No-go for the formation of heavy mass Primordial Black Holes in Single Field Inflation

Abstract We examine the possibility of Primordial Black Holes (PBHs) formation in single field models of inflation. Using the adiabatic or wave function renormalization scheme in the short range modes, we show that one-loop correction to the power spectrum is free from quadratic UV divergence. We consider a framework in which PBHs are produced during the transition from Slow Roll (SR) to Ultra Slow Roll (USR) followed by the end of inflation. We demonstrate that the renormalized power spectrum soften the contribution of the logarithmic IR divergence and severely restricts the possible mass range of produced PBHs in the said transition, namely, $$M_\textrm{PBH}\sim 10^{2}\,\textrm{g}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mtext>PBH</mml:mtext> </mml:msub> <mml:mo>∼</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>2</mml:mn> </mml:msup> <mml:mspace/> <mml:mtext>g</mml:mtext> </mml:mrow> </mml:math> $$\hat{\textrm{a}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mtext>a</mml:mtext> <mml:mo>^</mml:mo> </mml:mover> </mml:math> la a no-go theorem. In particular, we find that the produced PBHs are short lived ( $$t^\textrm{evap}_\textrm{PBH}\sim 10^{-20}\,\textrm{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>t</mml:mi> <mml:mtext>PBH</mml:mtext> <mml:mtext>evap</mml:mtext> </mml:msubsup> <mml:mo>∼</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>20</mml:mn> </mml:mrow> </mml:msup> <mml:mspace/> <mml:mtext>s</mml:mtext> </mml:mrow> </mml:math> ) and the corresponding number of e-folds in the USR region is restricted to $$\Delta {\mathcal {N}}_\textrm{USR}\approx 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>N</mml:mi> <mml:mtext>USR</mml:mtext> </mml:msub> <mml:mo>≈</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> .