Chapter 6

Deceleration Parameter

Now we transition to observations. The universe, it turns out, is large — and a little messy. Measuring $a(t)$ is no trivial task, especially when dark energy is added to the mix. In simpler times, no one gave dark energy any mind, though, and $a(t)$ could be determined entirely by local measurements. This is because, without dark energy, curvature is determined by the current matter density (since radiation is so much less important), which leads to only 3 options: open, closed, or flat/coasting universes. These possibilities can be distinguished by measuring the current value of the deceleration of the expansion of space, or $\ddot{a}$. Without knowing the exact equation for $a(t)$, the local ($t=t_0$) form of the equation can be approximated using a Taylor expansion. That leads to this expression (only valid for $t \approx t_0$, becoming less correct for other times):

$q_0$ is the deceleration parameter (so named because it is dimensionless and is the negative of acceleration). This definition can also be put in terms of density parameters:

Proper Distance

Using this approximation of $a(t)$ (or comparable expansion for $1/a(t)$) and inserting this into the expression for redshift $z = 1/a(t_e) - 1$, the proper distance can be expressed in terms of $z$, $H_0$, and $q_0$:

Here we see that Hubble's law ($cz=H_0d$) gets modified (to second order in $z$) by $q_0$ (the classical Hubble law is just the first term in this approximation, and more terms could get added $\propto z^2, \propto z^3,$ etc. For this approximation to be valid, $z \ll 1$ (otherwise the higher order terms aren't negligible).

Measurable Distances

This is great, but proper distances are not what we measure as they are instantaneous distances — light travels at a finite speed, during which $a$ changes, affecting the ways in which we can estimate distances with observations. There are various ways to estimate distances, the best of which are geometric (e.g., parallax), assuming we know (or can precisely calculate) the size of the short side of a triangle (Earth's semimajor axis in the case of parallax). These cases are rare in practice; most are based are knowing the intrinsic brightness, or luminosity, of an object and measuring its flux.

Luminosity Distance

Distances can be estimated if we know how intrinsically luminous an object is, i.e., if it is a standard candle. This distance is just that assumed by the inverse square law:

The observed flux from a source, however, depends on the curvature of the universe, or $S_\kappa(r)$. This is because the surface area of a sphere is different depending on the curvature ($A_{\rm sph} = 4\pi S_\kappa(r)^2$) AND because light is redshifted, losing energy $\propto (1+z)^{-1}$, AND because the distance between photons increase with the expansion of space, also $\propto (1+z)^{-1}$. So the actual measured flux (energy/time/area) is then

so $\boxed{d_L = S_\kappa(r)(1+z)}$ by definition. In a flat universe (like ours), $S_\kappa(r) = r = d_p(t_0)$, so $d_L = d_p(t_0) (1+z)$. Using the approximate expression for $d_p(t_0)$ above,

Angular-diameter Distance

Distances can also be estimated if we know the intrinsic size of an object, i.e., it is a standard ruler. This distance is just that assumed by the small-angle formula:

where $\ell$ is the length of the object and $\delta \theta$ is its observed angular size. The angular size depends on when the light was emitted, and $\ell = a(t_e) S_\kappa(r) \delta \theta$, so $\boxed{d_A = S_\kappa (r) / (1+z)}$. In a flat universe, $S_\kappa(r) = r = d_p(t_0)$, so $d_A = d_p(t_0) / (1+z)$. Using the approximate expression for $d_p(t_0)$ above,

As can be seen comparing $d_L$ and $d_A$ in a flat universe, they are related to each other by

Luminosity distances, but using magnitudes (ugh!)

Astronomers still use magnitudes, first defined by ancient Greek astronomers to rank star brightnesses. Due to the semi-logarithmic sensitivity of human eyes, this means you have to deal with a garbage relation between fluxes (a sane unit of measure) and observed magnitudes, which are properly called apparent magnitudes:

where $f_x = 2.53 \times 10^{-8}~{\rm W~m}^{-2}$ is just a constant defined by convention (and can be different in different magnitude systems). Similarly, a corresponding equation for luminosities can be written, which are called absolute magnitudes:

The constant $L_x = 78.7~{\rm L}_\odot$ is defined as the luminosity a star (or whatever) with a flux of $f_x$ would have if its luminosity distance $d_L = 10~{\rm pc}$. Using our definition of $d_L$ above,

Since galaxies are separated by Mpc, not pc, it's convenient to change the units of $d_L$, equivalent to multiplying the second term by a factor of $\log_{10}(10^5) = 5$, so to keep the equation balanced, we get a version of this equation called the distance modulus:

This is nothing more than the inverse square law, but in a semi-logarithmic form for no good reason. Think of it like a secret handshake in a secret society — an overly-complicated way to say hi to a fellow astronomer. Plugging this into our approximate expression for $d_L$ above yields the truly awful equation (but the one used IRL):