Hubble constant – Cosmic Reflections

The Hidden Universe

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

Thesis B6: Observational horizons limit our ability to observationally determine the very large scale geometry of the universe.
We can only see back to the time of decoupling of matter and radiation, and so have no direct information about earlier times; and unless we live in a “small universe”, most of the matter in the universe is hidden behind the visual horizon. Conjectures as to its geometry on larger scales cannot be observationally tested. The situation is completely different in the small universe case: then we can see everything there is in the universe, including our own galaxy at earlier times.

What an intriguing idea. If the entire universe (or the self-contained section we find ourselves in) is substantially smaller than the distance light has traveled since the universe became transparent to radiation (“decoupling”, about 380,000 years after the Big Bang), we might be able to see our Milky Way galaxy (and other galaxies) at various points in the past.

The key point here is that unless we live in a small universe, the universe itself is much bigger than the observable universe. There are many galaxies—perhaps an infinite number—at a greater distance than the horizon, that we cannot observe by any electromagnetic radiation. Furthermore, no causal influence can reach us from matter more distant than our particle horizon—the distance light can have travelled since the creation of the universe, so this is the furthest matter with which we can have had any causal connection. We can hope to obtain information on matter lying between the visual horizon and the particle horizon by neutrino or gravitational radiation observatories; but we can obtain no reliable information whatever about what lies beyond the particle horizon. We can in principle feel the gravitational effect of matter beyond the horizon because of the force it exerts (for example, matter beyond the horizon may influence velocities of matter within the horizon, even though we cannot see it). This is possible because of the constraint equations of general relativity theory, which are in effect instantaneous equations valid on spacelike surfaces. However we cannot uniquely decode that signal to determine what matter distribution outside the horizon caused it: a particular velocity field might be caused by a relatively small mass near the horizon, or a much larger mass much further away. Claims about what conditions are like on very large scales—that is, much bigger than the Hubble scale—are unverifiable, for we have no observational evidence as to what conditions are like far beyond the visual horizon. The situation is like that of an ant surveying the world from the top of a sand dune in the Sahara desert. Her world model will be a world composed only of sand dunes—despite the existence of cities, oceans, forests, tundra, mountains, and so on beyond her horizon.

Let us now define some terms that Ellis uses above.

visual horizon – the distance beyond which the universe was still opaque to photons due to high temperature and density

particle horizon – the distance beyond which light has not yet had time to reach us in all the time since the Big Bang; our particle horizon is, therefore, farther away than our visual horizon

spacelike surface – a three-dimensional surface in four-dimensional space-time where no event on the surface lies in the past or future of any other event on that surface; every point on the surface as it exists at one instant of time

Hubble scale – a cosmological distance unit equal to the reciprocal of the Hubble constant times the speed of light; see derivation below

A reasonable value for the Hubble constant H₀ is 70 km/s/Mpc. A galaxy one megaparsec distant has a cosmological recession velocity of 70 km/s, two megaparsecs distant 140 km/s, and so on.

You may notice that there are two units of distance in H₀: kilometers and megaparsecs. We can thus rewrite H₀in units of s^-1 (reciprocal seconds of time) as follows:

The Hubble time is defined as the inverse of the Hubble constant:

Converting this into more convenient units of years, we get

The Hubble scale is now simply the Hubble time multiplied by the speed of light.

Converting this into more convenient distance units of light years, and then parsecs, we get

As Ellis says, we are like ants in the Sahara desert that cannot see their Earth-universe beyond the sand dunes. Like the ant, is there a limit to our intellect as well?

References
Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.
[http://arxiv.org/abs/astro-ph/0602280]

Cosmologically Distant Objects Appear Magnified

George F. R. Ellis writes in section 2.3.3 of his outstanding survey paper, Issues in the Philosophy of Cosmology:

…there is a minimum apparent size for objects of ﬁxed physical size at some redshift z_c = z_∗ depending on the density parameter and the cosmological constant. The past light cone of the observer attains a maximum area at z_∗; the entire universe acts as a gravitational lens for further oﬀ objects, magnifying their apparent size so that very distant objects can appear to have the same angular size as nearby ones. For the Einstein-de Sitter universe, the minimum angular diameter is at z_∗ = 1.25; in low density universes, it occurs at higher redshifts.

Electromagnetic radiation such as visible light that we observe from a source that is in motion relative to us (the observer) experiences a change in wavelength that is given by

This is called redshift and is positive for a source that is moving away from us and negative for a source that is moving towards us. The higher the relative speed toward or away from us, the greater the magnitude of the redshift. Superimposed upon the kinematic velocities of individual galaxies relative to our Milky Way galaxy, since 1929 we have known that there is a cosmological redshift (called the Hubble flow) that is always positive and increasing in magnitude with increasing distance between any two galaxies. In the nearby universe, the redshift (or blueshift) from kinematic velocities (often referred to as “peculiar velocities”) swamp the contribution from the Hubble flow, so some galaxies are actually approaching each other. A good example of this is M31 and the Milky Way galaxy. For more distant galaxies, however, the cosmological redshift swamps any contribution from the kinematic velocities. Thus, redshift becomes a useful proxy for distance at cosmological distances.

From our everyday experience, we know that the further away an object is, the smaller is its angular size. However, there comes a point where the angular size of an object reaches a minimum, and at even greater distances, its angular size increases! As George Ellis states above, the entire universe acts as a gravitational lens to magnify distant objects.

Michael Richmond presents an equation for angular size as a function of redshift (based on some classical assumptions about the structure of the universe). In his equation, the angular size of an object also depends upon the value we choose for H₀, the Hubble constant, the matter density parameter, Ω_M, and, of course, the physical size of the object of interest.

Let’s work through an example using this equation. The visible part of the Andromeda Galaxy is estimated to be about 220,000 light years across. In megaparsecs, that is 0.0675. This is the value we will use for S.

For the Hubble constant, H₀, let use a recent result: 71.9 km/s/Mpc.

And, for the matter density parameter, Ω_M, let’s use 1.0. This indicates that we live in a universe that has just enough matter for the universe to eventually recollapse, were it not for dark energy. Though Richmond’s equation above only applies to a matter-dominated universe where the dark energy density parameter Ω_Λ is zero, as George Ellis indicates above, a minimum angular diameter is still reached in a universe with dark-energy (i.e. low density universe), only this occurs at a higher redshift than that presented here.

I have not been able to find or derive a more general equation for angular size as a function of redshift that will work for a dark-energy-dominated universe (perhaps a knowledgeable reader will post a comment here providing some insight into this issue), but it will be a useful exercise to continue with the calculation assuming the matter-dominated Einstein-de Sitter universe.

Casting Michael Richmond’s equation into the following SAS program, I was able to find that the Andromeda galaxy would reach a minimum angular size of 11.3 arcseconds at z = 1.25, as shown below.

In principle, measuring the angular size of a “standard” object at various redshifts could allow us to determine what kind of universe we live in. But there’s a problem. As we look further out into space we are also looking further back in time, so there is no guarantee that a “standard” object in today’s universe (say, a spiral galaxy such as M31) would have looked the same or even existed billions of years ago.

Richmond, Michael, Two classic cosmological tests
[https://web.archive.org/web/20180909221238/http://spiff.rit.edu/classes/phys443/lectures/classic/classic.html]