Modulated Reheating Mechanism

• Modulated reheating is a mechanism where a light spectator field causes spatial variations in the inflaton decay rate, leading to nonuniform reheating and curvature perturbations.
• The δN formalism is used to translate primordial isocurvature fluctuations into observable signatures in the power spectrum and local non-Gaussianity.
• Variants, such as modulated preheating and curvaton decay, alter energy-transfer efficiency and yield distinct predictions for dark radiation, isocurvature modes, and gravitational waves.

Modulated reheating is a primordial-perturbation mechanism in which the inflaton decay rate is not spatially homogeneous but depends on a light spectator field, usually denoted $\sigma$, whose super-horizon fluctuations were generated during inflation. The local reheating condition $H \simeq \Gamma(\sigma)$ therefore occurs on different time slices in different Hubble patches, so the post-inflationary expansion history acquires a patch-dependent e-folding number and a curvature perturbation $\zeta = \delta N$. In the fluid language this is an inhomogeneous energy-transfer problem between an oscillating inflaton sector and radiation, and in the $\delta N$ language it is closely related to the curvaton scenario, the inhomogeneous end of inflation, and modulated preheating (Alabidi et al., 2010, Enqvist et al., 2012).

1. Background setup and origin of the modulation

After inflation, the standard perturbative realization assumes an oscillating inflaton fluid $\phi$ with $w_\phi \approx 0$ and a radiation fluid $r$ with $w=1/3$. Their homogeneous energy-transfer equations are

$$\dot\rho_\phi + 3H\rho_\phi = -Q, \qquad \dot\rho_r + 4H\rho_r = +Q,$$

with

$Q=\Gamma(\sigma)\rho_\phi,$

and the Friedmann equation

$H \simeq \Gamma(\sigma)$0

The essential ingredient is that $H \simeq \Gamma(\sigma)$1 depends on a light field $H \simeq \Gamma(\sigma)$2, so the decay hypersurface is modulated by the inflationary fluctuation $H \simeq \Gamma(\sigma)$3 (Alabidi et al., 2010).

The modulating field is taken to be effectively massless during inflation, so it acquires nearly scale-invariant fluctuations with

$H \simeq \Gamma(\sigma)$4

or equivalently $H \simeq \Gamma(\sigma)$5 at horizon exit (Kamada et al., 2010, Kobayashi et al., 2011). Spatial dependence in $H \simeq \Gamma(\sigma)$6 can arise from explicit modulus dependence of the inflaton coupling, for example

$H \simeq \Gamma(\sigma)$7

which gives $H \simeq \Gamma(\sigma)$8, or in the more general form

$H \simeq \Gamma(\sigma)$9

Reheating is then defined locally by $\zeta = \delta N$0, and the resulting curvature perturbation is computed as the fluctuation in the integrated expansion from an initial flat slice to a final uniform-density slice (Kamada et al., 2010, Kobayashi et al., 2011).

This setup makes the mechanism conceptually distinct from single-field slow-roll generation of $\zeta = \delta N$1. The origin of the adiabatic mode is not the inflaton potential fluctuation itself, but the conversion of an isocurvature fluctuation in $\zeta = \delta N$2 into curvature through a spatially modulated decay history.

2. $\zeta = \delta N$3 derivation and transfer efficiency

In the sudden-decay treatment with matter-dominated inflaton oscillations before reheating and radiation domination afterward, the number of e-folds from the end of inflation to a late time after reheating can be written, up to overall constants, as

$\zeta = \delta N$4

Hence

$\zeta = \delta N$5

Expanding

$\zeta = \delta N$6

one obtains

$\zeta = \delta N$7

This immediately gives the linear conversion of the spectator fluctuation into the curvature perturbation (Kamada et al., 2010).

A closely related $\zeta = \delta N$8 derivation writes the local e-folding number as

$\zeta = \delta N$9

when the final hypersurface is taken just after the decay time $\delta N$0 defined by $\delta N$1. In that formulation the linear and second-order coefficients are

$\delta N$2

A more careful treatment introduces the transfer-efficiency parameter

$\delta N$3

so that

$\delta N$4

For a nonlinear transfer law

$\delta N$5

the same structure survives with

$\delta N$6

Thus the mechanism can be expressed as the transfer of curvature between fluids, and the efficiency of that transfer becomes an explicit control parameter (Alabidi et al., 2010).

The physical content of $\delta N$7 is straightforward. If $\delta N$8, decay is slow compared with the Hubble rate, so the linear Gaussian piece $\delta N$9 is suppressed. The non-linear contribution is suppressed less strongly in the ratio defining $\phi$0, which is why inefficient transfer is associated with enhanced local non-Gaussianity.

