Cosmologia

Introduction to Cosmology
Thus, in the early stages of our universe, when radiation was strongly dominant, the energy density, measured in units of the Planck density (EP / 3 ∼ P 3 × 10133 eV m−3 ), was comparable to one over the square of the cosmic time, measured in units of the Planck time (tP ∼ 5×10−44 s). Using the blackbody relation between energy density and temperature, given inequations (2.26) and (2.27), we may assign a temperature to a universe dominated by blackbody radiation: T (t) = 45 32π 2
1/4

TP

t tP

−1/2

≈ 0.61TP

t tP

−1/2

.

(5.68)

Here TP is the Planck temperature, TP = 1.4 × 1032 K. The mean energy per photon in a radiation-dominated universe is then Emean (t) ≈ 2.70kT (t) ≈ 1.66EP t tP
−1/2

,

(5.69)

and the number density ofphotons is (combining equations 5.67 and 5.69) εr (t) 0.057 t n(t) = ≈ 3 Emean (t) tP P
−3/2

.

(5.70)

Note that in a flat, radiation-only universe, as t → 0, εr → ∞ (equation 5.67). Thus, at the instant t = 0, the energy density of our own universe (well approximated as a flat, radiation-only model in its early stages) was infinite, according to this analysis; this infinite energy density wasprovided by an infinite number density of photons (equation 5.70), each of infinite energy (equation 5.69). Should we take these infinities seriously? Not really, since the assumptions of general relativity, on which the Friedmann equation is based, break down at t ≈ tP . Thus, extrapolating the results of this chapter to times earlier than the Planck time is not physically justified. Why can’t generalrelativity be used at times earlier than the Planck time? General relativity is a classical theory; that is, it does not take into account the effects of quantum mechanics. In cosmological contexts, general relativity assumes that the energy content of the universe is smooth down to arbitrarily

5.6. LAMBDA ONLY

97

small scales, instead of being parceled into individual quanta. As long asa radiation-dominated universe has many, many quanta, or photons, within a horizon distance, then the approximation of a smooth, continuous energy density is justifiable, and we may safely use the results of general relativity. However, if there are only a few photons within the visible universe, then quantum mechanical effects must be taken into account, and the classical results of generalrelativity no longer apply. In a flat, radiation-only universe, the horizon distance grows linearly with time: dhor (t) = 2ct = 2
P

t tP t tP

,

(5.71)

so the volume of the visible universe at time t is Vhor (t) = 4π 3 d ≈ 34 3 hor
3 P 3

.

(5.72)

Combining equations (5.72) and (5.70), we find that the number of photons inside the horizon at time t is N (t) = Vhor (t)n(t) ≈ 1.9 t tP3/2

.

(5.73)

The quantization of the universe can no longer be ignored when N (t) ≈ 1, equivalent to a time t ≈ 0.7tP . In order to accurately described the universe at its very earliest stages, prior to the Planck time, a theory of quantum gravity is needed. Unfortunately, a complete theory of quantum gravity does not yet exist. Consequently, in this book, I will not deal with timesearlier than the Planck time, t ∼ tP ∼ 5 × 10−44 s, when the number density of photons was n ∼ −3 ∼ P 2 × 10104 m−3 , and the mean photon energy was Emean ∼ EP ∼ 1 × 1028 eV.

5.6

Lambda only

As seen in Section 5.3, a spatially flat, single-component universe with w = −1 has a power-law dependence of scale factor on time: a ∝ t2/(3+3w) . (5.74) Now, for the sake of completeness, consider thecase with w = −1; that is, a universe in which the energy density is contributed by a cosmological

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CHAPTER 5. SINGLE-COMPONENT UNIVERSES

constant Λ.6 For such a flat, lambda-dominated universe, the Friedmann equation takes the form 8πGεΛ 2 a , (5.75) a2 = ˙ 3c2 where εΛ is constant with time. This equation can be rewritten in the form a = H0 a , ˙ where 8πGεΛ 1/2 . H0 = 3c2 The...