How far, in space, can we see? And can we see an object whose Hubble recessional velocity exceeds the speed of light? Maybe you’ve thought about these questions before, or perhaps you’ve seen them discussed in the literature or mentioned in the media. With the recent popularity of inflation and Big Bang cosmology, they’re hard to avoid. The discussion that follows is an attempt to resolve some common misconceptions—often seen in the popular literature—concerning the above two questions, and to do so in a way that appeals to kinematical intuition. A simple thought experiment will be used to initiate the discussion and to answer the question, “Can we see objects with faster-than-light recessional velocity?” Hubble’s law, along with a simple assumption about the kinematics of light in expanding space, will be used to derive expressions, customarily derived in a general relativistic context, that allow cosmologists to determine our observational limits and define our cosmological horizons. Some of the results may surprise you. Before we delve into the topic fully, though, let’s first lay some theoretical groundwork.

1.
Georges Lemaître was the first to publish research deriving this velocity-distance relationship, commonly referred to as Hubble’s law.
2.
“Distance” as used in this article corresponds to what a standard cosmology text would refer to as “proper distance.” By the “velocity” of a galaxy or photon, we mean the current rate at which its proper distance is increasing, and by “acceleration” we mean the current rate at which its velocity is increasing.
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In special relativity, velocity addition, for physical objects moving through space, is affected by a maximum and invariant universal speed, c, and takes the form u=u+v1+uv/c2.If there were no such speed limit, or if it were infinitely large (c = ∞), velocity addition would reduce to the simple Galilean form u = u′ + v. Because Hubble recessional velocity is not assumed to result from motion through space, and because the space of our universe is not assumed to expand within some other space—with its own maximum and invariant speed limit—there is no need to assume a special relativistic gamma factor limiting the rate at which spatial expansion can change the distance between two points (or objects), and, subsequently, there is no need to introduce a gamma factor when adding either Hubble recessional velocities together (velocities due merely to spatial expansion), or a peculiar velocity (a velocity through space) with a Hubble recessional velocity. A discussion of how observational cosmological redshift supports general relativistic spatial expansion, as opposed to a special relativistic model, can be found in Ref. 15.
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21.
Assuming a flat spatial geometry.
22.
R˙ means the rate of change of R with respect to time, or dR/dt.
23.
The most widely accepted model for spatial expansion—ΛCDM—is of this form. Specifically, R(t) ∝ sinh2/3 (t/tΛ). This model accounts for the gravitational influence of cold dark matter (CDM) and the repulsive influence of a cosmological constant, Λ.
24.
Rewrite Eq. (6) as RD˙R˙DR2=cR, then use the quotient rule from calculus to rewrite the lefthand side as. ddt DR . So, Eq. (6) becomes ddt DR =cR. Multiply both sides by dt and integrate from te to t0 to get D(t0)R(t0)D(te)R(te)=ctet01Rdt. Set D(te) = 0—distance of photon is 0 at time of emission—and solve for D(t0).
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W. M.
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Can galaxies exist within our particle horizon with Hubble recessional velocities greater than c?
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26.
Mathematically, the conventional approach, mentioned above, may require fewer steps, but a grasp of several general relativistic concepts is required.
27.
See Ref. 18, p. 448.
28.
Regions of space, actually. The things themselves may no longer exist.
29.
Keeping in mind that an event horizon in between us and the particle horizon will define a different kind of causal radius, a radius beyond which no object can receive light emitted from us right now.
30.
See Ref. 18, pp. 446, 447.
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33.
The expansion of space is said to “accelerate” if R¨ is positive.
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