[Paper Review] Dark Matter Primordial Black Holes and Inflation Models

Manuel Drees, Encieh Erfani|arXiv (Cornell University)|May 17, 2012

Cosmology and Gravitation Theories1 references3 citations

TL;DR

This paper investigates whether primordial black holes (PBHs) formed in the early universe could constitute dark matter, focusing on single-field inflation models. It finds that only the 'running-mass' inflation model—featuring a positive 'running of the running' of the spectral index—can produce sufficiently blue power spectra at small scales to enable long-lived PBH formation, while all large-field models fail due to insufficient spectral tilt evolution.

ABSTRACT

A broad range of single field models of inflation are analyzed in light of all relevant recent cosmological data, checking whether they can lead to the formation of long--lived Primordial Black Holes (PBHs) as candidate for dark matter. To that end we calculate the spectral index of the power spectrum of primordial perturbations as well as its first and second derivatives. PBH formation is possible only if the spectral index $n_S(k_0)$ increases significantly at small scales. Since current data indicate that the first derivative $α_S$ of the spectral index is negative at the pivot scale, PBH formation is only possible in the presence of a sizable and positive second derivative ("running of the running") $β_S$. Among the three small-field and five large-field models we analyze, only the "running-mass" model allows PBH formation, for a narrow range of parameters.

Motivation & Objective

• To assess whether primordial black holes (PBHs) formed during inflation can account for dark matter in the context of current cosmological data.
• To determine which inflationary models can generate the required blue-tilted power spectrum at small scales for PBH formation.
• To evaluate the role of higher-order spectral index derivatives—particularly 'running of the running'—in enabling PBH production.
• To compare predictions of small-field and large-field inflation models against observational constraints from CMB, BAO, H₀, and SPT data.
• To identify viable inflation models that can produce long-lived PBHs with masses >10¹⁵ g, consistent with dark matter abundance.

Proposed method

• The study analyzes 8 single-field inflation models: 3 small-field and 5 large-field, using their potential forms to compute the scalar spectral index and its derivatives.
• The spectral index $ n_S(k) $, its first derivative $ \alpha_S = dn_S/d\ln k $, and second derivative $ \beta_S = d^2n_S/d(\ln k)^2 $ (running of the running) are computed as key observables.
• The Press–Schechter formalism is applied to estimate PBH formation probability from Gaussian, spherically symmetric density perturbations with threshold $ \delta_{\rm th} = 1/3 $ during radiation domination.
• Observational constraints from WMAP7, BAO, $ H_0 $, SPT, and cluster data are used to fix the pivot scale $ k_{\rm pivot} $, with $ n_S(k_{\rm pivot}) = 0.9751 $ as central value.
• A model-independent analysis is performed to assess whether $ \beta_S $ can compensate for a negative $ \alpha_S $ at CMB scales to reach $ n_S(k_{\rm PBH}) \approx 1.37 $, required for significant PBH formation.
• The analysis focuses on PBH masses >10¹⁵ g, whose lifetimes exceed the age of the universe, making them viable dark matter candidates.

Experimental results

Research questions

• RQ1 Can inflationary models generate a sufficiently blue-tilted power spectrum at small scales to produce long-lived primordial black holes (PBHs) with masses >10¹⁵ g?
• RQ2 Is a positive 'running of the running' of the spectral index ($ \beta_S $) necessary to achieve the required spectral index $ n_S(k_{\rm PBH}) \approx 1.37 $ for PBH formation?
• RQ3 Which specific inflation models—among small-field and large-field types—can support a large positive $ \beta_S $, enabling PBH formation?
• RQ4 Can current cosmological data constrain the second derivative $ \beta_S $ well enough to rule out or support PBH dark matter scenarios?
• RQ5 Why do large-field inflation models fail to produce significant PBHs despite their popularity in fitting CMB data?

Key findings

• Only the 'running-mass' inflation model—a small-field model—produces a sufficiently large positive 'running of the running' ($ \beta_S $) to allow the spectral index to rise to $ n_S(k_{\rm PBH}) \approx 1.37 $ at small scales.
• All large-field inflation models studied predict either small or negative values of $ \beta_S $, making them incapable of generating the required blue tilt for significant PBH formation.
• The current central value of the first derivative $ \alpha_S $ is $ -0.020 $, which is more than 3σ below the value needed to achieve $ n_S(k_{\rm PBH}) \approx 1.37 $ alone.
• The second derivative $ \beta_S $ is currently weakly constrained by data, allowing it to compensate for a negative $ \alpha_S $, thus enabling PBH formation in models with favorable $ \beta_S $.
• The threshold for PBH formation is $ \delta_{\rm th} = 1/3 $, and only models with $ n_S(k_{\rm PBH}) \approx 1.37 $ at the relevant scale can produce PBHs with lifetimes exceeding the age of the universe.
• Among the 8 models analyzed, only the running-mass model allows PBH formation in a narrow but viable parameter range, making it the only candidate for PBHs as cold dark matter within this framework.