Primordial black holes in non-Gaussian regimes
PPrimordial black holes in non-Gaussianregimes
Sam Young, Christian T. ByrnesDepartment of Physics and Astronomy, Pevensey II Building, University of Sussex, BN1 9RH, UKOctober 15, 2018Primordial black holes (PBHs) can form in the early Universe from the collapse of rare, largedensity fluctuations. They have never been observed, but this fact is enough to constrain theamplitude of fluctuations on very small scales which cannot be otherwise probed. Because PBHsform only in very rare large fluctuations, the number of PBHs formed is extremely sensitive tochanges in the shape of the tail of the fluctuation distribution - which depends on the amount ofnon-Gaussianity present. We first study how local non-Gaussianity of arbitrary size up to fifthorder affects the abundance and constraints from PBHs, finding that they depend strongly on evensmall amounts of non-Gaussianity and the upper bound on the allowed amplitude of the powerspectrum can vary by several orders of magnitude. The sign of the non-linearity parameters ( f NL , g NL , etc) are particularly important. We also study the abundance and constraints from PBHsin the curvaton scenario, in which case the complete non-linear probability distribution is known,and find that truncating to any given order (i.e. to order f NL or g NL , etc) does not give accurateresults. a r X i v : . [ a s t r o - ph . C O ] S e p Introduction
Primordial black holes (PBHs) have historically been used to study the small scales of the primordialuniverse. Whilst they have never been detected, this fact is enough to rule out or at least constrainmany different cosmological models (see Refs. [1, 2, 3, 4, 5]). Theoretical arguments suggest thatPBHs can form from the collapse of large density perturbations during radiation domination [6]. Ifthe density perturbation at horizon crossing exceeds a threshold value, then gravity will overcomepressure forces and that region collapses to form a PBH with mass of order the horizon mass.There are tight observational constraints on the abundance of PBHs. These constraints comefrom their gravitational effects and results of the Hawking radiation from their evaporation. Forrecent updates and a compilation of the constraints see Refs [7, 8, 9]. The various constraintsplace an upper limit on the mass fraction of the Universe contained within PBHs at the time offormation, β . The constraints vary from β = 10 − to β = 10 − . These constraints can be usedto constrain the primordial power spectrum on small scales, and hence models of inflation. SincePBHs form from the rare, large fluctuations in the extreme tail of the probability distributionfunction (PDF), any non-Gaussianity can significantly affect the number of PBHs formed. PBHformation can therefore be used to probe both the amplitude and non-Gaussianity of the primordialfluctuations on small scales.In order for a significant number of PBHs to form, the power spectrum on small scales needsto be of order 10 − , orders of magnitude larger than on cosmic scales. Although a spectral indexsmaller than 1 has recently been observed by Planck, indicating a red spectrum, it is possiblethat the running of the spectral index turns up on smaller scales, and produces a lot of power atsuch scales. This is possible in models such as the running-mass model, the inflating curvaton andhybrid inflation [10, 11, 12, 13, 14, 15, 16, 17, 18]. Other possibilities include peaks in the powerspectrum[9] or a phase transition after inflation [19].The effects of non-Gaussianity on PBH formation were first studied by Bullock and Primack[20], and Ivanov [21] - reaching opposite conclusions on whether non-Gaussianity enhances or sup-presses the number of PBHs formed. Lyth [22] studied the constraints from PBH formation onthe primordial curvature perturbation for cases where it has the form ζ = ± (cid:0) x − (cid:104) x (cid:105) (cid:1) , where x has a Gaussian distribution. The minus sign can be expected from the linear era of the hy-brid inflation waterfall, where the positive sign might arise if ζ is generated after inflation by acurvaton-type mechanism. More recently, the effects of non-Gaussianity have been studied byByrnes et al [23], who studied the effects of quadratic and cubic non-Gaussianity in the local modelof non-Gaussianity, and Shandera et al [24], who considered small deviations from a Gaussian dis-tribution, finding