Collapsed Modified Power Law Inflation

• Collapsed Modified Power Law Inflation is a framework that deforms the standard power-law paradigm through modified potentials, background energy decay, or CSL-induced quantum collapse.
• It employs mechanisms like plateau potentials and anomalous conformal couplings to produce a finite minimum and address the graceful-exit problem while matching observational data.
• The models predict distinct spectral tilts and tensor-to-scalar ratios, enabling differentiation between nearly de Sitter and modified power-law regimes based on CMB observations.

Searching arXiv for the core papers and related formulations of collapsed/modified power-law inflation. Collapsed Modified Power Law Inflation denotes a family of inflationary constructions in which a power-law, inverse-power-law, or power-law-like sector is deformed so that the background expansion, the inflaton potential, or the quantum dynamics of primordial perturbations departs from vanilla power-law inflation. The literature surveyed here indicates that the expression is not used in a single uniform sense. In one usage, an inverse power-law potential is “collapsed” into a plateau with a finite minimum; in another, a power-law phase is modified by anomalous conformal coupling or by an appended tail; in a third, the term “collapsed” refers to Continuous Spontaneous Localization (CSL) acting on primordial perturbations while the background remains vanilla power-law inflation (Lu, 2013, Paul et al., 2020, Das, 20 Aug 2025).

1. Terminological scope and recurrent meanings

Across the relevant literature, “collapsed modified power law inflation” is best understood as an umbrella expression for several technically distinct mechanisms that alter the standard power-law paradigm. The common theme is that a simple power-law structure is retained at leading order but is reshaped to improve phenomenology, obtain a graceful exit, or encode quantum-to-classical collapse.

Usage	Representative construction	Defining feature
Plateau collapse of a power law	$$V(\phi)=V_0\left(1-\mu(M_{\rm pl}/\phi)^\beta\right)^2$$	finite minimum and large-field plateau
Background-level modification	$a(t)=a_0 t^q$ with anomalous conformal coupling	$\Lambda_{\rm eff}\propto t^{-2}$
Collapse of perturbations	vanilla power-law inflation plus CSL	spectral shift $\delta$, zero running

This taxonomy suggests that “collapse” may refer to the potential collapsing to zero at a finite field value, the effective cosmological term decaying during inflation, or the wavefunctional collapse of quantum perturbations. A further usage appears in entropic and modified-gravity realizations, where observational constraints force the deformation parameter toward a standard limit, effectively collapsing the modification back to conventional inflationary behavior (Luciano, 2023, Khodam-Mohammadi, 2024).

2. Plateau-type collapse of inverse power-law inflation

A concrete potential-level realization is the generalized inverse power-law model

$$V(\phi)=V_0\left(1-\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right)^2,\qquad \beta>0,\ \mu>0,$$

introduced as a generalization of the intermediate-inflation inverse power-law potential $V_{\rm int}(\phi)\propto (M_{\rm pl}/\phi)^\beta$. Its defining structural change is the addition of a constant plateau and the squared form, which guarantee $V(\phi)\ge 0$ and generate a finite minimum at

$$\phi_{\rm min}=\mu^{1/\beta}M_{\rm pl},\qquad V(\phi_{\rm min})=0.$$

Inflation occurs in the large-field regime $M_{\rm pl}\ll \phi$, where

$$V(\phi)\approx V_0\left(1-2\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right),$$

so the model behaves as a quasi-de Sitter plateau with an inverse-power correction (Lu, 2013).

