A novel pre-inflationary model in view of the lack of angular correlation of CMB

Abstract: In this paper we propose a novel unified cosmological model that connects a pre-inflationary epoch, starting at the Planckian time, with the onset of inflation within a single scalar-field framework. The pre-inflationary phase is characterized by a decelerated expansion with an increasing comoving Hubble horizon, followed by a gradually transition to an accelerated inflationary regime. This early dynamics leads to a modified causal structure that naturally accounts for the suppression of large-angle $(θ\gtrsim 60<sup>\circ)$ correlations in the cosmic microwave background (CMB) reported by the satellite PLANCK. We study the quantum fluctuations of the scalar field using the Mukhanov-Sasaki formalism and a canonical quantization procedure based on energy minimization. We find that the vacuum state is well-defined only for sub-horizon modes at the onset of inflation, which induces a natural cutoff in the primordial power spectrum. The resulting spectrum exhibits a suppression at large scales while remaining nearly scale-invariant at small scales. In the appropriate limit, the model recovers the standard de Sitter result, in agreement with current observational constraints. These results highlight the relevance of pre-inflationary dynamics for addressing large-scale anomalies within a consistent inflationary framework.

• The paper proposes a unified scalar-field model linking a pre-inflationary decelerated phase with inflation to naturally induce a cutoff in the large-scale power spectrum.
• It employs a power-law transition and canonical quantization to reconcile Planck-era energy conditions with the observed suppression of CMB angular correlations.
• The model offers practical testability through refined CMB analyses and parameter constraints that probe early-universe vacuum selection and dynamics.

Unified Pre-Inflationary–Inflationary Cosmological Dynamics and CMB Large-Scale Anomaly

Introduction and Motivation

The angular two-point correlation function from CMB observations, particularly from Planck, reveals an anomalous suppression of correlations at large angular scales (θ ≳ 60°). Standard single-field slow-roll inflation stretching from the Planck scale predicts scale-invariant correlations on large angular scales, thus failing to naturally explain this anomaly. Recent investigations indicate these anomalies may result from the initial quantum state’s selection, early universe dynamics, or a truncated primordial power spectrum due to a delayed onset of inflation. Further, the requirement that the energy scale for successful slow-roll inflation with plateau-like potentials is significantly below the Planck scale gives rise to a theoretical inconsistency, undermining the assumption that inflation begins immediately after the Planck era.

This work proposes a unified scalar-field framework connecting a pre-inflationary decelerated expansion era (beginning at Planck time) to canonical inflation. The aim is to model both the observed CMB suppression at large scales and the natural cutoff in the primordial power spectrum as consequences of early-universe physics and consistent vacuum selection.

Pre-Inflationary–Inflationary Model Construction

Dynamical Framework

The model postulates background dynamics dominated by a single scalar field, with the pre-inflationary phase featuring decelerated expansion ($\beta < 0$) and an increasing comoving Hubble radius, which transitions smoothly (through a critical parameter $\beta = 0$) to an accelerated inflationary (quasi-de Sitter) regime ($0 < \beta < 1$). The prized scale factor is a power-law form:

$$a(t) = a_\mathrm{pl} \left[ \frac{1 + (1-\beta)\alpha t}{1 + (1-\beta)\alpha t_\mathrm{pl}} \right]^{1/(1-\beta)}$$

with a transition from $\beta < 0$ (pre-inflation, decelerated, increasing comoving Hubble horizon) to $0 < \beta < 1$ (inflation, accelerated, decreasing horizon). At $\beta = 1$, this reduces asymptotically to de Sitter expansion. The construction ensures energy densities at Planck time match the Planck energy density, preserving consistency with quantum gravity initial conditions.

Through standard Friedmann equations, the associated scalar-field potential interpolates between forms appropriate for kinetic (decelerated) and potential (vacuum-dominated, inflationary) energy regimes:

$$U_b(\phi_b) = \frac{\alpha^2 M_\mathrm{pl}^2 (\beta+2)}{(1+\alpha(1-\beta)t_p)^2} \exp\left( -\frac{\sqrt{2(1-\beta)}\,\phi_b}{M_\mathrm{pl}} \right)$$

with crucial dependencies on $\beta$ determining the cosmic acceleration.

