Primordial Black Holes and Slow-Roll Violation

TL;DR

This paper discusses how violating the slow-roll approximation in single-field inflation is necessary for primordial black holes to account for dark matter, and how to accurately predict the power spectrum in such scenarios.

Contribution

It demonstrates that an optimized second-order evaluation of Hubble slow-roll parameters accurately models PBH formation even with large power spectrum amplification.

Findings

Slow-roll violation is essential for PBH dark matter models.

Second-order Hubble slow-roll parameters effectively predict power spectrum.

Accurate modeling remains valid for large amplification over multiple e-folds.

Abstract

For primordial black holes (PBH) to be the dark matter in single-field inflation, the slow-roll approximation must be violated by at least ${\cal O}(1)$ in order to enhance the curvature power spectrum within the required number of efolds between CMB scales and PBH mass scales. Power spectrum predictions which rely on the inflaton remaining on the slow-roll attractor can fail dramatically leading to qualitatively incorrect conclusions in models like an inflection potential and misestimate the mass scale in a running mass model. We show that an optimized temporal evaluation of the Hubble slow-roll parameters to second order remains a good description for a wide range of PBH formation models where up to a $10^7$ amplification of power occurs in $10$ efolds or more.

Equations

$$\beta$$

$$=\mathrm{erfc}\left(\frac{{\zeta }_c}{\sqrt{2}{\Delta }_{\zeta }}\right)\approx \sqrt{\frac{2}{\pi }}\frac{{\Delta }_{\zeta }}{{\zeta }_c}{e}^{-{\zeta }_c^2/(2{\Delta }_{\zeta }^2)},$$

$${\zeta }_c=\frac{9}{2\sqrt{2}}{\delta }_c.$$

$${\Delta }_{\zeta }^2(k_0)\approx 2.2\times 10^{-9}$$

$${\Delta }_{\zeta }^2=\frac{H_I^2}{8{\pi }^2{ϵ}_H},$$

$$\left|\frac{\Delta \ln {ϵ}_H}{\Delta N}\right|\ll 1$$

$$M_H\equiv \frac{4\pi \rho }{3H^3}=\frac{1}{2GH}$$

$$H^2={\left(\frac{g_{\ast }}{g_{\ast 0}}\right)}^{-1/3}{\Omega }_r{H}_0^2{a}^{-4},$$

$$M=\frac{\gamma }{2GH}=4.84\times 10^{24}\gamma {\left(\frac{g_{\ast }}{g_{\ast 0}}\right)}^{1/6}{a}_H^2{M}_{\odot }.$$

$$\frac{d{\Omega }_{\mathrm{PBH}}{h}^2}{d\ln M}=\frac{1}{2}\beta (M){\left(\frac{g_{\ast }}{g_{\ast 0}}\right)}^{-1/3}{\Omega }_r{h}^2{a}_H^{-1},$$

$$\bar{\beta }=\frac{a_H}{2}{\int }_M^{\infty }\frac{d{M}^{\prime }}{M^{\prime }}\frac{\beta }{a_H},$$

$${\Omega }_{\mathrm{PBH}}{h}^2=\bar{\beta }{\left(\frac{g_{\ast }}{g_{\ast 0}}\right)}^{-1/3}{\Omega }_r{h}^2{a}_H^{-1},$$

$$\bar{\beta }=1.3\times 10^{-9}{\gamma }^{-1/2}{\left(\frac{g_{\ast }}{g_{\ast 0}}\right)}^{1/4}\left(\frac{{\Omega }_{\mathrm{PBH}}{h}^2}{0.12}\right){\left(\frac{M}{M_{\odot }}\right)}^{1/2}.$$

$$N$$

$$=18.4-\frac{1}{12}\ln \frac{g_{\ast }}{g_{\ast 0}}+\frac{1}{2}\ln \gamma -\frac{1}{2}\ln \frac{M}{M_{\odot }}.$$

$$t_{\mathrm{ev}}=\frac{5120\pi G^2{M}^3}{\hslash c^4}=6.6\times 10^{74}{\left(\frac{M}{M_{\odot }}\right)}^3\mathrm{s}$$

$$M_{\min }=1.5\times 10^{-21}{\left(\frac{{\Omega }_m{h}^2}{0.14}\right)}^{-2/3}{M}_{\odot }.$$

