Part III: Cosmic Inflation | Chapter 1

Cosmic Inflation

The theory of exponential expansion in the very early universe: its motivation from the horizon, flatness, and monopole problems; the dynamics of the inflaton scalar field; slow-roll approximation; quantum fluctuations as the origin of cosmic structure

1. Introduction

Historical Context

By the late 1970s, the standard hot Big Bang cosmology had achieved spectacular successes — the prediction of the cosmic microwave background, primordial nucleosynthesis, and the Hubble expansion — yet it suffered from several deeply troubling fine-tuning problems. In 1981, Alan Guth proposed that a phase of exponential expansion in the very early universe, which he called inflation, could solve the horizon, flatness, and magnetic monopole problems simultaneously.

Guth's original “old inflation” model relied on a first-order phase transition and suffered from a “graceful exit” problem: bubble nucleation could not reheat the universe homogeneously. In 1982, Andrei Linde proposed “new inflation” (also independently by Albrecht & Steinhardt), where the inflaton rolls slowly along a nearly flat potential, providing a smooth end to inflation and efficient reheating.

Linde further developed “chaotic inflation” (1983), showing that inflation could occur for a wide class of potentials without requiring special initial conditions. Today, inflation is a cornerstone of modern cosmology, with its predictions for nearly scale-invariant, Gaussian, adiabatic perturbations spectacularly confirmed by COBE, WMAP, and Planck.

The inflationary paradigm asserts that the universe underwent a brief period of quasi-exponential expansion at $t \sim 10^{-36}$ s, during which the scale factor grew by at least a factor of $e^{60} \sim 10^{26}$. This chapter derives the theoretical framework rigorously, starting from the problems that motivate inflation, through the scalar field dynamics and slow-roll approximation, to the quantum generation of primordial perturbations.

2. The Horizon Problem

The most striking feature of the cosmic microwave background is its extraordinary uniformity: the temperature is $T = 2.7255 \pm 0.0006$ K in every direction, with fluctuations of order $\Delta T/T \sim 10^{-5}$. This uniformity demands an explanation, because in the standard Big Bang cosmology, widely separated patches of the CMB sky were never in causal contact.

2.1 The Particle Horizon

The particle horizon is the maximum comoving distance from which a signal traveling at the speed of light could have reached an observer by time t. From the FLRW metric with $ds^2 = 0$ along a radial null geodesic:

$$d_H(t) = a(t) \int_0^t \frac{c\,dt'}{a(t')}$$

The physical (proper) distance to the particle horizon. In terms of conformal time:$d_H = a(t) \cdot c\eta(t)$.

For a universe dominated by a single component with equation of state $p = w\rho c^2$, where $a(t) \propto t^{2/[3(1+w)]}$:

$$d_H(t) = a(t) \int_0^t \frac{c\,dt'}{a_0 \left(t'/t_0\right)^{2/[3(1+w)]}} = \frac{3(1+w)}{1+3w}\,ct$$

valid for $w > -1/3$, so that the integral converges at $t' = 0$

Radiation Domination

With $w = 1/3$ and $a \propto t^{1/2}$:

$$d_H = 2ct$$

The particle horizon is twice the light-travel distance. At the time of last scattering ($t_{\rm ls} \approx 380{,}000$ yr), $d_H \approx 2c \cdot t_{\rm ls}$.

Matter Domination

With $w = 0$ and $a \propto t^{2/3}$:

$$d_H = 3ct$$

In either case, the particle horizon grows linearly with time and is always of order the Hubble radius $c/H$.

2.2 Causal Disconnection on the CMB Sky

The angular size of the particle horizon at the last scattering surface, as seen today, is:

$$\theta_H \approx \frac{d_H(t_{\rm ls})}{d_A(z_{\rm ls})} \approx \frac{d_H(t_{\rm ls})}{d_C(z_{\rm ls})/(1+z_{\rm ls})}$$

For the standard matter+radiation cosmology (no inflation), this gives:

$$\theta_H \approx 1.6° \approx 0.03 \;\text{rad}$$

This means that patches of the CMB sky separated by more than about 2 degrees should be causally disconnected — they have never been in causal contact. The number of causally disconnected regions on the full CMB sky is approximately:

$$N_{\rm disconnected} \sim \frac{4\pi}{\pi \theta_H^2} \sim \frac{4}{\theta_H^2} \sim 4{,}000$$

Yet the CMB temperature is uniform to $10^{-5}$ across all ~4,000 of these independent patches. Without a mechanism to establish thermal equilibrium across the entire observable universe, this uniformity is a monumental coincidence.

3. The Flatness Problem

The first Friedmann equation can be rewritten in terms of the total density parameter$\Omega = \rho/\rho_c$:

$$|\Omega(t) - 1| = \frac{|k|c^2}{a^2(t) H^2(t)}$$

The departure from flatness is governed by the comoving Hubble radius $c/(aH)$.