3. Power spectrum, local non-Gaussianity, and the role of modulus dynamics

At leading order, the scalar power spectrum generated by the modulating field is

$\phi$1

and the local non-linearity parameter is

$\phi$2

If modulated reheating dominates the scalar perturbation, the tensor-to-scalar ratio is

$\phi$3

These expressions show that a large scalar signal does not require inflaton fluctuations to dominate, and the usual single-field relation $\phi$4 need not hold (Kamada et al., 2010).

In the efficiency language, the bispectrum amplitude scales inversely with the transfer efficiency: $\phi$5 For the example

$\phi$6

this gives

$\phi$7

which makes the $\phi$8 enhancement explicit (Alabidi et al., 2010).

The static-modulus approximation is not generally reliable. When the modulus evolves after horizon exit according to

$\phi$9

or, in the light-field slow-roll regime,

$w_\phi \approx 0$0

with $w_\phi \approx 0$1 during inflation and $w_\phi \approx 0$2 during the matter-dominated reheating era, the mapping from $w_\phi \approx 0$3 to $w_\phi \approx 0$4 alters both $w_\phi \approx 0$5 and $w_\phi \approx 0$6. A key quantity is

$w_\phi \approx 0$7

In this case,

$w_\phi \approx 0$8

For $w_\phi \approx 0$9, one finds $r$0 in a power-law potential; for $r$1, a varying $r$2 produces a new term proportional to $r$3. The numerical examples in the rolling-modulus analysis show that neglecting $r$4-motion can lead to $r$5 errors in $r$6, and the sudden-decay approximation can fail if $r$7 is nonmonotonic (Kobayashi et al., 2013).

Later non-perturbative treatments of Higgs-modulated reheating reached the same conclusion by different methods. Using the period-averaging method, an exact method, and a non-perturbative $r$8 method, the non-perturbative $r$9 approach was found to provide a reliable estimate across a wide range of reheating time and Higgs field values, including regimes where the Higgs oscillates significantly after inflation. In that setup, smaller Higgs self-coupling $w=1/3$0 leads to a larger curvature perturbation, and the non-Gaussianity is predominantly local (Deng et al., 23 Nov 2025).

4. Variants and adjacent mechanisms

The perturbative mechanism has a non-perturbative analogue in modulated preheating. There the inflaton oscillates in

$w=1/3$1

and energy transfer to the preheat field $w=1/3$2 proceeds through parametric resonance rather than perturbative decay. The comoving $w=1/3$3-occupancy grows as

$w=1/3$4

until backreaction shuts off the resonance at a time $w=1/3$5 determined by

$w=1/3$6

Because $w=1/3$7, the shutoff time is spatially modulated, and the curvature perturbation becomes

$w=1/3$8

By contrast, perturbative modulated reheating gives

$w=1/3$9

The extra factor $$\dot\rho_\phi + 3H\rho_\phi = -Q, \qquad \dot\rho_r + 4H\rho_r = +Q,$$0 can be $$\dot\rho_\phi + 3H\rho_\phi = -Q, \qquad \dot\rho_r + 4H\rho_r = +Q,$$1, so modulated preheating can produce a larger power spectrum and much larger non-Gaussianity than perturbative reheating (Enqvist et al., 2012).

Another extension is modulated curvaton decay, in which a third field $$\dot\rho_\phi + 3H\rho_\phi = -Q, \qquad \dot\rho_r + 4H\rho_r = +Q,$$2 modulates the curvaton decay rate $$\dot\rho_\phi + 3H\rho_\phi = -Q, \qquad \dot\rho_r + 4H\rho_r = +Q,$$3. In the sudden-decay expansion one finds

$$\dot\rho_\phi + 3H\rho_\phi = -Q, \qquad \dot\rho_r + 4H\rho_r = +Q,$$4

along with mixed second derivatives such as

$$\dot\rho_\phi + 3H\rho_\phi = -Q, \qquad \dot\rho_r + 4H\rho_r = +Q,$$5

This construction recovers the standard curvaton and pure modulated-reheating limits, and at tree level satisfies the Suyama–Yamaguchi inequality

$$\dot\rho_\phi + 3H\rho_\phi = -Q, \qquad \dot\rho_r + 4H\rho_r = +Q,$$6

with saturation when only one field sources $$\dot\rho_\phi + 3H\rho_\phi = -Q, \qquad \dot\rho_r + 4H\rho_r = +Q,$$7 or in the purely Gaussian case (Assadullahi et al., 2013).

