Cosmology
Slow-Roll Inflation
A scalar field rolling slowly down a flat potential — and the dimensionless parameters that Planck has measured directly
Slow-roll inflation is the standard implementation of cosmic inflation: a scalar field, the inflaton, rolls slowly down a nearly flat region of its potential V(φ). Two dimensionless slow-roll parameters control everything — ε = (M_Pl²/2)(V′/V)² for the slope, η = M_Pl² V″/V for the curvature — and both must be much less than one for the de Sitter approximation to hold. Slow-roll gives nearly constant Hubble parameter, w ≈ −1, and 50–60 e-folds of exponential expansion. Planck has measured the spectral index n_s = 0.9649 ± 0.0042, directly fixing the slope of V at horizon crossing. Slow-roll's job ends when ε grows past 1; reheating begins.
- ε = (M_Pl/2)·(V′/V)²<0.01 (Planck-allowed)
- η = M_Pl²·V″/V|η| ≪ 1
- e-folds N_*50–60
- Spectral index n_s0.9649 ± 0.0042 (Planck 2018)
- Tensor-to-scalar r< 0.036 (BICEP+Planck 2021)
- Inflation H_inf≲ 6 × 10¹³ GeV (from r bound)
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The setup: a scalar field on a potential
The inflaton φ is a real scalar field whose Lagrangian is
L = ½ ∂_μ φ ∂^μ φ − V(φ)In a homogeneous Friedmann-Lemaître-Robertson-Walker universe, the field equation is
φ̈ + 3 H φ̇ + V′(φ) = 0and the Friedmann equation is
H² = (8πG / 3) · [½ φ̇² + V(φ)]The first equation is Newton's second law with Hubble friction; the second is the energy budget. Two extreme cases bracket the dynamics: kinetic-energy-dominated, where ½φ̇² ≫ V, and potential-energy-dominated, where V ≫ ½φ̇². The latter gives w = p/ρ = −1, equation of state of vacuum energy, and exponential expansion. Slow-roll inflation is the regime where this holds.
The slow-roll conditions
Two simultaneous conditions ensure the universe stays in the potential-dominated regime for many e-folds:
ε = (M_Pl² / 2) · (V′ / V)² ≪ 1
η = M_Pl² · V″ / V ≪ 1 in absolute valuewhere M_Pl = (8πG)^(−1/2) ≈ 2.4 × 10¹⁸ GeV is the reduced Planck mass. The first slow-roll parameter ε measures the slope of V relative to V itself: small ε means the potential is nearly flat. The second slow-roll parameter η measures the curvature: small |η| means the slope is not changing much. The two conditions together justify dropping the φ̈ term in the field equation, giving
3 H φ̇ ≈ −V′(φ)
H² ≈ V / (3 M_Pl²)This is the slow-roll approximation. Friction balances gravity; the field drifts down its potential at a terminal velocity determined by V and H.
Number of e-folds: the integral that counts
The number of e-folds between two field values φ_1 and φ_2 is
N(φ_1 → φ_2) = ∫_{φ_2}^{φ_1} (V / V′) dφ / M_Pl²
= ∫_{φ_2}^{φ_1} (1 / √(2ε)) dφ / M_PlEach e-fold multiplies the scale factor by e ≈ 2.72. To solve the horizon problem — to bring our observable universe inside a single causally connected pre-inflation patch — we need the modes that left the horizon at some moment during inflation to have re-entered the horizon today. Working through the cosmological history backwards yields N_* between 50 and 60 e-folds, depending on the reheating temperature and the inflation scale.
Inflation ends when ε grows past 1. Tracking ε as φ decreases gives the field value φ_end at which inflation ends. Then we work backwards by 60 e-folds to find the field value φ_* at which the modes corresponding to today's observable scale exited the horizon. Both ε and η at φ_* determine the spectral index and tensor-to-scalar ratio that Planck observes.