$$N\approx 42+\frac{1}{2}\ln \gamma$$

$${\Delta }_{\zeta }^2=0.021,$$

$$\left|\frac{\Delta \ln {ϵ}_H}{\Delta N}\right|>\mathrm{0.38.}$$

$${ϵ}_V=\frac{1}{2}{\left(\frac{V^{\prime }}{V}\right)}^2\to 0.$$

$$\frac{{ϵ}_V}{{ϵ}_H}={\left(1+\frac{1}{2(3-{ϵ}_H)}\frac{d\ln {ϵ}_H}{dN}\right)}^2$$

$${ϵ}_H=\frac{1}{2}{\left(\frac{d\varphi }{dN}\right)}^2\approx {ϵ}_V$$

$$\frac{d\varphi }{dN}=-\frac{V^{\prime }}{V}.$$

$$\Delta_{\zeta}^{2}=\frac{V}{24\pi^{2}\epsilon_{V}}\Big{|}_{k\eta(\phi)=1},$$

$$V(\varphi )=\frac{\lambda v^4}{12}\frac{x^2(6-4ax+3x^2)}{(1+b{x}^2{)}^2},$$

$$\beta$$

$$\Delta N_{\mathrm{SR}}$$

$$V(\varphi )=V_0+\frac{1}{2}{m}^2(\ln \varphi ){\varphi }^2,$$

$$V(\varphi )=V_0\left[1+\frac{\tilde{c}}{8}{\varphi }^2\left(-1+2L+\tilde{g}{L}^2\right)\right],$$

$$\tilde{c}=-0.1711,\tilde{g}=0.09648,L_0=-0.756,$$

$$\ln {ϵ}_H\approx C_1+C_{2}N,(N\approx 0).$$

Full text

I Introduction

With the detection of black hole mergers by LIGO Abbott et al. (2016a, b, 2017) and the continuing lack of direct detection of WIMPs, interest in primordial black holes (PBHs) Zel’dovich and Novikov (1966); Hawking (1971); Carr (1975) as candidates for dark matter has recently received renewed attention Bird et al. (2016); Clesse and García-Bellido (2017); Sasaki et al. (2016); Kawasaki et al. (2016); Carr et al. (2016); Inomata et al. (2017); Nakama et al. (2017); Kühnel and Freese (2017); Chiba and Yokoyama (2017); Carr et al. (2017). PBHs form from density fluctuations that are comparable to or exceed order unity at horizon crossing. For PBHs to be the dark matter therefore requires a large amplification of the inflationary power spectrum between cosmic microwave background (CMB) scales and PBH mass scales.

In §II, we show that independently of the detailed model of inflation, this amplification requires at least an ${\cal O}(1)$ violation of slow roll in canonical single-field inflation. In the recent literature, this has lead to confusion as to which inflaton potentials support PBH formation Garcia-Bellido and Ruiz Morales (2017); Ezquiaga et al. (2017) and how best to calculate the power spectrum for cases that do Drees and Erfani (2011).

Conversely a much larger than ${\cal O}(1)$ violation is not required and so the various slow-roll (SR) and optimized slow-roll (OSR) approximations for the power spectrum reviewed in Appendix A perform differently. Using the inflection (§III), running mass (§IV) and slow-roll step (§V) models as examples, we show that the potential slow-roll formalism can yield qualitatively incorrect results whereas the optimized Hubble slow-roll hierarchy Motohashi and Hu (2015, 2017) can provide accurate results in most of the cases relevant to PBH formation where slow roll is not grossly violated.

II No go for slow roll

Following the pioneering work of Carr (1975), we model PBH formation using the Press-Schechter (PS) approach where the PBH fraction is determined by a collapse threshold $\delta_{c}$ in the density field smoothed with a Gaussian at the horizon reentry scale. In order to relate the collapse fraction directly to the curvature power spectrum, we approximate the result with the analogous probability of Gaussian curvature fluctuations lying above a threshold $\zeta_{c}$ Drees and Erfani (2011); Harada et al. (2013); Young et al. (2014)

[TABLE]

where the approximation assumes $\zeta_{c}\gg\Delta_{\zeta}$ so that PBHs form from rare peaks. Here the density and curvature thresholds are related by assuming a nearly scale-invariant curvature power spectrum for a few efolds around horizon crossing Drees and Erfani (2011); Young et al. (2014)

[TABLE]

In the following estimates, we take $\zeta_{c}=1.3$ based on studies that show a range of $\delta_{c}=0.4-0.5$ Musco and Miller (2013); Harada et al. (2013). The additional factor 2 in (II) is the usual PS bookkeeping that accounts for locally under threshold regions collapsing in globally over threshold regions. Note that $\Delta_{\zeta}^{2}$ is the variance of the curvature field per logarithmic interval in $k$ and hence $\beta$ represents the collapse fraction per $d\ln k$ in the spectrum.

