Cosmic inflation

Cosmic inflation is an early period of accelerated expansion, often modeled by a scalar field whose potential energy dominates its kinetic energy.

The leading relations $r=16{ϵ}_V$ and $n_s-1=-6{ϵ}_V+2{\eta }_V$ assume a canonical single slow-roll scalar, Einstein gravity and the usual vacuum initial fluctuations. More general inflation models need not obey these relations.

A family of inflationary scenarios in which a scalar initially away from the minimum of a potential can drive accelerated expansion without an initial thermal equilibrium near that minimum. For the quadratic example, a large slowly rolling field gives ${ϵ}_V=1/(4\pi G{\varphi }^2)$; its predictions depend on the potential and subsequent trajectory.

In flat general relativity with no separate cosmological constant, a canonical massive scalar has $\rho +3P=2{\dot{\varphi }}^2-m^2{\varphi }^2$. Accelerated expansion therefore requires ${\dot{\varphi }}^2<m^2{\varphi }^2/2$. Initial potential dominance is an instantaneous condition and does not alone fix the duration.

A graceful exit from inflation ends sustained accelerating expansion and connects it to a hot radiation-dominated universe. Crossing ${ϵ}_H=1$ ends acceleration in a homogeneous expanding model; reheating requires an additional mechanism to transfer inflaton energy into other species. Pure exponential-potential power-law inflation keeps ${ϵ}_H$ constant and its inflaton potential has no finite minimum, so neither an automatic endpoint nor oscillatory reheating follows from that potential alone.

Power-law inflation is cosmic inflation with a scale factor proportional to $t^p$ for constant $p>1$. Its Hubble parameter is $H=p/t$ and its first Hubble slow-roll parameter is ${ϵ}_H=1/p$. The comoving Hubble radius decreases as $t^{1-p}$. A canonical inflaton with an exponential inflaton potential admits an exact example. Because ${ϵ}_H$ is constant, an indefinitely maintained power law has no graceful exit from inflation.

Use natural units, the unreduced Planck mass $m_P=G^{-1/2}$, $V_0>0$ and $c\ne 0$. In a flat universe dominated by a canonical inflaton, $\varphi =(2m_P/c)\log (\beta t)$ and $a\propto t^p$ solve the inflaton equation of motion and Friedmann equation exactly when $p=16\pi /c^2$ and ${\beta }^2=V_0{c}^2/[2m_P^2(3p-1)]$. The positive-potential scaling solution requires $p>1/3$, while cosmic inflation requires $p>1$. Its constant equation-of-state parameter is $w=-1+2/(3p)$. In terms of the reduced Planck mass $M=m_P/\sqrt{8\pi }$, the exponential slope is $\lambda =c/\sqrt{8\pi }$ and the same exponent is $p=2/{\lambda }^2$.

For $V=V_0{e}^{-A\varphi /m_{\mathrm{pl}}}$, neglecting acceleration of the field and kinetic energy gives $3H\dot{\varphi }=AV/m_{\mathrm{pl}}$ and $H^2=8\pi V/(3m_{\mathrm{pl}}^2)$. Integration yields the displayed field with $a\propto (t+b{)}^{16\pi /A^2}$. The exact flat scalar-dominated solution has the same power but $C^2=A^4{V}_0/[2m_{\mathrm{pl}}^2(48\pi -A^2)]$. The two normalizations agree at leading slow-roll order. Inflation requires $A^2<16\pi$; the stronger slow-roll conditions require $A^2/(8\pi )\ll 1$.

The bilinear hybrid superpotential couples an inflaton $S$ to two waterfall fields $u,v$. In global supersymmetry, $V=|g{|}^2(|uv-\zeta {|}^2+|S{|}^2(|u{|}^2+|v{|}^2))$. The branch $u=v=0$ breaks supersymmetry and is transversely stable for $|S{|}^2>|\zeta |$, while the zero-energy branch $S=0$, $uv=\zeta$ is supersymmetric. A canonical Kähler potential in supergravity slightly lifts the inflationary valley. It must not be described as two isolated stationary minima of the full potential.

