Closed Quadratic Inflation

• Closed quadratic inflation denotes a family of models where a quadratic inflaton potential is combined with either a closed FLRW background or minimal kinetic deformations.
• The framework modifies standard predictions by incorporating non-canonical kinetic terms, higher-order corrections, and supergravity effects to achieve tensor-to-scalar ratios in line with CMB data.
• Advanced treatments employ gauge-invariant perturbation theory and multifield extensions to reconcile discrete hyperspherical modes and improve fits to Planck and lensing anomalies.

Searching arXiv for the specified topic and recent related papers. Closed quadratic inflation denotes a family of inflationary constructions centered on the quadratic potential $V(\phi)=\frac12 m^2\phi^2$, but embedded in additional structure that either modifies the background geometry or deforms the effective dynamics. In one usage, the model is genuinely a closed FLRW cosmology with $\Omega_{\mathcal K}<0$, discrete hyperspherical modes, and a primordial spectrum derived consistently in curved space. In another, the quadratic potential is “closed” or completed by higher-order kinetic, curvature, polynomial, or supergravity effects that alter the standard prediction $n_s \approx 0.967$, $r \approx 0.13$, which is in strong tension with current CMB bounds. A recent minimal deformation based on canonical normalization of a weakly non-canonical scalar produces $$V(\chi)=\frac12 m^2\left(\chi-\frac{\gamma}{14}\chi^7\right)^2$$, yielding $n_s \approx 0.965$ and $r \approx 0.036$, while a closed-universe treatment of $m^2\phi^2$ inflation shifts CMB curvature constraints toward spatial flatness when the primordial spectrum is computed in a gauge-invariant manner (Djeha, 7 Nov 2025, Specogna et al., 30 Sep 2025).

1. Terminology and the quadratic benchmark

The common starting point is the textbook single-field quadratic model,

$V(\phi)=\frac12 m^2\phi^2,$

whose slow-roll predictions are rigidly tied to the field value at horizon exit. In the formulations summarized here, this rigidity is precisely what motivates extensions: in one case to a spatially closed background, in another to controlled deformations of the kinetic sector, and in others to supergravity, monodromy, non-minimal gravity couplings, or multifield completions. This suggests that “closed quadratic inflation” is not a uniquely fixed technical term, but a context-dependent label whose meaning is set by the underlying construction (Specogna et al., 30 Sep 2025, Oikonomou et al., 2022).

Usage	Defining structure	Representative papers
Spatially closed quadratic inflation	Closed FLRW background with $V(\phi)=\frac12 m^2\phi^2$	(Specogna et al., 30 Sep 2025, Harigaya et al., 2014)
Minimal deformation of quadratic inflation	Canonical quadratic model modified by higher-order effective terms	(Djeha, 7 Nov 2025, Oikonomou et al., 2022)
Supergravity or UV completion of quadratic inflation	Quadratic dynamics realized through SUGRA, monodromy, or non-minimal coupling	(Ellis et al., 2014, Li et al., 2014, Pallis et al., 2014, Pallis et al., 2014)

For the unmodified quadratic model, one quoted benchmark is $\Omega_{\mathcal K}<0$0 and $\Omega_{\mathcal K}<0$1, while in a standard supergravity presentation with $\Omega_{\mathcal K}<0$2 one finds $\Omega_{\mathcal K}<0$3 and $\Omega_{\mathcal K}<0$4. In both cases, the tension is driven mainly by the tensor prediction, not by the scalar tilt (Djeha, 7 Nov 2025, Harigaya et al., 2014).

2. Closed FLRW geometry and perturbation theory

In the genuinely geometric usage, closed quadratic inflation is formulated on a closed FLRW background. One explicit line element is

$\Omega_{\mathcal K}<0$5

with $\Omega_{\mathcal K}<0$6 for the closed case, while an earlier closed-universe treatment writes

$\Omega_{\mathcal K}<0$7

The essential structural difference from flat space is that scalar perturbations are expanded in hyperspherical harmonics on $\Omega_{\mathcal K}<0$8, so the mode labels are discrete rather than continuous. In the older notation, the scalar Laplacian eigenfunctions satisfy $\Omega_{\mathcal K}<0$9; in the more recent notation, the physical scalar modes begin at $n_s \approx 0.967$0, while $n_s \approx 0.967$1 are non-dynamical or pure gauge (Specogna et al., 30 Sep 2025, Ratra, 2017).

