Relativity, Cosmology, and Time

Theory of Cosmic Inflation and Acceleration

Astronomical observations over the last 20 years have indicated that the universe has experienced two periods of acceleration (see Figure 10j). The first one was a very rapid expansion soon after the Big Bang. It is usually referred to as inflation. The dark energy accelerated expansion is gentler, and has occurred only recently in the current epoch. The inflation is explained by an as yet un-identified scalar field (inflaton) undergoing a phase transition. While there are three categories of theory for the accelerated expansion: modifications to general relativity perhaps with extra dimensions, a cosmological constant, and a universal evolving scalar field. None of these offers a satisfactory explanation (see a 2015 update on "Vacuum Energy Density").

Figure 10j Cosmic Inflation [view large image]

The followings provide a mathematical description of cosmic inflation and expansion without invoking a detailed mechanism.

• If we assume a homogeneous and isotropic universe, with the total mass to be constant as expressed in Eq.(18c), then Eq.(18a) can be recast into a form that shows the acceleration / deceleration explicitly:
  
  (d²R/dt²) / R = - 4G / 3 ---------- (20c)
  
  This formula shows that the matter dominated universe in general would experience deceleration as the term on the right is always negative.


• In the presence of other forms of matter-energy, there would be a pressure term p in the the equation of continuity :
  
  + 3[(dR/dt)/R]( + p/c²) = 0. Then Eq.(20c) becomes :
  
  (d²R/dt²) / R = - (4G / 3) ( + 3p/c²) --------- (20d)


• The matter-energy density and the pressure p are usually related by the equation of state, which can be expressed as p = wc², thus Eq.(20d) can be rewritten as:
  
  (d²R/dt²) / R = - (4G / 3) (1 + 3w) ---------- (20e)
  
  

  For relativistic matter, w = 1/3. For non-relativistic matter, w = 0. For the de Sitter universe, w = -1. The expansion of the universe will accelerate for w < -1/3, when the right-hand side in Eq.(20e) becomes positive. The negative pressure is a characteristic of expansion under constant density as shown by a simple example. It is also known as false vacuum. In the simplest version of the inflationary paradigm a single scalar field (inflaton) dominates the energy density of the universe. To achieve the acceleration condition w < -1/3 and the observed properties of our universe, the inflation must evolve slowly such that the potential energy dominates over the kinetic energy for a sufficient part of the inflation. Figure 10ka shows a theory (the old one) that doesn't work because the scalar field evolves rapidly. While the newer theory is just right creating an universe as we see it today. This

  Figure 10ka Theories of Inflation [view large image]

  scenario remains ambiguous as the precise form of the scalar field is unknown, and there is still nagging doubt about the occurrence of inflation at the early epoch of the universe.

• Another way to achieve acceleration (in later cosmic epoch) can be derived from the cosmological constant as shown in Eq.(20a). Assuming constant mass (including both baryonic and dark matter) within the expanding universe, and k=0 Eq.(20a) can be simplified to:
  
  dR/dt = R(_m/R³ + )^1/2,
  
  which can be recast to express the cosmic acceleration explicitly :
  
  (d²R/dt²) = R[- _m/(2R³) + ] ---------- (20f),
  
  where the time is in unit of 1/H₀=13.7x10⁹ years. This is a very simple second order differential equation suitable for numerical calculation on a home computer using the Basic computer language. For example, the effect of the density parameters 's on the age of the universe can be estimated by running the computer program with different combinations of the parameters.
  
  

  Even without solving the equation numerically, just by equating this equation to zero, i.e., at the moment when deceleration reverted to acceleration, yields the change over epoch at z = (1-R)/R = 0.8 (or about 6.9 billion years after the Big Bang) for m = 0.26, and = 0.74. Figure 10kb shows the evolution of the distribution of cosmic energy density among ordinary matter, the dark matter, and dark energy.

  Figure 10kb Evolution of Cosmic Energy Density Distribution

  
  Effort to identify the cosmological constant with the vacuum energy of the various quantum fields is not very successful. The simplest versions of quantum theory predict far too much energy - 10¹²⁰ higher than the observed value by one estimate. See "Vacuum Energy Density" for a 2015 update.


• The explanation of the current acceleration in term of an evolving scalar field presents a tremendous hierarchy problem. The very high energy (10¹⁶ Gev) associated with the scalar field in the inflation period requires an unreasonable amount of fine-tuning to account for the extraordinarily low energy (10^-48 Gev) of the scalar field in the current epoch.


• A different explanation was proposed by a research team in 2005, they links the dark energy acceleration to the gigantic ripples in space-time created in the epoch of early inflation. In effect, it is like hobbling amid a huge undulating wave in the ocean, you don't actually see the wave (the wavelength is too long) - all you feel is the swell moving up and down.

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