cosmological_redshift.html.
    File: Cosmology file: cosmological_redshift.html.