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- The space-time interval is associated with the Robertson-Walker metric. It has three forms depending on the curvature of the 3 dimensional space:

ds^{2}= c^{2}dt^{2}- R(t)^{2}(dr^{2}+ w^{2}(d^{2}+ sin^{2}d^{2})) ---------- (17)

where w = sin(k^{1/2}r) / k^{1/2}, k (in unit of cm^{-2}) can be considered as the total energy of the universe, and R(t) is called the scale factor. It expands the coordinate grid as a whole with the expansion of the universe. Such grid is referred to as comoving coordinates. Thus if we denote dr to be the comoving distance between two points, then dx = R(t)dr would be the physical distance actually measured by a ruler.

For flat space with zero curvature, k = 0, and w = r, where r ranges from 0 to infinity.

For closed space with positive curvature, k > 0, w = sin(r), where r ranges from 0 to 2.

For open space with negative curvature, k < 0, w = sinh(r), where r ranges from 0 to infinity. - In term of the Robertson-Walker metric and with the energy-momentum tensor T
_{00}= , the gravitational field equation can be written in a form (called Friedmann Equation) similar to the energy equation (K.E.+P.E.=E):

(dR/dt)^{2}- 8GR^{2}/ 3 = - kc^{2}---------- (18a)

which can be re-arranged into the form:

H^{2}- H_{0}^{2}= -k^{2}c^{2}/R^{2}---------- (18b)

where H(t) = (dR/dt) / R is the Hubble's parameter, = /_{c}, and_{c}= (3 H_{0}^{2}) / (8G) ~ 0.9x10^{-29}gm/cm^{3}is the critical density, which corresponds to the total energy density for a flat universe.

for flat space, =_{c}, = 1, k = 0;

for closed space, >_{c}, > 1, k > 0;

for open space, <_{c}, < 1, k < 0. - More generally, when several types of matter-energy coexist in the universe, the following consistency relation must be satisfied:

_{j}_{j}+_{k}= 1

where the sum is over the various types of matter-energy and_{k}= - kc^{2}/(R_{0}H_{0})^{2}. Present cosmological observations yield:

_{b}~ 0.04 for brayons (ordinary visible and nonluminous matter),

_{d}~ 0.26 for exotic dark matter,

_{}~ 0.7 for dark energy,

_{}~ 5 x 10^{-5}for photons (radiation), and

_{k}= 0. - For a homogeneous and isotropic universe = M / (4R
^{3}r_{0}^{3}/3) ---------- (18c)

where M may be interpreted as the total mass and r_{0}the radius of the universe, Eq.(18a) becomes:

(dR/dt)^{2}- 2GM / Rr_{0}^{3}= -kc^{2}---------- (18d)

The solution for this equation is:

for k = 0, R = (9GM/2r_{0}^{3})^{1/3}t^{2/3}, t = 2/3H ---------- (19a)

for k > 0, R = R_{o}(1 - cos(k^{½}µ)), ct = R_{o}(µ - k^{-½}sin(k^{½}µ)) ---------- (19b)

where R_{o}= GM / |k|c^{2}r^{3}, and µ is defined by c dt = R(µ) dµ.

for k < 0, R = R_{o}(cosh(|k|^{½}µ) - 1), ct = R_{o}(|k|^{-½}sinh(|k|^{½}µ) - µ) ---------- (19c) - The red shift caused by the cosmic expansion can be expression in terms of the scale factor R [see Figure 10a, which shows the red shift of the Balmer series of the hydrogen line spectra (dark lines, denoted by H
_{}, ...) superimposed on the continuous spectra (colored), the arrows indicate the direction of increasing values for the corresponding variables]:

z =_{1}/_{2}- 1 = R(t_{2}) / R(t_{1}) - 1 ---------- (19d)

where t _{2}denotes the current epoch,_{2}is the original wavelength (as measured on Earth),_{1}is the wavelength of a spectral line that was emitted at time t_{1}(red shifted) in the distant past, R(t_{2}) is often taken to be R_{0}= 1 for convenience, then R(t_{1}) is less than 1 in the past and is often just denoted as R(t). The quantity (z + 1) can be considered as the amount of stretching during the intervening time for the light to travel from there to here. It is a curious relativistic effect that the velocity and distance remain finite approaching the velocity of light and the event horizon respectively even though the amount#### Figure 10a Cosmic Red Shift [view large image]

of stretching becomes infinity. The relativistic expression for z+1 (In terms of the recession velocity v) is: z + 1 = [(1+ v/c) / (1- v/c)] ^{1/2}= 1/R(t).

