Cosmology PHYS3170 The angular-diameter-redshift relation

Transcription

1 Cosmology PHYS3170 The angular-diameter-redshift relation John Webb, School of Physics, UNSW Purpose: A fun way of understanding a non-intuitive aspect of curved expanding spaces that the apparent angular-diameter of a distant object can increase with increasing distance. Props required: Some or all of: a balloon, some string, a felt-tip pen, sticky tape, a ruler, a match stick, some graph paper, a protractor. Theory There is a rather non-intuitive observational consequence of the non- Euclidean geometry of space-time. This concerns the appearance of an object of fixed physical size, as viewed at different distances from an observer. Our every day experience of perspective does not apply in curved space-times. However, it is not the curvature of space alone which lead to a new effect compared to the Euclidean case, it is also the expansion of space-time together with the finite speed of light. Suppose we take a galaxy of some fixed physical size (diameter d), and the ask what angle, θ, that galaxy subtends at a view on Earth when it is a distance r. In Figure 1, the galaxy is represented by the circle. The lines from the points A and B to O represent the light rays (null-geodesics) from the galaxy edges to the observer. They are drawn as straight lines for illustrative purposes. The proper distance between A and B is given by the Robertson-Walker line element: [ ] dr dl 2 = c 2 dt 2 + R 2 2 (t) 1 kr + 2 r2 (dθ 2 + sin 2 θdϕ 2 ) (1) which can be used to give the proper distance between A and B. The orientation of the axes in our spherical geometry coordinate system is arbitrary (the assumption of isotropy), so we are at liberty to choose the coordinate system such that θ = 0, so sin θ = 0 and dϕ 2 = 0. Also, the 2 light rays 1

2 Figure 1: The angle subtended at the observer O by a galaxy leave points A and B at the same time and travel the same distance, r, i.e. both t and r are constant in equation (1), so dt 2 = dr 2 = 0. Equation (1) therefore simplifies to dl 2 = r 2 R 2 (t)dθ 2 = d 2 (2) since the spacelike separation AB = d (the time component has gone). Putting θ = dθ and re-arranging, θ = d r g R(t g ) where we have introduced the subscript g to indicate reference to the galaxy in Figure 1. For convenience we proceed by considering the mathematically simplest illustration of the time dependence of R(t g ) for the Einstein-de Sitter model ( k = 0, q 0 = 1/2 ). The scale factor of the Universe at time t g, R(t g ) can be expressed in terms of the present day scale factor, R(t 0 ), and the redshift, z g, corresponding to time t g, i.e., R(t g ) = R(t 0) (1 + z g ) = R(t 0) ( tg t 0 (3) ) 2/3 (4) Equation (2) shows how the result differs from the Euclidean result; R(t g ) 2

3 is the additional factor, which would be unity for Euclidean space. As the observer O looks at further and further distances, r g increases. However, as r g increases, the look-back time increases (because of the finite speed of light), and the scale factor decreases. Thus the two terms in the denominator of equation (2) compete, and to understand the effect on the observed angular size we must investigate how R(t g ) behaves. The interval for light rays (null geodesics) is, by definition, zero, so dl 2 on the left hand side of equation (1) is zero. Also, photons propagate along a constant spatial direction, so that θ, dθ 2, dϕ 2 = 0. Thus equation (1) reduces to cdt = ± R(t g)dr (5) (1 krg) 2 1/2 Since the photons are approaching the observer, r decreases as t increases along the null geodesic, i.e. dt/dr < 0, thus the negative sign in equation (5) is the only relevant solution. For the Einstein-de Sitter case, equation (5) therefore simplifies to dr = cdt R(t g ) Integrating between the appropriate time limits, the radial coordinate of the galaxy in Figure 1 is (6) r g = and using equation (4), this becomes t0 t g cdt R(t g ) (7) r g = c t0 R(t 0 ) t g If we carry out the integration above, we get ( ) 2/3 t0 dt (8) t r g = 3ct 0 R(t 0 ) [1 (1 + z) 1/2 ] (9) 2c = [1 (1 + z) 1/2 ] R(t 0 )H 0 3