In this regime, the slow-roll parameters simplify to

$a(t)=a_0 t^q$0

with $a(t)=a_0 t^q$1, so the relevant portion of the plateau is concave down. Inflation ends when $a(t)=a_0 t^q$2, namely when the field approaches the finite minimum rather than running along an infinite inverse-power tail. In terms of the horizon-crossing e-fold number $a(t)=a_0 t^q$3,

$a(t)=a_0 t^q$4

which yields

$a(t)=a_0 t^q$5

and, in the regime $a(t)=a_0 t^q$6,

$a(t)=a_0 t^q$7

while

$a(t)=a_0 t^q$8

A notable feature is that $a(t)=a_0 t^q$9 is independent of $\Lambda_{\rm eff}\propto t^{-2}$0, whereas $\Lambda_{\rm eff}\propto t^{-2}$1 retains a mild $\Lambda_{\rm eff}\propto t^{-2}$2-dependence (Lu, 2013).

This model interpolates between a trivial de Sitter limit and a Starobinsky-like plateau limit. As $\Lambda_{\rm eff}\propto t^{-2}$3, one has $\Lambda_{\rm eff}\propto t^{-2}$4, $\Lambda_{\rm eff}\propto t^{-2}$5, and $\Lambda_{\rm eff}\propto t^{-2}$6, which is observationally excluded. As $\Lambda_{\rm eff}\propto t^{-2}$7,

$\Lambda_{\rm eff}\propto t^{-2}$8

so the scalar tilt approaches the Starobinsky $\Lambda_{\rm eff}\propto t^{-2}$9 form while the tensor amplitude becomes even smaller. For $\delta$0, explicit examples include $\delta$1, which gives $\delta$2 and $\delta$3, and the model can generate $\delta$4 with $\delta$5, in contrast to the original intermediate inverse power-law model whose predictions lie outside Planck’s $\delta$6 confidence region for any $\delta$7 (Lu, 2013).

3. Background-level deformations of power-law expansion

A distinct realization modifies power-law inflation at the level of the action rather than by reshaping a potential minimum. In the anomalously coupled model, gravity is described by $\delta$8, while the inflaton sector lives in a conformally related metric

$\delta$9

For a monomial inflaton potential in the inflaton frame,

$$V(\phi)=V_0\left(1-\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right)^2,\qquad \beta>0,\ \mu>0,$$0

the slow-roll effective action contains an exponentially damped potential $$V(\phi)=V_0\left(1-\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right)^2,\qquad \beta>0,\ \mu>0,$$1. The field equations admit an exact modified power-law solution

$$V(\phi)=V_0\left(1-\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right)^2,\qquad \beta>0,\ \mu>0,$$2

with

$$V(\phi)=V_0\left(1-\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right)^2,\qquad \beta>0,\ \mu>0,$$3

The effective cosmological term

$$V(\phi)=V_0\left(1-\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right)^2,\qquad \beta>0,\ \mu>0,$$4

then decays as $$V(\phi)=V_0\left(1-\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right)^2,\qquad \beta>0,\ \mu>0,$$5, so inflation ends when the slow-roll hierarchy fails and the conformal field freezes to $$V(\phi)=V_0\left(1-\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right)^2,\qquad \beta>0,\ \mu>0,$$6. In this sense the background itself “collapses” through an internal decay of vacuum energy, rather than through a finite potential minimum (Paul et al., 2020).

The perturbation-sector predictions of this anomalously coupled model are

$$V(\phi)=V_0\left(1-\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right)^2,\qquad \beta>0,\ \mu>0,$$7

so the model can produce small $$V(\phi)=V_0\left(1-\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right)^2,\qquad \beta>0,\ \mu>0,$$8 for $$V(\phi)=V_0\left(1-\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right)^2,\qquad \beta>0,\ \mu>0,$$9, especially $V_{\rm int}(\phi)\propto (M_{\rm pl}/\phi)^\beta$0 with $V_{\rm int}(\phi)\propto (M_{\rm pl}/\phi)^\beta$1. The same framework also addresses the graceful-exit problem of conventional exponential power-law inflation, because the effective vacuum energy is no longer constant in time but decays as the universe expands (Paul et al., 2020).