A large collapse fraction requires a large amplitude of $\Delta_{\zeta}^{2}$, much larger than

[TABLE]

measured by the CMB at $k_{0}=0.05$ Mpc*-1* Ade et al. (2016a). Since the leading-order SR approximation gives the curvature power spectrum as

[TABLE]

where $\epsilon_{H}=-d\ln H_{I}/dN$ with $H_{I}$ as the Hubble parameter during inflation and $N$ as the efolds of inflation, the amplification of the power spectrum required for PBH formation can in principle be achieved by making $\epsilon_{H}$ small.

One might think that the slow-roll approximation still holds so long as $\epsilon_{H}$ is small and the expansion is nearly de Sitter $H_{I}\approx$ const. However, even if $\epsilon_{H}$ itself is small, if its fractional variation per efold violates

[TABLE]

then the slow-roll condition is broken. The various slow-roll approximations for the power spectrum reviewed in Appendix A all fail if (5) is strongly violated for a sustained period and perform differently for ${\cal O}(1)$ or transient violation  Kinney (2005); Dvorkin and Hu (2010); Namjoo et al. (2013); Martin et al. (2013); Motohashi et al. (2015); Motohashi and Starobinsky (2017a, b). Indeed without specifying a specific inflationary model, we shall now show that for PBHs to be all of the dark matter there must be at least an ${\cal O}(1)$ violation of the slow-roll condition (5) in single-field inflation.

The collapse fraction required for all of the dark matter to be PBHs depends on their mass. Smaller masses enter the horizon earlier during radiation domination and redshift more slowly than radiation once they collapse. We define the PBH mass as $M=\gamma M_{H}$, where

[TABLE]

is the horizon mass and $\gamma$ accounts for the efficiency of collapse. During radiation domination, the Hubble parameter $H$ is given by

[TABLE]

where we have assumed that the effective degrees of freedom in the entropy and energy densities approximately coincide $g_{*S}\approx g_{*}$ as they do before electron-positron annihilation in the standard model. Here $g_{*0}=3.36$ is the value of the latter today and the radiation energy density today is given by $\Omega_{r}h^{2}=4.18\times 10^{-5}$ for $T_{\rm CMB}=2.725\,$K and $N_{\rm eff}=3.046$. We then obtain $M$ as a function of the horizon crossing epoch $a_{H}$

[TABLE]

After formation, the PBH density dilutes as matter so the relic density today in a mass range around $M$ of $|d\ln M|=2d\ln k$ is given by

[TABLE]

where we ignore mass accretion and evaporation. If we define

[TABLE]

we obtain the cumulative abundance $>M$

[TABLE]

Note that if $\beta(M)=$ const.  as it is for a scale-invariant power spectrum, then $\bar{\beta}=\beta$.

We can invert (11) to find the value of $\bar{\beta}$ required to produce a given relic density

[TABLE]

Given that the cold dark matter density $\Omega_{c}h^{2}=0.1199\pm 0.0022$ Ade et al. (2016a), this gives the $\bar{\beta}$ required for all of the dark matter to be in PBHs above $M$. We can then set (12) equal to (II) to obtain necessary local value of $\Delta_{\zeta}^{2}$ corresponding to the chosen mass $M$.

In order to determine whether the slow-roll condition (5) is violated, we next need to estimate the change in efolds $\Delta N$ during inflation over which this enhancement occurs. The comoving scale corresponding to $M$ is $a_{H}H=a_{\rm exit}H_{I}$ and it exited the horizon during inflation $N$ efolds after the CMB scale $k_{0}=0.05$ Mpc${}^{-1}=a_{0}H_{I}$ exited. Assuming that $H_{I}\approx$ const., we can estimate the efolds by the ratio of comoving scales

[TABLE]

This sets an upper bound on the duration of the change $\Delta N\leq N$.