A worked example: Starobinsky inflation
The Starobinsky (R²) model has the potential
V(φ) = (3/4) M² M_Pl² · (1 − exp(−√(2/3) φ/M_Pl))²where M is a mass scale set by observation. Computing the slow-roll parameters at the φ_* corresponding to N_* = 60 e-folds gives
| Quantity | Slow-roll formula | Starobinsky at N_* = 60 |
|---|---|---|
| ε | (M_Pl²/2)(V′/V)² | ~ 3/(4 N²) ≈ 2 × 10⁻⁴ |
| η | M_Pl² V″/V | ~ −1/N ≈ −0.017 |
| n_s | 1 − 6ε + 2η | ≈ 0.967 |
| r | 16 ε | ≈ 0.003 |
| Inflation scale H_inf | √(V/3M_Pl²) | ~ 10¹³ GeV |
| Number of e-folds N_* | (3/4)·exp(√(2/3)φ_*/M_Pl) | 60 |
The predicted n_s = 0.967 matches Planck's measurement (n_s = 0.9649 ± 0.0042) within experimental error. The predicted r = 0.003 is below current upper limits (r < 0.036). Starobinsky inflation sits in the centre of the Planck-allowed (n_s, r) region — making it the canonical inflation model today.
Inflation models ranked by Planck-compatibility
| Model | Potential V(φ) | n_s | r | Status |
|---|---|---|---|---|
| Starobinsky / R² | (1 − e^(−√(2/3)φ/M_Pl))² | 0.967 | 0.003 | Preferred — matches data |
| α-attractors (small α) | tanh²(φ/√(6α) M_Pl) | ~0.965 | ~10⁻³ | Preferred — family fits Planck |
| Plateau (Higgs-like) | V_0(1 − ε·exp(−φ))² | 0.965 | ~10⁻² | Compatible |
| Power-law φ⁴ | (λ/4) φ⁴ | 0.940 | 0.27 | Ruled out by Planck (r too high) |
| Power-law φ² | (½) m² φ² | 0.967 | 0.13 | Ruled out by BICEP+Planck (r too high) |
| Hilltop V_0(1 − φⁿ/μⁿ) | n = 4, μ small | 0.94 | ~10⁻³ | Marginal |
| Natural V_0(1 + cos(φ/f)) | f > M_Pl | 0.95 | 0.05 | Ruled out by r |
The general lesson: simple polynomial potentials (chaotic inflation) are ruled out by tensor-to-scalar limits, while plateau-type potentials with effective M_Pl-scale flatness (Starobinsky, α-attractors) survive. The next generation of CMB B-mode experiments aims for r-sensitivity ~10⁻³, deep enough to detect or definitively rule out Starobinsky itself.
The spectral index, observationally
During slow-roll, quantum fluctuations of the inflaton produce primordial curvature perturbations whose power spectrum is
P_R(k) = (H² / 8π² M_Pl² ε) · (k/k_*)^(n_s − 1)The amplitude A_s = (H²)/(8π²M_Pl²ε) is measured directly by COBE, WMAP, Planck: A_s ≈ 2.1 × 10⁻⁹ at k_* = 0.05 Mpc⁻¹. The tilt n_s − 1 follows from differentiation:
n_s − 1 = −6ε + 2ηThis is a non-trivial prediction: scale-invariant fluctuations (n_s = 1) would require ε = η = 0, which would imply infinite inflation. Slow-roll instead predicts a slight tilt — almost always toward n_s < 1, because ε must eventually grow toward 1 to end inflation. The measured Planck value n_s = 0.9649 directly constrains the combination −6ε + 2η ≈ −0.035 at φ_*.
Combined with a tensor-to-scalar ratio bound r = 16ε < 0.036, we get ε < 0.002 at the relevant φ_*. The corresponding |η| ≤ 0.014. These are surprisingly tight constraints on the inflaton potential at a specific field value — the slope is very small (V′/V < 0.06/M_Pl) and the curvature is small (V″/V < 0.014/M_Pl²). The inflaton is, as advertised, rolling very slowly down a very flat slope.
Quantum fluctuations: the origin of structure
Slow-roll inflation predicts not just the homogeneous background but also primordial perturbations from quantum fluctuations of the inflaton. The argument goes:
- The inflaton field has quantum vacuum fluctuations of magnitude δφ ~ H/(2π) on any horizon-sized region.
- Exponential expansion stretches these modes until their physical wavelength exceeds the Hubble radius. Once a mode has 'exited the horizon', it freezes — friction H is too strong for the mode to evolve.
- The frozen mode imprints a curvature perturbation R = (H/φ̇) δφ on the comoving slice. This perturbation persists through reheating.