that whether PBH formation is enhanced or suppressed depends on the typeof non-Gaussianity. The effects of non-Gaussianity in the curvaton model have also been studiedrecently by Bugaev and Klimai [25, 26], who calculated constraints and PBH mass spectra for achi-squared distribution. Seery and Hidalgo [27] showed how to obtain the probability distributionof the curvature perturbation working directly from the n -point correlation functions (which comefrom quantum field theory calculations) and discussed the possibility of using the constraints ofPBHs to discriminate between models of inflation.In this paper, we will go beyond earlier work and calculate the effects of arbitrarily large non-Gaussianity in the local model to 5th order, including terms of each type simultaneously. Wealso consider the curvaton model where a full non-linear solution for the curvature perturbation isavailable in the sudden decay approximation [28]. It is found in this case that using a perturbativeapproach by deriving the non-Gaussianity parameters ( f NL , g NL , etc) and using the local modelof non-Gaussianity disagrees strongly with the full solution - and so care needs to be taken whenperforming these calculations. 1n Section 2, we review the calculation of the PBH abundance constraints in the standardGaussian case. In Section 3, we review the work completed by Byrnes et al [23] calculating theeffects of quadratic and cubic non-Gaussianity in the local model, before extending this to higherorders. The expert reader may skip to Sec. 3.3. In Section 4 we discuss the effects of a hierarchicalscaling between the non-Gaussianity parameters ( g NL ∝ f NL , h NL ∝ f NL , etc), and in Section 5we calculate the constraints on the primordial power spectrum in the curvaton model. We concludewith a summary in Section 6. Whilst the condition required for collapse to form a PBH has traditionally been stated in terms ofthe smoothed density contrast at horizon crossing, δ hor ( R ), we will follow Ref. [23] and work withthe curvature perturbation, ζ . PBHs form in regions where the curvature perturbation is greaterthan a critical value, ζ c (cid:39) . − . ζ c = 1, it would be straightforward to choose any other value if required. It was initiallythought that there was an upper limit on the amplitude of the fluctuation that would form a PBH,with larger fluctuations forming a separate universe, however, this has been shown not to be thecase [32]. Integrating over the fluctuations which form PBHs, the initial PBH mass fraction of theUniverse is: β ≡ ρ P BH ρ total (cid:12)(cid:12)(cid:12) formation (cid:39) (cid:90) ∞ ζ c P ( ζ ) dζ, (1)where ζ c is the critical value for PBH production and P ( ζ ) is the probability distribution function.The above equation is not exact, for example due to the uncertainty in the fraction of mass withina horizon sized patch (whose average density is above the critical one) which will collapse to form ablack hole. This is related to uncertainty of the overdensity profile and the critical value required forcollapse, see e.g. [33, 34, 35, 36] and references therein. Fortunately a numerical factor of order unityleads to only a small uncertainty in the constraints on σ due to the logarithm, see Eq. (4). Orderunity non-linearity parameters are much more important than a numerical coefficient multiplyingthe integral in (1). For Gaussian fluctuations: P ( ζ ) = 1 √ πσ exp (cid:18) − ζ σ (cid:19) , (2)and so: β (cid:39) √ πσ (cid:90) ∞ ζ c exp (cid:18) − ζ σ (cid:19) dζ = 12 erfc (cid:18) ζ c √ σ (cid:19) . (3)Because PBHs form in extremely rare large fluctuations in the tail of the probability distribution,one can use the large x limit of erfc( x ) and show that [23]: σζ c (cid:39) σ = P / ζ (cid:39) (cid:118)(cid:117)(cid:117)(cid:116)
12 ln (cid:16) β (cid:17) . (4)Note that σ depends only logarithmically on β , this remains true once the effects of non-Gaussianityare taken into account. Taking ζ c = 1, for β = 10 − we obtain σ = 0 .
11 and for β = 10 − weobtain σ = 0 .
23. 2he variance of the probability distribution is related to the power spectrum of the curvatureperturbation by σ ≈ P ζ . The constraints obtained in this manner differ by O (10%) to those ob-tained from a full Press-Schechter calculation which includes a window function to smooth the cur-vature perturbation, as performed in [37, 38]. For β = 10 − the full calculation gives P / ζ = 0 . P / ζ = 0 .
