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Observations show that the universe is fairly homogeneous and isotropic at scales larger than about 150 h-1 Mpc where 1 Mpc appeq 3 × 1024 cm and h approx 0.7 is a parameter related to the expansion rate of the universe. The conventional - and highly successful - approach to cosmology separates the study of large scale (l gtapprox 150 h-1 Mpc) dynamics of the universe from the issue of structure formation at smaller scales. The former is modeled by a homogeneous and isotropic distribution of energy density; the latter issue is addressed in terms of gravitational instability which will amplify the small perturbations in the energy density, leading to the formation of structures like galaxies. In such an approach, the expansion of the background universe is described by the metric (We shall use units with with c = 1 throughout, unless otherwise specified):

Equation 1 (1)

with Sk(chi) = (sinchi, chi, sinhchi) for the three values of the label k = (1,0,-1). The function a(t) is governed by the equations:

Equation 2 (2)

The first one relates expansion rate of the universe to the energy density rho and k = 0, ± 1 is a parameter which characterizes the spatial curvature of the universe. The second equation, when coupled with the equation of state p = p(rho) which relates the pressure p to the energy density, determines the evolution of energy density rho = rho(a) in terms of the expansion factor of the universe. In particular if p = wrho with (at least, approximately) constant w then, rho propto a-3(1+w) and (if we further assume k = 0, which is strongly favoured by observations) the first equation in Eq.(2) gives a propto t2/[3(1+w)]. We will also often use the redshift z(t), defined as (1 + z) = a0 / a(t) where the subscript zero denotes quantities evaluated at the present moment. in a k=0 universe, we can set a0 = 1 by rescaling the spatial coordinates.

It is convenient to measure the energy densities of different components in terms of a critical energy density (rhoc) required to make k = 0 at the present epoch. (Of course, since k is a constant, it will remain zero at all epochs if it is zero at any given moment of time.) From Eq.(2), it is clear that rhoc = 3H02 / 8pi G where H0 ident (dot{a} / a)0 - called the Hubble constant - is the rate of expansion of the universe at present. Numerically

Equation 3 Equation 3 Equation 3
Equation 3 Equation 3 (3)

The variables Omegai ident rhoi / rhoc will give the fractional contribution of different components of the universe (i denoting baryons, dark matter, radiation, etc.) to the critical density. Observations then lead to the following results:

(1) Our universe has 0.98 ltapprox Omegatot ltapprox 1.08. The value of Omegatot can be determined from the angular anisotropy spectrum of the cosmic microwave background radiation (CMBR; see Section 6) and these observations (combined with the reasonable assumption that h > 0.5) show [1] that we live in a universe with critical density, so that k = 0.

(2) Observations of primordial deuterium produced in big bang nucleosynthesis (which took place when the universe was about few minutes in age) as well as the CMBR observations show [2] that the total amount of baryons in the universe contributes about OmegaB = (0.024 ± 0.0012) h-2. Given the independent observations [3] which fix h = 0.72 ± 0.07, we conclude that OmegaB cong 0.04-0.06. These observations take into account all baryons which exist in the universe today irrespective of whether they are luminous or not. Combined with previous item we conclude that most of the universe is non-baryonic.

(3) Host of observations related to large scale structure and dynamics (rotation curves of galaxies, estimate of cluster masses, gravitational lensing, galaxy surveys ..) all suggest [4] that the universe is populated by a non-luminous component of matter (dark matter; DM hereafter) made of weakly interacting massive particles which does cluster at galactic scales. This component contributes about OmegaDM cong 0.20-0.35 and has the simple equation of state pDM approx 0. The second equation in Eq.(2), then gives rhoDM propto a-3 as the universe expands which arises from the evolution of number density of particles: rho = nmc2 propto n propto a-3.

(4) Combining the last observation with the first we conclude that there must be (at least) one more component to the energy density of the universe contributing about 70% of critical density. Early analysis of several observations [5] indicated that this component is unclustered and has negative pressure. This is confirmed dramatically by the supernova observations (see Ref. [6]; for a critical look at the current data, see Ref. [7]). The observations suggest that the missing component has w = p / rho ltapprox -0.78 and contributes OmegaDE cong 0.60-0.75. The simplest choice for such dark energy with negative pressure is the cosmological constant which is a term that can be added to Einstein's equations. This term acts like a fluid with an equation of state pDE = -rhoDE; the second equation in Eq.(2), then gives rhoDE = constant as universe expands.

