[Paper Review] Inflaton potential on a Riemann surface

Keisuke Harigaya, Masahiro Ibe|arXiv (Cornell University)|Apr 14, 2014

Cosmology and Gravitation Theories9 citations

TL;DR

This paper proposes that the inflaton potential can be multivalued on a Riemann surface, allowing trans-Planckian field variations during inflation while keeping the effective field theory within the Planck scale. By constructing a single-valued Lagrangian in 4D spacetime, the model realizes a multivalued potential via complex geometry, satisfying the Lyth bound without violating EFT constraints.

ABSTRACT

The observation of the B-mode in the cosmic microwave background radiation combined with the so-called Lyth bound suggests the trans-Planckian variation of the inflaton field during inflation. Such a large variation generates concerns over inflation models in terms of the effective field theory below the Planck scale. If the inflaton resides in a Riemann surface and the inflaton potential is a multivalued function of the inflaton field when it is viewed as a function on a complex plane, the Lyth bound can be satisfied while keeping field values in the effective field theory within the Planck scale. We show that a multivalued inflaton potential can be realized starting from a single-valued Lagrangian of the effective field theory below the Planck scale.

Motivation & Objective

• To resolve tensions between trans-Planckian inflaton field variations and effective field theory (EFT) consistency.
• To address concerns about large field inflation violating EFT bounds below the Planck scale.
• To show that a multivalued inflaton potential can emerge from a single-valued EFT Lagrangian.
• To demonstrate that the Lyth bound can be satisfied without requiring super-Planckian field values in the EFT framework.

Proposed method

• Model the inflaton as a field on a Riemann surface, introducing complex structure to the scalar potential.
• Construct a single-valued Lagrangian in 4D spacetime that gives rise to a multivalued potential when analytically continued to the complex plane.
• Utilize the monodromy structure of the Riemann surface to generate non-trivial potential behavior across branches.
• Apply techniques from complex geometry and effective field theory to ensure consistency with low-energy physics.
• Ensure that physical field values remain within the EFT regime despite large field range on the Riemann surface.
• Use the Lyth bound as a constraint to validate that the model supports observable B-mode polarization in the CMB.

Experimental results

Research questions

• RQ1 Can a multivalued inflaton potential be consistently derived from a single-valued effective field theory Lagrangian?
• RQ2 Does the use of a Riemann surface allow for trans-Planckian field variations without violating EFT unitarity?
• RQ3 How does the monodromy structure of the Riemann surface influence the shape of the inflaton potential?
• RQ4 Can the Lyth bound be satisfied while keeping the inflaton field values within the EFT regime?
• RQ5 What are the cosmological implications of a complex, multivalued inflaton potential for CMB B-mode signals?

Key findings

• A multivalued inflaton potential can be consistently realized from a single-valued effective field theory Lagrangian via complex geometry on a Riemann surface.
• The model satisfies the Lyth bound, enabling observable B-mode polarization in the CMB, even with sub-Planckian field values in the EFT framework.
• The Riemann surface structure allows for large field variations in field space without requiring super-Planckian values in the physical theory.
• The potential's multivaluedness arises naturally from the analytic structure of the complex scalar field, not from ad hoc constructions.
• The construction preserves unitarity and low-energy consistency of the effective field theory.
• The model provides a viable mechanism for large-field inflation that avoids the typical EFT breakdown associated with trans-Planckian fields.