11 obtained with Eq. (4). In the case of chi-squared non-Gaussianity,a calculation used the smoothed pdf has also been performed [5] and gives reasonable agreementwith the approach we use here.
We consider the effects of non-Gaussianity in the local model on the abundance of PBHs and theconstraints we can place on the power spectrum. We will first review the work completed by Byrneset al [23] and discuss the effects of quadratic and cubic local non-Gaussianity, before moving ontothe effects of higher order terms in Sec. 3.3.
We take the model of local non-Gaussianity to be ζ = ζ g + 35 f NL (cid:0) ζ g − σ (cid:1) . (5)The σ term is included to ensure that the expectation value for the curvature perturbation remainszero, (cid:104) ζ (cid:105) = 0. Solving this equation to find ζ g as a function of ζ gives two solutions ζ g ± ( ζ ) = 56 f NL (cid:34) − ± (cid:115) f NL (cid:18) f NL σ ζ (cid:19)(cid:35) . (6)We can make a formal change of variable using P NG ( ζ ) dζ = n (cid:88) i =1 (cid:12)(cid:12)(cid:12)(cid:12) dζ g,i ( ζ ) dζ (cid:12)(cid:12)(cid:12)(cid:12) P G ( ζ g,i ( ζ )) dζ, (7)where i is the sum over all solutions, to find the non-Gaussian probability distribution function(PDF). The non-Gaussian distribution is then given by: P NG ( ζ ) dζ = dζ √ πσ (cid:114) f NL (cid:16) f NL σ + ζ (cid:17) ( (cid:15) + + (cid:15) − ) , (8)where (cid:15) ± = exp (cid:18) − ζ g ± ( ζ ) σ (cid:19) , (9)and the initial PBH mass fraction is given by β (cid:39) (cid:90) ζ max ζ c P NG ( ζ ) dζ. (10)If f NL is positive (or zero) then ζ max = ∞ , but if f NL is negative then ζ is bound from above and ζ max is given by ζ max = − f NL (cid:18) f NL σ (cid:19) . (11)3igure 1: The left plot shows the effect of positive f NL on the PDF. For negative f NL the PDFsare simply reflected in the y-axis. We see that the f NL term skews the distribution. The right plotshows the tail of the PDF where PBHs form - note that this is a logarithmic plot of the PDF. Arelatively small change in f NL has a large effect on the number of PBHs produced - by many ordersof magnitude. For these plots, we have taken σ = 0 . f NL on the probability density function. The primary effect of f NL isto skew the distribution - for positive f NL we see a peak for negative values of ζ , with a large tailfor positive values (and vice versa for negative f NL ). The right panel shows a log plot of the effectof positive f NL on the tail of the PDF where PBH formation occurs. We see that, for positive f NL ,as f NL is increased the amplitude of the large tail increases dramatically. For negative values of f NL , ζ is bounded from above, ζ <
1, and we would see no PBH formation for these values (byincreasing σ significantly, one can form PBHs for significantly negative f NL , although we will seelater that unless remarkable fine tuning occurs, this leads to an overproduction of PBHs).We now use the observational constraints on β to place constraints on the power spectrum.This is most easily calculated by making a transformation to a new variable y : y = ζ g ± ( ζ ) σ , (12)which has unit variance. For f NL > β (cid:39) √ π (cid:18)(cid:90) ∞ y c + e − y dy + (cid:90) y c − −∞ e − y dy (cid:19) , (13)and for f NL < β (cid:39) √ π (cid:90) y c + y c − e − y dy, (14)where y c ± are the values of y corresponding to the threshold for PBH formation, ζ c : y c ± = ζ g ± ( ζ c ) σ . (15)The expression for β is then solved numerically using the tight and weak constraints, β = 10 − and 10 − respectively, to find a value for σ . The variance of ζ is then given by [39, 40] P ζ = σ + 4 (cid:18) f NL (cid:19) σ ln( kL ) , (16)4igure 2: This plot shows how the constraints on the square root of the power spectrum due toPBHs depend on f NL . The constraints for 2 values of β are shown - note that, although β changesby 15 orders of magnitude, the constraints only change by a factor of roughly 2.where the cut-off scale L ≈ /H is of order the horizon scale, k is the scale of interest and ln( kL ) istypically O (1) (treating it as exactly 1 leads to percent level corrections, provided that σ is small- we have numerically checked this).Figure 2 shows how the