# 17.10 Measuring Space Curvature

Astronomers have devised techniques to measure the curvature of space. General relativity predicts that the universe will have a global curvature due to the total density of matter in it. This curvature is subtle enough that it can be detected only with observations over a significant fraction of the observable universe. To test the curvature of our universe we cannot measure the angles of gigantic triangles in space. However, we look for distant objects and compare them with nearby objects. These measurements are extremely difficult. Remember that in the big bang model, space curvature is related to both the mean cosmic density and the rate of deceleration of the expanding universe. Positive curvature corresponds to a dense universe with a large deceleration.

Examples of curved spaces in everyday life. Click here for original source URL.

One test of the geometry of the universe involves the way that the density of objects varies with distance from the Milky Way. Astronomers can use galaxies as markers of expanding space. Using the two-dimensional analogies of a sphere, a plane, and a saddle-shaped surface, consider these surfaces with galaxies randomly scattered on them. Now imagine flattening the curved surfaces onto a plane so that we can measure the linear distance between any two points. The flat surface is a familiar type of Euclidean geometry — the area out to any distance R is ∝R^{2}. Therefore the number of galaxies out to a distance R increases proportional to R^{2}, or N ∝ R^{2}. To force a positively curved surface (sphere) onto a plane, the edge must be stretched, which thins out the density of galaxies far from the center. Therefore the number of galaxies out to a distance R increases more slowly than N ∝ R^{2}. To force a negatively curved surface (saddle-shape) onto a flat plane, the edge must be compressed, which increases the density of galaxies far from the center. Therefore the number of galaxies out to a distance R increases faster than N ∝ R^{2}. In three dimensions, the analogy holds. In other words, the curvature of the three-dimensional universe is revealed by whether the number of galaxies per cubic mega parsec increases more slowly or more quickly than N ∝ R^{3}.

Astronomers can actually count the number of galaxies per cubic mega parsec at different distances (or red shifts) from the Earth to measure the curvature of space. A research group who made one such count derived a density parameter of one or less. This density parameter is equivalent to a flat or open universe and a geometry that may be negatively curved. The results are promising, but there is a serious complication in applying this test that can lead to large systematic errors.

Any survey limited by apparent brightness will be far more sensitive to luminous objects than intrinsically dim objects. For example, though the brightest stars in the sky are giants and super giants, the most common stars in any volume of space are dwarfs. Exactly the same situation arises with galaxies. The most common type of galaxy is a dwarf galaxy, but giant galaxies can be seen at much larger distances, so they will be over-represented in any census. In fact, the bias in favor of luminous galaxies is worse than the analogous bias in favor of luminous stars. Stars dim by the inverse square of the distance. Galaxies also dim according to the geometry of space, proportional to (R/R_{0})^{2}. By manipulating the expression z = (R_{0}/R)^{-1} we can see that (R/R_{0})^{2} = (1+z)^{-2}. However, galaxies are dimmed by an additional factor of 1+z, due the slowing of the arrival rate of photons. They are dimmed by yet another factor of 1+z due to the stretching of the spectral range — the range of galaxy wavelengths we observe corresponds to a smaller wavelength range at the time the light was emitted.

The effect of cosmology is that light from a distant galaxy is dimmed by a factor of (1+z)^{4}. So a galaxy at a red shift of z = 1 appears 2^{4} = 16 times fainter than an identical galaxy in the nearby universe. A distant galaxy at a red shift of z = 3 appears 4^{4} = 256 times fainter than an identical local galaxy. When astronomers look for high red shift galaxies, they tend to select more luminous galaxies than when they look locally. The bottom line is that it is very difficult to compare identical samples of galaxies at different red shifts. This underlines how difficult it is to test the homogeneity assumption in the cosmological principle — distant objects are seen as they were when they were younger so they may not be the same as objects nearby that we see as they are now.

Another test of the universe’s geometry relies on the way that angles change in curved space. In flat, Euclidean space, more distant objects have smaller angular sizes on the sky, with the angular diameter inversely proportional to distance — the familiar small-angle equation. The curvature of space can distort the images of distant galaxies, and since it is mass that bends light, the distortion increases with the density of the universe. The angular size of a distant object in a closed universe actually increases with red shift! How can an object appear larger as it gets farther away? If you think of the universe as a gigantic lens, then the gravity in a high-density universe can actually magnify the image of a distant object. You can also understand this bizarre effect by recalling that the universe was smaller at a high red shift. Galaxies, however, have remained the same size during the expansion. At high red shift, a galaxy of a particular size would subtend a larger angle in what was then a smaller universe.

Astronomers have not succeeded in accurately measuring the curvature of space using galaxies. The cosmological models only begin to diverge at z= 0.2 to 0.3, which corresponds to a distance of d ∝ cz/H_{0}, or 1000 to 1500 Mpc. Multiplying by 3.3 to convert from parsecs to light years, these are look-back times of 3.3 to 5 billion years. Therefore, the only way to detect space curvature is to compare old and young objects. The evolution issue cannot be dodged; it is an inevitable consequence of the size of the universe and the finite speed of light. Currently, the most promising cosmological tools are supernovae. These "standard bombs" can be seen out to z = 1 and beyond, well beyond the distance where space curvature should show itself. Despite all the problems, we can draw two conclusions from existing measurements. First, models with a large amount of positive curvature (Ω_{0} > 1) are ruled out. This kind of universe is too small, too young, and too curved to fit the observations. Second, the curvature is small enough that it may be consistent with a flat model and a universe of critical density.

The most direct way to measure space curvature is to use the microwave background radiation. As a relic from the first 2% of the age of the universe, these waves have been traveling through space for billions of years. The most prominent ripples in the microwave background have an angular size of about 1º. At a time 300,000 years after the big bang, this represented the size of the first structures that were going to form in the still-hot universe. In a positively curved universe, angles subtended by distant objects are larger than they would be in flat space. You can think of this as positive magnification of a beam of photons. In a negatively curved universe, angles subtended by distant objects are smaller than they would be in flat space. You can think of this as negative magnification. Results from the Boomerang balloon experiment in 1999, and several others that soon followed, showed that the size of the most prominent microwave fluctuations exactly matched the predictions for no curvature. These results were confirmed and refined by the WMAP and Planck space missions. No space curvature is detected at a 1% level; space seems to be flat.

Microwave background measurements indicate that space is flat. This result is surprising because astronomers have not found enough matter in the universe, light or dark, to create flat space-time. The result is renewed interest in big bang models that incorporate vacuum energy in the form of a cosmological constant. This extra repulsive force can act to smooth out space. In other words, the definition of Omega, the density parameter, should be extended to include other components of the universe that can affect space curvature. The best interpretation of all the available evidence is that Ω_{tot} = 1 and space is flat. However, there is only enough normal and dark matter on any scale to add up to Ω_{matter} = 0.3. The observations of Type I supernovae indicate that the other major constituent of the universe is vacuum energy, also referred to as the cosmological constant, with a value of Ω_{vacuum} = 0.7. Radiation, in the form of the cosmic microwave background, is a negligible contributor. As you can see, the sum of these two components corresponds to flat space.