Bare mass effects on the reheating process after inflation

We consider the effects of a bare mass term for the inflaton, when the inflationary potential takes the form <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>V</a:mi><a:mo stretchy="false">(</a:mo><a:mi>ϕ</a:mi><a:mo stretchy="false">)</a:mo><a:mo>=</a:mo><a:mi>λ</a:mi><a:msup><a:mrow><a:mi>ϕ</a:mi></a:mrow><a:mrow><a:mi>k</a:mi></a:mrow></a:msup></a:mrow></a:math> about its minimum with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>k</e:mi><e:mo>≥</e:mo><e:mn>4</e:mn></e:math>. We concentrate on <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>k</g:mi><g:mo>=</g:mo><g:mn>4</g:mn></g:math>, but discuss general cases as well. Further, we assume <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mrow><i:mi>λ</i:mi><i:msubsup><i:mrow><i:mi>ϕ</i:mi></i:mrow><i:mrow><i:mi>end</i:mi></i:mrow><i:mrow><i:mn>2</i:mn></i:mrow></i:msubsup><i:mo>≫</i:mo><i:msubsup><i:mrow><i:mi>m</i:mi></i:mrow><i:mrow><i:mi>ϕ</i:mi></i:mrow><i:mrow><i:mn>2</i:mn></i:mrow></i:msubsup></i:mrow></i:math>, where <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>ϕ</k:mi><k:mrow><k:mi>end</k:mi></k:mrow></k:msub></k:math> is the inflaton field value when the inflationary expansion ends. We show that the presence of a mass term (which may be present due to radiative corrections or supersymmetry breaking) can significantly alter the reheating process, as the equation of state of the inflaton condensate changes from <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>w</m:mi><m:mi>ϕ</m:mi></m:msub><m:mo>=</m:mo><m:mfrac><m:mn>1</m:mn><m:mn>3</m:mn></m:mfrac></m:math> to <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>w</o:mi><o:mi>ϕ</o:mi></o:msub><o:mo>=</o:mo><o:mn>0</o:mn></o:math> when <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>λ</q:mi><q:msup><q:mi>ϕ</q:mi><q:mn>2</q:mn></q:msup></q:math> drops below <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msubsup><s:mi>m</s:mi><s:mi>ϕ</s:mi><s:mn>2</s:mn></s:msubsup></s:math>. We show that, for a mass <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:msub><u:mrow><u:mi>m</u:mi></u:mrow><u:mrow><u:mi>ϕ</u:mi></u:mrow></u:msub><u:mo>≳</u:mo><u:mn>3</u:mn><u:msup><u:mrow><u:mi>λ</u:mi></u:mrow><u:mrow><u:mfrac><u:mrow><u:mn>1</u:mn></u:mrow><u:mrow><u:mn>4</u:mn></u:mrow></u:mfrac></u:mrow></u:msup><u:msub><u:mrow><u:mi>T</u:mi></u:mrow><u:mrow><u:mi>RH</u:mi></u:mrow></u:msub></u:mrow></u:math>, the mass term will dominate at reheating. The value of <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>λ</w:mi></w:math> is relatively model independent as it is normalized by the cosmic microwave background perturbation spectrum. For T models of inflation, this leads to <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mrow><y:msub><y:mrow><y:mi>m</y:mi></y:mrow><y:mrow><y:mi>ϕ</y:mi></y:mrow></y:msub><y:mo>≳</y:mo><y:msub><y:mrow><y:mi>T</y:mi></y:mrow><y:mrow><y:mi>RH</y:mi></y:mrow></y:msub><y:mo>/</y:mo><y:mn>250</y:mn></y:mrow></y:math>. We compute the effects on the reheating temperature for cases where reheating is due to inflaton decay (to fermions, scalars, or vectors) or to inflaton scattering (to scalars or vectors). For scattering to scalars and in the absence of a decay, there is always a residual inflaton background that acts as cold dark matter. In this case, we derive a strong upper limit to the inflaton bare mass which for T models is <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mrow><ab:msub><ab:mrow><ab:mi>m</ab:mi></ab:mrow><ab:mrow><ab:mi>ϕ</ab:mi></ab:mrow></ab:msub><ab:mo><</ab:mo><ab:mn>350</ab:mn><ab:mtext> </ab:mtext><ab:mtext> </ab:mtext><ab:mi>MeV</ab:mi><ab:mo stretchy="false">(</ab:mo><ab:msub><ab:mrow><ab:mi>T</ab:mi></ab:mrow><ab:mrow><ab:mi>RH</ab:mi></ab:mrow></ab:msub><ab:mo>/</ab:mo><ab:msup><ab:mrow><ab:mn>10</ab:mn></ab:mrow><ab:mrow><ab:mn>10</ab:mn></ab:mrow></ab:msup><ab:mtext> </ab:mtext><ab:mtext> </ab:mtext><ab:mi>GeV</ab:mi><ab:msup><ab:mrow><ab:mo stretchy="false">)</ab:mo></ab:mrow><ab:mrow><ab:mn>3</ab:mn><ab:mo>/</ab:mo><ab:mn>5</ab:mn></ab:mrow></ab:msup></ab:mrow></ab:math>. We also consider the effect of the bare mass term on the fragmentation of the inflaton condensate.