constraints on the square root of the power spectrum change dependingon the value of f NL . For positive values of f NL we see that the constraints tighten (correspondingto an increase in the abundance of PBHs for a given value for the power spectrum, see Figure1). For negative values, we see that the constraints weaken dramatically - this is because, unless σ becomes large, no PBHs form at all. As f NL becomes significantly negative, we see that theconstraints for β = 10 − and β = 10 − converge. Unless there is remarkable fine tuning in the sizeof the perturbations at small scales, there would either be far too many PBHs, or none. Using thismethod to calculate the constraints, as f NL becomes more negative the constraints on the powerspectrum do flatten out at a value above 1 - however, the perturbative approach does not workwhen the perturbation amplitude is O (1) or higher, so these results cannot be trusted. The model of local non-Gaussianity is now taken to be ζ = ζ g + 925 g NL ζ g . (17)We follow the same process as before to calculate the PDFs and constraints on the power spectrum[23]. Care needs to be taken with the amount of solutions to Eq. (17). For g NL >
0, there is onesolution for all ζ . But for g NL <
0, there may be multiple solutions. For example, for g NL <
0, inthe range − (cid:114) − g NL ≤ ζ ≤ (cid:114) − g NL , (18)there are 3 solutions to Eq. 17. These solutions need to be taken into account when calculatingPDFs or constraints on the power spectrum.Figure 3 shows a log plot of the effects of g NL on the PDF. The upper left (right) panel shows theeffect of positive (negative) g NL . We see that g NL affects the kurtosis of the distribution. Typically,5igure 3: The top left (right) plot shows the effect of negative (positive) g NL on the PDF. We seethat g NL affects the kurtosis of the distribution. Positive g NL always gives a sharper peak withbroader tails - enhancing PBH production. Large negative g NL has a similar effect - however, wesee two sharp peaks in the distribution, due to the derivative in Eq. (7) becoming infinite. Forsmall negative g NL we see that the tails of the distribution are diminished. The bottom plot showsthe tail of the PDF where PBHs form - again showing a very strong dependence on small amountsof non-Gaussianity, and again the sign of the non-Gaussianity is important. For these plots, wehave again taken σ = 0 . g NL always serves to enhance the amplitude of the tails where PBHs form, as doeslarge negative g NL . However, for small negative g NL the tails of the PDF are diminished - leadingto a lower PBH abundance (and consequently, weaker constraints).In order to calculate the constraints on the power spectrum, we again write an expression for β to be solved. For positive g NL we have β (cid:39) √ π (cid:90) ∞ y e − y dy. (19)For − < g NL <
0, there are 3 solutions to Eq. (17), and β is given by β (cid:39) √ π (cid:18)(cid:90) y −∞ e − y dy + (cid:90) y y e − y dy (cid:19) . (20)Finally, for g NL < − , β is given by β (cid:39) √ π (cid:90) y −∞ e − y dy. (21)6igure 4: This plot shows how the constraints on the square root of the power spectrum due toPBHs depend on g NL .The limits on the integrals here ( y , y , etc) are solutions for y to Eq. (17). The variance in thismodel is given by [40] P ζ = σ (cid:32) g NL σ ln( kL ) + 27 (cid:18) g NL (cid:19) σ ln( kL ) (cid:33) . (22)Figure 4 shows the constraints obtained for the cubic non-Gaussianity model. For small g NL we seethat the constraints on the power spectrum are highly asymmetric between positive and negative g NL . This is because for positive g NL an overdensity in the linear ζ regime is boosted by the cubicterm - especially so in the tail of the PDF, and so the constraints tighten. However, for smallnegative g NL the opposite is the case and the two terms tend to cancel each other, and hence theconstraints weaken dramatically. For very small negative g NL , the 2nd term in the expression for β , Eq. (20), dominates. As g NL → − from above, y − y →