3.1 Growth of the Curvature Term

In a universe dominated by matter or radiation, $a^2 H^2$ decreases with time, so $|\Omega - 1|$ grows:

$$\text{Radiation era:} \quad a^2 H^2 \propto a^{-2} \quad \Rightarrow \quad |\Omega - 1| \propto a^2 \propto t$$

$$\text{Matter era:} \quad a^2 H^2 \propto a^{-1} \quad \Rightarrow \quad |\Omega - 1| \propto a \propto t^{2/3}$$

Tracing backward from the present, where Planck measures $|\Omega_0 - 1| < 0.005$, the departure from flatness at earlier epochs was extraordinarily small:

At nucleosynthesis ($T \sim 1$ MeV, $t \sim 1$ s):

$$|\Omega - 1|_{\rm BBN} \lesssim 10^{-16}$$

At the electroweak scale ($T \sim 100$ GeV, $t \sim 10^{-12}$ s):

$$|\Omega - 1|_{\rm EW} \lesssim 10^{-27}$$

At the Planck time ($T \sim 10^{19}$ GeV, $t \sim 10^{-43}$ s):

$$|\Omega - 1|_{\rm Pl} \lesssim 10^{-60}$$

The initial conditions must be flat to 60 decimal places. Without a dynamical mechanism, this constitutes an extreme fine-tuning problem.

4. The Monopole Problem

Grand Unified Theories (GUTs) predict that at temperatures $T \sim T_{\rm GUT} \sim 10^{16}$ GeV ($t \sim 10^{-36}$ s), the unified gauge symmetry breaks to the Standard Model gauge group. The Kibble mechanism guarantees that this phase transition produces topological defects, including magnetic monopoles with mass:

$$m_M \sim \frac{M_{\rm GUT}}{\alpha_{\rm GUT}} \sim 10^{17}\;\text{GeV}/c^2 \sim 10^{-8}\;\text{g}$$

The number density of monopoles produced is approximately one per correlation volume:

$$n_M \sim \xi^{-3} \sim T_{\rm GUT}^3$$

where $\xi \sim (H_{\rm GUT})^{-1}$ is the correlation length at the phase transition.

The resulting monopole energy density at the present epoch would be:

$$\Omega_M h^2 \sim 10^{11}$$

This exceeds the observed total density by 11 orders of magnitude. The universe would have recollapsed long before the present epoch. No monopoles have ever been detected, placing upper bounds of $n_M/n_\gamma < 10^{-30}$ from the Parker bound on galactic magnetic fields.

5. The Inflationary Solution

Inflation postulates that at very early times ($t \sim 10^{-36}$ s), the universe underwent a phase of accelerated expansion driven by a nearly constant vacuum energy density. During inflation, the scale factor grows quasi-exponentially:

Inflationary Expansion

$$\boxed{a(t) = a_i \exp\!\left(\int_{t_i}^{t} H(t')\,dt'\right) \approx a_i\,e^{H(t - t_i)}}$$

where $H \approx \text{const}$ during inflation, and the approximation holds for exact de Sitter expansion.

5.1 Solving the Horizon Problem

During inflation, the comoving Hubble radius $(aH)^{-1}$ shrinks because$a$ grows exponentially while H stays roughly constant. This means that the particle horizon grows enormously:

$$d_H^{\rm (inflation)} = a(t)\int_0^{t} \frac{c\,dt'}{a(t')} \supset a(t_f)\int_{t_i}^{t_f} \frac{c\,dt'}{a_i e^{H(t'-t_i)}}$$

$$= \frac{c}{H}\left(e^{N} - 1\right) \approx \frac{c}{H}\,e^{N}$$

where $N = H(t_f - t_i)$ is the number of e-folds of inflation.

With $N \gtrsim 60$ e-folds, a region much larger than our entire observable universe today was in causal contact before inflation. The thermal equilibrium established in the pre-inflationary era is stretched to superhorizon scales, explaining the uniformity of the CMB.

5.2 Solving the Flatness Problem

During inflation, since $H \approx \text{const}$ and $a \propto e^{Ht}$:

$$|\Omega - 1| = \frac{|k|c^2}{a^2 H^2} \propto a^{-2} \propto e^{-2Ht}$$

$$|\Omega - 1|_{\rm end} = |\Omega - 1|_{\rm start} \cdot e^{-2N}$$

For $N = 60$: $e^{-120} \sim 10^{-52}$. Even if$|\Omega - 1|_{\rm start} \sim \mathcal{O}(1)$, inflation drives the universe exponentially close to flatness.

5.3 Solving the Monopole Problem

If inflation occurs after (or during) the GUT phase transition, monopoles produced at$T_{\rm GUT}$ are diluted by the exponential expansion:

$$n_M \to n_M \cdot e^{-3N}$$

For $N = 60$: $e^{-180} \sim 10^{-78}$. The monopole density is diluted to far less than one per observable universe, consistent with the null observational result.

5.4 The Minimum Number of e-Folds

The minimum number of e-folds required to solve the horizon problem is set by requiring that the entire observable universe today was within one causal patch at the onset of inflation. The observable universe has comoving size $\sim c/H_0$, and the comoving Hubble radius at the start of inflation is $c/(a_i H_I)$. Requiring the latter to exceed the former:

$$N \gtrsim \ln\!\left(\frac{T_{\rm reheat}}{T_0}\right) \approx \ln\!\left(\frac{10^{15}\;\text{GeV}}{2.35 \times 10^{-13}\;\text{GeV}}\right) \approx 60\text{--}70$$

The exact number depends on the energy scale of inflation and the reheating temperature. For GUT-scale inflation: $N_{\rm min} \approx 60$.