- From the observed value of the Hubble constant H
_{o}= 71 km/sec-Mpc for the current epoch, and by substituting the speed of light c for v in the Hubble law: v = H_{o}d, it follows that the rate of cosmic expansion would be exceeding the speed of light beyond a distance D_{H}= c/H_{0}= 10^{28}cm. However, there is no contradiction, because special relativity states that no signal can propagate faster than the speed of light, but the expansion of the universe cannot be used to propagate a signal although it can move with superluminal speed. We will not be able to see this region anyway because the event horizon is located right at the boundary (Figure 10b1), if we take the Hubble time T = 1/H_{0}= 13.7 billion years (from WMAP modeling) to be the age of the observable universe.#### Figure 10b1 Event Horizon

[view large image]

- From the observed value of the Hubble constant H
- The cosmological model in the last section is the oldest and over-simplified before the discovery of dark matter and dark energy. Since then a number of revised models have been investigated and can be summarized in a "Cosmic Triangle" (Figure 10b2). Its three sides are identified with the scale of the density parameters :
_{mass}(for all matter - dark and brayonic),_{dark energy}, and_{curv}(associated with the curvature of space). The location of the various models are depicted in Figure 10b2 and explained briefly below in order of antiquity. - There are various kinds of cosmic distance according to the way it is measured (see Distance Measures in Cosmology by David Hogg).
They have different dependence on the redshift z, and yield different limiting values as z . Table 03 below is a summary of all these distances. The formulas in terms of z are derived from the "matter only" flat-space model, which can be expressed in close (analytic) form. The more realistic solution including dark energy has to be evaluated numerically as shown in Figures 10c. Figure 10d depicts the lookback dis-tance and time etc pictorially (not to scale). The Hubble distance in Table 03 is defined as D _{H}= cT = 13.7x10^{9}ly#### Figure 10c Types of Cosmic Distance [view large image]

#### Figure 10d Lookback Distance and Lookback Time [view large image]

The comoving distance is the fundamental distance measure in cosmology since many others can be derived in terms of it. Within the framework of general relativity, it can be expressed as:

D_{C}= D_{H}_{}dz' / [_{M}(1+z')^{3}+_{k}(1+z')^{2}+_{}]^{1/2}

where_{M}= 8G_{0}/3 H_{0}^{2},_{k}= -kc^{2}/ H_{0}^{2}, and_{}= c^{2}/3 H_{0}^{2}.

Thus, a theoretical comoving distance can be computed as a function of the redshift z from the density parameters 's resulting in many cosmological models with different combinations of these parameters. Similarly, the lookback or light travel distance is given by:

D_{T}= D_{H}_{}dz' / (1+z')[_{M}(1+z')^{3}+_{k}(1+z')^{2}+_{}]^{1/2}

The difference between D_{C}and D_{T}is that the former is derived from the conformal time , while the latter is related to the regular time t. The conformal time is introduced because of a peculiar feature in cosmic expansion. It has been mentioned that the physical distance dx is related to the comoving distance dr by the formula dx = R(t)dr. It follows that the velocity dr/dt = (1/R)dx/dt, then for dx/dt = c, dr/dt > c when R < 1. The conformal time defined by dt = Rd is invented to address this problem of exceeding the velocity of light. Since by using the conformal time dr/d = dx/dt, it would not have the problem when dx/dt = c. In the comoving frame, the mutual recession of two objects is viewed as the expansion of the coordinate grids (see Figure 10d).

Table 03 lists the various types of cosmic distances together with some indications of their dependence on the red shift z assuming an all matter universe and for k = 0. More realistic derivation for D_{C}and D_{T}can be derived from Eq.(20f).