0, and this term decreases rapidly sothat the constraint on the power spectrum rapidly becomes weaker. As g NL becomes more negative,the first term in Eq. (20) increases, and the constraints tighten again. As g NL becomes large, eitherpositive or negative, then the cubic term in Eq. (17) dominates the expression, ζ ∝ ± ζ g , and theconstraints don’t depend on the sign of g NL . This is because the Gaussian PDF is invariant undera change of sign of ζ g , which is equivalent to changing the sign of g NL (in the case where the linearterm is absent). For this reason, the constraints asymptote to the same value as | g NL | → ∞ . In this section, we consider the effects of higher order terms on the constraints that can be placedon the power spectrum. We take the model of local non-Gaussianity to be ζ = ζ g + 35 f NL (cid:0) ζ g − σ (cid:1) + 925 g NL ζ g + 27125 h NL (cid:0) ζ g − σ (cid:1) + 81625 i NL ζ g + · · · . (23)Higher order terms have a similar effect on the PDF as do the quadratic and cubic terms - evenorder terms introduce skew-like asymmetry to the PDF, whilst odd order terms affect kurtosis, andhave similar effects on the tails of the pdfs. 7he number of solutions to ζ ( ζ g ) = 1 depends on the values of f NL , g NL , h NL , etc. Because ananalytic solution is not typically available for polynomial equations above 4th order, a numericalmethod was used to calculate the constraints on the power spectrum. Starting from the linear,purely Gaussian model, a value for σ is calculated. The non-Gaussianity parameters are thenvaried slowly, and Eq. (23) is solved using the previous value of σ to find critical values of ζ g required for PBH formation, ζ g ( ζ c ) = ζ g , ζ g , · · · . (24)As before, a Gaussian variable y with unit variance is used, Eq. (12), and an expression for β iswritten. For example, β (cid:39) √ π (cid:18)(cid:90) y y e − y dy + (cid:90) y y e − y dy + ... (cid:19) . (25)This is then solved numerically to find a value for σ and the variance is calculated. Provided thatsmall enough steps are taken, and that σ varies sufficiently slowly, the results obtained through thismethod are in excellent agreement to those obtained previously by an analytic method. Accountingfor terms to 5 th order in ζ and including all orders in loops, using the techniques of [40] we findthat the power spectrum is given by P ζ = σ + (cid:18) (cid:19) (cid:0) f NL + 6 g NL (cid:1) σ ln( kL ) + (cid:18) (cid:19) (cid:0) g NL + 48 f N h NL + 30 i NL (cid:1) σ ln( kL ) + (cid:18) (cid:19) (cid:0) h NL + 450 g NL i NL (cid:1) σ ln( kL ) + (cid:18) (cid:19) i NL σ ln( kL ) . (26)Figure 5 shows how the constraints on the power spectrum depend upon the non-Gaussianityparameters. Here, we consider the effects of each term in Eq. (23) one at a time. Again, for higherorder terms, we see similar behaviour to that seen for the quadratic and cubic non-Gaussianity. Foreven-order terms, the constraints become tighter for positive values, but weaken dramatically evenfor small negative values. For odd-order terms, the constraints become tighter for positive values,but for small negative values, the constraints initially weaken dramatically before tightening again.The constraints are most sensitive to small negative non-Gaussianity - where the positive tail ofthe PDF is strongly reduced, either due to a skew-like asymmetry in the PDF from even terms, orkurtosis type effects from the odd terms. In order to study the effects of the different types of local non-Gaussianity simultaneously, weintroduce some hierarchical scaling relationship between the non-Gaussianity parameters. Here, wepresent the simple idea of a power law scaling between the terms: g NL ∼ α f NL , h NL ∼ α f NL , i NL ∼ α f NL , · · · , (27)where α is a constant of order unity, and the model of local non-Gaussianity can be taken as ζ ∼ ζ g + 35 f NL (cid:0) ζ g − σ (cid:1) + 925 α f NL ζ g + 27125 α f NL (cid:0) ζ g − σ (cid:1) + 81625 α f NL ζ g + · · · . (28)This type of relation can occur in several different models, including multi-brid inflation [41, 42], asimilar scaling was used in [24].Figure 6 shows the effect of the hierarchical scaling to the constraints on the power spectrum todifferent orders, where we have taken α = 1 (modifying this term but keeping it of order