6. Scalar Field Dynamics

The physical mechanism driving inflation is a scalar field $\varphi$ — the inflaton — evolving in a potential $V(\varphi)$. We derive the equations of motion from the action principle.

6.1 Action and Stress-Energy Tensor

The action for a minimally coupled scalar field in curved spacetime is:

$$S_\varphi = \int d^4x\,\sqrt{-g}\left[-\frac{1}{2}g^{\mu\nu}\partial_\mu\varphi\,\partial_\nu\varphi - V(\varphi)\right]$$

The stress-energy tensor, obtained by varying with respect to $g^{\mu\nu}$, is:

$$T_{\mu\nu} = \partial_\mu\varphi\,\partial_\nu\varphi - g_{\mu\nu}\!\left[\frac{1}{2}g^{\alpha\beta}\partial_\alpha\varphi\,\partial_\beta\varphi + V(\varphi)\right]$$

For a homogeneous field $\varphi = \varphi(t)$ in the FLRW background (so that$\partial_i\varphi = 0$), the stress-energy takes the perfect fluid form with:

Inflaton Energy Density and Pressure

$$\boxed{\rho_\varphi = \frac{1}{2}\dot\varphi^2 + V(\varphi)}$$

$$\boxed{p_\varphi = \frac{1}{2}\dot\varphi^2 - V(\varphi)}$$

The kinetic term $\dot\varphi^2/2$ contributes positively to both $\rho$ and$p$, while the potential $V$ contributes with opposite signs.

6.2 The Klein-Gordon Equation in FLRW

The equation of motion for $\varphi$ follows from the Euler-Lagrange equation (equivalently, from $\nabla_\mu T^{\mu\nu} = 0$ or from the continuity equation):

Starting from the covariant Klein-Gordon equation $\Box\varphi - V'(\varphi) = 0$:

$$\frac{1}{\sqrt{-g}}\partial_\mu\!\left(\sqrt{-g}\,g^{\mu\nu}\partial_\nu\varphi\right) - \frac{dV}{d\varphi} = 0$$

For the FLRW metric, $\sqrt{-g} = a^3(t)\,r^2\sin\theta / \sqrt{1-kr^2}$ and with$\varphi = \varphi(t)$:

$$\frac{1}{a^3}\frac{d}{dt}\!\left(a^3\,\dot\varphi\right) + \frac{dV}{d\varphi} = 0$$

Klein-Gordon Equation in FLRW Spacetime

$$\boxed{\ddot\varphi + 3H\dot\varphi + \frac{dV}{d\varphi} = 0}$$

The $3H\dot\varphi$ term is Hubble friction: the expansion of the universe acts as a damping force on the rolling scalar field.

6.3 Condition for Inflation

Accelerated expansion requires $\ddot a > 0$, which from the second Friedmann equation demands:

$$\ddot a > 0 \quad \Leftrightarrow \quad \rho + 3p/c^2 < 0 \quad \Leftrightarrow \quad \dot\varphi^2 < V(\varphi)$$

When the potential energy dominates over the kinetic energy:

$$\dot\varphi^2 \ll V(\varphi) \quad \Rightarrow \quad p_\varphi \approx -\rho_\varphi \quad \Rightarrow \quad w \approx -1$$

The scalar field behaves like a cosmological constant, driving exponential expansion.

The Friedmann equations for the inflaton-dominated universe become:

$$H^2 = \frac{8\pi G}{3}\left[\frac{1}{2}\dot\varphi^2 + V(\varphi)\right] = \frac{1}{3M_P^2}\left[\frac{1}{2}\dot\varphi^2 + V(\varphi)\right]$$

$$\dot H = -\frac{4\pi G}{c^2}\left(\rho_\varphi + p_\varphi\right) = -\frac{\dot\varphi^2}{2M_P^2}$$

where we define $M_P = (8\pi G)^{-1/2} = 2.435 \times 10^{18}$ GeV is the reduced Planck mass (using natural units $c = \hbar = 1$).

7. The Slow-Roll Approximation

The slow-roll approximation formalizes the condition for sustained inflation. It requires that the inflaton rolls slowly enough that both its kinetic energy and its acceleration are negligible.

7.1 Slow-Roll Parameters

Define the Hubble slow-roll parameters:

$$\epsilon_H \equiv -\frac{\dot H}{H^2} = \frac{\dot\varphi^2}{2M_P^2 H^2}$$

$$\eta_H \equiv -\frac{\ddot\varphi}{H\dot\varphi}$$

Inflation occurs when $\epsilon_H < 1$; it ends when $\epsilon_H = 1$.

It is often more convenient to work with the potential slow-roll parameters, which depend only on the shape of $V(\varphi)$:

Potential Slow-Roll Parameters

$$\boxed{\epsilon \equiv \frac{M_P^2}{2}\left(\frac{V'}{V}\right)^2}$$

$$\boxed{\eta \equiv M_P^2\,\frac{V''}{V}}$$

where primes denote derivatives with respect to $\varphi$. The slow-roll conditions are $\epsilon \ll 1$ and $|\eta| \ll 1$.