The closed-universe perturbation framework predates recent quadratic analyses and is not intrinsically tied to a quadratic potential. A notable example studies closed de Sitter inflation driven by

$n_s \approx 0.967$2

not by $n_s \approx 0.967$3, and derives the late-time matter power spectrum from inflation, radiation, and matter epochs matched at nonsingular transitions. Its inflationary vacuum is fixed by Hawking’s prescription that only field configurations regular on the Euclidean de Sitter sphere are included, which in the mode solution selects $n_s \approx 0.967$4, $n_s \approx 0.967$5 up to a phase. The resulting spectrum is not a pure power law and reduces to the flat Harrison–Zel’dovich form only in the large-$n_s \approx 0.967$6 limit (Ratra, 2017).

This distinction is conceptually important. A closed universe does not by itself specify the inflaton potential, and the closed-universe perturbation problem must be solved in a gauge-consistent manner independently of whether the background potential is quadratic, nearly constant, or otherwise deformed.

3. Gauge-invariant closed quadratic inflation and CMB constraints

A recent reassessment of Planck constraints treats closed quadratic inflation as a physically consistent alternative to the usual phenomenological curvature parametrization. The inflationary sector is the standard chaotic potential

$n_s \approx 0.967$7

but the primordial scalar spectrum is derived in a gauge-invariant way using the conserved variable $n_s \approx 0.967$8, following the closed-universe formalism of Kiefer and Vardanyan. The flat-space reference spectrum is written as

$n_s \approx 0.967$9

while the closed spectrum takes the practical form

$r \approx 0.13$0

with $r \approx 0.13$1 and $r \approx 0.13$2 (Specogna et al., 30 Sep 2025).

The central comparison is between this inflation-derived spectrum and the phenomenological ansatz

$r \approx 0.13$3

which is commonly used to mimic curvature effects at large scales but is not a complete inflationary calculation. When the closed quadratic model is confronted with Planck PR3 $r \approx 0.13$4 and PR4 $r \approx 0.13$5 likelihoods, the inferred preference for negative curvature weakens. In $r \approx 0.13$6, the preference for $r \approx 0.13$7 decreases from $r \approx 0.13$8 to $r \approx 0.13$9; in $$V(\chi)=\frac12 m^2\left(\chi-\frac{\gamma}{14}\chi^7\right)^2$$0, it reduces to $$V(\chi)=\frac12 m^2\left(\chi-\frac{\gamma}{14}\chi^7\right)^2$$1 (Specogna et al., 30 Sep 2025).

Dataset	Standard curvature treatment	Closed quadratic inflation
PL18	$$V(\chi)=\frac12 m^2\left(\chi-\frac{\gamma}{14}\chi^7\right)^2$$2	$$V(\chi)=\frac12 m^2\left(\chi-\frac{\gamma}{14}\chi^7\right)^2$$3
PL18 + lensing	$$V(\chi)=\frac12 m^2\left(\chi-\frac{\gamma}{14}\chi^7\right)^2$$4	$$V(\chi)=\frac12 m^2\left(\chi-\frac{\gamma}{14}\chi^7\right)^2$$5
CamSpec	$$V(\chi)=\frac12 m^2\left(\chi-\frac{\gamma}{14}\chi^7\right)^2$$6	$$V(\chi)=\frac12 m^2\left(\chi-\frac{\gamma}{14}\chi^7\right)^2$$7
CamSpec + lensing	$$V(\chi)=\frac12 m^2\left(\chi-\frac{\gamma}{14}\chi^7\right)^2$$8	$$V(\chi)=\frac12 m^2\left(\chi-\frac{\gamma}{14}\chi^7\right)^2$$9

The fit behavior is scale dependent. At low multipoles, the model naturally suppresses power for $n_s \approx 0.965$0, improving the fit to the quadrupole anomaly. At high multipoles, however, the model permits less strongly negative curvature than the phenomenological treatment, so it cannot account as efficiently for the $n_s \approx 0.965$1 lensing anomaly. The result is a mild deterioration for PR3 $n_s \approx 0.965$2, where the anomaly is stronger, but a mild improvement for PR4 $n_s \approx 0.965$3, where the anomaly is less pronounced (Specogna et al., 30 Sep 2025).

