PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

CHAPTER 1. INTRODUCTION
the spatially flat case of our interest, reduce to the following two Friedmann equations:
H 2 = 8πG ρ,
3
(1.2a)
ä
a = −4πG (ρ+3p),
3
(1.2b)
where H = ȧ/a is the Hubble parameter and the overdots represent differentiation with
respect to the cosmic time coordinate t. The quantities ρ and p denote the total energy
density and the pressure of the constituents of the universe that are responsible for its
expansion. We can write the first of the above two Friedmann equations as
H 2
H 2 0
( a0
) 4 ( a0
) 3
= Ω r +Ωm +ΩΛ , (1.3)
a a 3
where a 0 denotes the scale factor today and H 0 is the Hubble constant, i.e. the present
value of the Hubble parameter which is usually expressed as h100kms −1 Mpc −1 . The
quantities Ω r and Ω m represent the dimensionless density parameters (which we shall
soon define) corresponding to radiation and matter (i.e. baryons plus the dark matter),
respectively. The quantity Ω Λ is the density parameter associated with the cosmological
constant, which is a specific type of dark energy, whose energy density is strictly a
constant, while its pressure is exactly the negative of its energy density. The density parameterΩ
S
for the speciesS is defined in terms of its density today, say,ρ 0 S , asΩ S = ρ0 S /ρ c,
where ρ c is the critical density that is given by ρ c = 8πG/(3H0 2 ). The radiation energy
density in the universe is dominated by the CMB, whose current temperature determines
the value of Ω r . It is straightforward to show that Ω r h 2 ≃ 2.56 × 10 −5 [6]. Observations
towards determining distances to a variety of galaxies by the Hubble Key Project point
to h = 0.73±0.05 [21, 22]. Note that Ω m = Ω b +Ω c , where Ω b and Ω c are the density parameters
corresponding to the baryons and the Cold Dark Matter (CDM). The measurements
of the primordial abundances of the light elements such as helium or deuterium
and their comparison with detailed models of nucleosynthesis during the radiation dominated
epoch lead to the constraint that Ω b h 2 ≃ 0.02 (see, for example, Refs. [23, 24]).
The determination of the luminosity distances to galaxies at high redshifts using type Ia
supernovae as standard candles by the Supernova Cosmology Project [25, 26] and the
Supernova Legacy Survey [27, 28, 29] suggest that Ω m ≃ 0.3 and Ω Λ ≃ 0.7. It should be
added that each of these constraints have been corroborated independently by other sets
of observations. These allow us to arrive at the concordant, background, ΛCDM model
of the universe, with about 70% of its energy density today being contributed by the cosmological
constant, roughly 25% by cold dark matter, and the rest by the baryons.
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