AR-2019-2020

10 20 30 40 50 60 N e 10 − 10 10 − 9 P R n S 0 . 974 r 0 . 002 CMB Normalization Inflation without PBH feature Slow-roll Approx Mukhanov-Sasaki 10 20 30 40 50 60 N e 10 − 11 10 − 9 10 − 7 10 − 5 10 − 3 P R CMB Normalization n S 0 . 965 r 0 . 0025 Inflation with PBH feature Slow-roll approx Mukhanov-Sasaki Figure 6: Left panel : The scalar power spectrum P R is determined: (a) by using the slow-roll approx- imation (10) (solid green), and (b) by numerically solving the Mukhanov-Sasaki equation (red dots) for the base KKLT inflation potential (11). P R is plotted as a function of the number of e-folds before the end of inflation N e . Note that both methods give identical results for a smoothly varying potential, in which case P R decreases monotonically with decreasing N e . Right panel : Shows the plot of the scalar power spectrum during the formation of 10 − 13 M PBHs in our model. This panel demonstrates that the slow-roll formula (10), shown in solid green, miscalculates the amplitude as well as the peak position of P R . Therefore, one must numerically solve the Mukhanov-Sasaki equation (dotted red) in order to compute P R accurately. where H , η H 1 , (9) during the slow-roll regime, in which scalar field perturbations are usually quantified in terms of the comoving curvature perturbation R and its power spectrum: P R = 1 8 π 2 H m p 2 1 H . (10) One finds that a decrease in the value of the slow roll parameter H as the scalar field surmounts the bump (speed-breaker) in its potential leads to a corresponding increase in the amplitude of the power spectrum (10). This has been illustrated in Fig. 6. A more accurate determination of P R is provided by solving the Mukhanov-Sasaki equation. Mishra and Sahni have solved this equation to determine the black hole abundance in a class of inflationary models including KKLT inflation, which has the base potential: V b ( φ ) = V 0 φ n φ n + M n , (11) supplemented by a speed-breaker in the form of a local Gaussian bump: ε ( φ ) = A exp − 1 2 ( φ − φ 0 ) 2 σ 2 , (12) which is characterised by its height A , position φ 0 and width σ . Note that V ( φ ) is characterized by 4 parameters { V 0 , A, φ 0 , σ } . Since V 0 fixes the overall CMB normalization, only three parameters { A, φ 0 , σ } are relevant for PBH formation. A speed-breaker consisting of a tiny bump of height A 1 slows down the inflaton field sufficiently to enhance the scalar power spectrum relevant for PBH formation as shown in Figs. 6 and 7. Thus, the existence of black holes in the universe need not be attributed entirely to dying massive stars, black holes of a primordial origin could also contribute substantially to the black hole abundance today.