To establish a lower bound of the variation $|\Delta\ln\epsilon_{H}/\Delta N|$, let us consider the largest $\Delta N$ or the smallest PBH mass that can compose the dark matter. Low mass PBHs evaporate by Hawking radiation. Even if the mass of PBHs grows after formation by merging and accretion, in order to be the dark matter they must at least survive until matter radiation equality. We therefore equate the time scale for evaporation by Hawking radiation

[TABLE]

to the time of equality using the radiation dominated estimate of (7) and $t\approx 1/2H$ to obtain

[TABLE]

With the Planck constraint $\Omega_{m}h^{2}=0.1426\pm 0.0020$ Ade et al. (2016a), this scale left the horizon at

[TABLE]

relative to CMB scales where we have used $g_{*}=106.75$ which is appropriate for $M<1.2\times 10^{-6}\gamma M_{\odot}$ in the standard model.

We can now put this together to place a lower bound on the level of slow-roll violation. Since $\gamma<1$ requires larger $\beta$ and larger $\Delta^{2}_{\zeta}$, we set $\gamma=1$ to be conservative and from (II) and (12) obtain

[TABLE]

as the level of locally scale-invariant power ($\bar{\beta}=\beta$) required at this mass. Given $\Delta_{\zeta}^{2}(k_{0})$ on CMB scales (3),

[TABLE]

We conclude that for PBHs to be the dark matter in single-field inflation, the slow-roll condition must be violated at least at ${\cal O}(1)$.

Although the various approximations employed in this bound carry large uncertainties, including the choice of $\gamma$, the Gaussian approximation for rare peaks, etc., they all enter into $\Delta\ln\epsilon_{H}/\Delta N$ logarithmically. Even orders of magnitude changes in the mass scale and power spectrum amplification would not qualitatively change this result.

III Inflection model

The inflection model was employed in Garcia-Bellido and Ruiz Morales (2017) to attempt to produce a large peak in the curvature power spectrum with a near-inflection point in the potential $V(\phi)$, at which $V^{\prime}$ and $V^{\prime\prime}$ are close to zero. This means that

[TABLE]

The field equation for the inflaton $\phi$ and the Friedmann equation can be written as

[TABLE]

and so given the slow-roll condition (5) $\epsilon_{V}\approx\epsilon_{H}$. To map the potential and its derivative onto the power spectrum in $k$ space, the slow-roll approximation uses

[TABLE]

to solve for $N(\phi)$ on the slow-roll attractor

[TABLE]

In order to distinguish this solution from the exact relation we call this $N_{\rm SR}(\phi)$. Since $V\approx 3H^{2}$, the scalar power spectrum is given by the slow-roll potential form (SR-V)

[TABLE]

where $\eta$ is the conformal time to the end of inflation and $\eta\approx k_{0}^{-1}e^{-N_{\rm SR}(\phi)}$ under SR-V. An inflection model where $\epsilon_{V}\rightarrow 0$ would then predict a large enhancement of the power spectrum under SR-V but violations of (5) can prevent this from occurring in practice.

The inflection potential explored in Garcia-Bellido and Ruiz Morales (2017) is given by

[TABLE]

where $x=\phi/v$. They define

[TABLE]

where $\beta=0$ means the existence of an inflection point $V^{\prime\prime}=0$ at real $x$ and $\Delta N_{\rm SR}$ is associated with the efolds required to cross the inflection under the slow-roll approximation (22). The parameter $\lambda$ is determined by the CMB normalization (3). Given $\lambda$, the phenomenological parameters $(a,\beta,\Delta N_{\rm SR})$ fully define the potential. Note that a similar inflection potential is also considered in the context of critical Higgs inflation Ezquiaga et al. (2017).

Following Garcia-Bellido and Ruiz Morales (2017), we choose $(a,\beta,\Delta N_{\rm SR})=(3/2,4\times 10^{-5},35)$. As shown in Fig. 1, the potential has a plateau around the near-inflection point $\phi=v$. Under the slow-roll approximation the inflaton should slow down and take $\Delta N_{\rm SR}=35$ efolds to cross this point, which would significantly amplify the power spectrum. The SR-V quantities are shown in Fig. 1 by red dashed curves. Here we have assumed that the CMB normalization scale $k_{0}=0.05\,{\rm Mpc}^{-1}$ exits the horizon at $\phi=\phi_{0}\approx 3.71$ when $N=0$.

