AR-2019-2020

φ V ( φ ) φ i φ ∗ φ end k ∗ = 0 . 05 ( Mpc ) − 1 CMB Reheating Inflaton potential with PBH feature PBH bump φ PBH Figure 5: Inflaton potential with a primordial black hole feature in the form of a local bump superimposed on it. The feature arises at an intermediate scalar field value φ PBH before the end of inflation φ end . Note that the bump size is shown significantly amplified for the purposes of illustration. to ‘near inflection point’ scenarios which have difficulty in producing large mass PBHs without introducing a significant red tilt into the primordial perturbation spectrum on CMB scales. Interestingly, a tiny local dip-like feature, which originates when a term such as V b ( φ ) ε ( φ ) ( ε 1), localised at φ = φ 0 , is subtracted from the base inflationary potential V b ( φ ), also serves the purpose of PBH formation. Therefore, a general potential capable of generating PBHs becomes: V ( φ ) = V b ( φ ) [1 ± ε ( φ )] , (5) where V b is the base inflationary potential. The inflaton slows down while surmounting the bump/dip, resulting in the amplification of the scalar power spectrum and the production of PBHs. As demonstrated by Mishra and Sahni , both bumps and dips in the inflaton potential can suc- cessfully generate PBH’s in a variety of mass-ranges (see Fig. 5). In the standard single field inflationary paradigm, inflation is sourced by a minimally coupled canonical scalar field φ with a suitable potential V ( φ ). The background evolution of the scalar field is given by : ¨ φ + 3 H ˙ φ + V ( φ ) = 0 . (6) The extent of inflation is indicated by the total number of e-foldings during inflation: Δ N e = N i e − N end e = log e a end a i = t end t i H ( t ) dt, (7) where H ( t ) is the Hubble parameter during inflation. N e denotes the number of e-foldings before the end of inflation so that N e = N i e corresponds to the beginning of inflation, while N e = N end e = 0 corresponds to the end of inflation. The slow-roll phase of inflation, ensured by the presence of the Hubble friction term in the equation (6), is usually characterised by the first two Hubble slow-roll parameters H , η H , H = − ˙ H H 2 , η H = − ¨ φ H ˙ φ (8)