4. Minimal deformations that revive quadratic inflation

A distinct line of work keeps the inflationary background spatially flat but rescues quadratic inflation by a minimal deformation of the scalar sector. One recent realization begins with a non-canonical kinetic function

$n_s \approx 0.965$4

with $n_s \approx 0.965$5, together with a bare quadratic potential $n_s \approx 0.965$6. After the field redefinition

$n_s \approx 0.965$7

the canonically normalized potential becomes

$n_s \approx 0.965$8

Its leading correction is a negative $n_s \approx 0.965$9 term,

$r \approx 0.036$0

which flattens the potential at large field values and suppresses the slow-roll parameter $r \approx 0.036$1, hence the tensor amplitude $r \approx 0.036$2 (Djeha, 7 Nov 2025).

This deformation leaves the scalar tilt nearly unchanged while moving the tensor prediction into the Planck-allowed region. The paper quotes the transition from the uncorrected values

$r \approx 0.036$3

to corrected values around

$r \approx 0.036$4

A representative numerical point is

$r \approx 0.036$5

for $r \approx 0.036$6, $r \approx 0.036$7, $r \approx 0.036$8, and $r \approx 0.036$9. The same deformation changes reheating from the matter-like quadratic value $m^2\phi^2$0 to $m^2\phi^2$1, raises the reheating temperature by a factor of about $m^2\phi^2$2 at fixed $m^2\phi^2$3, and extends the reheating duration from $m^2\phi^2$4 to $m^2\phi^2$5 at fixed $m^2\phi^2$6 GeV (Djeha, 7 Nov 2025).

A related but conceptually different rescue mechanism adds an $m^2\phi^2$7 correction in the Jordan/string frame,

$m^2\phi^2$8

while retaining the quadratic potential

$m^2\phi^2$9

Here the potential is not deformed directly; instead the modified gravity sector changes the Raychaudhuri equation,

$V(\phi)=\frac12 m^2\phi^2,$0

and thereby alters the slow-roll hierarchy. For $V(\phi)=\frac12 m^2\phi^2,$1, $V(\phi)=\frac12 m^2\phi^2,$2, and $V(\phi)=\frac12 m^2\phi^2,$3, the model gives

$V(\phi)=\frac12 m^2\phi^2,$4

That paper explicitly states that it does not study a closed spatial geometry, so here “closed” can only be understood as a modified or completed quadratic-inflation scenario rather than a $V(\phi)=\frac12 m^2\phi^2,$5 universe (Oikonomou et al., 2022).

5. Supergravity, monodromy, and non-minimal completions

Quadratic inflation has also been embedded in supergravity frameworks where the central difficulty is the $V(\phi)=\frac12 m^2\phi^2,$6-problem and the control of trans-Planckian motion. In no-scale supergravity, one construction based on

$V(\phi)=\frac12 m^2\phi^2,$7

with suitable stabilization yields a quadratic inflaton direction identified with $V(\phi)=\frac12 m^2\phi^2,$8. The reported predictions are $V(\phi)=\frac12 m^2\phi^2,$9 for $V(\phi)=\frac12 m^2\phi^2$0 and $V(\phi)=\frac12 m^2\phi^2$1 for $V(\phi)=\frac12 m^2\phi^2$2. An alternative explicit no-scale model with

$V(\phi)=\frac12 m^2\phi^2$3

gives $V(\phi)=\frac12 m^2\phi^2$4 for $V(\phi)=\frac12 m^2\phi^2$5 and $V(\phi)=\frac12 m^2\phi^2$6 for $V(\phi)=\frac12 m^2\phi^2$7, and the same work proposes identifying the inflaton with a right-handed sneutrino whose decay can generate a lepton asymmetry via leptogenesis (Ellis et al., 2014).

A different supergravity realization identifies the inflaton with the phase of a complex field, not with a canonically rolling real scalar. With Kähler potential chosen so that the phase enjoys a global $V(\phi)=\frac12 m^2\phi^2$8 symmetry, and superpotential

$V(\phi)=\frac12 m^2\phi^2$9

the scalar potential becomes

$\Omega_{\mathcal K}<0$00

At the radial minimum $\Omega_{\mathcal K}<0$01, the effective inflaton potential is

$\Omega_{\mathcal K}<0$02

so the model reproduces the standard quadratic predictions

$\Omega_{\mathcal K}<0$03

The same paper characterizes this as a new type of monodromy inflation in supersymmetric field theory, with the large excursion realized by the phase while the field norm remains sub-Planckian (Li et al., 2014).

