# Measuring the spectral running from cosmic microwave background and primordial black holes

## Abstract

We constrain the spectral running by combining cosmic microwave background (CMB) data, baryon acoustic oscillation (BAO) data and the constraint from primordial black holes (PBHs). We find that the constraint from PBHs has a significant impact on the running of running of scalar spectral index, and a power-law scalar power spectrum without running is consistent with observational data once the constraint from PBHs is taken into account. In addition, from the constraints on the slow-roll parameters, the derived tensor spectral index in the single-field slow-roll inflation model is quite small, namely \(|n_t|\lesssim 9.3\times 10^{-3}\) which will be very difficult to be measured by CMB data only in the future, and the absolute value of derived running of tensor spectral index is not larger than \(2.1\times 10^{-4}\) at 95% confidence level.

## 1 Introduction

In 1971, Hawking proposed that the highly overdense region of inhomogeneities would eventually cease expanding and collapse into a black hole in [8]. A large positive value of the running of running may lead to the formation of primordial black holes (PBH) at small scales in the early universe. Different from the astrophysical black holes which should be heavier than a particular mass (around 3 solar mass \(M_\odot \)), PBHs can have very small masses. However, the PBHs with masses smaller than \(10^{-18} M_\odot \) would have completely evaporated by now due to the Hawking radiation. In particular, the quantum emissions from PBHs with mass around \(10^{-18} M_\odot \) can generate a \(\gamma \)-ray background which should be observed today. Conversely, the non-detection of such a \(\gamma \)-ray background in our Universe provides a stringent constraint on the PBHs with such a mass. Recently various constraints on the abundance of PBHs are present in [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Ones expect that the absence of PBHs should tightly constrain the running of running of scalar spectral index. See some recent related works in [23, 24].

On the other hand, inflation [25, 26, 27, 28] is taken as the leading paradigm for the physics in the very early universe. The initial inhomogeneities and spatial curvature are supposed to be stretched away by the quasi-exponential expansion of inflation. Because the Hubble parameter is roughly a constant during inflation, it predicts a nearly scale-invariant power spectrum of the curvature (scalar) perturbations seeded by the quantum fluctuations of inflaton field during inflation. Typically, the scalar spectral index \(n_s\) is related to the number of e-folds *N* before the end of inflation by \(n_s-1\sim -1/N\), and then \(\beta _s\sim -1/N^3\sim (n_s-1)^3\sim -\mathcal{O}(10^{-5})\). In this sense, the running of running of scalar spectra index given in Eqs. (2) and (3) seems too large to fit in the typical inflation models.

In this paper we adopt two methods to compare inflationary predictions with current cosmological datasets, in particular including the constraint from the PBHs. The first method consists of a phenomenological parameterization of the primordial spectra of both scalar and tensor perturbations, and the second exploits the analytic slow-roll-parameter dependence of primordial perturbations.

This paper is organized as follows. In Sect. 2, we explain the datasets used in this paper. In Sects. 3 and 4, we present the constraints on the phenomenological parameters and the slow-roll parameters with the publicly available codes CosmoMC [29], respectively. A brief summary will be given in Sect. 5.

## 2 Data

The full-mission Planck observes the temperature and polarization anisotropies of the cosmic microwave background (CMB) radiation. In this paper we also combine all the data taken by the BICEP2 and Keck Array CMB polarization experiments up to and including the 2014 observing season [3] with the Planck data.

Baryon Acoustic Oscillation (BAO) detections measure the correlation function and power spectrum in the clustering of galaxies. Measuring the position of these oscillations in the matter power spectra at different redshifts removes degeneracies in the interpretation of the CMB anisotropies. The BAO data adopted in this paper include 6dFGS [5], MGS [6], BOSS \(\mathrm {DR11}\_{\mathrm {Ly}\alpha }\) [30], BOSS DR12 with nine anisotropic measurements [31], and eBOSS DR14 [32].

Actually Eq. (7) is a very conservative constraint. The ranges of *k* corresponding to the \(\gamma \)-ray background extend from \(10^{14}\) Mpc\(^{-1}\) to \(10^{15}\) Mpc\(^{-1}\), and the upper limit of scalar power spectrum is generally in the range of 0.1 to 0.01. Therefore a more restricted constraint might be \(P_s(k=10^{15} \hbox {Mpc}^{-1})<10^{-2}\) which is denoted by “PBH*”.