7.2 Slow-Roll Equations of Motion

Under the slow-roll conditions $\dot\varphi^2/2 \ll V(\varphi)$ and$|\ddot\varphi| \ll |3H\dot\varphi|$, the equations simplify dramatically:

Friedmann equation (slow-roll):

$$H^2 \approx \frac{V(\varphi)}{3M_P^2}$$

Klein-Gordon equation (slow-roll):

$$3H\dot\varphi \approx -V'(\varphi)$$

Combined:

$$\dot\varphi \approx -\frac{M_P^2\,V'}{3H} \approx -\frac{V'}{V}\,M_P^2\,H$$

7.3 Number of e-Folds

The number of e-folds of inflation between field values $\varphi_*$ (horizon crossing) and $\varphi_{\rm end}$ (end of inflation) is:

Number of e-Folds

$$\boxed{N = \int_{t_*}^{t_{\rm end}} H\,dt = -\frac{1}{M_P^2}\int_{\varphi_*}^{\varphi_{\rm end}} \frac{V}{V'}\,d\varphi}$$

where we used $dt = d\varphi/\dot\varphi$ and the slow-roll equations. Inflation ends when $\epsilon(\varphi_{\rm end}) = 1$.

Example: Chaotic Inflation $V = m^2\varphi^2/2$

Slow-roll parameters: $\epsilon = 2M_P^2/\varphi^2$, $\eta = 2M_P^2/\varphi^2 = \epsilon$

End of inflation: $\epsilon = 1 \Rightarrow \varphi_{\rm end} = \sqrt{2}\,M_P$

Number of e-folds: $N = \frac{\varphi_*^2 - \varphi_{\rm end}^2}{4M_P^2} \approx \frac{\varphi_*^2}{4M_P^2}$

For $N = 60$: $\varphi_* \approx \sqrt{240}\,M_P \approx 15.5\,M_P$ (super-Planckian field excursion)

8. Inflationary Models

Different choices of the inflaton potential $V(\varphi)$ give different models, each predicting specific values of the spectral index $n_s$ and the tensor-to-scalar ratio$r$. Planck 2018 constraints ($n_s = 0.965 \pm 0.004$,$r < 0.06$) strongly discriminate among models.

Model	Potential $V(\varphi)$	$n_s$	$r$	Status
Chaotic ($\varphi^2$)	$\frac{1}{2}m^2\varphi^2$	$1 - 2/N$	$8/N$	Disfavoured
Natural inflation	$\Lambda^4\!\left[1 + \cos(\varphi/f)\right]$	$1 - 1/N$ (for $f \gg M_P$)	$4/N$	Marginal
Starobinsky $R^2$	$\frac{3M_P^2}{4}\lambda\!\left(1 - e^{-\sqrt{2/3}\,\varphi/M_P}\right)^2$	$1 - 2/N$	$12/N^2$	Favoured
Hilltop (quadratic)	$V_0\!\left[1 - (\varphi/\mu)^2 + \cdots\right]$	$1 - 1/N$ (approx.)	Small	Favoured
Power-law ($\varphi^{2/3}$)	$\lambda\varphi^{2/3}$	$1 - 5/(6N)$	$8/(3N)$	Marginal

General Predictions for Monomial Potentials $V \propto \varphi^p$

$$n_s = 1 - \frac{p+2}{2N}, \qquad r = \frac{4p}{N}$$

For $N = 60$: $\varphi^2$ predicts $n_s \approx 0.967$,$r \approx 0.13$ (too large); Starobinsky predicts $n_s \approx 0.967$,$r \approx 0.003$ (well within bounds).

9. Quantum Fluctuations During Inflation

Perhaps the most remarkable prediction of inflation is that quantum vacuum fluctuations of the inflaton field, stretched to macroscopic scales by the exponential expansion, become the seeds for all cosmic structure — galaxies, clusters, and the CMB anisotropies.

9.1 Scalar (Curvature) Perturbations

During inflation, the inflaton field experiences quantum fluctuations of amplitude:

$$\delta\varphi \sim \frac{H}{2\pi}$$

on scales comparable to the Hubble radius. This is the de Sitter vacuum fluctuation of a massless scalar field: each mode is frozen once it exits the Hubble radius ($k = aH$).

These field fluctuations translate into curvature perturbations via $\mathcal{R} = -(H/\dot\varphi)\delta\varphi$. The dimensionless power spectrum of curvature perturbations is:

Scalar Power Spectrum

$$\boxed{\mathcal{P}_\mathcal{R}(k) = \left.\frac{1}{2\epsilon}\,\frac{H^2}{(2\pi)^2 M_P^2}\right|_{k = aH} = \left.\frac{1}{24\pi^2}\,\frac{V}{M_P^4\,\epsilon}\right|_{k=aH}}$$

Evaluated at the moment when scale k exits the Hubble radius ($k = aH$). Planck 2018: $\mathcal{P}_\mathcal{R} = (2.10 \pm 0.03) \times 10^{-9}$ at the pivot scale $k_* = 0.05\;\text{Mpc}^{-1}$.

9.2 Spectral Index

Since H and $\dot\varphi$ vary slowly during inflation, the power spectrum isnearly but not exactly scale-invariant. The deviation is characterized by the spectral index:

Spectral Index

$$\boxed{n_s - 1 \equiv \frac{d\ln\mathcal{P}_\mathcal{R}}{d\ln k} = -6\epsilon + 2\eta}$$

A perfectly scale-invariant (Harrison-Zel'dovich) spectrum has $n_s = 1$. Slow-roll inflation predicts $n_s < 1$ (red tilt), confirmed by Planck:$n_s = 0.9649 \pm 0.0042$.

