Limits on Stellar-Mass Compact Objects as Dark Matter from Gravitational Lensing of Type Ia Supernovae

The nature of dark matter (DM) remains unknown despite very precise knowledge of its abundance in the Universe. An alternative to new elementary particles postulates DM as made of macroscopic compact halo objects (MACHO) such as black holes formed in the very early Universe. Stellar-mass primordial black holes (PBHs) are subject to less robust constraints than other mass ranges and might be connected to gravitational-wave signals detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO). New methods are therefore necessary to constrain the viability of compact objects as a DM candidate. Here we report bounds on the abundance of compact objects from gravitational lensing of type Ia supernovae (SNe). Current SNe data sets constrain compact objects to represent less than 35.2% (Joint Lightcurve Analysis) and 37.2% (Union 2.1) of the total matter content in the Universe, at 95% confidence level. The results are valid for masses larger than $\sim 0.01M_{\odot }$ (solar masses), limited by the size SNe relative to the lens Einstein radius. We demonstrate the mass range of the constraints by computing magnification probabilities for realistic SNe sizes and different values of the PBH mass. Our bounds are sensitive to the total abundance of compact objects with $M\gtrsim 0.01M_{\odot }$ and complementary to other observational tests. These results are robust against cosmological parameters, outlier rejection, correlated noise, and selection bias. PBHs and other MACHOs are therefore ruled out as the dominant form of DM for objects associated to LIGO gravitational wave detections. These bounds constrain early-Universe models that predict stellar-mass PBH production and strengthen the case for lighter forms of DM, including new elementary particles.

Figure 1

Probability of lensing magnification $\mathrm{\Delta }\mu$ [Eq. (1)] and its dependence on SNe redshift $z$ and compact object abundance $\alpha ={\mathrm{\Omega }}_{\mathrm{PBH}}/{\mathrm{\Omega }}_M$. A sizable compact object population displaces the maximum of the PDF towards the empty-beam distance, compensated with a probability tail for high magnification. Both effects grow with redshift, we show only $z\ge 0.2$. Left panel: Probability density function [Eq. (2)], normalized to unity. Histograms show to residuals of JLA (blue, solid) and Union 2.1 (green, dashed) data in the redshift ranges shown. Curves show theoretical predictions for negligible PBH (red, solid), 50% PBH (black, dotted) and PBH-only (black, dashed) universes at the mean redshift $\overline{z}$ of the subsample. A fiducial Gaussian scatter with typical SNe uncertainties ${\sigma }_{\mu }=0.15$ has been assumed to facilitate comparison of theory and data. Right panel: Tail distribution (cumulative) in logarithmic scale to highlight the enhanced probability of high magnification in PBH models. Histograms are normalized to the number of SNe in each redshift interval and theory predictions are normalized to the number of JLA SNe. Horizontal lines correspond to 1–4 events and vertical lines mark where $3{\sigma }_{\mu }$, $5{\sigma }_{\mu }$ outliers are expected, relative to the SNe measurement uncertainty.

Figure 2

Magnification PDF dependence on the PBH mass for extended SNe (compare with Fig. 1). The curves for each PBH mass assume a SNe radius $R_S\approx 115\mathrm{AU}$ (see Supplemental Material, Sec. I). Right panel: PDF without noise. For $M\lesssim {10}^{-2}{M}_{\odot }$ the result converges the analytical fit used in the analysis [21]. Note that introducing realistic noise (as in Fig. 1) would render both curves practically indistinguishable around the peak. Left panel: cumulative PDF, convolved with noise. The high-magnification tail decays faster than $(\mathrm{\Delta }\mu {)}^{-3}$ but a significant fraction of highly magnified SNe are predicted even for low PBH masses.

Figure 3

Bounds on the abundance of PBHs as a function of the mass (95% C.L.). The analysis of SNe lensing using the JLA (solid) and Union 2.1 compilations (dashed) constrain the PBH fraction in the range $M\gtrsim 0.01M_{\odot }$. This range includes the masses of black hole events observed by LIGO (gray), only weakly constrained by previous data including microlensing (EROS [41]), the stability of stellar compact systems (Eridanus II [42, 43]) and CMB [44, 45]. The CMB excluded regions correspond to Planck-TT (solid), Planck-full (dotted) for the limiting cases of collisional (red), and photoionization (orange) (see Ref. [45] for details).