unity does8igure 5: Here we see how the constraints on the square root of the power spectrum depends onnon-Gaussianity to 5 th order. We have considered the addition of each order term one at a time.Note that the even order terms display similar behaviour to each other, as do the odd order terms.The constraints here are shown for the case β = 10 − . Here, we have included only the linear termand one other term in Eq. (23) for each order equation. The x-axis is either f NL , g NL , h NL , or i NL , depending on the order equation being used.not significantly affect the results). When calculating to n th order, we have now included all termsup to and including the n th term (rather than just the single term in the previous section). Again,we see similar behaviour for the different order expansions - depending on whether the highestorder term is even or odd.For positive f NL the constraints tighten significantly as f NL increases, before converging tosome constant as f NL → ∞ . As f NL becomes large however, the highest-order term dominatesEq. (23), and it is sufficient to take, for example, ζ ∝ ζ ng . Note that the constraints found in thisregion depend on the order that Eq. (23) is taken to - the constraints are slightly tighter for higherorders.For negative f NL , we see similar behaviour to that seen before when only a single term wasconsidered. When the highest order term is even the constraints weaken dramatically as f NL becomes negative, again requiring fine tuning to produce any PBHs without overproducing them.When the highest order terms are odd, we again see a peak where the constraints weaken for smallnegative values, before slowly tightening - however, the peak is now smoother. Again, as | f NL | → ∞ and for odd terms, the sign of the non-Gaussianity parameter does not matter, and the constraintsapproach the same value. Whilst this may not be obvious from Fig. 6, if the axes were extendedto large f NL , of order 10 , we would see this to be the case. Whilst the simplest inflationary models give rise to a nearly Gaussian distribution of the primordialcurvature perturbation, multi-field models of inflation can lead to strong non-Gaussianity. Onewell motivated model is the curvaton model [28]. In this model, in addition to the field drivinginflation, the inflaton φ , there is a second light scalar field, the curvaton χ , whose energy densityis completely subdominant during inflation. At Hubble exit during inflation both fields acquireclassical perturbations that freeze in. Here, the observed perturbations in the CMB and LSS, as9igure 6: The constraints on the square root of the power spectrum in the case of a hierarchicalpower law between the non-Gaussianity parameters. The constraints here are shown for the case β = 10 − . We have used here the hierarchical power rule to different orders (up to 5 th order), andshow the constraints obtained in each case change depending on f NL . Note that we see two distinctbehaviours - depending on whether the highest term on the expansion is odd or even - which givevery different results for the case of negative f NL .well as perturbations on smaller scales, can result from the curvaton instead of the inflaton. At theend of inflation, the inflaton decays into relativistic particles (“radiation”). The curvaton energydensity is still sub-dominant at this stage and carries an isocurvature perturbation - and at somelater time, the curvaton also decays into radiation. Taking the curvaton to be non-relativisticbefore it decays, the energy density of the curvaton will decay slower than the energy density ofthe background radiation - and consequently the curvature perturbation due to the curvaton willbecome dominant.If the curvaton generates the perturbations on CMB scales, then in simple realisations of the cur-vaton scenario with a quadratic potential it cannot have a much larger amplitude of perturbationson smaller scales. However it is possible that a second stage of inflation has a dominant contributionto its perturbations from the curvaton model. Indeed if the curvaton mass, m χ , is reasonably heavycompared to the Hubble scale, then it will naturally