We study a scalar particle of mas decaying into two particles of mass during the radiation and matter dominated epochs of a standard cosmological model. An adiabatic approximation is introduced that is valid for degrees of freedom (d.o.f.) with typical wavelengths much smaller than the particle horizon ( Hubble radius) at a given time. We implement a nonperturbative method that includes the cosmological expansion and obtain a cosmological Fermi’s Golden Rule that enables one to compute the decay law of the parent particle with mass , along with the build up of the population of daughter particles with mass . The survival probability of the decaying particle is with being an effective momentum and time dependent decay rate. It features a transition timescale between the relativistic and nonrelativistic regimes and for is always smaller than the analogous rate in Minkowski spacetime, as a consequence of (local) time dilation and the cosmological redshift. For the decay law is a “stretched exponential” , whereas for the nonrelativistic stage with , we find , with the Minkowski space time decay width at rest. The Hubble timescale introduces an energy uncertainty which relaxes the constraints of kinematic thresholds. This opens new decay channels into heavier particles for , with the (local) comoving energy of the decaying particle. As the expansion proceeds this channel closes and the usual two particle threshold restricts the decay kinematics.
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"value": "We study a scalar particle of mas <math><msub><mi>m</mi><mn>1</mn></msub></math> decaying into two particles of mass <math><msub><mi>m</mi><mn>2</mn></msub></math> during the radiation and matter dominated epochs of a standard cosmological model. An adiabatic approximation is introduced that is valid for degrees of freedom (d.o.f.) with typical wavelengths much smaller than the particle horizon (<math><mo>\u221d</mo></math> Hubble radius) at a given time. We implement a nonperturbative method that includes the cosmological expansion and obtain a cosmological Fermi\u2019s Golden Rule that enables one to compute the decay law of the parent particle with mass <math><msub><mi>m</mi><mn>1</mn></msub></math>, along with the build up of the population of daughter particles with mass <math><msub><mi>m</mi><mn>2</mn></msub></math>. The survival probability of the decaying particle is <math><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msup><mi>e</mi><mrow><mo>\u2212</mo><msub><mover><mi>\u0393</mi><mo>\u02dc</mo></mover><mi>k</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mi>t</mi></mrow></msup></math> with <math><msub><mover><mi>\u0393</mi><mo>\u02dc</mo></mover><mi>k</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> being an effective momentum and time dependent decay rate. It features a transition timescale <math><msub><mi>t</mi><mrow><mi>nr</mi></mrow></msub></math> between the relativistic and nonrelativistic regimes and for <math><mi>k</mi><mo>\u2260</mo><mn>0</mn></math> is always smaller than the analogous rate in Minkowski spacetime, as a consequence of (local) time dilation and the cosmological redshift. For <math><mi>t</mi><mo>\u226a</mo><msub><mi>t</mi><mrow><mi>nr</mi></mrow></msub></math> the decay law is a \u201cstretched exponential\u201d <math><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msup><mi>e</mi><mrow><mo>\u2212</mo><mo>(</mo><mi>t</mi><mo>/</mo><msup><mi>t</mi><mo>*</mo></msup><msup><mo>)</mo><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></msup></math>, whereas for the nonrelativistic stage with <math><mi>t</mi><mo>\u226b</mo><msub><mi>t</mi><mrow><mi>nr</mi></mrow></msub></math>, we find <math><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msup><mi>e</mi><mrow><mo>\u2212</mo><msub><mi>\u0393</mi><mn>0</mn></msub><mi>t</mi></mrow></msup><mo>(</mo><mi>t</mi><mo>/</mo><msub><mi>t</mi><mrow><mi>nr</mi></mrow></msub><msup><mo>)</mo><mrow><msub><mi>\u0393</mi><mn>0</mn></msub><msub><mi>t</mi><mrow><mi>nr</mi></mrow></msub><mo>/</mo><mn>2</mn></mrow></msup></math>, with <math><msub><mi>\u0393</mi><mn>0</mn></msub></math> the Minkowski space time decay width at rest. The Hubble timescale <math><mo>\u221d</mo><mn>1</mn><mo>/</mo><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> introduces an energy uncertainty <math><mi>\u0394</mi><mi>E</mi><mo>\u223c</mo><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> which relaxes the constraints of kinematic thresholds. This opens new decay channels into heavier particles for <math><mn>2</mn><mi>\u03c0</mi><msub><mi>E</mi><mi>k</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>\u226b</mo><mn>4</mn><msubsup><mi>m</mi><mn>2</mn><mn>2</mn></msubsup><mo>\u2212</mo><msubsup><mi>m</mi><mn>1</mn><mn>2</mn></msubsup></math>, with <math><msub><mi>E</mi><mi>k</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> the (local) comoving energy of the decaying particle. As the expansion proceeds this channel closes and the usual two particle threshold restricts the decay kinematics."
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