Type of Distance Definition Function of z

(all matter)Distance as z Comoving / Conformal (D _{C})Distance scale expanding with the universe at the current epoch 2[1-1/(1+z) ^{1/2}] D_{H}2D _{H}= 27.4x10^{9}ly

( = 47.2x10^{9}ly for more realistic model)Coordinate / Proper (D _{P})Distance scale expanding with the universe at redshift z D _{C}/(1+z)0 Angular Diameter (D _{A})Intrinsic linear size L / angular size D _{C}/(1+z)0 Luminosity (D _{L})Calculated from apparent brightness (1+z) D _{C}Light Travel / Lookback (D _{T})Light travel distance from source to here (2/3)[1-1/(1+z) ^{3/2}] D_{H}(2/3)D _{H}= 9.1x10^{9}ly

( = 13.7x10^{9}ly for more realistic model)Light Travel Time /

Lookback Time (t_{l})Time difference between source & here (2/3)[1-1/(1+z) ^{3/2}] T(2/3)T = 9.1x10 ^{9}yrs

( = 13.7x10^{9}yrs for more realistic model)#### Table 03 Types of Cosmic Distance

The diagram (Figure 10e) below lists some cosmic parameters as a function of the red shift z. It includes :

H - Hubble constant in km/sec-Mpc,

r-comov - distance (to us) according to Hubble's law in Mpc,

dm - density of matter in % of the total cosmic mass-energy,

age - age of the astronomical object in Gyr,

time - traveling time to reach us in Gyr,

size - angular size scaling in kpc/1^{"},

angle - angular size of 1 kpc astronomical object in arcseconds;

1 Gly = 1,000,000,000 light years = 9.461x10^{26}cm,

1 Mpc = 1,000,000 pc = 3,261,566 light years = 3.08568x10^{24}cm,

1 Gyr = 1,000,000,000 years.

The above data were computed with H#### Figure 10e Relation of Cosmic Parameters to the Red Shift z [view large image]

_{o}= 67.15 km/sec-Mpc,_{m}= 0.317,_{}= 0.683 according to the latest observation from ESA/Planck. The program to calculate cosmic models with various input data is available online known as "Cosmic Calculator".

- The diagram on the left of Figure 10f shows the Type Ia supernova raw data up to a redshift of about 1.5 (or down to 0.4 in term of the scale factor). After various adjustments for these observations, the data points agree remarkably with a theoretical curve as shown in the diagram on the right. This model is computed with matter-energy composition and spatial curvature estimated by WMAP.
#### Figure 10f Type Ia SN Data [view large image]

Note that (by flipping the curve to the left) Figure 10f represents a short segment of one of the cosmic models in Figure 10g. - The various data reductions include:
- Change redshift to scale factor.
- Correction for over estimate of luminosity distance by expansion-induced dimming of light.
- Conversion from coordinate distance to light travel time or lookback time.
- Average over multiple galaxies.

Note: For closed space the sum of the angles in a triangle would be greater than 180- Figure 10g shows the three types of universe - open, flat, and closed. Universes that are too far above the critical divide expand too fast for matter to condense into stars and galaxies; such universes therefore remain devoid of life. Those that fall too far below the critical divide collapse before stars have a chance to form. The shaded area indicates the range of cosmological expansions and epochs in which observers could evolve. By measuring the sum of the angles in a huge narrow triangle with one vertex on Earth and the other two at the points of emission of CMBR (see insert in Figure 10g), the WMAP data yields a value of 180
^{o}indicating that the universe is#### Figure 10g Types of Universe [view large image]

flat. Such conclusion is further supported by the acceleration of Type Ia supernovae, which imply a precise amount of dark energy needed to make the universe flat (with the total matter-energy density equals to the critical density). ^{o}; it is smaller than 180^{o}for open space; it equals to 180^{o}exactly for flat space.

While the Hubble's parameter H(t) depends on the time t, its value at the current epoch t

For the radiation (relativistic matter, i.e., particles with kinetic energy higher than the rest mass energy) dominated universe, the radiation pressure p = /3. Starting from the thermodynamic relation dU = -pdV, where U = V, and V is the volume of the system (the universe in this case), it can be shown that 1/R

1. Model B - This model includes only the baryonic matter. The universe seems to be open before 1970s, and nobody knew the cosmic expansion is accelerating. 2. OCDM - This is the model when cold dark matter was favored by observation in the 1970s. The "O" represents "Open" universe, which was still embraced by many astronomers. The cosmic age is constrained by the age of star cluster. 3. SCDM - It stands for "Standard Cold Dark Matter" with no dark energy but now they got a hint that the universe is flat. 4. CDM - This is the most current model constructed after the discovery of dark energy in the 1990s by observing the Type Ia supernova explosion. It fits all the astronomical observations including the CMBR. | |

## Figure 10b2 Cosmological Models [view large image] |

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