A related but technically different rescue of power-law inflation uses a non-canonical kinetic term,

$V_{\rm int}(\phi)\propto (M_{\rm pl}/\phi)^\beta$2

with an inverse power-law potential. The model retains

$V_{\rm int}(\phi)\propto (M_{\rm pl}/\phi)^\beta$3

but changes the tensor amplitude to

$V_{\rm int}(\phi)\propto (M_{\rm pl}/\phi)^\beta$4

and predicts

$V_{\rm int}(\phi)\propto (M_{\rm pl}/\phi)^\beta$5

An extended potential

$V_{\rm int}(\phi)\propto (M_{\rm pl}/\phi)^\beta$6

provides a minimum at $V_{\rm int}(\phi)\propto (M_{\rm pl}/\phi)^\beta$7, thereby resolving the graceful-exit problem while preserving effective power-law behavior over the CMB-relevant range (Unnikrishnan et al., 2013).

4. Collapse dynamics of primordial perturbations

In the CSL-based formulation, “collapsed modified Power Law Inflation” means vanilla power-law inflation with exponential potential,

$V_{\rm int}(\phi)\propto (M_{\rm pl}/\phi)^\beta$8

supplemented by CSL dynamics acting on scalar and tensor perturbations. Vanilla power-law inflation has constant slow-roll parameters

$V_{\rm int}(\phi)\propto (M_{\rm pl}/\phi)^\beta$9

hence

$V(\phi)\ge 0$0

with zero running, but it suffers from a graceful-exit problem and predicts $V(\phi)\ge 0$1 that is too large for the observed $V(\phi)\ge 0$2 (Das, 20 Aug 2025).

CSL modifies the mode Schrödinger equation through a stochastic Wiener term and a nonlinear collapse term,

$V(\phi)\ge 0$3

with a scale- and time-dependent collapse rate

$V(\phi)\ge 0$4

This induces a constant spectral shift

$V(\phi)\ge 0$5

so that power-law inflation predicts

$V(\phi)\ge 0$6

with

$V(\phi)\ge 0$7

The model thereby preserves the exact power-law background but modifies the perturbation spectra through objective collapse (Das, 20 Aug 2025).

The observationally relevant parameter window is sharp. From $V(\phi)\ge 0$8, one obtains

$V(\phi)\ge 0$9

Using $$\phi_{\rm min}=\mu^{1/\beta}M_{\rm pl},\qquad V(\phi_{\rm min})=0.$$0, consistency requires

$$\phi_{\rm min}=\mu^{1/\beta}M_{\rm pl},\qquad V(\phi_{\rm min})=0.$$1

In the allowed region, the tensor spectrum is more red-tilted than in vanilla power-law inflation, while both scalar and tensor runnings remain zero. A central caveat is that CSL does not solve the graceful-exit problem at the background level; the paper explicitly notes that one still needs an additional mechanism such as a modified potential, a perturbed power-law scenario, or a waterfall transition (Das, 20 Aug 2025).

5. Modified-gravity and entropic realizations

A major class of modified power-law inflation models is gravitational rather than scalar-potential based. In power-law Starobinsky inflation,

$$\phi_{\rm min}=\mu^{1/\beta}M_{\rm pl},\qquad V(\phi_{\rm min})=0.$$2

current data allow small but nonzero departures from the $$\phi_{\rm min}=\mu^{1/\beta}M_{\rm pl},\qquad V(\phi_{\rm min})=0.$$3 limit. A numerical MCMC analysis finds

$$\phi_{\rm min}=\mu^{1/\beta}M_{\rm pl},\qquad V(\phi_{\rm min})=0.$$4

at $$\phi_{\rm min}=\mu^{1/\beta}M_{\rm pl},\qquad V(\phi_{\rm min})=0.$$5 confidence level, with derived

$$\phi_{\rm min}=\mu^{1/\beta}M_{\rm pl},\qquad V(\phi_{\rm min})=0.$$6

so the model is a viable deformation of Starobinsky inflation rather than a return to strict $$\phi_{\rm min}=\mu^{1/\beta}M_{\rm pl},\qquad V(\phi_{\rm min})=0.$$7 behavior (Saini et al., 2023).

