Cosmological redshift and recession velocities

In a recent BBC4 documentary, 'The Beginning and End of the Universe', nuclear physicist and broadcaster Jim Al Khalili visits the Telescopio Nazionale Galileo (TNG). There, he performs some nifty arithmetic to calculate that the redshift $z$ of a selected galaxy is:
$$
z = \frac{\lambda_o - \lambda_e}{\lambda_e} =
\frac{\lambda_o}{\lambda_e} - 1 \simeq 0.1\,,
$$ where $\lambda_o$ denotes the observed wavelength of light and $\lambda_e$ denotes the emitted wavelength. He then applies the following formula to calculate the recession velocity of the galaxy:
$$
v = c z = 300,000 \; \text{km s}^{-1} \cdot 0.1 \simeq 30,000 \; \text{km s}^{-1} \,,
$$ where $c$ is the speed of light.

After pausing for a moment to digest this fact, Jim triumphantly concludes with an expostulation normally reserved for use by people under the mental age of 15, and F1 trackside engineers:

"Boom.....science!"

It's worth noting, however, that the formula used here to calculate the recession velocity is only an approximation, valid at low redshifts, as Jim undoubtedly explained in a scene which hit the cutting-room floor. So, let's take a deeper look at the concept of cosmological redshift to understand what the real formula should be.

In general relativistic cosmology, the universe is represented by a Friedmann-Roberston-Walker (FRW) spacetime. Geometrically, an FRW model is a $4$-dimensional Lorentzian manifold $\mathcal{M}$ which can be expressed as a 'warped product' (Barrett O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, 1983):
$$
I \times_R \Sigma \,.
$$ $I$ is an open interval of the pseudo-Euclidean manifold $\mathbb{R}^{1,1}$, and $\Sigma$ is a complete and connected $3$-dimensional Riemannian manifold. The warping function $R$ is a smooth, real-valued, non-negative function upon the open interval $I$, otherwise known as the 'scale factor'.

If we denote by $t$ the natural coordinate function upon $I$, and if we denote the metric tensor on $\Sigma$ as $\gamma$, then the Lorentzian metric $g$ on $\mathcal{M}$ can be written as
$$
g = -dt \otimes dt + R(t)^2 \gamma \,.
$$ One can consider the open interval $I$ to be the time axis of the warped product cosmology. The $3$-dimensional manifold $\Sigma$ represents the spatial universe, and the scale factor $R(t)$ determines the time evolution of the spatial geometry.

Now, a Riemannian manifold $(\Sigma,\gamma)$ is equipped with a natural metric space structure $(\Sigma,d)$. In other words, there exists a non-negative real-valued function $d:\Sigma \times \Sigma
\rightarrow \mathbb{R}$ which is such that

$$\eqalign{d(p,q) &= d(q,p) \cr
d(p,q) + d(q,r) &\geq d(p,r) \cr
d(p,q) &= 0 \; \text{iff} \; p =q}$$ The metric tensor $\gamma$ determines the Riemannian distance $d(p,q)$ between any pair of points $p,q \in \Sigma$. The metric tensor $\gamma$ defines the length of all curves in the manifold, and the Riemannian distance is defined as the infimum of the length of all the piecewise smooth curves between $p$ and $q$.

In the warped product space-time $I \times_R \Sigma$, the spatial distance between $(t,p)$ and $(t,q)$ is $R(t)d(p,q)$. Hence, if one projects onto $\Sigma$, one has a time-dependent distance function on the points of space,
$$
d_t(p,q) = R(t)d(p,q) \,.
$$Each hypersurface $\Sigma_t$ is a Riemannian manifold $(\Sigma_t,R(t)^2\gamma)$, and $R(t)d(p,q)$ is the distance between $(t,p)$ and $(t,q)$ due to the metric space structure $(\Sigma_t,d_t)$.

The rate of change of the distance between a pair of points in space, otherwise known as the 'recession velocity' $v$, is given by
$$\eqalign{
v = \frac{d}{dt} (d_t(p,q)) &= \frac{d}{dt} (R(t)d(p,q)) \cr &= R'(t)d(p,q) \cr &=
\frac{R'(t)}{R(t)}R(t)d(p,q) \cr &= H(t)R(t)d(p,q) \cr &=
H(t)d_t(p,q)\,. }
$$ The rate of change of distance between a pair of points is proportional to the spatial separation of those points, and the constant of proportionality is the Hubble parameter $H(t) \equiv R'(t)/R(t)$.

Galaxies are embedded in space, and the distance between galaxies increases as a result of the expansion of space, not as a result of the galaxies moving through space. Where $H_0$ denotes the current value of the Hubble parameter, and $d_0 = R(t_0)d$ denotes the present 'proper' distance between a pair of points, the Hubble law relates recession velocities to proper distance by the simple expresssion $v = H_0d_0$.