- After inflation ends, modes re-enter the horizon one by one as the universe expands. Each mode re-enters at a different cosmic time, with its phase still set by inflation.
- Modes that re-enter during recombination contribute to CMB anisotropies. Modes that re-enter much later become large-scale structure.
The predicted δT/T ~ 10⁻⁵ for CMB anisotropies is exactly what is observed. The predicted Gaussianity (because quantum fluctuations of a free field are Gaussian to lowest order) is exactly what is observed (Planck constraint on local non-Gaussianity: |f_NL^local| < 5). The predicted adiabaticity (single field → single mode → all components fluctuating together) is exactly what is observed (Planck rules out isocurvature contamination above ~1%). Slow-roll inflation is the only known model that explains these observations from first principles, with no fitted parameters.
History: from Guth to Planck
Guth's original 1981 paper had a problem. He proposed inflation as a delayed first-order phase transition: the universe gets stuck in a metastable vacuum, inflates exponentially while supercooled, then tunnels to the true vacuum. But the bubbles of true vacuum would collide chaotically, leaving the universe inhomogeneous and full of bubble walls — incompatible with the smooth CMB.
The fix came almost simultaneously from Andrei Linde and from Andreas Albrecht and Paul Steinhardt in 1982: 'new inflation' replaced the first-order transition with a slow-roll scenario. The inflaton sits near the top of a smooth potential and rolls slowly downhill, giving a continuous classical evolution rather than a quantum tunneling event. The slow-roll approximation and the slow-roll parameters were introduced explicitly to characterise the dynamics.
Throughout the 1980s, models proliferated: chaotic inflation (Linde 1983), hybrid inflation (Linde 1991), natural inflation (Freese, Frieman, Olinto 1990), α-attractors (Kallosh, Linde 2013). All shared the slow-roll core. The 2002 WMAP results gave the first ~10% measurement of n_s. The 2013 Planck results gave the first sub-percent measurement, definitively below 1.
The 2014 BICEP2 announcement of r ≈ 0.2 made global headlines: it would have meant chaotic inflation, with a Planck-scale tensor signal. Within a year the signal was traced to galactic dust contamination. Subsequent BICEP/Keck + Planck joint analyses set r < 0.036 in 2021. The current observational consensus is that slow-roll inflation is real, plateau-type potentials are favoured, and we still don't see the smoking-gun B-mode signal that would establish r definitively.
Common pitfalls
- Treating slow-roll as exact de Sitter. Slow-roll is quasi-de-Sitter: H is slowly varying, not exactly constant. The variation is what produces n_s < 1 and gives a specific spectral tilt.
- Confusing ε with the slope V′. ε depends on the ratio V′/V, not on V′ alone. A steep potential with high V can still have small ε.
- Forgetting M_Pl. Slow-roll parameters are dimensionless because of M_Pl factors. A potential is 'flat' compared to the Planck scale, not in absolute units.
- Assuming the slow-roll approximation is always valid. Specific scenarios (ultra-slow-roll, fast-roll) violate the usual ε,η ≪ 1 conditions and require different formulas.
- Treating Planck-measured n_s as a single-field result. Multi-field inflation can produce additional features: isocurvature modes, non-Gaussianity, runs in the spectral index. Planck's tight bounds favour single-field slow-roll but don't uniquely require it.
Frequently asked questions
Why does the inflaton have to roll slowly?
To stay close to de Sitter. The condition for exponential expansion is that the potential energy V(φ) dominates the total energy density, with kinetic energy ½φ̇² ≪ V. If φ rolls too fast, kinetic energy spoils the de Sitter equation of state w ≈ −1, and the universe behaves more like dust or radiation than vacuum. Slow-roll is the only way to get a long stretch of nearly exponential expansion from a scalar field. The slow-roll parameter ε = ½(V′/V)² M_Pl² quantifies this: ε ≪ 1 keeps kinetic energy negligible. Inflation ends precisely when ε grows past 1.
What are the slow-roll parameters?
Two dimensionless numbers built from the inflaton potential V(φ). The first slow-roll parameter ε = (M_Pl²/2)(V′/V)² controls the slope of the potential. The second slow-roll parameter η = M_Pl² V″/V controls the curvature. Both must be much less than one for the slow-roll approximation to apply. Together they determine how many e-folds inflation lasts and the spectrum of primordial perturbations it generates. Planck's measurement n_s = 0.9649 ± 0.0042 corresponds to combinations like n_s − 1 ≈ −6ε + 2η at horizon crossing, which directly constrains the shape of V near the relevant φ value.