For nonzero $g$ and $\zeta$, these field values give $W=0$ and all Kähler covariant derivatives of a superpotential zero, hence zero supergravity auxiliary fields and zero vacuum energy. With only the supplied chiral fields, the product condition gives a complex vacuum manifold rather than one isolated point. A specified gauged opposite-charge symmetry can add a D-flatness condition and a gauge quotient, but no such gauge sector should be invented from the field labels alone.

In the small-field global approximation, the waterfall fluctuations on $u=v=0$ have the displayed masses. One becomes negative when $|S{|}^2<|\zeta |$, driving the fields towards the supersymmetric vacuum branch. In the canonical supergravity model, the exact masses follow from expanding the exponential and Kähler-covariant derivatives; their Planck-suppressed corrections should not be discarded unless the corresponding field and symmetry-breaking scales are small.

The supergravity F-term potential on the inflationary branch has the displayed exact form. Its expansion is $|g\zeta {|}^2(1+x^2/2+O(x^3))$, so the inflaton quadratic mass cancels at tree level for this canonical model. For $x>0$ its derivative is $|g\zeta {|}^2{e}^{x}x(1+x)>0$; the branch is therefore a transverse valley, not a stationary supersymmetry-breaking vacuum.

Approximately constant positive Hubble parameter gives a nearly de Sitter spacetime expansion in flat slicing. Its scale factor can be approximated by $a=-1/(H\tau )$ with negative conformal time increasing toward zero. Slow variation of the expansion rate limits the accuracy of this approximation.

Reheating transfers the inflaton energy into other particles after cosmic inflation, eventually establishing a hot thermal bath. It requires interactions in addition to a background scalar potential. The duration and averaged equation-of-state parameter of reheating affect the number of e-folds between a chosen scale's horizon exit and the end of inflation.

Nonperturbative particle production after cosmic inflation, for example by parametric resonance caused by an oscillating inflaton. Large occupation numbers can grow before a thermal bath exists. Backreaction, rescattering and subsequent thermalization affect the outcome, so an effective perturbative decay rate need not describe every stage.^[ref]

A coarse-grained description in which quadratic inflaton oscillations have zero mean pressure and lose energy at constant effective decay rate $\Gamma$ to a relativistic component. Adding the equations preserves total energy accounting with the appropriate expansion work. Efficient transfer occurs around $H\sim \Gamma$, but producing particles and achieving thermal equilibrium are different steps. The potential alone does not specify the decay rate or thermalization interactions.

The reheating temperature characterizes the thermal radiation bath after energy from a nonthermal component has been transferred and thermalized. For sudden decay of a dominant nonrelativistic relic, $T_R\simeq [3M_{\mathrm{Pl}}^2{\Gamma }^2/(c_2{g}_{\star }){]}^{1/4}$. It differs from the residual radiation temperature immediately before decay.

For a quartic inflaton potential, an endpoint field of order the unreduced Planck mass leaves density of order $\lambda M_P^4$. If prompt, efficient reheating converts it to equilibrium radiation, $\rho =({\pi }^2/30)g_{\ast }{T}_R^4$ gives the displayed scaling. With the potential-only endpoint estimate ${\varphi }_e=M_P/\sqrt{\pi }$, the coefficient is $[15/(2{\pi }^4){]}^{1/4}$. End-of-inflation kinetic energy changes that coefficient by order one, and delayed transfer or incomplete thermalization requires a reheating model rather than the potential alone.

The sudden-decay approximation converts a dominant cold relic's rest-mass density into radiation at $H\simeq \Gamma$, neglecting expansion during the conversion. Its pre-decay radiation temperature obeys $T_{-}^3\simeq 3M_{\mathrm{Pl}}^2{\Gamma }^2/(c_1{g}_{\star }Ym)$, while the reheating temperature is set by the post-decay radiation energy density. Constant relativistic counts and prompt thermalization are additional assumptions.

