Inflaton Field Dynamics
The inflaton is a hypothetical scalar field $\phi(t, \mathbf{x})$ whose vacuum energy drives exponential expansion in the early universe. In this chapter we derive the complete dynamical equations governing its evolution — the Klein-Gordon equation in an expanding FLRW background, the energy-momentum tensor, the modified Friedmann equations, and the phase-space structure that reveals why inflation is an attractor.
1. Introduction: Scalar Fields in Cosmology
Scalar fields are the simplest type of quantum field — they carry no spin and transform trivially under Lorentz transformations. Despite this simplicity, scalar fields play a central role in modern physics: the Higgs field discovered at the LHC in 2012 is a scalar, and the inflaton that drives cosmic inflation is modeled as a scalar field evolving in a potential $V(\phi)$.
In classical mechanics, a field $\phi(t, \mathbf{x})$ is described by a Lagrangian density $\mathcal{L}$ that depends on $\phi$ and its derivatives. For a real scalar field minimally coupled to gravity, the Lagrangian density takes the canonical form:
$\mathcal{L} = \frac{1}{2}g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi - V(\phi)$
The first term is the kinetic energy (generalized to curved spacetime through the metric $g^{\mu\nu}$), while $V(\phi)$ is the potential energy. Physically, the field's dynamics are analogous to a ball rolling in the potential $V(\phi)$, but with the crucial addition of a friction term arising from the expansion of the universe.
Why scalars? In a homogeneous and isotropic universe, vector fields would single out a preferred direction, violating isotropy. Tensor fields (beyond the metric) introduce additional complications. A scalar field naturally respects the symmetries of the FLRW metric and provides the simplest realization of a slowly-varying vacuum energy that can drive exponential expansion.
Key Assumptions
- ●Homogeneity: The inflaton depends only on time at the background level: $\phi = \phi(t)$. Spatial perturbations $\delta\phi(t,\mathbf{x})$ are treated later.
- ●Minimal coupling: The field couples to gravity only through the metric — no direct $R\phi^2$ coupling (the non-minimal case is discussed in the Higgs inflation chapter).
- ●Classical treatment: We treat $\phi$ as a classical field. Quantum fluctuations around this background generate primordial perturbations (Chapter 4).
- ●Single field: One scalar field dominates the energy budget during inflation. Multi-field extensions exist but are not needed here.
With these assumptions, the full action for gravity plus the inflaton is:
$S = \int d^4x\,\sqrt{-g}\left[\frac{M_P^2}{2}R + \frac{1}{2}g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi - V(\phi)\right]$
where $M_P = (8\pi G)^{-1/2} \approx 2.4 \times 10^{18}$ GeV is the reduced Planck mass and $R$ is the Ricci scalar. We now derive the field equation by varying this action with respect to $\phi$.
2. Derivation: Klein-Gordon Equation in FLRW Spacetime
We begin with the scalar field action in curved spacetime:
$S_\phi = \int d^4x\,\sqrt{-g}\left[\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi - V(\phi)\right]$
Step 1: Vary the Action
To find the equation of motion, we require $\delta S_\phi / \delta\phi = 0$. Under a variation $\phi \to \phi + \delta\phi$:
$\delta S_\phi = \int d^4x\,\sqrt{-g}\left[g^{\mu\nu}\partial_\mu\phi\,\partial_\nu(\delta\phi) - V'(\phi)\,\delta\phi\right]$
where $V'(\phi) \equiv dV/d\phi$. The first term involves derivatives of $\delta\phi$, so we integrate by parts. In curved spacetime, integration by parts uses the covariant divergence:
$\int d^4x\,\sqrt{-g}\,g^{\mu\nu}\partial_\mu\phi\,\partial_\nu(\delta\phi) = -\int d^4x\,\sqrt{-g}\,\frac{1}{\sqrt{-g}}\partial_\mu\!\left(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\phi\right)\delta\phi$
where we dropped the boundary term (assuming $\delta\phi$ vanishes at the boundaries). The combination $\frac{1}{\sqrt{-g}}\partial_\mu(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\phi)$ is precisely the covariant d'Alembertian $\Box\phi \equiv \nabla_\mu\nabla^\mu\phi$.