How many e-folds does slow-roll inflation produce?
Typically 50 to 60. The number of e-folds N between a given moment and the end of inflation is N = ∫ V/V′ dφ / M_Pl² (in slow-roll). To solve the horizon problem and flatness problem, the observable universe needs to have left the horizon at least 50 e-folds before inflation ended. The exact number depends on the reheating temperature: lower T_RH requires more e-folds because of the additional expansion needed to redshift modes back into the horizon today. Standard slow-roll models target N_* between 50 and 60 for the modes corresponding to the current Hubble scale.
What is the spectral index n_s and what does Planck measure?
The spectral index n_s parameterises the tilt of the primordial scalar power spectrum: P_R(k) = A_s (k/k_*)^(n_s − 1). For exactly scale-invariant fluctuations (Harrison-Zeldovich), n_s = 1. Slow-roll inflation predicts a slight tilt: n_s − 1 ≈ −6ε + 2η, and almost always n_s < 1 because ε must end inflation by growing to one. Planck 2018 measured n_s = 0.9649 ± 0.0042 — definitively below one at the 8σ level. This is one of the cleanest predictions of slow-roll inflation confirmed by observation, and it directly constrains the inflaton potential at the φ value that corresponds to today's horizon scale.
What is the tensor-to-scalar ratio?
The ratio r = A_t / A_s of the amplitudes of primordial tensor and scalar perturbations. In slow-roll, r = 16ε at horizon crossing, so a non-zero r directly measures the slope of the inflaton potential — or equivalently, the energy scale of inflation. Current observational limits (BICEP/Keck + Planck) give r < 0.036 at 95% confidence, bounding the inflation Hubble scale at H_inf ≲ 6 × 10¹³ GeV. Future surveys — LiteBIRD, CMB-S4, Simons Observatory — target r ~ 10⁻³, deep enough to detect or rule out the simplest single-field slow-roll models like Starobinsky inflation, which predicts r ≈ 0.003.
Why does inflation end?
When the slow-roll parameter ε grows past 1, the kinetic energy of the inflaton becomes comparable to its potential energy and the de Sitter approximation breaks. Physically, the inflaton rolls into the steep part of its potential — the cliff — and starts accelerating. ε measures the slope of V; on the plateau ε is tiny, but as φ descends toward the minimum the slope steepens and ε grows. The exact φ value where ε = 1 marks the end of inflation. After this point the field oscillates around the minimum and reheating begins.
What are typical slow-roll potential shapes?
Several families are observationally viable. (1) Starobinsky inflation: V ∝ (1 − exp(−√(2/3) φ/M_Pl))², a natural shape from R² gravity, predicts n_s ≈ 0.965 and r ≈ 0.003 — perfectly consistent with Planck. (2) α-attractor models: V ∝ tanh²(φ/√(6α) M_Pl), a class that captures many inflation scenarios. (3) Plateau models: V ∝ φ²/(1 + cφ²), with broad flat regions. (4) Power-law (chaotic) inflation V ∝ φ²ⁿ: Linde's classic 1983 proposal, mostly ruled out today because it predicts r too large. (5) Hilltop inflation: V ≈ V_0(1 − (φ/μ)ⁿ), useful in some symmetry-breaking scenarios. The Planck preferred zone in the (n_s, r) plane sharply constrains which shapes are alive.
How are quantum fluctuations imprinted in the CMB?
Quantum vacuum fluctuations of the inflaton field have wavelength ~1/H during inflation. As inflation stretches space exponentially, these modes are stretched until their wavelength exceeds the Hubble radius — they 'exit the horizon' and freeze. The amplitude at horizon exit is δφ ~ H/(2π), giving rise to a curvature perturbation R ~ Hδφ/φ̇ that persists. Sixty e-folds later, in the late universe, these modes re-enter the horizon as classical density perturbations. CMB anisotropies of order δT/T ~ 10⁻⁵ are exactly the predictions of slow-roll inflation. Planck has measured the full angular power spectrum and finds spectacular agreement with this picture, including the predicted Gaussianity and adiabaticity.