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.

## References

*c*, and takes the form $u=u\u2032+v1+u\u2032v/c2.$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.

*R*with respect to time, or

*dR*/

*dt*.

*R(t*) ∝ sinh

^{2/3}(

*t*/

*t*). This model accounts for the gravitational influence of cold dark matter (CDM) and the repulsive influence of a cosmological constant,

_{Λ}*Λ*.

*quotient rule*from calculus to rewrite the lefthand side as. $ddt\u230a DR \u230b.$ So, Eq. (6) becomes $ddt\u230a DR \u230b=cR.$ Multiply both sides by

*dt*and integrate from

*t*

_{e}to

*t*

_{0}to get $D(t0)R(t0)\u2212D(te)R(te)=c\u222btet01Rdt.$ Set

*D(t*

_{e}) = 0—distance of photon is 0 at time of emission—and solve for

*D(t*

_{0}).

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