Other completions flatten the quadratic model by coupling it non-minimally to gravity. In one case the Jordan-frame action uses

$\Omega_{\mathcal K}<0$04

so that the Einstein-frame potential is

$\Omega_{\mathcal K}<0$05

For sufficiently large $\Omega_{\mathcal K}<0$06, the model yields $\Omega_{\mathcal K}<0$07 in the non-SUSY case, and in the no-scale SUGRA limit gives $\Omega_{\mathcal K}<0$08, $\Omega_{\mathcal K}<0$09, with $\Omega_{\mathcal K}<0$10. Beyond no-scale SUGRA, the same framework allows $\Omega_{\mathcal K}<0$11 with $\Omega_{\mathcal K}<0$12 GeV (Pallis et al., 2014).

The BICEP2-era literature also contains “hybrid-to-quadratic” and polynomial versions of this completion logic. One supergravity model based on the standard hybrid superpotential

$\Omega_{\mathcal K}<0$13

and a high-scale soft mass for the singlet inflaton yields an effective quadratic regime with $\Omega_{\mathcal K}<0$14 and $\Omega_{\mathcal K}<0$15, while still ending through a waterfall transition at a gauge-symmetry-breaking scale $\Omega_{\mathcal K}<0$16 GeV (Pallis et al., 2014). Another polynomial model,

$\Omega_{\mathcal K}<0$17

was presented as a small quartic correction to the quadratic backbone and, for $\Omega_{\mathcal K}<0$18, was reported to give $\Omega_{\mathcal K}<0$19 and $\Omega_{\mathcal K}<0$20 for $\Omega_{\mathcal K}<0$21 in the BICEP2 context (Kobayashi et al., 2014).

6. Initial conditions, multifield rescues, and conceptual limits

Closed-universe initial conditions place additional constraints on nearly quadratic models. In a supergravity construction with

$\Omega_{\mathcal K}<0$22

the quadratic inflaton potential is modified by shift-symmetry breaking in the Kähler potential,

$\Omega_{\mathcal K}<0$23

Near the CMB scales this remains nearly quadratic, but at larger field values it becomes steeper because of the $\Omega_{\mathcal K}<0$24 dependence of the supergravity potential. The paper analyzes a closed FRW background,

$\Omega_{\mathcal K}<0$25

and argues that if the deformation parameter is too large, a closed universe of Planckian initial size collapses before inflation can begin. The resulting bound is approximately $\Omega_{\mathcal K}<0$26, which translates into a lower bound $\Omega_{\mathcal K}<0$27, with a numerical scan giving a slightly weaker but consistent $\Omega_{\mathcal K}<0$28 for the specific coefficients used there (Harigaya et al., 2014).

A separate line of work rescues quadratic inflation not by changing the geometry or the single-field potential, but by adding extra quadratic degrees of freedom. In a model with potential

$\Omega_{\mathcal K}<0$29

two inflatons and one curvaton decouple the rigid single-field relation $\Omega_{\mathcal K}<0$30. The paper finds that one extra curvaton can lower $\Omega_{\mathcal K}<0$31 but usually pushes $\Omega_{\mathcal K}<0$32 too high, whereas two inflatons plus one curvaton can satisfy Planck constraints. A three-sneutrino example with

$\Omega_{\mathcal K}<0$33

gives

$\Omega_{\mathcal K}<0$34

while also fitting neutrino mass-squared differences and mixing angles (Ellis et al., 2013).

Closed-universe dynamics can also be discussed independently of any quadratic potential. An exact deflation–inflation construction with Robertson–Walker metric

$\Omega_{\mathcal K}<0$35

shows that a nonsingular minimum radius requires $\Omega_{\mathcal K}<0$36, and in the constant-vacuum-energy case yields

$\Omega_{\mathcal K}<0$37

That construction uses a generic scalar potential $\Omega_{\mathcal K}<0$38, not necessarily $\Omega_{\mathcal K}<0$39, and therefore serves mainly as a reminder that “closed inflation” and “quadratic inflation” are logically distinct inputs even when they are combined in a single model (Mashkevich, 2009).

In aggregate, the modern literature presents closed quadratic inflation less as a single model than as a structured research program. One branch studies a genuinely closed universe with gauge-invariant curvature perturbations and discrete spectra; another seeks minimal deformations that flatten the $\Omega_{\mathcal K}<0$40 potential and modify reheating; others construct supergravity, monodromy, non-minimal, or multifield completions. Across these variants, the shared aim is to retain the calculational simplicity of quadratic inflation while relaxing the rigid tensor prediction and, in the closed-universe branch, enforcing consistency between curvature, perturbation theory, and CMB inference (Specogna et al., 30 Sep 2025, Djeha, 7 Nov 2025).