Effects of radiative corrections on Starobinsky inflation
John Ellis, Tony Gherghetta, Kunio Kaneta, Wenqi Ke, Keith A. Olive
Abstract
We analyze radiative corrections to the predictions of Starobinsky-like models of inflation arising from self-interactions of the inflaton, and from its Yukawa couplings, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>y</a:mi> </a:math> , to matter fermions, and dimensionful trilinear couplings, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>κ</c:mi> </c:math> , to scalar fields, which could be responsible for reheating the Universe after inflation. The inflaton self-interactions are found to be of higher order in the Hubble expansion rate during inflation, and hence unimportant for CMB observations. In contrast, Einstein-frame matter couplings to an inflaton generating Starobinsky-like inflation can have significant effects on the spectral index of scalar CMB perturbations, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>n</e:mi> <e:mi>s</e:mi> </e:msub> </e:math> , and on the tensor-to-scalar ratio, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>r</g:mi> </g:math> . Using a renormalization-group improved analysis of the effective inflationary potential, we find that the measurement of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>n</i:mi> <i:mi>s</i:mi> </i:msub> </i:math> constrains the inflaton coupling to light fermions in the Einstein frame; <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:mi>y</k:mi> <k:mo><</k:mo> <k:mn>4.5</k:mn> <k:mo>×</k:mo> <k:msup> <k:mrow> <k:mn>10</k:mn> </k:mrow> <k:mrow> <k:mo>−</k:mo> <k:mn>4</k:mn> </k:mrow> </k:msup> </k:mrow> </k:math> , corresponding to an upper limit on the reheating temperature <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>T</m:mi> <m:mrow> <m:mi>RH</m:mi> </m:mrow> </m:msub> <m:mo><</m:mo> <m:mn>2</m:mn> <m:mo>×</m:mo> <m:msup> <m:mn>10</m:mn> <m:mn>11</m:mn> </m:msup> <m:mtext> </m:mtext> <m:mtext> </m:mtext> <m:mi>GeV</m:mi> </m:math> , whereas the ACT DR6 measurement of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msub> <o:mi>n</o:mi> <o:mi>s</o:mi> </o:msub> </o:math> corresponds to <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mn>3.8</q:mn> <q:mo>×</q:mo> <q:msup> <q:mn>10</q:mn> <q:mrow> <q:mo>−</q:mo> <q:mn>4</q:mn> </q:mrow> </q:msup> <q:mo><</q:mo> <q:mi>y</q:mi> <q:mo><</q:mo> <q:mn>5.6</q:mn> <q:mo>×</q:mo> <q:msup> <q:mn>10</q:mn> <q:mrow> <q:mo>−</q:mo> <q:mn>4</q:mn> </q:mrow> </q:msup> </q:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mn>1.7</s:mn> <s:mo>×</s:mo> <s:msup> <s:mn>10</s:mn> <s:mn>11</s:mn> </s:msup> <s:mtext> </s:mtext> <s:mtext> </s:mtext> <s:mi>GeV</s:mi> <s:mo><</s:mo> <s:msub> <s:mi>T</s:mi> <s:mrow> <s:mi>RH</s:mi> </s:mrow> </s:msub> <s:mo><</s:mo> <s:mn>2.8</s:mn> <s:mo>×</s:mo> <s:msup> <s:mn>10</s:mn> <s:mn>11</s:mn> </s:msup> <s:mtext> </s:mtext> <s:mtext> </s:mtext> <s:mi>GeV</s:mi> </s:math> , while the upper limits on <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>r</u:mi> </u:math> provide weaker constraints. data also imply a constraint on a trilinear inflaton coupling to light scalars in the Einstein frame: <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mi>κ</w:mi> <w:mo>≤</w:mo> <w:mn>4</w:mn> <w:mo>×</w:mo> <w:msup> <w:mn>10</w:mn> <w:mn>12</w:mn> </w:msup> <w:mtext> </w:mtext> <w:mtext> </w:mtext> <w:mi>GeV</w:mi> </w:math> , corresponding to <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:msub> <y:mi>T</y:mi> <y:mrow> <y:mi>RH</y:mi> </y:mrow> </y:msub> <y:mo>≤</y:mo> <y:mn>4.2</y:mn> <y:mo>×</y:mo> <y:msup> <y:mn>10</y:mn> <y:mn>13</y:mn> </y:msup> <y:mtext> </y:mtext> <y:mtext> </y:mtext> <y:mi>GeV</y:mi> </y:math> . We further present constraints on inflaton couplings to massive fermions and scalars, and analyze constraints on couplings in the Jordan frame.