Cosmology texts often introduce what they call 'comoving' spatial coordinates $(\theta,\phi,r)$. In these coordinates, galaxies which are not subject to proper motion due to local inhomogeneities in the distribution of matter, retain the same spatial coordinates at all times.

In effect, comoving spatial coordinates are merely coordinates upon $\Sigma$ which are lifted to $I \times \Sigma$ to provide spatial coordinates upon each hypersurface $\Sigma_t$. The radial coordinate $r$ of a point $q \in \Sigma$ is chosen to coincide with the Riemannian distance in the metric space $(\Sigma,d)$ which separates the point at $r=0$ from the point $q$. Hence, assuming the point $p$ lies at the origin of the comoving coordinate system, the distance between $(t,p)$ and $(t,q)$ can be expressed in terms of the comoving coordinate $r(q)$ as $R(t)r(q)$.

If light is emitted from a point $(t_e,p)$ of a warped product space-time and received at a point $(t_0,q)$, then the integral,
$$
d(t_e) = \int^{t_0}_{t_e}\frac{c}{R(t)} \, dt \, ,
$$ expresses the Riemannian distance $d(p,q)$ in $\Sigma$, (equivalent to the comoving coordinate distance), travelled by the light between the point of emission and the point of reception. The distance $d(t_e)$ is a function of the time of emission, $t_e$, a concept which will become important further below.

The present spatial distance between the point of emission and the point of reception is:
$$
R(t_0)d(p,q) = R(t_0) \int^{t_0}_{t_e}\frac{c}{R(t)} \, dt \,.
$$ The distance which separated the point of emission from the point of reception at the time the light was emitted is:
$$
R(t_e)d(p,q) = R(t_e) \int^{t_0}_{t_e}\frac{c}{R(t)} \, dt \,.
$$ The following integral defines the maximum distance in $(\Sigma,\gamma)$ from which one can receive light by the present time $t_0$:
$$
d_{max}(t_0) = \int^{t_0}_{0}\frac{c}{R(t)} \, dt \,.
$$ From this, cosmologists define something called the 'particle horizon':
$$
R(t_0) d_{max}(t_0) = R(t_0) \int^{t_0}_{0}\frac{c}{R(t)} \, dt
\,.
$$ We can only receive light from sources which are presently separated from us by, at most, $R(t_0) d_{max}(t_0)$. The size of the particle horizon therefore depends upon the time-dependence of the scale factor, $R(t)$.

Under the FRW model which currently has empirical support, (the 'concordance model', with cold dark matter, a cosmological constant $\Lambda$, and a mass-energy density equal to the critical density), the particle horizon is approximately 46 billion light years. This is the conventional definition of the present radius of the observable universe, before the possible effect of inflationary cosmology is introduced...

To obtain an expression which links recession velocity with redshift, let us first return to the Riemannian/ comoving distance travelled by the light that we detect now, as a function of the time of emission $t_e$:
$$
d(t_e) = \int^{t_0}_{t_e}\frac{c}{R(t)} \, dt \,.
$$ We need to replace the time parameter here with redshift, and to do this we first note that the redshift can be expressed as the ratio of the scale-factor at the time of reception to the time of emission:
$$
1+ z = \frac{R(t_0)}{R(t)} \,.
$$ Taking the derivative of this with respect to time (Davis and Lineweaver, p19-20), and re-arranging obtains:
$$
\frac{dt}{R(t)} = \frac{-dz}{R(t_0) H(z)} \,.
$$ Substituting this in and executing a change of variables in which $t_o \rightarrow z' = 0$ and $t_{e} \rightarrow z' = z$, we obtain an expression for the Riemannian/comoving distance as a function of redshift:
$$
d(z) = \frac{c}{R(t_0)} \int^{0}_{z}\frac{dz'}{H(z')} \, .
$$ From our general definition above of the recession velocity between a pair of points $(p,q)$ separated by a Riemannian/comoving distance $d(p,q)$ we know that:
$$
v =  R'(t)d(p,q) \,.
$$ Hence, we obtain the following expression (Davis and Lineweaver Eq. 1) for the recession velocity of a galaxy detected at a redshift of $z$:
$$
v = R'(t) d(z) = \frac{c}{R(t_0)} R'(t) \int^{0}_{z}\frac{dz'}{H(z')} \, .
$$ To obtain the present recession velocity, one merely sets $t = t_0$:
$$
v = R'(t_0) d(z) = \frac{c}{R(t_0)} R'(t_0) \int^{0}_{z}\frac{dz'}{H(z')} \, .
$$ At low redshifts, such as the case of $z \simeq 0.1$, the integral reduces to:
$$
 \int^{0}_{z}\frac{dz'}{H(z')} \approx \frac{z}{H(0)} =  \frac{z}{H(t_0)} \, .
$$ Hence, recalling that $H(t) \equiv R'(t)/R(t)$, at low redshifts one obtains Jim Al Khalili's:
$$
v = cz \,.
$$ Boom...mathematics!