4 Finally, we can substitute equations (4) and (9) into equation (3) and see how the apparent angular diameter of a galaxy depends on redshift θ = H 0d 2c (1 + z) 3/2 [(1 + z) 1/2 1] (10) If we now differentiate equation (10), we find that there exists a minimum value of θ, θ min, which, for the Einstein de-sitter model, occurs at a redshift, z min = 1.25, given by θ min = 6.75 H 0d c (11) Figure 2: Angular diameter-redshift relation Figure 2 gives a sketch of equation (10), showing that the apparent angular diameter of a galaxy decreases with increasing redshift out to z = 1.25, goes through a minimum at that redshift, and then increases towards higher redshift. In the preceding paragraphs, we have only derived the result for k = 0 (q 0 = 1/2). The results for other values are sketched in Figure 2. The rest of this text describes an entertaining way of obtaining the form of this relation in a qualitative way. 4

5 Experiment 1: A practical demonstration of the angular diameterredshift relation in a curved but non-expanding space Consider the surface of a balloon as the surface of a pretend universe (in 2D). Imagine an observer on that surface who observes a 2D object of fixed linear size (a matchstick, viewed perpendicular to the line of sight). As the matchstick is moved away from the observer, perpendicular to the line of sight, great circles passing through the observer and matchstick ends denote the paths (null geodesics) taken by light rays to the observer. As the matchstick approaches the equator, the angle subtended at the observer by the matchstick decreases, as intuition would tell you. However, look what happens as the matchstick passes below the equator (taking the observer at the North pole). The angle subtended at the observer now begins to increase. We can make simple measurements to confirm what happens as follows: 1. Blow up a balloon. 2. Mark the position of the observer somewhere on the surface. 3. Place a matchstick on the surface of the balloon perpendicular to the line of sight somewhere near the observer. 4. Then, using great circles for measurements: measure the angle subtended at the observer by the match stick. 5. Measure the distance to the matchstick, expressed as a fraction of the circumference of the balloon. 6. Repeat the above for a range of distances between observer and matchstick. 7. Plot on a sheet of graph paper the fractional increase in distance for the different measurements made in (c) (x axis) against the observed angle (y axis). This is a nice analogy of the effect of curved space. However, there is another property of the universe which is not allowed for in the simple experiment above, but which is included in the theoretical derivation of the previous section, and which can enhance the effect: the expansion of spacetime. 5

6 Experiment 2: A practical demonstration of the angular diameterredshift relation in a curved and expanding space. Now repeat the experiment above, but this time, allowing in the following way for the expansion of space and the finite speed of light: 1. blow up a balloon to, say, 1/3 or 1/2 maximum size. The surface area of the balloon is analogous to the volume of the universe at some early time, t. 2. Now place the matchstick somewhere near the observer, perpendicular to the observer sight-line, mark its position and draw on the balloon the great circles passing through the ends of the matchstick to the observer at the pole. 3. If you now blow the balloon up, space is expanding. The great circles are moving apart. The angle subtended at the observer increases as you blow the balloon up. The matchstick does not increase in size, but the light rays move apart as space expands. Therefore, proceed as follows: 4. Measure the distance to the matchstick before you blow it up further from its original size. 5. Measure the distance to the matchstick after you blow it further. 6. Measure the angle subtended at the observer. 7. Repeat the above for a range of distances between observer and matchstick. 8. Plot on a sheet of graph paper the fractional increase in distance (corresponding to the ratio of the scale factors at the two different epochs ) (x axis) against the observed angle (y axis). By placing the matchstick at different distances from the observer, you are effectively placing the object being observed at different redshifts and hence different epochs. The further the distance from the observer, the more you have to blow the balloon up to allow for the greater elapsed time due to the finite speed of light. 6