Variants also exist in which the modulation is kinematic rather than purely coupling-driven. In velocity modulation, the rest-frame decay width of a daughter species is constant but the laboratory-frame rate fluctuates because the daughter Lorentz factor fluctuates: $$\dot\rho_\phi + 3H\rho_\phi = -Q, \qquad \dot\rho_r + 4H\rho_r = +Q,$$8 The induced curvature perturbation is maximized when the daughter is semi-relativistic at decay (Nakayama et al., 2011). In indirect modulation, the spectator field need not couple directly to the inflaton at all: it can modulate the phase space of the inflaton decay by generating masses for the decay products,

$$\dot\rho_\phi + 3H\rho_\phi = -Q, \qquad \dot\rho_r + 4H\rho_r = +Q,$$9

For $Q=\Gamma(\sigma)\rho_\phi,$0, $Q=\Gamma(\sigma)\rho_\phi,$1, and $Q=\Gamma(\sigma)\rho_\phi,$2, the induced spectrum was found to be

$Q=\Gamma(\sigma)\rho_\phi,$3

which is $Q=\Gamma(\sigma)\rho_\phi,$4 times larger than the observed $Q=\Gamma(\sigma)\rho_\phi,$5 (Karam et al., 2020).

These variants show that “modulated reheating” is better understood as a conversion class: any post-inflationary process that makes the decay hypersurface depend on a fluctuating spectator can realize the same $Q=\Gamma(\sigma)\rho_\phi,$6 logic, even when the microphysics differs substantially.

5. Isocurvature, dark sectors, baryogenesis, asymmetry, and small-scale signals

If the modulating field later becomes cold dark matter, its fluctuation carries an isocurvature mode

$Q=\Gamma(\sigma)\rho_\phi,$7

Writing

$Q=\Gamma(\sigma)\rho_\phi,$8

current CMB+LSS bounds impose $Q=\Gamma(\sigma)\rho_\phi,$9 for perfectly correlated or anti-correlated modes, and $H \simeq \Gamma(\sigma)$00 if uncorrelated. In the limit $H \simeq \Gamma(\sigma)$01, appropriate to dominant modulated preheating, $H \simeq \Gamma(\sigma)$02, which is of order unity for realistic $H \simeq \Gamma(\sigma)$03 and is therefore ruled out. The same bounds imply that if $H \simeq \Gamma(\sigma)$04 is CDM, modulated preheating is unlikely to give the dominant contribution to the curvature perturbation, and they also constrain $H \simeq \Gamma(\sigma)$05 and the primordial tensor-to-scalar ratio $H \simeq \Gamma(\sigma)$06 (Enqvist et al., 2012).

The modulus produced by reheating can also be dark radiation. If the dominant inflaton-decay channel produces $H \simeq \Gamma(\sigma)$07 modulus quanta and $H \simeq \Gamma(\sigma)$08 Standard Model quanta, then at reheating

$H \simeq \Gamma(\sigma)$09

and after neutrino decoupling the extra relativistic component is

$H \simeq \Gamma(\sigma)$10

For $H \simeq \Gamma(\sigma)$11, one finds $H \simeq \Gamma(\sigma)$12. The corresponding dark-radiation isocurvature $H \simeq \Gamma(\sigma)$13 vanishes at linear order if one decay channel dominates and the subdominant channels satisfy

$H \simeq \Gamma(\sigma)$14

If the same modulus instead becomes dominant cold dark matter through coherent oscillations, then

$H \simeq \Gamma(\sigma)$15

is required to suppress dark-matter isocurvature (Kobayashi et al., 2011).

A separate observational constraint comes from baryogenesis. In Affleck–Dine baryogenesis with modulated reheating, if

$H \simeq \Gamma(\sigma)$16

then

$H \simeq \Gamma(\sigma)$17

Current Planck bounds on a totally correlated baryon mode are roughly

$H \simeq \Gamma(\sigma)$18

Therefore the branches with $H \simeq \Gamma(\sigma)$19 are the viable ones if $H \simeq \Gamma(\sigma)$20 is dominantly generated by modulated reheating (Kamada et al., 2010).

Subdominant modulated reheating has also been used to model the hemispherical CMB power asymmetry. With a dominantly linear modulation of $H \simeq \Gamma(\sigma)$21 and a red-tilted $H \simeq \Gamma(\sigma)$22-spectrum generated by tachyonic growth, the total spectrum takes the form

$H \simeq \Gamma(\sigma)$23

and the asymmetry scales as

$H \simeq \Gamma(\sigma)$24

For the choice $H \simeq \Gamma(\sigma)$25, the model yields $H \simeq \Gamma(\sigma)$26, $H \simeq \Gamma(\sigma)$27, $H \simeq \Gamma(\sigma)$28, $H \simeq \Gamma(\sigma)$29, $H \simeq \Gamma(\sigma)$30, $H \simeq \Gamma(\sigma)$31, and $H \simeq \Gamma(\sigma)$32 (McDonald, 2013).

