## Relativity, Cosmology, and Time

### Angular-Size Redshift Relation The angular-size redshift relation describes the relationship between the angular size observed on the sky of an object of given physical size, and the object's redshift. In elementary Euclidean geometry the relation between angular size , linear size l and distance d (from the Earth) would simply be given by the equation: = l / d

In an expanding universe the distance d is a function of the redshift z:

d = (c/Ho) {qoz + (qo - 1)[(1 + 2qoz)½ - 1]} / [qo2(1 + z)2]

#### Figure 10m Angular-Size / Redshift [view large image]

where qo = (4 G / 3) / H02 is the deceleration parameter with the substitution by
Eq.(20e). In standard cosmology qo = 0.5 for flat space, qo > 0.5 for closed space, and qo < 0.5 for open space (see Figure 10m, angular size in unit of mas = milliarcseconds).
As z 0, d (c/Ho) z , and  1/z
while for z , d (c/Ho) / qoz, and  z . These limiting cases clearly demonstrate the curious effect that the angular size of an object becomes larger as it is further away from Earth. It appears to decrease with distance only for nearby objects. Figure 10n shows the infrared blobs produced by the first stars (high z objects). It is suggested that the appearance of the puffy blobs with large angular size is caused by the expansion of the universe with z > 1.6 as shown in Figures 10m and 10n.

#### Figure 10n Infrared Blobs [view large image] In principle, the angular-size redshift relation can be used to select the type of space for our universe. However, it is notoriously difficult to collect reliable data in practice because the astronomical yardstick can vary in size and in luminosity over time (the evolutionary and selection effects). In addition, we can only measure the projections on the celestial surface according to the orientation of the objects. All past attempts using data from galaxies, the separation of the lobes of radio sources, quasars, and radio galaxies produced inconclusive results. The observational data in Figure 10oa are based on selective compact radio sources. The best fitting regression analysis gives a value of qo 0.21. More recent study in 2004

#### Figure 10oa Model with Cosmological Constant [view large image]

with ultra-compact radio sources find close match for the model of an universe with cosmological constant, m = 0.24,  = 0.76, and spatially flat. The observational data range between 0.6 < z < 2.7 with a population mean for the linear size l ~ 6.2 pc. It indicates a "switch over" from deceleration to acceleration at z = 0.85 (see Figure 10oa). A novel method to circumvent the small size of astronomical objects is to observe the orientations of pair of galaxies. It is suggested that by correcting for redshift and angular size with a correct geometry of the universe, the orientations of distant pairs of galaxies should be completely random, as shown in Diagram a, Figure 10ob. Otherwise, there would be a preferred direction as shown in Diagram b, Figure 10ob. A 2010 study on distant galaxies with z ~ 0.5 indicates that the geometry of the universe is consistent with the standard cosmological model, with its flat curvature. It also shows that the dark energy

#### Figure 10ob Geometry of the Universe [view large image]

is in agreement with being a vacuum energy, which can be represented by Einstein's famous cosmological constant.

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