## 3 Constraints on the spectral running from CMB and PBHs

*r*, to quantify the tensor amplitude compared to the scalar amplitude at the pivot scale:

The \(68\%\) limits on the cosmological parameters in the \(\Lambda \)CDM + \(r~+~\alpha _s~+~\beta _s\) model from the data combinations of CMB + BAO and CMB + BAO + PBH respectively

Parameter | CMB+BAO | CMB+BAO+PBH |
---|---|---|

\(\Omega _bh^2\) | \(0.02232\pm 0.00015\) | \(0.02238\pm 0.00014\) |

\(\Omega _ch^2\) | \(0.1180\pm 0.0008\) | \(0.1179\pm 0.0008\) |

\(100\theta _{\mathrm {MC}}\) | \(1.0410\pm 0.0003\) | \(1.0410\pm 0.0003\) |

\(\tau \) | \(0.079\pm 0.012\) | \(0.073\pm 0.012\) |

\(\ln \left( 10^{10}A_s\right) \) | \(3.088\pm 0.023\) | \(3.077\pm 0.022\) |

\(n_s\) | \(0.9660\pm 0.0040\) | \(0.9693\pm 0.0038\) |

\(\alpha _s\) | \(0.0077^{+0.0104}_{-0.0103}\) | \(-0.0063^{+0.0073}_{-0.0072}\) |

\(\beta _s\) | \(0.019\pm 0.013\) | \(-0.0035^{+0.0065}_{-0.0023}\) |

| \(<0.086 \) | \(<0.079\) |

*r*. We constrain all of these 9 parameters in the \(\Lambda \)CDM+\(r+\alpha _s+\beta _s\) model by adopting two different data combinations, namely CMB+BAO and CMB+BAO+PBH, respectively. The results are given in Table 1 and Fig. 1. The constraint on the running of running of scalar spectral index is

## 4 Constraints on the slow-roll parameters

*t*and the inflaton field \(\phi \), respectively. The inflaton field slowly rolls down its potential if \(\epsilon \ll 1\) and \(|\eta |\ll 1\), where

*r*, \(n_t\) and \(\alpha _t\) are all taken as the derived parameters. The constraints on the slow-roll parameters \(\{\epsilon , \eta , \xi , \sigma \}\) and the contour plots of these slow-roll parameters are illustrated in Table 2 and Fig. 2. We see that the constraint on the fourth slow-roll parameter \(\sigma \) is improved once the constraint from PBHs is included because the running of running is dominated by it in the slow-roll inflation model.

The 95% limits on the slow-roll parameters and the derived parameters from the data combinations of CMB + BAO and CMB + BAO + PBH respectively

Parameter | CMB + BAO | CMB + BAO + PBH |
---|---|---|

\(\epsilon \) | \(<0.0053\) | \(<0.0047\) |

\(\eta \) | \(-\,0.0112^{+0.0122}_{-\,0.0116}\) | \(-\,0.0104^{+0.0111}_{-\,0.0102}\) |

\(\xi \) | \(-\,0.0043^{+0.0106}_{-\,0.0107}\) | \(0.0031^{+0.0077}_{-\,0.0074}\) |

\(\sigma \) | \(0.0104^{+0.0134}_{-\,0.0133}\) | \(-\,0.0018^{+0.0045}_{-\,0.0062}\) |

| \(<0.082\) | \(<0.073\) |

\(-n_t\ (\times 10^{-2})\) | \(<1.1\) | \(<0.93\) |

\(-\alpha _t\ (\times 10^{-4})\) | \(<2.6\) | \(<2.1\) |

## 5 Summary

In this paper we use two methods to constrain the slow-roll inflation models by combining CMB, BAO and the constraint from PBHs. Even though a positive running of running of scalar spectral index is slightly preferred by the data combination of CMB and BAO datasets, a power-law scalar power spectrum without running is consistent with the data once the constraint from PBHs is taken into account.

We can also directly constrain the slow-roll parameters from the observational data. An advantage of this method is that we can work out the predictions of single-field slow-roll inflation model by using these constrained slow-roll parameters. For example, we illustrate the predictions of the parameters characterizing the tensor power spectrum, and find that both the tensor spectral index and its running are negative and their absolute values are not larger than \(9.3\times 10^{-3}\) and \(2.1\times 10^{-4}\) at 95% CL, respectively. Our results imply that it is very difficult to measure these two parameters in the future.