Adding the $$\phi_{\rm min}=\mu^{1/\beta}M_{\rm pl},\qquad V(\phi_{\rm min})=0.$$8-attractor deformation yields the power-law $$\phi_{\rm min}=\mu^{1/\beta}M_{\rm pl},\qquad V(\phi_{\rm min})=0.$$9-Starobinsky model, whose Einstein-frame potential combines the $M_{\rm pl}\ll \phi$0 correction with an $M_{\rm pl}\ll \phi$1-modified exponential. Its preferred parameters are

$M_{\rm pl}\ll \phi$2

with

$M_{\rm pl}\ll \phi$3

and a Bayes factor that mildly favors this two-parameter deformation over pure Starobinsky inflation (Saini et al., 22 May 2025).

A Jordan-frame rectification of power-law $M_{\rm pl}\ll \phi$4 inflation reaches a related conclusion from a different route. Instead of identifying $M_{\rm pl}\ll \phi$5 with a strict power-law scale factor, the rectified framework parameterizes observables through

$M_{\rm pl}\ll \phi$6

leading at leading order to

$M_{\rm pl}\ll \phi$7

For the constant-$M_{\rm pl}\ll \phi$8 class, compatibility with ACT and Planck is obtained for

$M_{\rm pl}\ll \phi$9

and a representative value $$V(\phi)\approx V_0\left(1-2\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right),$$0 gives

$$V(\phi)\approx V_0\left(1-2\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right),$$1

This disentangles a power-law Lagrangian from a power-law background and effectively collapses the dynamics toward a quasi-de Sitter, Starobinsky-like regime (Odintsov et al., 8 Sep 2025).

Entropy-based realizations display an explicit “collapse of the modification” under data constraints. In Barrow entropy cosmology, slow-roll power-law inflation with $$V(\phi)\approx V_0\left(1-2\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right),$$2 yields

$$V(\phi)\approx V_0\left(1-2\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right),$$3

but observational consistency forces

$$V(\phi)\approx V_0\left(1-2\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right),$$4

so Barrow corrections are effectively negligible during inflation (Luciano, 2023). By contrast, non-extensive entropic cosmology with Tsallis, Rényi, and Sharma–Mittal entropies broadens the viable parameter space of monomial power-law inflation: Tsallis entropy admits $$V(\phi)\approx V_0\left(1-2\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right),$$5 to $$V(\phi)\approx V_0\left(1-2\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right),$$6 in alignment with Planck 2018, whereas the Rényi and Sharma–Mittal models require an extremely small entropy parameter,

$$V(\phi)\approx V_0\left(1-2\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right),$$7

to achieve successful power-law inflation with $$V(\phi)\approx V_0\left(1-2\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right),$$8 around $$V(\phi)\approx V_0\left(1-2\mu\left(\frac{M_{\rm pl}}{\phi}\right)^\beta\right),$$9–$a(t)=a_0 t^q$00 (Khodam-Mohammadi, 2024).

No-scale supergravity also furnishes a microscopic origin for a power-law Starobinsky deformation. A power-law $a(t)=a_0 t^q$01 model can be obtained from a modified Kähler potential containing a $a(t)=a_0 t^q$02 term, and it is equivalent to generalized non-minimal curvature-coupled models of the form

$a(t)=a_0 t^q$03

In that framework, $a(t)=a_0 t^q$04 yields $a(t)=a_0 t^q$05, showing how small departures from $a(t)=a_0 t^q$06 can raise the tensor amplitude substantially while breaking global Weyl symmetry (Chakravarty et al., 2014).

