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Inflationary Theory

Field: Cosmology

A theory of exponential expansion of space in the early universe.

Sequence of Expressions

Let ϕ\phi be the scalar field (inflaton) and V(ϕ)V(\phi) be its potential energy density. The dynamics of ϕ\phi in a Friedmann-Lemaître-Robertson-Walker (FLRW) background are governed by the Klein-Gordon equation in curved spacetime:\n1−g∂μ(−ggμν∂νϕ)=−V′(ϕ)\frac{1}{\sqrt{-g}} \partial_\mu (\sqrt{-g} g^{\mu\nu} \partial_\nu \phi) = -V'(\phi)
Consider a homogeneous and isotropic universe described by the metric ds2=−c2dt2+a(t)2(dx2+dy2+dz2)ds^2 = -c^2 dt^2 + a(t)^2 (dx^2 + dy^2 + dz^2). The first Friedmann equation relates the Hubble parameter H=a˙/aH = \dot{a}/a to the total energy density ρ\rho and the spatial curvature kk: \n(a˙a)2=8πG3c2ρ−kc2a2\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3c^2} \rho - \frac{kc^2}{a^2}
Define the scale factor a(t)a(t) as the function describing the relative expansion of the spatial metric over cosmic time tt. It is normalized such that a(t0)=1a(t_0) = 1. The evolution of a(t)a(t) is governed by the Friedmann equations and relates the physical distance Dphys(t)D_{phys}(t) to the comoving distance Dcom(t)D_{com}(t) via:\nDphys(t)=a(t)Dcom(t)D_{phys}(t) = a(t) D_{com}(t)
Let dobsd_{obs} be the comoving distance between two points separated by the observed CMB temperature TCMBT_{CMB}. The particle horizon dH(t)d_H(t) at the time of decoupling tdect_{dec} is defined by the integral: dH(tdec)=a(tdec)×1a(tdec)×integraltinitialtdecdt′d_H(t_{dec}) = a(t_{dec}) \times \frac{1}{a(t_{dec})} \times \text{integral}_{t_{initial}}^{t_{dec}} dt' The Horizon Problem arises because the observed distance dobsd_{obs} for regions separated by angular scale θ\theta (corresponding to dobs≠0d_{obs} \neq 0) exceeds the causal horizon dH(tdec)d_H(t_{dec}) calculated using the standard Big Bang model, implying that these regions could not have been in causal contact prior to tdect_{dec}.
Consider the spatial curvature parameter kk in the Friedmann equation. The density parameter Ω\Omega is defined as Ω=ρρc\Omega = \frac{\rho}{\rho_c}. For the universe to be spatially flat, Ω\Omega must equal 1, implying k=0k=0. The evolution of the deviation from flatness, Ω−1\Omega - 1, is governed by the equation: dΩdt=−2a˙a(Ω−1)\frac{d\Omega}{dt} = -2\frac{\dot{a}}{a} \left( \Omega - 1 \right) For Ω\Omega to remain close to 1 over cosmological timescales, the initial value Ω(tinitial)\Omega(t_{initial}) must be fine-tuned such that ∣Ω(tinitial)−1∣≪1|\Omega(t_{initial}) - 1| \ll 1, a condition naturally satisfied by inflationary dynamics.
The Cosmic Microwave Background (CMB) radiation field I(ν,t)I(\nu, t) is characterized by a perfect blackbody spectrum at time tt, defined by the Planck function: I(ν,T)=2hν3c21ehν/(kBT)−1I(\nu, T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/(k_B T)} - 1} The temperature T(t)T(t) evolves with the scale factor a(t)a(t) according to the adiabatic cooling law: T(t)=T0(a0a(t))T(t) = T_0 \left( \frac{a_0}{a(t)} \right) where T0T_0 is the present-day temperature and a0a_0 is the scale factor today. The observed spectrum confirms this relationship, providing evidence for the thermal history of the early universe.
The density parameter Ω\Omega at any time tt is defined as the ratio of the total energy density ρ\rho to the critical density ρc(t)\rho_c(t): Ω(t)=ρ(t)ρc(t)=ρ(t)3H2/(8πG)\Omega(t) = \frac{\rho(t)}{\rho_c(t)} = \frac{\rho(t)}{3H^2/(8\pi G)} Assuming a perfect fluid description ρ(t)=∑iρi(t)\rho(t) = \sum_i \rho_i(t), the evolution of Ω\Omega is determined by the Friedmann equation: H2(t)=8πG3ρ(t)=8πG3∑iρi,0a(t)−3(1+wi)H^2(t) = \frac{8\pi G}{3} \rho(t) = \frac{8\pi G}{3} \sum_i \rho_{i,0} a(t)^{-3(1+w_i)} where wiw_i is the equation of state parameter for component ii.
Consider the linearized perturbation equations for the inflaton field ϕ\phi in an expanding background. The evolution of the perturbation δϕ\delta\phi is governed by a second-order differential equation. The Lyapunov exponents λ\lambda are defined by the asymptotic growth rate of the perturbation δϕ(t)\delta\phi(t): λ=lim⁡t→∞1tln⁡∣δϕ(t)δϕ(t0)∣\lambda = \lim_{t \to \infty} \frac{1}{t} \ln \left| \frac{\delta\phi(t)}{\delta\phi(t_0)} \right| For a general dynamical system defined by x˙=F(x,t)\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t), the exponents are found by analyzing the eigenvalues of the Jacobian matrix J=∂F∂x\mathbf{J} = \frac{\partial \mathbf{F}}{\partial \mathbf{x}} evaluated along the trajectory, quantifying the exponential divergence of nearby trajectories.
Let vv be the recession velocity of a galaxy, dd be its proper distance from the observer, and H0H_0 be the Hubble constant, defined as the derivative of the scale factor a(t)a(t) with respect to cosmic time tt evaluated at the present epoch (t0t_0). The relationship is given by:\nvc=H0d/c\frac{v}{c} = H_0 d / c
The inflationary epoch is characterized by a period of quasi-de Sitter expansion, where the scale factor a(t)a(t) grows exponentially with time tt. This rapid expansion is modeled by the condition a¨>0\ddot{a} > 0 and the slow-roll approximation, leading to:\na(t)∝eHinfta(t) \propto e^{H_{inf} t}