## Notes

### Acknowledgements

We thank the anonymous referee for valuable suggestions and comments. We acknowledge the use of HPC Cluster of ITP-CAS. This work is supported by grants from NSFC (Grant nos. 11335012, 11575271, 11690021, 11747601), Top-Notch Young Talents Program of China, and partly supported by the Strategic Priority Research Program of CAS and Key Research Program of Frontier Sciences of CAS.

## References

- 1.P.A.R. Ade et al., [Planck Collaboration], Astron. Astrophys.
**594**, A13 (2016). arXiv:1502.01589 [astro-ph.CO] - 2.P.A.R. Ade et al., [Planck Collaboration], Astron. Astrophys.
**594**, A20 (2016). arXiv:1502.02114 [astro-ph.CO] - 3.P.A.R. Ade et al., [BICEP2 and Keck Array Collaborations], Phys. Rev. Lett.
**116**, 031302 (2016). arXiv:1510.09217 [astro-ph.CO] - 4.H. Gil-Marin, Mon. Not. R. Astron. Soc.
**460**(4), 4210 (2016). arXiv:1509.06373 [astro-ph.CO]ADSCrossRefGoogle Scholar - 5.F. Beutler, Mon. Not. R. Astron. Soc.
**416**, 3017 (2011). arXiv:1106.3366 [astro-ph.CO]ADSCrossRefGoogle Scholar - 6.A.J. Ross, L. Samushia, C. Howlett, W.J. Percival, A. Burden, M. Manera, Mon. Not. R. Astron. Soc.
**449**(1), 835 (2015). arXiv:1409.3242 ADSCrossRefGoogle Scholar - 7.Q.G. Huang, K. Wang, S. Wang, Phys. Rev. D
**93**(10), 103516 (2016). arXiv:1512.07769 [astro-ph.CO]ADSCrossRefGoogle Scholar - 8.S. Hawking, Mon. Not. R. Astron. Soc.
**152**, 75 (1971)ADSCrossRefGoogle Scholar - 9.L. Chen, Q.G. Huang, K. Wang, JCAP
**1612**(12), 044 (2016). arXiv:1608.02174 [astro-ph.CO]ADSCrossRefGoogle Scholar - 10.A.M. Green, Phys. Rev. D
**94**(6), 063530 (2016). arXiv:1609.01143 [astro-ph.CO]ADSCrossRefGoogle Scholar - 11.K. Schutz, A. Liu, Phys. Rev. D
**95**(2), 023002 (2017). arXiv:1610.04234 astro-ph.CO]ADSCrossRefGoogle Scholar - 12.S. Wang, Y.F. Wang, Q.G. Huang, T.G.F. Li, Phys. Rev. Lett.
**120**(19), 191102 (2018). arXiv:1610.08725 [astro-ph.CO]ADSCrossRefGoogle Scholar - 13.D. Gaggero, G. Bertone, F. Calore, R.M.T. Connors, M. Lovell, S. Markoff, E. Storm, Phys. Rev. Lett.
**118**(24), 241101 (2017). arXiv:1612.00457 [astro-ph.HE]ADSCrossRefGoogle Scholar - 14.Y. Ali-Hamoud, M. Kamionkowski, Phys. Rev. D
**95**(4), 043534 (2017). arXiv:1612.05644 [astro-ph.CO]ADSCrossRefGoogle Scholar - 15.D. Aloni, K. Blum, R. Flauger, JCAP
**1705**(05), 017 (2017). arXiv:1612.06811 [astro-ph.CO]ADSCrossRefGoogle Scholar - 16.B. Horowitz, arXiv:1612.07264 [astro-ph.CO]
- 17.F. Kuhnel, K. Freese, Phys. Rev. D