Step 2: General Covariant Klein-Gordon Equation
Setting $\delta S_\phi = 0$ and using the arbitrariness of $\delta\phi$, we obtain:
$\Box\phi + V'(\phi) = 0$
General covariant Klein-Gordon equation
This is the curved-spacetime generalization of the flat-space Klein-Gordon equation. All the effects of gravity are encoded in the d'Alembertian operator $\Box$.
Step 3: Specialize to the FLRW Metric
The flat FLRW metric in Cartesian coordinates is:
$ds^2 = -dt^2 + a^2(t)\left(dx^2 + dy^2 + dz^2\right)$
The metric determinant is $g = -a^6$, so $\sqrt{-g} = a^3$. The inverse metric has components $g^{00} = -1$ and $g^{ij} = a^{-2}\delta^{ij}$. For a homogeneous field $\phi = \phi(t)$, spatial derivatives vanish. The d'Alembertian becomes:
$\Box\phi = \frac{1}{\sqrt{-g}}\partial_\mu\!\left(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\phi\right) = \frac{1}{a^3}\frac{d}{dt}\!\left(a^3 \cdot (-1) \cdot \dot{\phi}\right)$
Expanding the time derivative:
$\Box\phi = -\frac{1}{a^3}\frac{d}{dt}\!\left(a^3\dot{\phi}\right) = -\frac{1}{a^3}\!\left(3a^2\dot{a}\,\dot{\phi} + a^3\ddot{\phi}\right) = -\ddot{\phi} - 3\frac{\dot{a}}{a}\dot{\phi}$
Recognizing $H \equiv \dot{a}/a$ as the Hubble parameter:
$\Box\phi = -\ddot{\phi} - 3H\dot{\phi}$
Step 4: The Inflaton Equation of Motion
Substituting into $\Box\phi + V'(\phi) = 0$:
$\ddot{\phi} + 3H\dot{\phi} + V'(\phi) = 0$
Klein-Gordon equation in FLRW spacetime
Physical Interpretation of Each Term
- ●$\ddot{\phi}$ — Acceleration: The "inertia" of the field, analogous to $m\ddot{x}$ in Newtonian mechanics.
- ●$3H\dot{\phi}$ — Hubble Friction: This is the crucial new term. The factor of 3 arises from three spatial dimensions expanding. As the universe expands, the field's kinetic energy is redshifted away, exactly like the momentum of a particle in expanding space. This friction term is what enables slow roll: even on a steep potential, the field can roll slowly if $H$ is large enough.
- ●$V'(\phi)$ — Potential Gradient: The "force" driving the field downhill, analogous to $-dV/dx$ in mechanics.
The analogy is precise: the Klein-Gordon equation is identical to a damped harmonic oscillator with time-dependent damping coefficient $\gamma = 3H(t)$. During inflation,$H$ is nearly constant and large, providing strong friction that slows the field's descent down the potential.
3. Derivation: Energy-Momentum Tensor of the Scalar Field
The energy-momentum tensor is obtained by varying the matter action with respect to the metric:
$T_{\mu\nu} = -\frac{2}{\sqrt{-g}}\frac{\delta S_\phi}{\delta g^{\mu\nu}}$
Step 1: Variation of the Action with Respect to the Metric
The matter action is $S_\phi = \int d^4x\,\sqrt{-g}\left[\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi - V(\phi)\right]$. We need two standard variational identities:
$\frac{\delta(\sqrt{-g})}{\delta g^{\mu\nu}} = -\frac{1}{2}\sqrt{-g}\,g_{\mu\nu}$
$\frac{\delta g^{\alpha\beta}}{\delta g^{\mu\nu}} = \frac{1}{2}\!\left(\delta^\alpha_\mu\delta^\beta_\nu + \delta^\alpha_\nu\delta^\beta_\mu\right)$
Applying these to the integrand:
$\frac{\delta}{\delta g^{\mu\nu}}\!\left[\sqrt{-g}\left(\frac{1}{2}g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi - V(\phi)\right)\right]$
$= \sqrt{-g}\left[\frac{1}{2}\partial_\mu\phi\,\partial_\nu\phi - \frac{1}{2}g_{\mu\nu}\!\left(\frac{1}{2}g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi - V(\phi)\right)\right]$
Step 2: The Full Energy-Momentum Tensor
Multiplying by $-2/\sqrt{-g}$, we obtain the energy-momentum tensor for a scalar field:
$T_{\mu\nu} = \partial_\mu\phi\,\partial_\nu\phi - g_{\mu\nu}\!\left[\frac{1}{2}g^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi - V(\phi)\right]$
Step 3: Extract Energy Density and Pressure
For a homogeneous field $\phi(t)$ in FLRW, only $\partial_0\phi = \dot{\phi}$ is nonzero. The energy-momentum tensor takes the perfect fluid form $T^{\mu}_{\ \nu} = \text{diag}(-\rho_\phi, p_\phi, p_\phi, p_\phi)$.