9.3 Tensor Perturbations (Gravitational Waves)

Inflation also produces a stochastic background of primordial gravitational waves through quantum fluctuations of the metric tensor itself. The tensor (gravitational wave) power spectrum is:

$$\mathcal{P}_T(k) = \frac{2}{\pi^2}\,\frac{H^2}{M_P^2}\bigg|_{k=aH}$$

This is a pure prediction of quantum gravity during inflation — it depends only on $H$and fundamental constants.

9.4 The Tensor-to-Scalar Ratio

The ratio of tensor to scalar power is a key observable:

Tensor-to-Scalar Ratio

$$\boxed{r \equiv \frac{\mathcal{P}_T}{\mathcal{P}_\mathcal{R}} = 16\epsilon}$$

Combined with the spectral index, this gives the consistency relation:$r = -8n_T$, where $n_T = -2\epsilon$ is the tensor spectral index. This is a smoking gun prediction of single-field slow-roll inflation.

Observational Constraints (Planck 2018 + BICEP/Keck)

$n_s$

$0.9649 \pm 0.0042$

$r$

$< 0.036$ (95% CL)

$\mathcal{P}_\mathcal{R}$

$2.10 \times 10^{-9}$

The detection of $n_s < 1$ at high significance is a major triumph of inflationary cosmology. A detection of $r > 0$ would directly measure the energy scale of inflation: $V^{1/4} = (3\pi^2 r \mathcal{P}_\mathcal{R}/2)^{1/4} M_P \approx 1.06 \times 10^{16}(r/0.01)^{1/4}$ GeV.

9.5 The Lyth Bound

An important result connecting the tensor-to-scalar ratio to the field excursion during inflation is the Lyth bound:

$$\frac{\Delta\varphi}{M_P} = \int_0^N \sqrt{\frac{r}{8}}\,dN \approx \sqrt{\frac{r}{8}}\,N \approx \left(\frac{r}{0.01}\right)^{1/2}$$

If $r \gtrsim 0.01$, the inflaton must traverse a super-Planckian distance in field space ($\Delta\varphi > M_P$), which has deep implications for UV-complete theories of gravity and the “swampland” conjectures.

10. The Primordial Power Spectrum

The primordial power spectrum is conventionally parameterized as a power law about a pivot scale $k_*$:

$$\mathcal{P}_\mathcal{R}(k) = A_s \left(\frac{k}{k_*}\right)^{n_s - 1 + \frac{1}{2}\alpha_s\ln(k/k_*) + \cdots}$$

where $A_s = \mathcal{P}_\mathcal{R}(k_*)$ is the scalar amplitude and$\alpha_s = dn_s/d\ln k$ is the running of the spectral index.

A perfectly scale-invariant spectrum ($n_s = 1$, $\alpha_s = 0$) is the Harrison-Zel'dovich spectrum, first proposed on phenomenological grounds in the early 1970s. Inflation provides a physical mechanism for generating a nearly scale-invariant spectrum, with small deviations that encode information about the inflationary potential.

Why Nearly Scale-Invariant?

1. During exact de Sitter expansion ($H = \text{const}$), every mode exits the horizon in exactly the same way, giving $\mathcal{P}_\mathcal{R} = \text{const}$: perfect scale invariance.

2. In realistic slow-roll inflation, H decreases slightly as the inflaton rolls down the potential. Modes that exit the horizon earlier (larger scales) see a slightly larger H, giving them slightly more power: $n_s < 1$ (red tilt).

3. The deviation from scale invariance is of order $|n_s - 1| \sim \epsilon, \eta \sim 1/N \sim 0.02\text{--}0.03$, exactly as observed by Planck.

The running of the spectral index in slow-roll inflation is second order:

$$\alpha_s = \frac{dn_s}{d\ln k} = 16\epsilon\eta - 24\epsilon^2 - 2\xi^2$$

where $\xi^2 = M_P^4\,V'V'''/(V^2)$. Typically$|\alpha_s| \sim 10^{-3}\text{--}10^{-4}$, consistent with Planck's constraint $\alpha_s = -0.0045 \pm 0.0067$.

11. Numerical Example: Slow-Roll Evolution

The following Python code numerically integrates the slow-roll equations for the chaotic inflation potential $V = m^2\varphi^2/2$, computing the inflaton evolution, slow-roll parameters, and observable predictions as functions of the number of e-folds.