6. Exit mechanisms, transient tails, and scale-dependent deformations

One route to a “collapsed” power-law phase is to retain power-law behavior only over a finite interval. In warm inflation, exact $a(t)=a_0 t^q$07 can be made observationally viable through thermal dissipation, and the graceful-exit problem can be addressed by perturbing the Hubble rate to an affine form,

$a(t)=a_0 t^q$08

with small negative $a(t)=a_0 t^q$09. In the weak-dissipation regime this induces a modified potential

$a(t)=a_0 t^q$10

with

$a(t)=a_0 t^q$11

For $a(t)=a_0 t^q$12 and $a(t)=a_0 t^q$13, this deformation keeps the model inside the Planck 2018 and BICEP/Keck allowed region while providing a graceful exit and sufficient e-foldings; the exact power-law behavior is therefore only approximate and effectively collapses at late times (Alhallak et al., 2022).

A different transient construction appends a power-law inflationary tail to standard $a(t)=a_0 t^q$14 inflation. After the Starobinsky slow-roll era ends, a scalar with exponential potential and constant equation of state drives

$a(t)=a_0 t^q$15

For $a(t)=a_0 t^q$16, one has

$a(t)=a_0 t^q$17

so the tail is only slightly accelerating. Its principal effect is to shorten the effective slow-roll duration relevant for the pivot scale and to address the Trans-Planckian Censorship Conjecture while leaving CMB observables controlled mainly by the earlier $a(t)=a_0 t^q$18 phase (Odintsov et al., 6 Apr 2025).

Scale-dependent suppression can also be produced by a transition between two power-law pieces of the inflaton potential. In Whipped inflation,

$a(t)=a_0 t^q$19

the inflaton begins in a steeper, relatively fast-roll phase and then transitions to a lower-power slow-roll attractor. The transition from cubic to quadratic form produces a suppression of scalar power at large scales and improves the fit to Planck plus BICEP2 relative to a pure power-law primordial spectrum by

$a(t)=a_0 t^q$20

while keeping the bispectrum consistent with Planck bounds (Hazra et al., 2014).

7. Comparative assessment

The surveyed constructions share a common objective but differ in what exactly is being modified. Potential-level models collapse an inverse power law into a plateau with a finite vacuum expectation value; background-level models generate internal decay of an effective cosmological term; perturbation-level models use CSL to shift spectral indices without altering the background equations; modified-gravity and entropic models deform the relation between the inflaton sector and the Friedmann dynamics; transient-tail and affine-warm models turn exact power-law inflation into a finite, self-terminating phase (Lu, 2013, Paul et al., 2020, Das, 20 Aug 2025, Alhallak et al., 2022).

Several misconceptions are therefore best avoided. Collapsed Modified Power Law Inflation is not a unique model class with a standard Lagrangian, and “collapse” does not universally mean graceful exit. In the CSL realization, collapse refers to quantum state reduction and does not cure the background graceful-exit problem. In Barrow cosmology, by contrast, the modification effectively collapses back to standard behavior because $a(t)=a_0 t^q$21. In plateau-type inverse-power models, collapse refers to a bounded potential with a finite minimum. A plausible implication is that the phrase is most useful as a descriptor of a structural motif—power-law behavior deformed so as to become finite, observationally viable, or self-terminating—rather than as the name of a single canonical scenario.

From an observational standpoint, the viable branches cluster into three signatures. One class yields a Starobinsky-like red tilt with ultra-suppressed tensors, as in the generalized inverse power-law plateau. A second class yields $a(t)=a_0 t^q$22, as in power-law Starobinsky and power-law $a(t)=a_0 t^q$23-Starobinsky deformations. A third class predicts vanishing running together with a comparatively red tensor tilt that violates the standard consistency relation, as in the CSL-modified power-law framework. Future discrimination among these branches depends chiefly on improved measurements of $a(t)=a_0 t^q$24, tensor tilt $a(t)=a_0 t^q$25, and any nonzero running of scalar or tensor indices (Saini et al., 2023, Saini et al., 22 May 2025, Das, 20 Aug 2025).