The energy density is $\rho_\phi = -T^0_{\ 0}$. Computing:
$T_{00} = \dot{\phi}^2 - g_{00}\!\left[\frac{1}{2}(-1)\dot{\phi}^2 - V(\phi)\right] = \dot{\phi}^2 + \frac{1}{2}\dot{\phi}^2 + V(\phi) - \dot{\phi}^2$
Using $g_{00} = -1$, the $T_{00}$ component gives:
$\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V(\phi)$
Inflaton energy density = kinetic + potential
For the spatial components, using $g_{ij} = a^2\delta_{ij}$:
$T_{ij} = 0 - a^2\delta_{ij}\!\left[-\frac{1}{2}\dot{\phi}^2 - V(\phi)\right] = a^2\delta_{ij}\!\left[\frac{1}{2}\dot{\phi}^2 - V(\phi)\right]$
Since $p_\phi = T^i_{\ i}/(3) = T_{ij}g^{ij}/(3)$, using $g^{ij} = a^{-2}\delta^{ij}$:
$p_\phi = \frac{1}{2}\dot{\phi}^2 - V(\phi)$
Inflaton pressure = kinetic - potential
Step 4: Equation of State
The equation of state parameter $w$ is defined as:
$w \equiv \frac{p_\phi}{\rho_\phi} = \frac{\frac{1}{2}\dot{\phi}^2 - V(\phi)}{\frac{1}{2}\dot{\phi}^2 + V(\phi)}$
Limiting Cases
- ●Potential domination $V(\phi) \gg \frac{1}{2}\dot{\phi}^2$: $w \to \frac{-V}{+V} = -1$. This is a cosmological constant — exactly the condition needed for accelerated expansion (inflation).
- ●Kinetic domination $\frac{1}{2}\dot{\phi}^2 \gg V(\phi)$: $w \to +1$. This is called "kination" or "stiff matter." The energy density scales as $\rho \propto a^{-6}$, faster than radiation.
- ●Oscillating field: When $\phi$ oscillates rapidly in a quadratic potential, the time-averaged equation of state is $\langle w \rangle = 0$, mimicking pressureless matter (dust). This is crucial for reheating.
The condition for inflation is $w < -1/3$, which requires the potential energy to dominate:$\dot{\phi}^2 < V(\phi)$. This is the essential physical requirement that the slow-roll approximation formalizes.
4. Derivation: Friedmann Equations with the Inflaton
Einstein's field equations $G_{\mu\nu} = T_{\mu\nu}/M_P^2$ applied to the FLRW metric with the inflaton as the sole source of energy-momentum yield the Friedmann equations.