Python

inflation_slowroll.py161 lines

#!/usr/bin/env python3
"""
inflation_slowroll.py
Numerical integration of slow-roll inflation for V = m^2 phi^2 / 2.
Computes phi(N), epsilon(N), eta(N), and predicted n_s, r.
"""
import numpy as np
from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt

# ============================
# Setup: V = (1/2) m^2 phi^2
# Using Planck units (M_P = 1)
# ============================
M_P = 1.0  # Reduced Planck mass

def V(phi):
    """Inflationary potential: chaotic m^2 phi^2 / 2."""
    return 0.5 * phi**2  # m^2 absorbed into normalization

def dVdphi(phi):
    """First derivative of potential."""
    return phi

def d2Vdphi2(phi):
    """Second derivative of potential."""
    return 1.0

def epsilon(phi):
    """Slow-roll parameter epsilon = (M_P^2 / 2)(V'/V)^2."""
    return 0.5 * M_P**2 * (dVdphi(phi) / V(phi))**2

def eta(phi):
    """Slow-roll parameter eta = M_P^2 (V''/V)."""
    return M_P**2 * d2Vdphi2(phi) / V(phi)

def n_s(phi):
    """Scalar spectral index: n_s - 1 = -6*epsilon + 2*eta."""
    return 1.0 - 6.0 * epsilon(phi) + 2.0 * eta(phi)

def r_tensor(phi):
    """Tensor-to-scalar ratio: r = 16 * epsilon."""
    return 16.0 * epsilon(phi)

# ============================
# Find phi_end: epsilon(phi_end) = 1
# For V = m^2 phi^2 / 2: epsilon = 2/phi^2 => phi_end = sqrt(2)
# ============================
phi_end = np.sqrt(2.0) * M_P
print(f"phi_end = {phi_end:.4f} M_P  (where epsilon = 1)")

# ============================
# ODE: dphi/dN = -M_P^2 * V'/(V) = -M_P^2 / phi (for phi^2 potential)
# (N counts e-folds backward from end of inflation)
# We integrate from phi_end backward in e-folds.
# Using forward N from start: dphi/dN = +M_P^2 * V'/V (sign flip)
# Better: define N as e-folds before end, so N=0 at end, N=60 at CMB.
# Then dphi/dN = M_P^2 * V'/V = M_P^2 / phi (for phi^2)
# ============================

def dphi_dN(N, phi):
    """
    dphi/dN where N is e-folds before end of inflation.
    dphi/dN = M_P^2 * V'(phi) / V(phi)
    """
    return M_P**2 * dVdphi(phi) / V(phi)

# Integrate from N=0 (end) to N=75 (well before CMB scales)
N_span = [0, 75]
sol = solve_ivp(dphi_dN, N_span, [phi_end],
                rtol=1e-12, atol=1e-14,
                dense_output=True, max_step=0.1)

N_arr = np.linspace(0, 75, 1000)
phi_arr = sol.sol(N_arr)[0]

# Compute observables
eps_arr = np.array([epsilon(p) for p in phi_arr])
eta_arr = np.array([eta(p) for p in phi_arr])
ns_arr = np.array([n_s(p) for p in phi_arr])
r_arr = np.array([r_tensor(p) for p in phi_arr])

# ============================
# Print predictions at N = 50, 55, 60, 65
# ============================
print("\n" + "=" * 65)
print("  SLOW-ROLL PREDICTIONS: V = m^2 phi^2 / 2")
print("=" * 65)
print(f"  {'N':>5s}  {'phi/M_P':>10s}  {'epsilon':>10s}  {'eta':>10s}  {'n_s':>10s}  {'r':>10s}")
print("-" * 65)
for N_target in [50, 55, 60, 65, 70]:
    idx = np.argmin(np.abs(N_arr - N_target))
    p = phi_arr[idx]
    print(f"  {N_target:5d}  {p/M_P:10.4f}  {epsilon(p):10.6f}  "
          f"{eta(p):10.6f}  {n_s(p):10.6f}  {r_tensor(p):10.6f}")
print("=" * 65)
print(f"\n  Planck 2018:  n_s = 0.9649 +/- 0.0042")
print(f"  BICEP/Keck:   r < 0.036 (95% CL)")

# ============================
# Plotting
# ============================
fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle(r'Slow-Roll Inflation: $V = m^2\phi^2/2$',
             fontsize=16, fontweight='bold')

# Panel 1: phi(N)
ax = axes[0, 0]
ax.plot(N_arr, phi_arr / M_P, 'purple', lw=2)
ax.axvline(60, color='gray', ls='--', alpha=0.5, label='N = 60')
ax.set_xlabel('e-folds before end (N)')
ax.set_ylabel(r'$\phi / M_P$')
ax.set_title(r'Inflaton Field $\phi(N)$')
ax.legend()
ax.grid(alpha=0.3)

# Panel 2: epsilon(N)
ax = axes[0, 1]
ax.semilogy(N_arr, eps_arr, 'crimson', lw=2)
ax.axhline(1.0, color='black', ls=':', alpha=0.5, label=r'$\epsilon = 1$ (end)')
ax.axvline(60, color='gray', ls='--', alpha=0.5, label='N = 60')
ax.set_xlabel('e-folds before end (N)')
ax.set_ylabel(r'$\epsilon$')
ax.set_title(r'Slow-Roll Parameter $\epsilon(N)$')
ax.legend()
ax.grid(alpha=0.3)

# Panel 3: eta(N)
ax = axes[1, 0]
ax.semilogy(N_arr, np.abs(eta_arr), 'darkorange', lw=2)
ax.axhline(1.0, color='black', ls=':', alpha=0.5, label=r'$|\eta| = 1$')
ax.axvline(60, color='gray', ls='--', alpha=0.5, label='N = 60')
ax.set_xlabel('e-folds before end (N)')
ax.set_ylabel(r'$|\eta|$')
ax.set_title(r'Slow-Roll Parameter $\eta(N)$')
ax.legend()
ax.grid(alpha=0.3)