At fixed scale factor and unchanged relativistic counts, instantaneous conversion of a relic into thermal radiation gives the displayed entropy-density enhancement. It exceeds one because the gas receives energy from outside its prior equilibrium sector. This does not contradict cosmological entropy conservation, whose source-free reversible assumptions fail during decay and thermalization.

The number of e-folds during an interval of expansion is the logarithmic growth of the scale factor:

Start with an inflationary patch of physical radius $H_i^{-1}$ in units $c=1$. After $N$ e-folds and subsequent expansion, its present radius is $H_i^{-1}{e}^N{a}_0/a_{\mathrm{reh}}$. Requiring it to cover the present Hubble radius gives the displayed inequality. With instantaneous conversion to radiation, fixed relativistic degrees of freedom and a fixed ratio $H_i/H_{\mathrm{end}}$, energy conservation gives $H_i\propto T_{\mathrm{reh}}^2$, whereas adiabatic subsequent expansion gives $a_0/a_{\mathrm{reh}}\propto T_{\mathrm{reh}}$. Thus the required $N$ increases approximately as $\log T_{\mathrm{reh}}$. Holding $H_i$ fixed while varying the end energy independently is a different comparison and reverses that simple temperature-only trend. The inflationary energy history and post-inflationary entropy history must be specified.^[ref]

A primordial perturbation is an early-universe fluctuation that seeds later cosmic microwave background anisotropy and large-scale structure. Inflation generates scalar curvature perturbations and tensor perturbations from quantum fluctuations.

The dimensionless spectrum is the contribution to a statistically isotropic field variance per logarithmic wavenumber interval: $⟨X^2⟩=\int d\log k{\Delta }_X(k)$, with any necessary smoothing or regulators. Some conventions denote this same quantity ${\Delta }_X^2$. The definition, not the superscript convention, fixes its meaning.

The dimensionless spectrum is $\mathcal{P}(k)=k^{3}P(k)/(2{\pi }^2)$. If it rises as $k^4$ below equality and as ${\ln }^{2}k$ above equality, then the dimensional spectrum rises as $k$ and falls as $k^{-3}{\ln }^{2}k$. Thus the dimensional spectrum has a turnover while the dimensionless spectrum only bends to a much slower rise. A schematic transition must smoothly join the two asymptotic regimes.

A canonical homogeneous inflaton $\varphi (t)$ hasand obeys $\ddot{\varphi }+3H\dot{\varphi }+V^{\prime }(\varphi )=0$.

The inflaton potential is the potential energy density of an inflaton as a function of its field value. Its derivative enters the inflaton equation of motion, while its slope and curvature enter the potential slow-roll parameters.

A homogeneous canonical scalar field with positive scalar potential has energy density ${\rho }_{\varphi }={\dot{\varphi }}^2/2+V$ and pressure $P_{\varphi }={\dot{\varphi }}^2/2-V$. When its potential energy dominates its kinetic energy, the equation-of-state parameter approaches $-1$. If other components and spatial curvature are negligible, the Friedmann acceleration equation gives accelerating expansion. Small initial kinetic energy is not enough to ensure long cosmic inflation; the field must also remain on a sufficiently flat part of the scalar potential, as measured by the potential slow-roll parameters.

For a homogeneous canonical inflaton in an expanding FLRW metric,The expansion contributes the damping term called Hubble friction.

For a homogeneous canonical scalar field, the cosmological perfect-fluid continuity equation reduces to $\dot{\varphi }(\ddot{\varphi }+3H\dot{\varphi }+V^{\prime })=0$. It implies the inflaton equation of motion wherever $\dot{\varphi }\ne 0$, and by continuity at isolated turning points. It cannot by itself exclude a frozen field on a time interval. Indeed, choose $\varphi ={\varphi }_0$ with $V({\varphi }_0)>0$ but $V^{\prime }({\varphi }_0)\ne 0$; a de Sitter scale factor in flat slicing with $H^2=V({\varphi }_0)/(3M_{\mathrm{Pl}}^2)$ satisfies the homogeneous Friedmann equation and Friedmann acceleration equation, but not the scalar equation. Varying the inflaton action in an expanding universe supplies the missing equation also for a frozen field, requiring $V^{\prime }({\varphi }_0)=0$.