On much smaller scales, spectator-sourced modulated reheating can generate blue-tilted, strongly non-Gaussian curvature perturbations and scalar-induced stochastic gravitational waves. In the 2025 analysis with Higgs-like spectator couplings, Planck’s large-scale bound $H \simeq \Gamma(\sigma)$33 forces $H \simeq \Gamma(\sigma)$34 at $H \simeq \Gamma(\sigma)$35, while BBO/DECIGO-level signals require large couplings $H \simeq \Gamma(\sigma)$36. For “SM-like” $H \simeq \Gamma(\sigma)$37, the present-day induced background remains below $H \simeq \Gamma(\sigma)$38 at $H \simeq \Gamma(\sigma)$39, well under the quoted sensitivities (Benaco et al., 7 Oct 2025).

6. Explicit realizations in particle physics and UV constructions

Concrete realizations span effective field theory, supergravity, and string compactifications. In the LARGE Volume Scenario of type-IIB string flux compactifications, the inflaton is a fibre divisor and the modulaton is a blow-up mode made light by poly-instanton corrections. The visible-sector gauge bosons live on a D7 stack, and the inflaton decay rate into gauge bosons is

$H \simeq \Gamma(\sigma)$40

The resulting local bispectrum parameter is

$H \simeq \Gamma(\sigma)$41

and for generic values of the underlying parameters the model predicts a local bispectrum with $H \simeq \Gamma(\sigma)$42 of order “a few”. A moderate tuning of the parameters can raise $H \simeq \Gamma(\sigma)$43 to $H \simeq \Gamma(\sigma)$44. In the numerical example with $H \simeq \Gamma(\sigma)$45, $H \simeq \Gamma(\sigma)$46, $H \simeq \Gamma(\sigma)$47, $H \simeq \Gamma(\sigma)$48, $H \simeq \Gamma(\sigma)$49, $H \simeq \Gamma(\sigma)$50, and $H \simeq \Gamma(\sigma)$51, one finds

$H \simeq \Gamma(\sigma)$52

(Cicoli et al., 2012).

In Higgs-modulated reheating within RG-improved inflation motivated by asymptotically safe gravity, the Higgs field $H \simeq \Gamma(\sigma)$53 plays the role of the modulator, and the decay rate is expanded as

$H \simeq \Gamma(\sigma)$54

The universal part of the local non-Gaussianity is

$H \simeq \Gamma(\sigma)$55

and for the $H \simeq \Gamma(\sigma)$56 coupling used in that analysis, $H \simeq \Gamma(\sigma)$57 gives

$H \simeq \Gamma(\sigma)$58

However, the intrinsic Higgs self-interaction can generate

$H \simeq \Gamma(\sigma)$59

for $H \simeq \Gamma(\sigma)$60, which conflicts with the Planck bound unless the Higgs fraction is reduced or parameters are dialed (Cai et al., 2013).

A supergravity $H \simeq \Gamma(\sigma)$61-inflation realization instead uses gravitational reheating of conformally noninvariant fields whose masses depend on a light flat direction,

$H \simeq \Gamma(\sigma)$62

so that the scalaron decay rate becomes

$H \simeq \Gamma(\sigma)$63

The total perturbation is a mixture of the inflationary and modulated-reheating contributions. In this model the combined spectral index is

$H \simeq \Gamma(\sigma)$64

which lies between $H \simeq \Gamma(\sigma)$65 and $H \simeq \Gamma(\sigma)$66, the local non-Gaussianity can be $H \simeq \Gamma(\sigma)$67, and the tensor-to-scalar ratio satisfies

$H \simeq \Gamma(\sigma)$68

(Watanabe et al., 2013).

A related two-field construction lets the same light scalar act both as curvaton and as modulator of the inflaton decay rate,

$H \simeq \Gamma(\sigma)$69

The final linear perturbation contains both contributions,

$H \simeq \Gamma(\sigma)$70

Near the cancellation line

$H \simeq \Gamma(\sigma)$71

the linear $H \simeq \Gamma(\sigma)$72-piece is suppressed and the higher-order terms dominate, giving

$H \simeq \Gamma(\sigma)$73

with the possibility that both the tensor-to-scalar ratio and the non-linearity parameters are simultaneously large (Choi et al., 2012).

Across these realizations, the common structure is the same: a light field modulates a post-inflationary decay hypersurface, and the conversion of its isocurvature fluctuation into $H \simeq \Gamma(\sigma)$74 is governed by the local sensitivity of the decay history to that field. The differences between models lie in how $H \simeq \Gamma(\sigma)$75 depends on the spectator, how the spectator evolves between horizon exit and reheating, and whether additional sectors generate isocurvature, non-Gaussian, or small-scale signatures that constrain the mechanism.