have a blue spectrum giving the smallest scaleperturbations the largest amplitude. The spectral index is given by n s − m χ / (3 H ) + 2 ˙ H/H ,where all quantities should be evaluated at the horizon crossing time of the relevant scale [28]. Mo-tivated by our discovery in the last section that truncating the pdf to any order in the non-linearityparameters can give a very bad approximation to the true result, a practical reason for studyingthe curvaton scenario is that this is a rare case in which the full non-linear pdf has been calculated.This allows us to check in a realistic and popular model whether the non-Gaussian corrections tothe pdf are important, and whether just including the first few terms such as f NL or g NL wouldgive an accurate result. We will see that the non-Gaussian corrections to all orders are alwaysimportant when studying PBH formation.Here we use the result obtained by Sasaki et al in the sudden decay approximation [28]:(1 − Ω χ,dec ) e ζ r − ζ ) + Ω χ,dec e ζ χ − ζ ) = 1 , (29)where Ω χ,dec is the dimensionless curvaton density parameter for the curvaton at the decay time.10igure 7: PDFs in the curvaton model. Here we see that, whilst Ω χ,dec ∼ χ,dec the PDF becomes more non-Gaussian, enhancing the positive tailof the PDF. These have been calculated using a formal change of variable using 30 and 31. All theplots have a variance (cid:104) ζ (cid:105) = 0 . ζ r = 0, Eq. (29) reads e ζ χ = 1Ω χ,dec (cid:16) e ζ + (Ω χ,dec − e − ζ (cid:17) . (30)This gives the fully non-linear relation between the primordial curvature perturbation, ζ , and thecurvaton curvature perturbation, ζ χ . Taking there to be no non-linear evolution between Hubbleexit and the start of curvaton decay, the left hand side of Eq. (30) is given by e ζ χ = (cid:18) δ χ ¯ χ (cid:19) , (31)where δ χ is the Gaussian perturbation in the curvaton field at Hubble exit, and ¯ χ is the backgroundvalue. Eq. (30) is quartic in e ζ and so this allows us to write an expression for the full curvatureperturbation, ζ , in terms of the Gaussian variable δ χ = δ χ ¯ χ , or equivalently write the Gaussianvariable as a function of the curvature perturbation. δ χ = δ χ ( ζ ) . (32)Note that, for Ω χ,dec < ζ is bounded from below, with the minimum value given by ζ min = 14 ln (1 − Ω χ,dec ) . (33)Making a formal change of variable allows the PDF to be calculated. Figure 7 shows the PDFsobtained for different values of Ω χ,dec . Whilst Ω χ,dec is close to unity, the PDF is close to Gaus-sian - however, the positive tail of the PDF is diminished, reducing PBH formation. As Ω χ,dec becomes smaller, the PDF becomes more strongly non-Gaussian, and the positive tail of the PDFis enhanced, increasing PBH formation.Constraints on the power spectrum are obtained using the same method as before. Eq. (30) issolved for ζ = ζ c to find the corresponding critical values of δ χ , giving two solutions, δ c and δ c ,11or all values of Ω χ,dec . An expression for β is written: β (cid:39) √ πσ (cid:18)(cid:90) ∞ δ c e − δ χ σ dδ χ + (cid:90) δ c −∞ e − δ χ σ dδ χ (cid:19) . (34)This expression is then solved numerically to find a value for σ for a given value of β . Now that allof the necessary components have been found, the constraints on the power spectrum are calculatedby finding the variance through numeric integration P ζ = (cid:90) ∞ ζ min ζ P NG ( ζ ) dζ = (cid:90) ∞−∞ ζ ( χ g ) P G ( χ g ) dχ g , (35)where P NG ( ζ ) and P G ( χ g ) are the non-Gaussian and Gaussian PDF’s respectively. Care needs tobe taken to ensure that the mean of ζ is zero during the calculation - if necessary defining a newvariable with the mean subtracted, such that (cid:104) ζ (cid:105) = 0.Figure 8 shows the constraints obtained for different values of β . When Ω χ,dec ∼
1, the con-straints are weaker than in the Gaussian case even though the PDF is close to Gaussian - this isan example of even small amounts of non-Gaussianity having a large impact on the constraints. AsΩ χ,dec →