In a flat universe containing a homogeneous canonical scalar field, the ansatzhas the exact solutionIt interpolates between decelerating power-law expansion at early times and exponential accelerated expansion at late times when $2/3<{\varphi }_0^2/M_{\mathrm{Pl}}^2<2$.

For a canonical scalar field in a spatially flat expanding universe,The field Euler-Lagrange equation is

For nearly constant inflaton potential and constant $H_{\inf }$, the spatial Fourier transform obeys

For $a=-1/(H_{\inf }\tau )$, the rescaled modesatisfies

Neglecting metric perturbations, the rescaling $f=a\delta \varphi$ removes the expansion friction from the scalar perturbation action after a boundary term is discarded. Its canonical momentum is $f^{\prime }$, and each real Fourier oscillator has time-dependent squared frequency $k^2+a^2{V}^{\prime \prime }-a^{\prime \prime }/a$. The oscillator can become inverted on superhorizon scales; the canonical commutation relations still apply.

The Bunch-Davies vacuum selects the mode that approaches the positive-frequency ground-state mode of angular frequency $ck$ in the far past $\tau \to -\infty$.

For a minimally coupled massless scalar field in flat de Sitter spacetime, $a=-1/(H\tau )$ and the Canonically rescaled de Sitter scalar mode is $u_k=a{f}_k$. The equation $u_k^{\prime \prime }+(k^2-2/{\tau }^2)u_k=0$ and the positive-frequency condition $u_k\sim e^{-ik\tau }/\sqrt{2k\mathcal{V}}$ select the displayed mode. Direct differentiation verifies its Wronskian normalization, which ensures the equal-time canonical commutation relations with box-normalized creation and annihilation operators. A conformally coupled massless scalar has a different mode equation.

A Fourier mode exits the Hubble horizon during inflation when $k=aH$. On superhorizon scales $k\ll aH$, the growing mode of a sufficiently light scalar perturbation approaches a constant.

A cosmological mode is on a superhorizon scale when its physical wavenumber is smaller than the Hubble expansion rate, $k/a\ll H$, equivalently $k\ll aH$.

A Fourier-space variance $P_k$ in three dimensions is scale invariant when the dimensionless power per logarithmic wavenumber interval,is independent of $k$. For a massless inflaton mode at late times,so ${\Delta }^2(k)$ is constant.

An exact $P(k)=A/k^3$ has divergent unsubtracted point variance. Its finite isotropic covariance difference is the displayed logarithm, unchanged by simultaneous dilation of $r,r_0$. Thus the nonzero-mode scaling statement has a natural subtraction interpretation. Exact dilation invariance of a continuous finite-variance isotropic covariance itself would instead force a constant covariance, corresponding only to a zero-mode spectral measure. Cutoffs or a zero-mode subtraction must accompany the nontrivial spectrum.

For a light scalar in the Bunch-Davies vacuum with metric perturbations neglected, the mode $e^{-ik\tau }(1-i/(k\tau ))/\sqrt{2k}$ gives the displayed equal-time spectrum. The dimensionless power tends to $H^2/(4{\pi }^2)$ on superhorizon scales. The exact massless infinite-volume correlation has an infrared divergence; a finite-volume or finite-duration infrared regulator, and smearing for coincident products, makes the correlators well defined.

The second term in the equal-time two-point function of a de Sitter scalar is independent of time for constant $H$. It dominates on superhorizon scales and has constant dimensionless power spectrum, giving a scale-invariant inflationary power spectrum. The infrared qualification of a scale-invariant covariance still applies to the formally infinite-volume coincident variance.

The equal-time two-point function of a de Sitter scalar has a contribution $P_{\mathrm{vac}}=1/(2k{a}^2)$, giving the displayed dimensionless power spectrum per logarithmic wavenumber. At fixed comoving wavenumber it decreases as $a^{-2}$; on short physical scales it reproduces the vacuum fluctuations of a massless scalar field in Minkowski spacetime. Its coincident integral is ultraviolet divergent, so the fixed-mode decrease cannot be interchanged with an unregulated integral over all modes.