However, in reality the inflaton arrives at the near-inflection point with excess kinetic energy relative to the inflationary attractor (see Fig. 1). The maximum violation of slow roll $d\ln\epsilon_{H}/dN\approx-6$ occurs when $\epsilon_{V}\rightarrow 0$ in (20) and the field consequently cannot slow down sufficiently quickly to stay on the attractor. Instead the field rolls past this point in about an efold $\Delta N\approx 1$ and ends inflation soon thereafter. Consequently $\epsilon_{H}\gg\epsilon_{V}$ at the near-inflection point and the inflation never slows down below $\epsilon_{H}\sim 0.1$.

Figure 1 (bottom) compares the power spectra from the SR-V approximation and the exact numerical solution of the Mukhanov-Sasaki (MS) equation of motion for the curvature fluctuation. Whereas the SR-V approximation shows a large enhancement of the power spectrum, the true solution does not. In this case since even the SR approximation of (4) correctly predicts the lack of a power spectrum enhancement and no PBHs, we do not consider the other approximations of Appendix A further.

While the parameter set $(a,\beta,\Delta N_{\rm SR})=(3/2,4\times 10^{-5},35)$ does not sustain a long enough period for the inflaton to be trapped at the near-inflection point and reduce $\epsilon_{H}$, as shown in Fig. 2, increasing the parameter $\Delta N_{\rm SR}$ in (25) to $\Delta N_{\rm SR}=100,123,126$ allows the inflaton to traverse the near inflection in $\Delta N\sim 2,15,30$, specifically the efolds between entering into the inflection and the end of inflation. This does lead to $\epsilon_{H}$ reductions of $2.4,1.1\times 10^{4},3.2\times 10^{5}$ compared to CMB scales at $N=0$. There is therefore a small and fine-tuned window in which the power spectrum is substantially enhanced without the inflaton getting stuck so that there are too many efolds between when CMB scales left the horizon and the end of inflation. Moreover, the net reduction in $\epsilon_{H}$ between the nearly scale-invariant region before the inflection to the minimum rapidly saturates at a level that falls short of the required $10^{7}$ enhancement for PBHs to be the dark matter11*1 Reference Garcia-Bellido and Ruiz Morales (2017) aimed to produce $\sim 10^{4}$ amplification with the inflection model compared to the $\sim 10^{7}$ amplification taken here as required for PBHs, which originates from their threshold value $\zeta_{c}=0.086$..

Although further increasing $\Delta N_{\rm SR}$ can enhance the change from $N=0$ to the minimum by shifting the curves in Fig. 2 to the left, it does so by placing the CMB scales near the maximum of $\epsilon_{H}$ where the power spectrum has an unacceptably large red tilt. We conclude that at least for these values of $a$ and $\beta$, it is impossible to make PBHs be all of the dark matter in the inflection model.

IV Running mass model

The logarithmically running mass model provides a more phenomenological approach for designing a potential that produces PBHs. Let us assume the potential

[TABLE]

has a local extremum at $\phi=\phi_{*}$, namely, $V^{\prime}(\phi_{*})=0$. Using a Taylor expansion of $m^{2}(\ln\phi)$ around $\ln\phi=\ln\phi_{*}$, we can express the potential as

[TABLE]

where $L\equiv\ln(\phi/\phi_{*})$. Following Drees and Erfani (2011), we set

[TABLE]

where $L_{0}=\ln(\phi_{0}/\phi_{*})$ and the pivot scale $k_{0}$ exits the horizon at $\phi=\phi_{0}\approx 0.47\phi_{*}$. This choice provides a tilt on CMB scales of $n_{s}(k_{0})=0.964$ with an allowed running $\alpha_{s}(k_{0})=0.012$ and large running of the running of its value. This provides a sufficiently scale-invariant spectrum at CMB scales while allowing a large enhancement of the power spectrum at much higher $k$ for PBH formation.

Since this model is based on Taylor expansion around $\phi_{*}$, it only applies the vicinity of $\phi_{*}$ and does not describe how inflation ends. Indeed since $\epsilon_{H}$ is continuously decreasing, if this truncation were exact, the power spectrum would be continuously amplified on small scales and overproduce PBHs. Hence, we should regard this model as a phenomenological description of inflation only between CMB and the onset of PBH formation where $\Delta_{\zeta}^{2}={\cal O}(10^{-2})$.