# Panel 4: n_s vs r (with Planck contours schematic)
ax = axes[1, 1]
ax.plot(r_arr, ns_arr, 'purple', lw=2, label=r'$V = m^2\phi^2/2$')
# Mark specific e-fold values
for N_target in [50, 60, 70]:
    idx = np.argmin(np.abs(N_arr - N_target))
    ax.plot(r_arr[idx], ns_arr[idx], 'o', ms=8,
            label=f'N = {N_target}')
# Planck constraint
ax.axhline(0.9649, color='gray', ls='--', alpha=0.5, label=r'Planck $n_s$')
ax.axvline(0.036, color='red', ls='--', alpha=0.5, label='BICEP/Keck bound')
ax.set_xlabel(r'$r$ (tensor-to-scalar ratio)')
ax.set_ylabel(r'$n_s$ (spectral index)')
ax.set_title(r'$n_s$ vs $r$ Predictions')
ax.set_xlim(0, 0.25)
ax.set_ylim(0.93, 1.0)
ax.legend(fontsize=8)
ax.grid(alpha=0.3)

plt.tight_layout()
plt.savefig('inflation_slowroll.png', dpi=150, bbox_inches='tight')
plt.show()
print("\nPlot saved to inflation_slowroll.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Expected Output

phi_end = 1.4142 M_P (where epsilon = 1)

=================================================================

SLOW-ROLL PREDICTIONS: V = m^2 phi^2 / 2

=================================================================

N phi/M_P epsilon eta n_s r

-----------------------------------------------------------------

50 14.2127 0.009901 0.009901 0.960396 0.158416

55 14.8997 0.009009 0.009009 0.963964 0.144144

60 15.5563 0.008264 0.008264 0.966942 0.132231

65 16.1864 0.007634 0.007634 0.969512 0.122150

70 16.7929 0.007092 0.007092 0.971752 0.113475

=================================================================

The $\varphi^2$ model predicts $r \approx 0.13$ at $N = 60$, which is ruled out by BICEP/Keck ($r < 0.036$). This demonstrates the power of CMB observations to discriminate among inflationary models.

12. Summary of Key Results

Klein-Gordon Equation (FLRW)

$$\ddot\varphi + 3H\dot\varphi + V'(\varphi) = 0$$

Inflaton Stress-Energy

$$\rho_\varphi = \tfrac{1}{2}\dot\varphi^2 + V(\varphi), \qquad p_\varphi = \tfrac{1}{2}\dot\varphi^2 - V(\varphi)$$

Slow-Roll Parameters

$$\epsilon = \frac{M_P^2}{2}\!\left(\frac{V'}{V}\right)^2, \qquad \eta = M_P^2\,\frac{V''}{V}$$

Number of e-Folds

$$N = -\frac{1}{M_P^2}\int_{\varphi_*}^{\varphi_{\rm end}} \frac{V}{V'}\,d\varphi, \qquad \epsilon(\varphi_{\rm end}) = 1$$

Scalar Power Spectrum

$$\mathcal{P}_\mathcal{R}(k) = \frac{1}{24\pi^2}\,\frac{V}{M_P^4\,\epsilon}\bigg|_{k=aH}$$

Spectral Index and Tensor-to-Scalar Ratio

$$n_s - 1 = -6\epsilon + 2\eta, \qquad r = 16\epsilon$$

Lyth Bound

$$\frac{\Delta\varphi}{M_P} \approx \left(\frac{r}{0.01}\right)^{1/2}$$

The Logical Chain of Inflationary Cosmology

1. Hot Big Bang cosmology suffers from horizon, flatness, and monopole problems

2. A period of accelerated expansion ($\ddot a > 0$) solves all three simultaneously

3. Acceleration requires $w < -1/3$, achieved by a scalar field with $\dot\varphi^2 \ll V(\varphi)$

4. The slow-roll approximation ($\epsilon \ll 1$, $|\eta| \ll 1$) gives tractable dynamics

5. Quantum fluctuations of the inflaton $\delta\varphi \sim H/(2\pi)$ generate primordial perturbations

6. These produce a nearly scale-invariant power spectrum $\mathcal{P}_\mathcal{R}(k)$ with red tilt $n_s < 1$

7. Tensor modes (gravitational waves) are also produced, with amplitude $r = 16\epsilon$

8. Planck confirms $n_s = 0.965$ and constrains $r < 0.036$, strongly favouring plateau-like potentials (Starobinsky, Higgs inflation)

9. Future experiments (LiteBIRD, CMB-S4) will probe $r \sim 10^{-3}$, potentially detecting primordial gravitational waves

Bibliography

Textbooks & Monographs

Mukhanov, V. (2005). Physical Foundations of Cosmology. Cambridge University Press. — Rigorous treatment of inflation, slow-roll dynamics, and primordial perturbations.
Weinberg, S. (2008). Cosmology. Oxford University Press. — Comprehensive coverage of inflationary perturbation theory and observational predictions.
Dodelson, S. & Schmidt, F. (2020). Modern Cosmology, 2nd ed. Academic Press. — Clear pedagogical treatment of inflation, slow-roll parameters, and quantum fluctuations.
Liddle, A.R. & Lyth, D.H. (2000). Cosmological Inflation and Large-Scale Structure. Cambridge University Press. — Dedicated monograph on inflationary cosmology and its observational consequences.
Baumann, D. (2009). “TASI Lectures on Inflation,” arXiv:0907.5424. — Highly influential lecture notes covering the theoretical foundations of inflation.
Kolb, E.W. & Turner, M.S. (1990). The Early Universe. Addison-Wesley. — Classic treatment of the horizon and flatness problems and early inflationary models.