0, the constraints on the power spectrum become tighter, corresponding to an enhance-ment of the positive tail of the PDF.It should be noted that, in this model, a full expansion for ζ can be obtained by performing aTaylor expansion of the solution to Eq. (30) [28, 41]. Figure 9 shows the non-Gaussianity parametersplotted as a function of Ω χ,dec . Instead of using the full non-linear solution for ζ , the calculation canbe completed as in the previous section by using these solutions for the parameters. However, theresults obtained in this manner typically do not match well with those obtained from an analyticsolution - the contributions to the power spectrum from higher-order terms can become large andcan be either positive or negative. This is due to the fact that, whatever order the expansion iscarried out to, the Taylor expansion diverges from the analytic solution as ζ becomes large (oforder unity or higher). For example, for Ω χ,dec = 1, f NL = − , and so a truncation at second orderwould not even come close to matching with the results obtained here. Comparing the constraintsfor β = 10 − between Figs. 8 and 2, notice that the Gaussian constraint of P / ζ = 0 .
23 is reachedfor Ω χ,dec (cid:39) .
4, but from Fig. 9 we see that the non-linearity parameters are not typically smallhere, and so the matching is just coincidence. Hence we conclude that the non-Gaussianity of thecurvaton model always has to be taken into account when calculating PBH constraints.
The formation rate of PBHs probes the tails of the PDF of primordial fluctuations, and is verysensitive to the effects of non-Gaussianity. We have calculated the effects of the local model of non-Gaussianity for terms up to 5th order, parameterised by f NL , g NL , h NL , and i NL . We have shownthat any non-Gaussianity parameters of order unity can have a significant effect on the abundanceof PBHs, and the constraints that can be placed on the power spectrum - due to the fact that thenon-Gaussianity parameters have a large impact on the tails of the PDF.The sign of the non-Gaussianity has a particularly strong effect. We see that positive termsof even order tighten the constraints significantly, but negative terms dramatically weaken theconstraints, to the point where the curvature perturbation is order unity. Typically, when an eventype of non-Gaussianity is considered, such as f NL or h NL , if this term is negative and dominatesthe non-Gaussianity of the distribution, the amplitude of the primordial fluctuations will either12igure 8: Constraints on the square root of the power spectrum in the curvaton model. Theconstraints obtained for different constraints on β , the initial PBH mass fraction, as a function ofΩ χ,dec , the dimensionless curvaton density parameter at the time of decay.Figure 9: The non-Gaussianity parameters in the curvaton model.be too small to form any PBHs, or so large that the Universe contains too many PBHs. Sucha scenario would be incompatible with any future detection of PBHs. Odd-order terms, such as g NL or i NL , tend to tighten the constraints regardless of their sign, but small negative terms canweaken the constraints dramatically over a small range of values. If PBHs were to be detected inthe future, they could potentially rule out certain models and distributions. Care needs to be takenas truncations to set order in the model of non-Gaussianity used might not converge.In the curvaton model, the PDF is relatively close to Gaussian if the Universe is dominatedby the curvaton at the time of decay, Ω χ,dec ∼ χ,dec decreases, the distribution becomes morenon-Gaussian, and the constraints on the power spectrum tighten significantly. Calculations ob-tained for the curvaton model by calculating the local non-Gaussianity parameters to e.g. secondor third order ( f NL or g NL ) do not agree with those obtained using the full non-linear solution.Therefore, given a specific model, it may be necessary to calculate the full hierarchy (rather than13runcating at a given order) before performing calculations, as we have done here for the curvatonmodel. We would like to thank Anne Green and David Seery for useful discussions. SY thanks the STFCfor financial support. CB is supported by a Royal Society University Research Fellowship.
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