The potential, slow-roll parameters, and power spectrum are presented in Fig. 3. As before, we normalize efolds as $\phi(N=0)=\phi_{0}$. In this model $d\ln\epsilon_{H}/dN\approx-1.2$ at its extremum, indicating an ${\cal O}(1)$ violation of (5). Consequently, even though at the same field position $\epsilon_{V}(\phi)\approx\epsilon_{H}(N(\phi))$, the prolonged violation causes the slow-roll approximation (22) to misestimate the efold corresponding to a given field position $N_{\rm SR}(\phi)\neq N(\phi)$. In the middle panel of Fig. 3, we see that this leads to a strong deviation of $\epsilon_{V}(N_{\rm SR})$ from $\epsilon_{H}(N)$.

The bottom panel of Fig. 3 depicts the numerical solution for the curvature power spectrum and the relative error for the various formulae given in Appendix A. The curvature power spectrum is enhanced to a sufficient value to form PBHs $\Delta_{\zeta}^{2}=0.02$ by $\ln(k/k_{0})\approx 35$. We see that SR-V underestimates the power spectrum by a factor of $8.4$ ($-88$%) there and substantially more beyond this point.

The SR approximation of (4) corrects $\phi(N)$ and $\epsilon_{H}$ and performs better with a maximal overestimate of a factor of 1.7 (70%) at $N\approx 30$. Most of this improvement comes from the correction to $\phi(N)$ and note that the SR-V case integrates (22), an approach that is called “exact” in Drees and Erfani (2011). While this misestimate amounts mainly to a shift in efolds, the efolds from the CMB scale control the mass scale of the PBHs through (II). On the other hand, the errors for the optimized first and second order Hubble slow roll approximations of Appendix A, OSR1 and OSR2, peak at about $40\%$ and $10\%$, respectively. Since $d\ln\epsilon_{H}/dN$ never greatly exceeds unity, the higher approximations improve the accuracy without adding much to the computational cost.

V Slow-roll Step Model

In order to more systematically explore slow-roll violation in the PBH context, we can bypass the choice of a potential and directly model the critical function $\epsilon_{H}(N)$, as in an effective field theory of inflation approach. This approach also allows one to straightforwardly generalize these results to noncanonical inflation models in the Horndeski and Gleyzes-Langlois-Piazza-Vernizzi (GLPV) classes by replacing $\epsilon_{H}(N)$ with the general scalar source function Motohashi and Hu (2017).

CMB measurements place observational constraints on $\epsilon_{H}$ around $N=0$, the epoch when $k_{0}=0.05$ Mpc*-1* left the horizon during inflation. If the slow-roll approximation is satisfied on these scales $\epsilon_{H}$ should be well characterized locally by its Taylor expansion

[TABLE]

The scalar tilt and the tensor-scalar ratio are given by

[TABLE]

The Planck CMB data imply Ade et al. (2016b)

[TABLE]

For definiteness, let us take the upper bound on $r$ to fix $C_{1}$, and then use $n_{s}$ to fix $C_{2}$:

[TABLE]

Next, for PBH formation we must change $\ln\epsilon_{H}$ by at least $\ln(10^{7})\sim 16$ before the end of inflation. We therefore consider a steplike transition around $N_{s}$

[TABLE]

Similar to the running mass model, we assume that this form only parametrizes $N\lesssim N_{s}$ so that $\ln\epsilon_{H}$ undergoes another transition to end inflation near $N=60$. The step causes a change of $\Delta\ln\epsilon_{H}\sim-2C_{3}$ across $\sim d$ efolds. For definiteness, we take $C_{3}=8.7$ and $N_{s}=40$ so that $d$ determines the amount by which (5) is violated.

In Fig. 4, we first consider a fairly wide step

[TABLE]

Here $\ln\epsilon_{H}$ has its minimum at $N\approx 65.9$, and its difference from $N=0$ is $\Delta\ln\epsilon_{H}\approx 16.0$. The maximum amplitude of slow-roll violation is $d\ln\epsilon_{H}/dN\approx-0.85$. The relative errors for SR, OSR1, OSR2 are about $60\%$, $-20\%$, $-4\%$, respectively. For slow changes of $\ln\epsilon_{H}$, the optimized formula works extremely well.

We next increase the slow-roll violation by decreasing the step width in Fig. 5,

[TABLE]

for which $\ln\epsilon_{H}$ has its minimum at $N\approx 52.2$, and its difference from $N=0$ is $\Delta\ln\epsilon_{H}\approx 16.3$. The maximum amplitude of time variation is $d\ln\epsilon_{H}/dN\approx-2.2$. The relative errors for SR, OSR1, OSR2 are about $100\%$, $-80\%$, $-50\%$, respectively. While OSR2 still performs relatively well, the larger the violation of slow roll, the less the higher order optimizations improve the result. For the same level of slow-roll violation $C_{3}/d$, the approximations perform similarly if the amplitude $C_{3}$ is changed at fixed width $d$.