Key Papers

Guth, A.H. (1981). “Inflationary universe: A possible solution to the horizon and flatness problems,” Physical Review D 23, 347–356. — The original proposal of cosmic inflation.
Linde, A.D. (1982). “A new inflationary universe scenario,” Physics Letters B 108, 389–393. — “New inflation” with slow-roll in a symmetry-breaking potential.
Albrecht, A. & Steinhardt, P.J. (1982). “Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking,” Physical Review Letters 48, 1220–1223. — Independent formulation of new inflation.
Mukhanov, V.F. & Chibisov, G.V. (1981). “Quantum Fluctuations and a Nonsingular Universe,” JETP Letters 33, 532–535. — First calculation of quantum fluctuations during inflation as the origin of structure.
Starobinsky, A.A. (1980). “A new type of isotropic cosmological models without singularity,” Physics Letters B 91, 99–102. — The R² inflation model, the first viable inflationary model.
Lyth, D.H. (1997). “What would we learn by detecting a gravitational wave signal in the cosmic microwave background anisotropy?” Physical Review Letters 78, 1861–1863. — The Lyth bound relating r to the inflaton field excursion.
Planck Collaboration (2020). “Planck 2018 results. X. Constraints on inflation,” Astronomy & Astrophysics 641, A10. arXiv:1807.06211. — Observational constraints on inflationary models from CMB data.
BICEP/Keck Collaboration (2021). “Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season,” Physical Review Letters 127, 151301. arXiv:2110.00483. — Tightest upper bound on the tensor-to-scalar ratio r < 0.036.

← Thermal History CMB Physics →

Model	Potential \(V(\varphi)\)	\(n_s\)	\(r\)	Status
Chaotic (\(\varphi^2\))	\(\frac{1}{2}m^2\varphi^2\)	\(1 - 2/N\)	\(8/N\)	Disfavoured
Natural inflation	\(\Lambda^4\!\left[1 + \cos(\varphi/f)\right]\)	\(1 - 1/N\) (for \(f \gg M_P\))	\(4/N\)	Marginal
Starobinsky \(R^2\)	\(\frac{3M_P^2}{4}\lambda\!\left(1 - e^{-\sqrt{2/3}\,\varphi/M_P}\right)^2\)	\(1 - 2/N\)	\(12/N^2\)	Favoured
Hilltop (quadratic)	\(V_0\!\left[1 - (\varphi/\mu)^2 + \cdots\right]\)	\(1 - 1/N\) (approx.)	Small	Favoured
Power-law (\(\varphi^{2/3}\))	\(\lambda\varphi^{2/3}\)	\(1 - 5/(6N)\)	\(8/(3N)\)	Marginal

Cosmology

Cosmic Inflation

1. Introduction

Historical Context

2. The Horizon Problem

2.1 The Particle Horizon

Radiation Domination

Matter Domination

2.2 Causal Disconnection on the CMB Sky

3. The Flatness Problem

3.1 Growth of the Curvature Term

4. The Monopole Problem

5. The Inflationary Solution

Inflationary Expansion

5.1 Solving the Horizon Problem

5.2 Solving the Flatness Problem

5.3 Solving the Monopole Problem

5.4 The Minimum Number of e-Folds

6. Scalar Field Dynamics

6.1 Action and Stress-Energy Tensor

Inflaton Energy Density and Pressure

6.2 The Klein-Gordon Equation in FLRW

Klein-Gordon Equation in FLRW Spacetime

6.3 Condition for Inflation

7. The Slow-Roll Approximation

7.1 Slow-Roll Parameters

Potential Slow-Roll Parameters

7.2 Slow-Roll Equations of Motion

7.3 Number of e-Folds

Number of e-Folds

Example: Chaotic Inflation \(V = m^2\varphi^2/2\)

8. Inflationary Models

General Predictions for Monomial Potentials \(V \propto \varphi^p\)

9. Quantum Fluctuations During Inflation

9.1 Scalar (Curvature) Perturbations

Scalar Power Spectrum

9.2 Spectral Index

Spectral Index

9.3 Tensor Perturbations (Gravitational Waves)

9.4 The Tensor-to-Scalar Ratio

Tensor-to-Scalar Ratio

Observational Constraints (Planck 2018 + BICEP/Keck)

9.5 The Lyth Bound

10. The Primordial Power Spectrum

Why Nearly Scale-Invariant?

11. Numerical Example: Slow-Roll Evolution

Expected Output

12. Summary of Key Results

Klein-Gordon Equation (FLRW)

Inflaton Stress-Energy

Slow-Roll Parameters

Number of e-Folds

Scalar Power Spectrum

Spectral Index and Tensor-to-Scalar Ratio

Lyth Bound

The Logical Chain of Inflationary Cosmology

Bibliography