George, I and the curvaton

Technical Summary: "George, I and the curvaton"

Introduction and Context
This paper serves as a retrospective overview of the joint research conducted by Konstantinos Dimopoulos and the late George Lazarides, centered on the development and application of the curvaton hypothesis. The work addresses the problem of generating the observed large-scale structure and curvature perturbations in the Universe. While standard inflationary models attribute these perturbations solely to the inflaton field, the curvaton mechanism proposes that a secondary, light scalar field (the curvaton) dominates the generation of these perturbations. This approach liberates inflation model-building by removing the strict requirement for the inflaton to produce the curvature perturbation, thereby rendering many theoretically motivated inflation models viable that would otherwise be excluded.

Methodology and Key Projects
The authors detail four primary research projects developed in collaboration, employing theoretical frameworks within supersymmetry, supergravity, and effective field theory.

1. Curvaton Dynamics and Potential Analysis

   • Methodology: The authors analyzed a curvaton field with a potential containing both a mass term of order the Hubble parameter ($H$) and higher-order non-renormalisable terms: $V(\sigma, H) = \frac{1}{2}(m^2 \pm cH^2)\sigma^2 + \lambda\sigma^{n+4}/M^n$.
   • Results: They derived the evolution of the curvaton density parameter ($\Omega_\sigma$) under different dominance regimes.
     • When the quasi-quadratic term ($cH^2\sigma^2$) dominates, the density evolution depends on the barotropic parameter $w$ and the sign of $c$.
     • When the non-renormalisable term dominates, an attractor solution was found that drives the curvaton perturbation $\delta\sigma$ to zero if the power $n$ exceeds a critical value ($n > n_c$).
   • Significance: This work established the dynamical conditions under which a curvaton can successfully dominate the energy density and imprint its perturbations on the Universe.

2. The Peccei-Quinn (PQ) Field as Curvaton

   • Methodology: Moving away from ad hoc fields, the authors identified the PQ field (resolving the strong CP and $\mu$ problems) as a natural curvaton candidate. They constructed a superpotential involving fields $N, \bar{N}, h_1, h_2$ with specific R- and PQ-charges.
   • Results: The resulting scalar potential allows the PQ field to act as a curvaton. However, the model requires tachyonic amplification of PQ fluctuations. This occurs when the PQ field "loiters" on top of a local maximum of the scalar potential before rolling down.
   • Significance: This connects the curvaton mechanism to realistic Beyond-the-Standard-Model (BSM) physics, specifically solving the strong CP problem while simultaneously generating cosmological perturbations.

3. The Orthogonal Axion as Curvaton

   • Methodology: To address the constraint that simple curvaton models cannot accommodate low-scale inflation ($H_* > 10^7$ GeV), the authors explored a model where the curvaton is a Pseudo-Nambu-Goldstone Boson (PNGB) with a decay constant $f_a$ that grows during inflation.
   • Results: Using a toy model and a more realistic supergravity model involving fields $P, Q, S$, they constructed a scenario with two PNGBs: the QCD axion and an "orthogonal axion." The potential features two valleys (trivial and shifted). In the shifted valley, the decay constant is enlarged, enhancing the curvaton perturbations from $H_*/2\pi$ to a larger value.
   • Significance: This mechanism allows for successful curvaton cosmology even in low-scale inflation scenarios (e.g., $H_{inf} \sim 1$ TeV), which are difficult to achieve with standard scalar curvatons.

4. Vector Curvaton and the $\eta$-Problem in Supergravity

   • Methodology: The authors extended the curvaton paradigm to vector fields. They considered a Proca theory with a massive Abelian vector boson. The vector field is frozen during inflation and begins oscillating when $m \sim H$, behaving as pressureless matter ($\rho_A \propto a^{-3}$) that eventually dominates radiation.
   • Results:
     • Isotropy: Despite being a vector field, the oscillating vector curvaton imposes a scalar curvature perturbation without introducing anisotropy.
     • $\eta$-Problem Solution: In the context of supergravity hybrid inflation, the authors utilized a vector curvaton with a gauge kinetic function modulated by the inflaton ($f(\phi) \propto a^{-4}$). The backreaction from the vector field creates an effective "flattened" potential for the inflaton.
   • Significance: This offers a natural solution to the infamous $\eta$-problem in supergravity, where the slow-roll parameter $\eta$ is typically of order unity. The vector backreaction allows for steep potentials to sustain slow-roll inflation, yielding a scalar spectral index $n_s$ consistent with observations ($0.967 < n_s < 0.978$).

Overall Significance and Claims
The paper claims that the collaboration between Dimopoulos and Lazarides significantly advanced the curvaton hypothesis by:

• Demonstrating that the curvaton mechanism can be realized using realistic BSM fields (PQ field, axions) rather than ad hoc scalars.
• Extending the mechanism to vector fields, proving that scalar curvature perturbations can be generated without fundamental scalar fields (a relevant consideration prior to the discovery of the Higgs boson).
• Providing a robust solution to the $\eta$-problem in supergravity inflation through vector backreaction, making specific inflationary models naturally viable.

The work is presented as a tribute to George Lazarides, highlighting his role as a pivotal collaborator in developing these theoretical frameworks over a decade of joint research.