At leading slow-roll order, the dimensionless primordial curvature-perturbation amplitude at horizon exit is

The scalar spectral index describes the departure of the primordial scalar spectrum from scale invariance. To first order in potential slow-roll parameters,

The running describes the scale dependence of the scalar spectral index. Differentiating $n_s-1\simeq -6{ϵ}_V+2{\eta }_V$ with the potential slow-roll flow equations gives the displayed second-order slow-roll result. It includes the third potential slow-roll parameter and is generally smaller than the leading spectral tilt.

The leading scalar spectrum is proportional to $V^3/V^{\prime 2}$. Along a slow-roll trajectory, $d\varphi /d\ln a\simeq -M_{\mathrm{pl}}^2{V}^{\prime }/V$ and $d\ln k/d\ln a=1-{ϵ}_H$. Differentiating at horizon exit and keeping first order gives $n_s-1=-6{ϵ}_V+2{\eta }_V$. The derivative-factor correction contributes at second order.

For a canonical scalar field with potential $V(\varphi )$, slow roll reduces the field and Friedmann equations to

With approximately constant positive $H$ and potential slope $V^{\prime }$, a homogeneous scalar field initially at rest solves $\ddot{\varphi }+3H\dot{\varphi }=-V^{\prime }$ with the displayed velocity. The transient decays on time $1/(3H)$, leaving the slow-roll approximation $3H\dot{\varphi }\simeq -V^{\prime }$. The fractional potential change over a Hubble time is of order $M_{\mathrm{Pl}}^2(V^{\prime }/V{)}^2$. Sustained slow roll normally uses both a small slope parameter and a small absolute curvature parameter $|M_{\mathrm{Pl}}^2{V}^{\prime \prime }/V|$. Local derivative bounds must hold over the traversed field interval; initial small kinetic energy alone is not a global inflation-duration theorem.

Along cosmological horizon exit $k\propto aH$, differentiation gives $d\log k/d\log a=1-{ϵ}_H$, where ${ϵ}_H=-\dot{H}/H^2$. The chain rule gives the first expression. In reduced Planck units, the slow-roll equations $3H\dot{\varphi }\simeq -V^{\prime }$ and $3H^2\simeq V$ give the second expression to leading order. Applied to a quantity already of slow-roll order, the omitted factor changes the answer only at second order.

Single-field slow-roll inflation uses one canonical scalar clock whose potential evolves slowly enough for the slow-roll parameters to remain small. Its adiabatic long mode produces the single-field inflation squeezed-limit consistency relation.

Starobinsky inflation is an inflationary model whose Einstein-frame scalar potential can be writtenThe exponentially flat large-field plateau supports slow-roll inflation, while the potential is quadratic near its minimum.

A Hubble slow-roll parameter measures the fractional change of the Hubble rate or of a preceding slow-roll parameter per e-fold. Small values make the inflationary background evolve adiabatically.

For a canonical inflaton under the slow-roll approximation, $\dot{\varphi }/H\simeq -M_{\mathrm{Pl}}^2{V}^{\prime }/V$ and $ϵ\simeq {ϵ}_V=(M_{\mathrm{Pl}}^2/2)(V^{\prime }/V{)}^2$. Differentiating $\log {ϵ}_V$ along this trajectory givesHere $\eta =d\log ϵ/d\log a$, not the alternate convention $-\ddot{\varphi }/(H\dot{\varphi })$. The logarithmic definition requires $ϵ>0$; exact de Sitter expansion is treated as a limit.

The first Hubble slow-roll parameter isFor a canonical inflaton, $ϵ={\dot{\varphi }}^2/(2M_{\mathrm{Pl}}^2{H}^2)\ge 0$.

For flat expansion with a canonical scalar and no additional fluid, differentiating the Friedmann equation gives $\dot{H}=-4\pi G{\dot{\varphi }}^2\le 0$. Expansion can still accelerate if $-\dot{H}/H^2<1$. Equality at an isolated turning point does not imply eternal de Sitter expansion.

