The idea for this test is pretty simple:
pick objects which are identical in size **S**
and observe them at a range of distances.
The **angular diameter distance D _{A}**
will be related to the apparent angular size of the
objects

If we use radians as the units for the apparent angular diameter, and if the object is at a large distance, then

Now, in an ordinary Euclidean space, like a nice flat soccer pitch, the relationship between apparent angular size and distance is simple. If Joe stands at one end of the field while Alice carries a meterstick to the other end of the field, and Joe makes a series of measurements of the apparent angular size as a function of distance,

Q: What relationship between apparent angular size and distance will Joe find?

Well, this relationship is simple enough in Euclidean space.
But what will happen if we use sources at cosmological
distances in an expanding universe?
In this case,
the angular size depends on the diameter of the source
and on the **proper distance** between
the source and observer
at the time **t _{e}** when the light from the source was emitted.

In order to compare theory to observations more easily,
we should convert from time of emission **t _{e}**
to redshift

The scale factor at the time when light was emitted (long ago) is related to the scale factor when the light is absorbed by our telescopes (now) in a simple manner:

So the angular size as a function of redshift becomes

The comoving coordinate at the time the light was emitted
is a more complicated function of redshift.
If we continue to assume that the cosmological constant
**Ω _{Λ}** is zero,
then

Thus, we find that the angular size as a function of redshift is pretty complicated.

Let's look at the limiting cases:
what happens at small redshifts,
when **z ~ 0**?
What happens at large redshifts,
when **z >> 1**?

Q: Simplify the expression for θ as a function of redshift for z << 1. Q: Simplify the expression for θ as a function of redshift for z >> 1.

Evidently, the angular size of objects must
reach a minimum value and then start to
INCREASE with distance!
One can show that if
**Ω _{M} = 1**,
then the location of this minimum angular size
is at

Now, what can we use as our "standard yardsticks" in the sky? Ideally, we would find a type of object which

- is very luminous, so that we can see it at great distances
- is very large, so that we can resolve its angular extent at great distances
- is always exactly the same linear size

That last requirement is the tough one. Giant elliptical galaxies, for example, satisfy the first two requirements, but how exactly would one define the size of an object which gradually fades away? Moreover, why should all giant elliptical galaxies be identical in size?

In the nineteen-sixties and nineteen-seventies, astronomers turned to radio galaxies as a possibility. Some of the brightest radio sources in the sky were associated with giant galaxies (and giant ellipticals, in particular).

The observations of radio sources weren't quite as clean as one might have hoped. Consider this comparison of angular size and redshift from 1978:

Figure taken from
Hooley, Longair and Riley, MNRAS 182, 127 (1978)

Here's a subset of the same dataset, plotted together with two cosmological predictions.

Figure taken from
Hooley, Longair and Riley, MNRAS 182, 127 (1978)

As Hooley, Longair and Riley stated in their abstract,

This is indeed the Achille's heel of the angular-diameter
test:
are radio sources *really* constant in size at
different redshifts?
Or, if they do shrink (or grow) at high redshift,
just how quickly does their size change?
We must also keep in mind that BIG sources
are very likely to be MORE LUMINOUS than
small ones;
therefore, at high redshift, there is likely
to be a strong bias against detecting the smaller,
less luminous sources.

The angular diameter test, alas, seems to depend more upon our assumptions about the properties of radio sources than it does on the cosmological parameters.

Figure taken from
Okoye and Onuora, ApJ 260, 370 (1982)

In the figure to the left, there are predictions for the angular diameter test for a large range of possibilities.

- A, B, C, D, E
- The curves assume perfectly constant standard yardsticks: radio sources which are identical in every way. The different curves correspond to different cosmological parameters.
- O, Q, S, T, U
- The curves assume that the radio sources are changing both in size and luminosity with time.
- R
- This is the simple 1/z relationship.

Nine years later, Onuora (1991) was still struggling with ambiguities in the data. The graph at left shows galaxies as circular symbols, and quasars as crosses. The author concluded that orientation and evolution made it difficult to draw any conclusions.

The angular diameter test has fallen out of favor in the past decade or two. My guess is that astronomers have realized that the "standard yardsticks" of the skies -- radio sources -- suffer from too many complicating factors to be effective distance indicators. After all, now that astronomers can use radio interferometers to resolve these sources and map their properties clearly, we know that the radio emission from these distant galaxies is not coming from the same regions as the optical light. In many cases, the radio emission turns out to be due to a compact core surrounded by two lobes. A relativistic jet carries material from a central AGN outward until it smashes into the intergalactic medium, where it excavates giant "bubbles" of low density material. The three sources shown below were all included in the studies above.

Figure taken from
The NRAO Image Gallery

Figure taken from
The NRAO Image Gallery

Image taken from
The Atlas of DRAGNs

It is no surprise now that radio galaxies are not "standard yardsticks."

One of the other classic tests uses
"standard candles" rather than "standard yardsticks."
The idea is to measure the apparent brightness
of a set of objects which emit the same luminosity,
but are at a range of distances.
The relationship between apparent magnitude
and redshift depends on the
**luminosity distance**
as a function of redshift.
If we can measure that relationship precisely,
we can extract the cosmological parameters
from the equation for luminosity distance.