**95**(8), 083508 (2017). arXiv:1701.07223 [astro-ph.CO]ADSCrossRefGoogle Scholar - 18.Y. Inoue, A. Kusenko, JCAP
**1710**(10), 034 (2017). arXiv:1705.00791 [astro-ph.CO]ADSCrossRefGoogle Scholar - 19.B. Carr, M. Raidal, T. Tenkanen, V. Vaskonen, H. Veermae, Phys. Rev. D
**96**(2), 023514 (2017). arXiv:1705.05567 [astro-ph.CO]ADSMathSciNetCrossRefGoogle Scholar - 20.A.M. Green, Phys. Rev. D
**96**(4), 043020 (2017). arXiv:1705.10818 [astro-ph.CO]ADSCrossRefGoogle Scholar - 21.M. Zumalacarregui, U. Seljak, arXiv:1712.02240 [astro-ph.CO]
- 22.Z.C. Chen, Q.G. Huang, arXiv:1801.10327 [astro-ph.CO]
- 23.B. Carr, T. Tenkanen, V. Vaskonen, Phys. Rev. D
**96**(6), 063507 (2017). arXiv:1706.03746 [astro-ph.CO]ADSCrossRefGoogle Scholar - 24.K. Kohri, T. Terada, arXiv:1802.06785 [astro-ph.CO]
- 25.A.H. Guth, Phys. Rev. D
**23**, 347 (1981)ADSCrossRefGoogle Scholar - 26.A.A. Starobinsky, Phys. Lett. B
**91**, 99 (1980)ADSCrossRefGoogle Scholar - 27.A.D. Linde, Phys. Lett.
**108B**, 389 (1982)ADSCrossRefGoogle Scholar - 28.A. Albrecht, P.J. Steinhardt, Phys. Rev. Lett.
**48**, 1220 (1982)ADSCrossRefGoogle Scholar - 29.A. Lewis, S. Bridle, Phys. Rev. D
**66**, 103511 (2002). astro-ph/0205436ADSCrossRefGoogle Scholar - 30.T. Delubac et al., BOSS Collaboration, Astron. Astrophys.
**574**, A59 (2015). arXiv:1404.1801 [astro-ph.CO] - 31.Y. Wang et al., [BOSS Collaboration], Mon. Not. Roy. Astron. Soc
**469**(3), 3762 (2017). arXiv:1607.03154 [astro-ph.CO] - 32.M. Ata, Mon. Not. R. Astron. Soc.
**473**(4), 4773 (2018). arXiv:1705.06373 [astro-ph.CO]ADSMathSciNetCrossRefGoogle Scholar - 33.B.J. Carr, Astrophys. J.
**201**, 1 (1975)ADSCrossRefGoogle Scholar - 34.S.W. Hawking, Commun. Math. Phys.
**43**, 199 (1975)ADSCrossRefGoogle Scholar - 35.D.N. Page, S.W. Hawking, Astrophys. J.
**206**, 1 (1976)ADSCrossRefGoogle Scholar - 36.P. Sreekumar et al., [EGRET Collaboration], Astrophys. J.
**494**, 523 (1998). arXiv:astro-ph/9709257 - 37.B.J. Carr, K. Kohri, Y. Sendouda, J. Yokoyama, Phys. Rev. D
**81**, 104019 (2010). arXiv:0912.5297 [astro-ph.CO]ADSCrossRefGoogle Scholar - 38.B. Carr, F. Kuhnel, M. Sandstad, Phys. Rev. D
**94**(8), 083504 (2016). arXiv:1607.06077 [astro-ph.CO]ADSCrossRefGoogle Scholar - 39.A.R. Liddle, D.H. Lyth, Phys. Lett. B
**291**, 391 (1992). astro-ph/9208007ADSCrossRefGoogle Scholar - 40.E.J. Copeland, E.W. Kolb, A.R. Liddle, J.E. Lidsey, Phys. Rev. Lett.
**71**, 219 (1993). hep-ph/9304228ADSCrossRefGoogle Scholar - 41.Q.G. Huang, Sci. China Phys. Mech. Astron.
**58**, 11 (2015). arXiv:1407.4639 [astro-ph.CO]CrossRefGoogle Scholar - 42.E.D. Stewart, D.H. Lyth, Phys. Lett. B
**302**, 171 (1993). gr-qc/9302019ADSCrossRefGoogle Scholar - 43.S.M. Leach, A.R. Liddle, J. Martin, D.J. Schwarz, Phys. Rev. D
**66**, 023515 (2002). https://doi.org/10.1103/PhysRevD.66.023515. arXiv:astro-ph/0202094 - 44.Q.G. Huang, Phys. Rev. D
**76**, 043505 (2007). astro-ph/0610924ADSCrossRefGoogle Scholar - 45.Q.G. Huang, S. Wang, arXiv:1701.06115 [astro-ph.CO]

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