Causal Structure and Horizon Evolution

Decelerated pre-inflation allows the comoving Hubble horizon to grow. This temporal increase admits that some primordial fluctuations never come into causal contact prior to inflation, resolving tension between full-horizon causal correlations in standard inflation and the CMB’s observed large-angle suppression. The transition to an accelerated (inflationary) phase guarantees subsequent horizon “exit” and decoherence of the modes relevant for large-scale structure, as required by standard inflationary phenomenology.

Vacuum Selection and Quantum Fluctuation Dynamics

The quantum spectra of primordial perturbations are computed in the Mukhanov–Sasaki formalism. Unlike the canonical assumption of the Bunch–Davies vacuum (which is uniquely defined for high-$k$ in inflationary backgrounds with decreasing Hubble horizons), in this background the physical choice of vacuum becomes ambiguous for long-wavelength (super-horizon) modes due to the background’s dynamical evolution.

A canonical quantization is performed, yielding generalized mode solutions (Hankel functions) for the perturbations. The vacuum state is fixed by the requirement that the expectation value of the energy density $\beta = 0$0 is minimized at the onset of inflation. Crucially, this minimization is only well-defined for sub-horizon modes: for $\beta = 0$1, the vacuum normalization fails and the quantum state structure must be reconsidered. This naturally induces a cutoff in the primordial power spectrum for large-wavelength (low-$\beta = 0$2) modes, offering a physical basis for CMB angular correlation suppression at large scales.

Power Spectrum Characteristics and Observational Implications

The solution for the fluctuations across the pre-inflationary/inflationary background yields the following general form for the power spectrum in the IR (super-horizon limit) at the end of inflation:

$\beta = 0$3

with the spectral index $\beta = 0$4 and $\beta = 0$5 a function of $\beta = 0$6.

• Sub-horizon modes ($\beta = 0$7) yield a nearly scale-invariant spectrum if $\beta = 0$8
• Super-horizon modes are suppressed or undefined, leading to a natural large-scale cutoff

For $\beta = 0$9, the standard de Sitter result is recovered, achieving consistency with best-fit Planck inflationary constraints ($0 < \beta < 1$0).

Strong result: The emergence of a cutoff in the spectrum from the breakdown of the vacuum prescription at long wavelengths provides a self-consistent, physically motivated mechanism for the large-angle suppression in the CMB, without recourse to ad hoc regularization or initial condition engineering.

Implications and Future Directions

Practical implications: The model offers a viable and systematic approach for including pre-inflationary physics, potentially testable via refined analyses of the CMB at large angular scales or other primordial cosmological observables sensitive to horizon-scale physics.

Theoretical implications: The formalism indicates that inflation models invoking Planck-scale initial conditions can be embedded within a wider cosmological history, without requiring problematic initial energy densities orders of magnitude above those consistent with inflationary plateaus. The dynamic cutoff in the power spectrum is not only consistent with current observations, but may act as a sensitive probe for pre-inflationary physics and vacuum structure.

Future directions: Extensions could include more sophisticated renormalized vacuum prescriptions for super-horizon modes, inclusion of more complex scalar sectors, or confrontation with non-Gaussian statistics and other large-scale anomalies. Model parameters could, in principle, be constrained through precise measurement of CMB large-scale angular correlations beyond the predictions of canonical slow-roll inflation.

Conclusion

This unified pre-inflationary/inflationary scalar-field framework connects Planck-era physics to conventional inflation through a natural transition in equation-of-state dynamics. The model provides a consistent explanation for the suppression of large-angle correlations in the CMB by demonstrating that the vacuum state of primordial fluctuations is only well-defined for sub-horizon modes at the onset of inflation, naturally inducing a cutoff in the power spectrum at large scales. This approach preserves compatibility with standard inflationary predictions at small scales, offers a theoretical resolution to the mismatch between inflationary energy scales and Planckian initial conditions, and suggests new paths for probing the physics of the earliest moments of the universe.