Step 1: First Friedmann Equation
The $(0,0)$ component of Einstein's equations gives the first Friedmann equation. For a flat universe ($k=0$) with energy density $\rho_\phi$:
$H^2 = \frac{\rho_\phi}{3M_P^2} = \frac{1}{3M_P^2}\!\left[\frac{1}{2}\dot{\phi}^2 + V(\phi)\right]$
First Friedmann equation (energy constraint)
Step 2: Second Friedmann Equation (Raychaudhuri)
The spatial trace of Einstein's equations gives the acceleration equation. Combining the $(0,0)$ and $(i,j)$ components:
$\dot{H} = -\frac{\rho_\phi + p_\phi}{2M_P^2} = -\frac{\dot{\phi}^2}{2M_P^2}$
This is a remarkable result: $\dot{H}$ depends only on the kinetic energy $\dot{\phi}^2$. Since $\dot{\phi}^2 \geq 0$, we always have $\dot{H} \leq 0$ — the Hubble rate can only decrease (or remain constant in pure de Sitter space where $\dot{\phi} = 0$).
$\dot{H} = -\frac{\dot{\phi}^2}{2M_P^2}$
Raychaudhuri equation for inflaton-dominated universe
Step 3: Condition for Inflation
Inflation means accelerated expansion: $\ddot{a} > 0$. Using $H = \dot{a}/a$:
$\frac{\ddot{a}}{a} = \dot{H} + H^2 = H^2\!\left(1 + \frac{\dot{H}}{H^2}\right) = H^2(1 - \epsilon)$
where we defined the first Hubble slow-roll parameter $\epsilon \equiv -\dot{H}/H^2$. Thus $\ddot{a} > 0$ requires $\epsilon < 1$. Using the Friedmann equations:
$\epsilon = \frac{\dot{\phi}^2}{2M_P^2 H^2} = \frac{3\dot{\phi}^2}{\dot{\phi}^2 + 2V(\phi)}$
The condition $\epsilon < 1$ translates to:
$\ddot{a} > 0 \quad\Longleftrightarrow\quad \dot{\phi}^2 < V(\phi)$
Condition for accelerated expansion
Step 4: Continuity Equation and Consistency with Klein-Gordon
The conservation of the energy-momentum tensor $\nabla_\mu T^{\mu\nu} = 0$ gives the continuity equation in FLRW:
$\dot{\rho}_\phi + 3H(\rho_\phi + p_\phi) = 0$
Continuity equation
Let us verify this is consistent with the Klein-Gordon equation. Substituting $\rho_\phi = \frac{1}{2}\dot{\phi}^2 + V$ and $p_\phi = \frac{1}{2}\dot{\phi}^2 - V$:
$\dot{\rho}_\phi = \dot{\phi}\ddot{\phi} + V'(\phi)\dot{\phi}$
$\rho_\phi + p_\phi = \dot{\phi}^2$
$\Rightarrow\quad \dot{\phi}\!\left[\ddot{\phi} + V'(\phi)\right] + 3H\dot{\phi}^2 = 0$
Dividing by $\dot{\phi}$ (assuming $\dot{\phi} \neq 0$), we recover exactly the Klein-Gordon equation: $\ddot{\phi} + 3H\dot{\phi} + V'(\phi) = 0$. This confirms that the Friedmann equations and the Klein-Gordon equation form a consistent system — any two of the three equations imply the third.
5. Derivation: Phase Space Analysis
The inflaton equation of motion can be rewritten as a first-order dynamical system in the phase space $(\phi, \dot{\phi})$. This reveals the attractor nature of the slow-roll solution and explains why inflation occurs for generic initial conditions.
Step 1: Rewrite as a Dynamical System
Define $\pi \equiv \dot{\phi}$ as the field momentum. The Klein-Gordon equation becomes the system:
$\dot{\phi} = \pi$
$\dot{\pi} = -3H(\phi, \pi)\,\pi - V'(\phi)$
where $H$ itself depends on $\phi$ and $\pi$ through the Friedmann equation:
$H(\phi, \pi) = \frac{1}{\sqrt{3}\,M_P}\sqrt{\frac{1}{2}\pi^2 + V(\phi)}$
This is a closed, autonomous system: the right-hand sides depend only on $(\phi, \pi)$, not explicitly on time.