VI Conclusion

For PBHs to be the dark matter in single-field inflation, the slow-roll approximation must be violated by at least ${\cal O}(1)$ in order to enhance the curvature power spectrum within the required number of efolds between CMB and PBH scales. As a consequence, power spectrum predictions which rely on the inflaton remaining on the slow-roll attractor can fail dramatically. Models like the inflection potential which might seem to enhance the power spectrum under the potential slow-roll approximation when calculated properly provide no enhancement across most of its parameter space.

Since the slow-roll approximation must fail by ${\cal O}(1)$ but is not required to fail by much more, approximations based on an optimized temporal evaluation of Hubble slow-roll parameters at various orders (OSR1,2) can perform much better at little extra computational cost. In particular the OSR2 approximation remains a good description of the potential out to slow-roll violations of $|d\ln\epsilon_{H}/dN|<2$ which encompasses a wide range of PBH formation models where the $10^{7}$ amplification of power occurs in $\sim 10$ efolds or more.

Note added: The analysis of the inflection model in §III is based on the parameter set used in version 3 of Garcia-Bellido and Ruiz Morales (2017). In version 4 of Garcia-Bellido and Ruiz Morales (2017), which appeared after this work was completed, a different set of parameters is used instead to produce a large amplification, for which the difference between SR-V and exact calculation of curvature power spectrum is reduced. As mentioned in the last two paragraphs of §III, this type of modification requires tuning and still fails to produce the required $10^{7}$ amplification of power over the nearly scale-invariant portion of the curvature power spectrum.

While this work was being completed, Kannike et al. (2017) appeared where the violation of slow roll in inflection model was mentioned in Sec. 2.3. Also, Germani and Prokopec (2017) investigated the violation of slow-roll approximation and an amplification of the curvature power spectrum during ultra-slow-roll phase in the inflection models in Garcia-Bellido and Ruiz Morales (2017); Ezquiaga et al. (2017) reaching similar conclusions.

Acknowledgements.

H.M. was supported in part by MINECO Grant SEV-2014-0398, PROMETEO II/2014/050, Spanish Grant FPA2014-57816-P of the MINECO, and European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreements No. 690575 and 674896. W.H. was supported by grants NSF PHY-0114422, NSF PHY-0551142, U.S. Dept. of Energy contract DE-FG02-13ER41958, and NASA ATP NNX15AK22G.

Appendix A Power Spectrum Approximations

In this appendix, we summarize various power spectrum approximations compared in the main text. For the derivation of OSR, see Motohashi and Hu (2015) for details.

The Hubble flow slow-roll parameters are defined by

[TABLE]

whereas the potential slow-roll parameter is

[TABLE]

• •

SR-V

[TABLE]

with background evolution $\phi(N)$ solved on the slow-roll attractor $\epsilon_{V}\approx\epsilon_{H}$

[TABLE]

and $\eta(N)$ given by the Friedmann equation. In this approximation, the power spectrum can be simply estimated from the potential and $\phi(N)$ since $3H^{2}\approx V\approx$ const. so that $k\eta\approx e^{-N(\phi)}k/k_{0}$.

• •

SR

[TABLE]

The standard leading-order slow-roll approximation. It can be evaluated by solving background evolution equations.

• •

OSR1

[TABLE]

The leading order optimized slow-roll formula obtained from Taylor expansion of the generalized slow-roll approximation Motohashi and Hu (2015). The evaluation epoch is not horizon exit $k\eta=1$ but $k\eta=x_{1}$ where

[TABLE]

which corresponds to resumming the next-leading-order term in the source evolution. In Motohashi and Hu (2015) it was denoted as the optimized leading-order approximation (OLO).

• •

OSR2

[TABLE]

where

[TABLE]

The next-leading-order optimized slow-roll formula obtained from the Taylor expansion of the generalized slow-roll approximation. In Motohashi and Hu (2015) it was denoted as the optimized next-to-leading-order approximation (ONO).

• •

Numerical (MS)

[TABLE]

Numerical solution of Mukhanov-Sasaki equation of the curvature modefunction $\zeta_{k}$ with Bunch-Davies boundary condition at $k\eta\gg 1$.