(5) The universe also contains radiation contributing an energy density OmegaR h2 = 2.56× 10-5 today most of which is due to photons in the CMBR. The equation of state is pR = (1/3) rhoR; the second equation in Eq.(2), then gives rhoR propto a-4. Combining it with the result rhoR propto T4 for thermal radiation, it follows that T propto a-1. Radiation is dynamically irrelevant today but since (rhoR / rhoDM) propto a-1 it would have been the dominant component when the universe was smaller by a factor larger than OmegaDM / OmegaR appeq 4× 104 OmegaDM h2.

(6) Taking all the above observations together, we conclude that our universe has (approximately) OmegaDE appeq 0.7, OmegaDM appeq 0.26, OmegaB appeq 0.04, OmegaR appeq 5× 10-5. All known observations are consistent with such an - admittedly weird - composition for the universe.

Using rhoNR propto a-3, rhoR propto a-4 and rhoDE = constant we can write Eq.(2) in a convenient dimensionless form as

Equation 4 (4)

where tau = H0 t, a = a0 q(tau), OmegaNR = OmegaB + OmegaDM and

Equation 5 (5)

This equation has the structure of the first integral for motion of a particle with energy E in a potential V(q). For models with Omega = OmegaNR + OmegaDE = 1, we can take E = 0 so that (dq / dtau) = (V(q))1/2. Based on the observed composition of the universe, we can identify three distinct phases in the evolution of the universe when the temperature is less than about 100 GeV. At high redshifts (small q) the universe is radiation dominated and Img src="../../New_Gifs/qdot.gif" alt="dot{q}"> is independent of the other cosmological parameters. Then Eq.(4) can be easily integrated to give a(t) propto t1/2 and the temperature of the universe decreases as T propto t-1/2. As the universe expands, a time will come when (t = teq, a = aeq and z = zeq, say) the matter energy density will be comparable to radiation energy density. For the parameters described above, (1 + zeq) = OmegaNR / OmegaR appeq 4× 104 OmegaDM h2. At lower redshifts, matter will dominate over radiation and we will have a propto t2/3 until fairly late when the dark energy density will dominate over non relativistic matter. This occurs at a redshift of zDE where (1 + zDE) = (OmegaDE / OmegaNR)1/3. For OmegaDE approx 0.7, OmegaNR approx 0.3, this occurs at zDE approx 0.33. In this phase, the velocity dot{q} changes from being a decreasing function to an increasing function leading to an accelerating universe. In addition to these, we believe that the universe probably went through a rapidly expanding, inflationary, phase very early when T approx 1014 GeV; we will say more about this in Section 7. (For a textbook description of these and related issues, see e.g. Ref. [8].)

Before we conclude this section, we will briefly mention some key aspects of the background cosmology described by a Friedmann model.

(a) The metric in Eq.(1) can be rewritten using the expansion parameter a or the redshift z = (a0 / a)-1 -1 as the time coordinate in the form

Equation 6 (6)

This form clearly shows that the only dynamical content of the metric is encoded in the function H(a) = (dot{a} / a). An immediate consequence is that any observation which is capable of determining the geometry of the universe can only provide - at best - information about this function.

(b) Since cosmological observations usually use radiation received from distant sources, it is worth reviewing briefly the propagation of radiation in the universe. The radial light rays follow a trajectory given by

Equation 7 (7)

if the photon is emitted at rem at the redshift z and received here today. Two other quantities closely related to rem(z) are the luminosity distance, dL, and the angular diameter distance dA. If we receive a flux F from a source of luminosity L, then the luminosity distance is defined via the relation F ident L / 4pi dL2(z) . If an object of transverse length l subtends a small angle theta, the angular diameter distance is defined via (l = theta dA). Simple calculation shows that:

Equation 8 (8)

(c) As an example of determining the spacetime geometry of the universe from observations, let us consider how one can determine a(t) from the observations of the luminosity distance. It is clear from the first equation in Eq. (8) that

Equation 9 (9)

where the last form is valid for a k = 0 universe. If we determine the form of dL(z) from observations - which can be done if we can measure the flux F from a class of sources with known value for luminosity L - then we can use this relation to determine the evolutionary history of the universe and thus the dynamics.

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