Step 2: The Slow-Roll Attractor
The slow-roll approximation neglects $\ddot{\phi}$ in the Klein-Gordon equation, giving:
$\pi_\text{SR}(\phi) \approx -\frac{V'(\phi)}{3H(\phi)}$
where in the slow-roll regime $H \approx \sqrt{V/(3M_P^2)}$. This defines a one-dimensional curve in the $(\phi, \pi)$ plane — the slow-roll attractor.
To show this curve is an attractor, consider a perturbation $\delta\pi$ away from the slow-roll solution. The linearized equation for $\delta\pi$ is:
$\delta\dot{\pi} \approx -3H\,\delta\pi$
This has the solution $\delta\pi \propto e^{-3Ht}$, meaning perturbations decay exponentially on a timescale $\tau \sim 1/(3H)$. Since during inflation$H$ is large and nearly constant, any initial velocity is rapidly damped toward the slow-roll trajectory.
Step 3: Phase Portrait for $V = m^2\phi^2/2$
For the quadratic potential $V(\phi) = \frac{1}{2}m^2\phi^2$, the system becomes:
$\dot{\phi} = \pi$
$\dot{\pi} = -\frac{\sqrt{3}\,\pi}{M_P}\sqrt{\frac{1}{2}\pi^2 + \frac{1}{2}m^2\phi^2} - m^2\phi$
The slow-roll curve is $\pi_\text{SR} \approx -m\phi/\sqrt{3}$ (for $\phi > 0$, $M_P = 1$). The phase portrait has the following structure:
Phase Portrait Features
- ●Attractor: Regardless of initial conditions, trajectories rapidly converge to the slow-roll curve. High kinetic energy is quickly damped by Hubble friction.
- ●Slow-roll region: On the attractor, the field slowly rolls toward $\phi = 0$ while the universe inflates exponentially.
- ●Oscillation region: Near $\phi = 0$, the Hubble friction decreases and the field oscillates. In the phase portrait, trajectories spiral inward toward the origin, losing energy to expansion.
- ●Forbidden region: The Friedmann constraint $\frac{1}{2}\pi^2 + V(\phi) \geq 0$ is automatically satisfied for $V \geq 0$. There is no forbidden region for positive-definite potentials.
The attractor behavior is the dynamical explanation for why inflation does not require fine-tuned initial conditions for the field velocity — only for the field value (it must start sufficiently far from the minimum to generate enough e-folds).
6. Applications
de Sitter Space as an Exact Solution
When the inflaton sits at the minimum of its potential with $\dot{\phi} = 0$ and$V(\phi) = V_0 = \text{const}$, the Friedmann equation becomes:
$H^2 = \frac{V_0}{3M_P^2} = \text{const} \quad\Rightarrow\quad a(t) = a_0\,e^{Ht}$
This is de Sitter space — the maximally symmetric solution of Einstein's equations with a positive cosmological constant. It represents eternal exponential expansion with$w = -1$ exactly. Real inflation is "quasi-de Sitter": the inflaton rolls slowly, so $H$ decreases gradually and $w \gtrsim -1$.
The Inflaton as an Effective Description
The inflaton field is an effective low-energy description. In the context of quantum field theory, it represents the lightest degree of freedom relevant during inflation. All heavier fields (with masses $m \gg H$) are frozen and can be integrated out, leaving an effective potential $V(\phi)$ for the inflaton. This is why single-field inflation is such a robust framework: it follows from an effective field theory argument regardless of the UV completion.
Connection to Particle Physics
- ●Grand Unified Theories (GUTs): The original proposal by Guth used a first-order phase transition at the GUT scale ($\sim 10^{16}$ GeV). The inflaton was identified with the GUT Higgs field driving the symmetry breaking $SU(5) \to SU(3) \times SU(2) \times U(1)$.
- ●Standard Model Higgs: Bezrukov and Shaposhnikov (2008) showed that the Higgs field itself can drive inflation if non-minimally coupled to gravity via a term $\xi R\phi^2$ with $\xi \sim 10^4$. This "Higgs inflation" model makes predictions compatible with Planck data.
- ●String Theory: Moduli fields in string compactifications provide natural inflaton candidates. The potential landscape of string theory ($\sim 10^{500}$ vacua) may offer many possible inflationary trajectories.
