Inflation in Supergravity from Field Redefinitions

• Main Authors: Michał Artymowski, Ido Ben-Dayan
• Format: Article
• Language: English
• Published: MDPI AG 2020-05-01
• Series: Symmetry
• Subjects: supergravity; cosmic inflation; Kähler geometry
• Online Access: https://www.mdpi.com/2073-8994/12/5/806

Supergravity (SUGRA) theories are specified by a few functions, most notably the real Kähler function denoted by <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mrow> <mo>(</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mover accent="true"> <mi>T</mi> <mo>¯</mo> </mover> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>K</mi> <mo>+</mo> <mo form="prefix">log</mo> <msup> <mrow> <mo>|</mo> <mi>W</mi> <mo>|</mo> </mrow> <mn>2</mn> </msup> </mrow> </semantics> </math> </inline-formula>, where K is a real Kähler potential, and W is a holomorphic superpotential. A field redefinition <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>→</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> changes neither the theory nor the Kähler geometry. Similarly, the Kähler transformation, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>K</mi> <mo>→</mo> <mi>K</mi> <mo>+</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mover accent="true"> <mi>f</mi> <mo>¯</mo> </mover> <mn>2</mn> </msub> <mo>,</mo> <mi>W</mi> <mo>→</mo> <msup> <mi>e</mi> <mrow> <mo>−</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> </msup> <mi>W</mi> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <msub> <mi>f</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula> is holomorphic and leaves G and hence the theory and the geometry invariant. However, if we perform a field redefinition only in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>K</mi> <mrow> <mo>(</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mover accent="true"> <mi>T</mi> <mo>¯</mo> </mover> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>→</mo> <mi>K</mi> <mrow> <mo>(</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mover accent="true"> <mi>T</mi> <mo>¯</mo> </mover> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, while keeping the same superpotential <inline-formula> <math display="inline"> <semantics> <mrow> <mi>W</mi> <mo>(</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, we get a different theory, as G is not invariant under such a transformation while maintaining the same Kähler geometry. This freedom of choosing <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> allows construction of an infinite number of new theories given a fixed Kähler geometry and a predetermined superpotential W. Our construction generalizes previous ones that were limited by the holomorphic property of <i>W</i>. In particular, it allows for novel inflationary SUGRA models and particle phenomenology model building, where the different models correspond to different choices of field redefinitions. We demonstrate this possibility by constructing several prototypes of inflationary models (hilltop, Starobinsky-like, plateau, log-squared and bell-curve) all in flat Kähler geometry and an originally renormalizable superpotential <i>W</i>. The models are in accord with current observations and predict <inline-formula> <math display="inline"> <semantics> <mrow> <mi>r</mi> <mo>∈</mo> <mo>[</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>06</mn> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> spanning several decades that can be easily obtained. In the bell-curve model, there also exists a built-in gravitational reheating mechanism with <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mi>R</mi> </msub> <mo>∼</mo> <mi mathvariant="script">O</mi> <mrow> <mo>(</mo> <msup> <mn>10</mn> <mn>7</mn> </msup> <mi>G</mi> <mi>e</mi> <mi>V</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>.