Using the e-fold variable $N=\log a$, define

The comoving curvature perturbation is the scalar curvature perturbation on hypersurfaces orthogonal to the total matter flow. For single-field adiabatic inflation it is conserved on superhorizon scales.

For adiabatic scalar perturbations without anisotropic stress, define $A=4\pi G{a}^2(\bar{\rho }+\bar{P})={\mathcal{H}}^2-{\mathcal{H}}^{\prime }$ and $c_a^2={\bar{P}}^{\prime }/{\bar{\rho }}^{\prime }$. In conformal Newtonian gauge, $\mathcal{R}=-\varphi -\mathcal{H}({\varphi }^{\prime }+\mathcal{H}\varphi )/A$. Background conservation gives $A^{\prime }/A=-\mathcal{H}(1+3c_a^2)$. Differentiating the expression for $\mathcal{R}$ and using the scalar-potential equation gives the displayed identity. For regular adiabatic modes with nonzero background enthalpy, evolution is suppressed on scales $k\ll \mathcal{H}$.

For single-field adiabatic inflation, the Mukhanov-Sasaki variable rescales the comoving curvature perturbation by $z=a\sqrt{2ϵ}{M}_{\mathrm{Pl}}$, where $M_{\mathrm{Pl}}$ is the reduced Planck mass. Its quadratic bulk action is $\frac{1}{2}\int [(v^{\prime }{)}^2-(\nabla v{)}^2+(z^{\prime \prime }/z)v^2]$, after a boundary term is removed. A Fourier transform mode obeys $v_k^{\prime \prime }+(k^2-z^{\prime \prime }/z)v_k=0$ and has canonical Wronskian normalization. The mass scale and sound-speed factors depend on the underlying scalar theory; this definition assumes the canonical single-field action and positive slow-roll kinetic coefficient.

For an adiabatic single-field attractor, gradient terms are negligible well outside the Hubble radius and the growing comoving-curvature mode is constant. The separate-universe interpretation identifies it with the difference of local expansion histories. The conservation statement can fail for a non-attractor phase or an independent entropy mode.

On a single-field attractor, a field fluctuation changes the local point along the background trajectory. The corresponding time shift has magnitude $|\delta \varphi /\dot{\varphi }|$, and local expansion converts it into curvature of magnitude $H|\delta \varphi /\dot{\varphi }|$. On spatially flat slices a standard sign convention gives $\mathcal{R}=H\delta {\varphi }_{\mathrm{flat}}/\dot{\varphi }$. The vacuum mode amplitude $H/(2\pi )$ then determines the scalar curvature spectrum.

For approximately constant $H$ and $ϵ$, a vacuum-normalized curvature mode is $u_k(\tau )=H(1+ik\tau )e^{-ik\tau }/\sqrt{4ϵM_{\mathrm{Pl}}^2{k}^3}$. Its derivative is $u_k^{\prime }=H{k}^2\tau e^{-ik\tau }/\sqrt{4ϵM_{\mathrm{Pl}}^2{k}^3}$. Ordered unequal-time Wick contractions use the complex conjugate at the earlier vertex, consistently with the vacuum prescription for inflationary in-in integrals.

The non-adiabatic pressure perturbation isOn superhorizon scales it sources the evolution of the comoving curvature perturbation through a convention-dependent signed relation proportional to $H\delta P_{\mathrm{nad}}/(\bar{\rho }+\bar{P})$.

A cosmological entropy perturbation changes the relative composition of cosmic fluids while leaving the total curvature initially unchanged. For cold dark matter and photons, a common relative entropy variable is

At leading order in canonical single-field slow roll and evaluated at horizon exit $k=aH$,

At a fixed horizon-exit e-fold count and with the later cosmology fixed, rescaling a canonical single-field inflationary potential by a positive constant leaves its potential slow-roll parameters and field solution unchanged. The slow-roll curvature power spectrum is proportional to $V/{ϵ}_V$, so the matter power spectrum scales by the same factor while its leading tilt and transfer-function shape remain unchanged. A changed specified reheating history can also shift the horizon-exit count assigned to a fixed physical wavenumber.