In the previous lecture , we derived expressions for the luminosity distance. It depends in part on the comoving coordinate as a function of redshift,

and in part on the manner in which one chooses to measure the brightness of a source; if one uses a photon-counting device,

or, if one uses an energy-measuring device,

For the remainder of this discussion,
we will adopt the
**photon-counting luminosity distance.**
Many of the conclusions we reach would be the same
if we were to adopt the energy-measuring version,
although the detailed numbers would be slightly different.

Let's begin with a review of Mr. Hubble's work. If we can measure properties of the local universe alone -- as was the case in the 1930s and 1940s -- then what can we learn from the magnitude-redshift relationship? Well, to find out, we can look at how luminosity distance depends on the value of two parameters in particular.

Q: If we look at the nearby universe, which of the cosmological parameters can we measure? Q: Use the graph below to estimate the value of that parameter.

Notice the quantities plotted on the vertical axis of this graph: distance, in units of Mpc. In most cases, astronomers measure the APPARENT magnitude of some type of object -- supernovae or galaxies.

Figure taken from
Sandage and Tamman, ApJ 256, 339 (1982)

Figure taken from
Sandage and Tamman, ApJ 197, 265 (1975)

In order to compute the distance, we need to know the ABSOLUTE magnitude of that class of object. This is an important point:

- In order to measure
**H**, one needs to know the absolute magnitude of the source_{0}

If we are interested in the **Ω _{M}**
parameter,
then we must look much farther afield.
The difference between the apparent magnitudes of
some standard candle in different cosmologies
only becomes significant once we extend our
observations to redshifts greater than

Q: Which value ofΩproduces the largest distance modulus? Explain._{M}

Let's look at little more closely at some of the models.

Q: If one can measure objects out toz = 0.5, how 'standard' must the standard candles be to distinguish models withΩfrom models with_{M}= 0.5Ω? Q: If one can measure objects out to_{M}= 1.0z = 1.0, how 'standard' must the standard candles be to distinguish models withΩfrom models with_{M}= 0.5Ω? Q: If one wishes to measure_{M}= 1.0Ωto a precision of 10 percent, and one can detect objects out to_{M}z = 1.0, how 'standard' must the standard candles be?

Notice that the effect of the Hubble constant at these large scales is pretty simple: without changing the SHAPE of the relationship, it just shifts the distance values

or the distance modulus values.

For example, compare the graph above to the graph below.

This is actually a very useful feature of the redshift-magnitude relationship in this regime. Even if we don't know the ABSOLUTE value of some standard candle, we can still use it to investigate the cosmological parameters. As long as all instances of some particular type of object have exactly the same luminosity/absolute magnitude, the SHAPE of the redshift-magnitude graph will reveal the underlying parameters.

Let's pick our standard candle: Type Ia supernovae. There is good evidence that these objects do have the same absolute magnitude, with the caveats

- it's necessary to make corrections using the light curve shape
- some objects are just wierd, but can be recognized by their spectrum and discarded
- there is an intrinsic scatter of about 0.13 mag in their absolute mag values
- dust in the host galaxy can redden and extinguish the light from some events
- there may be evolutionary effects which cause small changes in absolute magnitude at high redshift

In the late nineteen-nineties, two groups of astronomers searched for high-redshift supernovae. They had to find the supernovae, measure the light curves well enough to determine peak magnitudes and the shape (in order to correct for shape-dependent effects), and -- this is the hard part -- acquire a spectrum to verify that the event really was a Type Ia and to measure its redshift.

Q: Why was getting a spectrum "the hard part"?

Let's look at their results, which were published in 1998 and 1999.

Figure taken from
Riess et al., ApJ 116, 1009 (1998)

Figure taken from
Perlmutter et al., ApJ 517, 565 (1999)

Now, as you can see, all the interesting
action is stuffed into the upper-right
corner of these graphs.
It might be easier to answer some of these questions
if we made a slightly different graph.
Let's change the vertical axis so that we plot
the difference between the distance modulus for
some particular model and the distance modulus
for some *benchmark* model;
say, **Ω _{M} = 1.0**.

So, again, here are the results from the supernova studies.

Figure taken from
Riess et al., ApJ 116, 1009 (1998)

Figure taken from
Perlmutter et al., ApJ 517, 565 (1999)

So, what do we learn?
What exactly is the value of **Ω _{M}**?

In the years since the first announcements, astronomers have measured additional SNe Ia at high redshift, and continued to investigate systematic effects which can complicate this test.

Figure taken from
Riess et al., ApJ 607, 665 (2004)

Figure taken from
Freedman et al., ApJ 704, 1036 (2009)

In order to understand these results,
we will need to go back to the cosmological
drawing board
and see what happens if we dare to
introduce a non-zero value for
**the cosmological constant Λ**.

- Okoye and Onuora, ApJ 260, 370 (1982) show how much the angular diameter test depends on one's understanding of the evolution of radio sources.
- Sandage and Tamman's series "Steps toward the Hubble Constant" is a set of ten papers which describes the many procedures one must follow to calibrate and use cosmological tests.
- Riess et al. (1998) was one of the earliest papers to announce the evidence for a non-zero cosmological constant, based on observations of supernovae.
- Perlmutter et al. (1999) is an early paper from another group studying high-redshift supernovae.

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.