A straightforward answer to this question is: "Find a good ruler of fixed length and use it! " All that is needed is to find a decent ruler, which can be used at large ("cosmological") distances.
Firstly, a few words about the meaning of curvature. If the Universe is curved, for example, a bit like the surface of the Earth which is more or less spherical, then the relationship between the distance to an object and the angle it subtends is not what we expect for a flat space (a flat surface).
If someone living at the North Pole draws a map of the world on a flat
piece of paper, with herself at the centre, then she would find that the
Antarctic appears to be an enormous continent forming a ring, and that
the apparent sizes of countries in the South are all much bigger than their
real sizes. If our cartographer knows the real sizes of both the northern
and southern continents, and if she is a good mathematician, then she would
deduce from this distorted map that the surface of our planet is not flat but
spherical.
In three dimensions, astronomers attempt the same map making experiment.
If one knows in advance the real sizes of distant objects, then if their
measured angular sizes turn out to be bigger, equal or smaller than those
expected for a flat geometry, one can deduce that space is "spherical",
flat or even hyperbolic, a bit like a surface curved like a horse's
saddle.
The curvature of the Universe is a function of two parameters: firstly, the density parameter, which is equal to 1 if the density of the Universe is just that which is necessary for the expansion of the Universe to slow down indefinitely and just about stop, and secondly, the cosmological constant, which represents and additional energy component of the Universe, whose nature remains mysterious, which contributes to an acceleration in the expansion of the Universe. Theoretical modelling of the Universe when it was very young suggests that the Universe may be nearly flat, which implies that the sum of these two parameters should be nearly equal to 1.
To measure the curvature, it is therefore necessary to have distant objects whose sizes are known and which can be used as cosmic standard rulers.
Two scientists, Boud Roukema, at IUCAA, in India (also a visitor at DARC, Observatoire de Paris-Meudon) and Gary Mamon, at IAP (also associated with the DAEC, Observatoire de Paris-Meudon), have recently proposed the use of the average size of « bubbles » of large scale structure in the Universe, traced by galaxies, as a statistical standard ruler for measuring the curvature of the Universe. Indeed, on very large scales of order of many hundreds of millions of light years, gravity has little influence on structures, so that very large structures simply follow the general expansion of the Universe. These structures thus serve as standard rulers in a reference frame that is comoving with the expansion of the Universe.
In the local Universe, which is representative of the present-day Universe, such structures formed by galaxies are well known since the discovery in the mid 80's of the Great Wall by Valérie de Lapparent (IAP) and her colleagues.
All that is needed is to detect similar structures far away by using bright enough objects, and then to find the values of the curvature parameters for which the distant bubbles of large scale structure have, on average, the same size as the local bubbles.
The picture below shows that the two scientists have discovered these bubbles in the distribution of distant quasars. For these bubbles to have the same size (the figures are compensated for the expansion of the Universe) as the bubbles in the local Universe, the density parameter must be roughly 0.3, whereas the cosmological constant is only weakly constrained. Hence, this approach to the curvature of the Universe gives the same value of the density parameter as the study of the speeds of galaxies in clusters and groups of galaxies.
If this analysis is combined with those of other teams who use supernovae (stars which explode) in distant galaxies and who deduce that the cosmological constant is non-zero, then this shows that the Universe is nearly flat and that the model of an accelerating Universe (with non-zero cosmological constant) seems to be the correct one.
The circles drawn on the picture of the nearby Universe (centre of the diagram, Las Campanas survey) show in a simplified way the spatial distribution of galaxies, in the form of bubbles and voids.
If the flat model with a cosmological constant is correct, then it must be possible to draw circles of the same size on the top picture (red dots, where the curvature model is that with the density parameter equal to 0.3 and the cosmological constant equal to 0.7) in order to trace out the large scale structures which are far away from us. This is indeed the case.
If the model (still fashionable very recently) of a flat universe without a cosmological constant were the correct one, then it would be possible to copy the circles from the nearby survey to the structures of the survey of distant quasars shown at the bottom of the diagram (black dots, where the density parameter equals 1 and the cosmological constant is absent). However, the circles are too big, or equivalently, the bubbles of large scale structure are too small under this hypothesis.
In other words, it's the top picture, not the bottom one, which gives the correct choice of the curvature parameters: the Universe is almost flat with a non-zero cosmological constant!
In the near future, French cosmologists hope to detect the same sort of distribution of galaxies, quasars and hot gas at great distances in new surveys, for example, in the VIRMOS deep survey, which is currently being carried out on a big European optical telescope, the VLT. With these data, we expect to deduce constraints on the curvature of the Universe and on the cosmological constant which are even more precise than those which we have just obtained.
Tangential large scale structure as a standard ruler: curvature parameters from quasars, Roukema & Mamon (2000a), Astronomy & Astrophysics 358, 395, astro-ph/9911413
Lifting cosmic degeneracy within a single quasar survey,
Roukema &
Mamon (2000b), Astronomy & Astrophysics, in press,
astro-ph/0010511