Differentiating the curvature spectrum at horizon exit givesto first order in Hubble slow-roll parameters.

The tensor spectrum is proportional to $H^2$, but the single-field slow-roll curvature spectrum is proportional to $H^2/{ϵ}_H$. Thus $n_s-1\simeq -2{ϵ}_H-d\ln {ϵ}_H/dN$. A decreasing $H$ gives a red leading tensor tilt, while the evolving scalar conversion factor can change the sign of the scalar tilt.

For the two graviton polarizations in canonical single-field inflation,at horizon exit.

The graviton power spectrum is the two-point spectrum of a tensor metric perturbation. A convention may quote it per polarization or summed over both polarizations, so that choice must accompany its normalization.

An adiabatic tensor mode is a long-wavelength transverse-traceless metric perturbation whose leading local effect can be produced by a large spatial diffeomorphism. Its effect on short modes is therefore constrained by a Ward identity.

For a single adiabatic long tensor mode, the squeezed scalar-scalar-graviton bispectrum is the anisotropic momentum response of the scalar power spectrum:

The tensor spectral index is the logarithmic slope $n_T=d\log {\Delta }_t^2/d\log k$. Canonical slow-roll inflation gives $n_T=-2ϵ$ to first order.

The tensor-to-scalar ratio is $r={\Delta }_t^2/{\Delta }_{\mathcal{R}}^2$. Canonical single-field slow roll gives $r=16ϵ$.

Combining $r=16ϵ$ with $n_T=-2ϵ$ gives the leading-order consistency relation

A primordial power spectrum is blue when its dimensionless power increases with wavenumber. Canonical single-field slow roll cannot produce a blue tensor spectrum because $ϵ\ge 0$ implies $n_T\le 0$, but its scalar spectrum can be blue when $\eta <-2ϵ$.

For $V(\varphi )=m^2{\varphi }^2$ in units where $3H^2\simeq V$ and for positive $\varphi$,and

For $V=m^2{\varphi }^2/2$ and unreduced Planck mass, the slow-roll e-fold count is $N=2\pi ({\varphi }_i^2-{\varphi }_f^2)/m_P^2$. A formal initial potential $V_i\sim m_P^4$ gives ${\varphi }_i^2\sim 2m_P^4/m^2$, hence the displayed large-field estimate and expansion factor $e^N$. The final-field contribution is order one in $N$. Without specifying $m/m_P$ there is no unique numerical expansion factor. The extrapolation to Planck density is a classical estimate, rather than a controlled treatment of quantum-gravity corrections.

For $V=m^2{\varphi }^2/2$ and unreduced Planck mass $m_P=G^{-1/2}$, the potential slow-roll parameter is ${ϵ}_V=m_P^2(V^{\prime }/V{)}^2/(16\pi )=m_P^2/(4\pi {\varphi }^2)$. Its conventional value one gives the displayed endpoint estimate. Exact accelerated expansion ends at ${ϵ}_H=1$, or ${\dot{\varphi }}^2=V$ for a canonical scalar alone. Substitution of slow-roll formulas into this exact condition gives a different order-one coefficient because slow roll is failing there; a precise endpoint requires integrating the full inflaton equation of motion and Friedmann equation. The large-field e-fold estimate is insensitive to this order-one endpoint uncertainty.

For a quadratic inflaton potential ending at the leading estimate ${ϵ}_V=1$, $\Phi /M_{\mathrm{Pl}}=\sqrt{4N+2}$. Cosmological horizon exit gives $d\log k/dN=-1+{ϵ}_H$. A narrow logarithmic interval therefore probes a field span $2\Delta \log k/(\sqrt{4N+2}(1-{ϵ}_H))$, to the accuracy of the slow-roll approximation. At sixty e-folds, five logarithmic wavenumber units probe about $0.64M_{\mathrm{Pl}}$ around a background field near $15.6M_{\mathrm{Pl}}$.