- ●Axions: Pseudo-Nambu-Goldstone bosons (axions) naturally have flat potentials of the form $V(\phi) = \Lambda^4[1 - \cos(\phi/f)]$, providing a natural realization of "natural inflation" (Freese et al., 1990).
7. Historical Context
The development of inflationary cosmology involved contributions from several groups working independently in the late 1970s and early 1980s.
Alan Guth (1981)
Guth proposed "old inflation" based on a first-order phase transition in a GUT theory. The universe supercools in a false vacuum, driving exponential expansion. However, this model suffered from the "graceful exit problem": bubbles of true vacuum nucleate but never percolate, leaving the universe in an inhomogeneous state. Despite this flaw, Guth's paper introduced the key insight that exponential expansion solves the horizon, flatness, and monopole problems.
Andrei Linde (1982)
Linde proposed "new inflation" (independently of Albrecht & Steinhardt), where the inflaton slowly rolls from near the top of a Coleman-Weinberg potential. This solved the graceful exit problem because inflation ends smoothly as the field reaches the minimum. Later, Linde introduced "chaotic inflation" (1983), where the field starts at large values with a simple $V \propto \phi^n$ potential, requiring no thermal phase transition. This is the framework we derive equations for in this chapter.
Albrecht & Steinhardt (1982)
Independently of Linde, Albrecht and Steinhardt proposed the "new inflation" scenario with a Coleman-Weinberg potential. Their analysis showed explicitly how the slow-roll phase transitions to rapid oscillation and thermalization, providing a complete cosmological history from inflation through reheating.
Mukhanov & Chibisov (1981)
Perhaps the most profound contribution: Mukhanov and Chibisov showed that quantum fluctuations of the inflaton field during a de Sitter phase generate a nearly scale-invariant spectrum of density perturbations. This connected the microphysics of the inflaton to the observed CMB anisotropies and large-scale structure, making inflation a testable theory. Their result, $\mathcal{P}_\mathcal{R} \propto H^2/\dot{\phi}^2$, remains the cornerstone of inflationary perturbation theory.
Starobinsky (1980)
Even before Guth, Starobinsky proposed a model of inflation based on quantum corrections to gravity — the $R + R^2$ model. This is equivalent to a scalar field with a specific potential and remains one of the best-fitting models to Planck CMB data ($n_s \approx 0.965$, $r \approx 0.004$).
8. Python Simulation: Full Inflaton Dynamics
We numerically solve the coupled Klein-Gordon and Friedmann equations for the quadratic potential $V(\phi) = \frac{1}{2}m^2\phi^2$ using a simple Euler method. The simulation tracks the field value $\phi(t)$, the velocity $\dot{\phi}(t)$, the Hubble parameter $H(t)$, the scale factor $a(t)$, the equation of state $w(t)$, and produces a phase portrait $(\phi, \dot{\phi})$. All quantities are in Planck units ($M_P = 1$).
Inflaton Dynamics: KG + Friedmann System for V = m²φ²/2
PythonClick Run to execute the Python code
Code will be executed with Python 3 on the server
Summary of Key Results
| Equation | Expression | Physics |
|---|---|---|
| Klein-Gordon | $\ddot{\phi} + 3H\dot{\phi} + V' = 0$ | Damped oscillator with Hubble friction |
| Energy density | $\rho = \frac{1}{2}\dot{\phi}^2 + V$ | Kinetic + potential energy |
| Pressure | $p = \frac{1}{2}\dot{\phi}^2 - V$ | Negative when potential dominates |
| Friedmann | $H^2 = \rho/(3M_P^2)$ | Expansion rate from energy |
| Raychaudhuri | $\dot{H} = -\dot{\phi}^2/(2M_P^2)$ | $H$ decreases monotonically |
| Inflation condition | $\dot{\phi}^2 < V(\phi)$ | Potential must dominate kinetic |
| Equation of state | $w = (\frac{1}{2}\dot{\phi}^2 - V)/(\frac{1}{2}\dot{\phi}^2 + V)$ | $w \to -1$ during inflation |