No LIGO MACHO∗ : Primordial Black Holes, Dark Matter and Gravitational Lensing of Type Ia Supernovae Miguel Zumalac´arregui1, 2, 3, † and Uroˇs Seljak1, 4, ‡ 1
arXiv:1712.02240v1 [astro-ph.CO] 6 Dec 2017
Berkeley Center for Cosmological Physics, LBNL and University of California at Berkeley, Berkeley, California 94720, USA 2 Institut de Physique Th´eorique, Universit´e Paris Saclay CEA, CNRS, 91191 Gif-sur-Yvette, France 3 Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden 4 Physics and Astronomy Department, LBNL, University of California at Berkeley, Berkeley, California 94720, USA (Dated: December 7, 2017) Black hole merger events detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO) have revived dark matter models based on primordial black holes (PBH) or other massive compact halo objects (MACHO). This macroscopic dark matter paradigm can be distinguished from particle physics models through their gravitational lensing predictions: compact objects cause most lines of sight to be demagnified relative to the mean, with a long tail of higher magnifications. We test the PBH model using the lack of lensing signatures on type Ia supernovae (SNe), modeling the effects of large scale structure, allowing for a non-gaussian model for the intrinsic SNe luminosity distribution and addressing potential systematic errors. Using current JLA (Union) SNe data, we derive bounds ΩPBH /ΩM < 0.346 (0.405) at 95% confidence, ruling out the hypothesis of MACHO/PBH comprising the totality of the dark matter at 5.01σ(4.28σ) significance. The finite size of SNe limits the validity of the results to MPBH & 10−2 M , fully covering the black hole mergers detected by LIGO and closing that previously open PBH mass range.
I.
INTRODUCTION
A major goal of cosmology is to characterize the dark components of the universe. The nature of Dark Matter (DM), the component sourcing the formation of large scale structure (LSS) and contributing 27% of the energy budget of the universe, remains highly elusive. The standard DM scenario postulates a new elementary particle, abundantly produced in the early universe and with a small cross section that makes it difficult to detect by current experiments or be produced in particle colliders. Although cosmology is insensitive to most microscopic details of DM scenarios, observations prefer cold dark matter (CDM) models in which DM behaves as a nonrelativistic fluid. An alternative to microscopic dark matter scenarios postulates that CDM is formed by Primordial Black Holes (PBH) or other macroscopic entities, generically known as massive compact halo objects (MACHO), that would have formed in the early universe [1–4]. PBHs behave as non-relativistic matter on sufficiently large scales, making them viable CDM candidates for cosmology. They leave no trace on particle searches, but can be probed by a series of small-scale effects that depend on the mass and other properties of the object (see Fig. 1 and [5, 6] for recent reviews). Interestingly, the weakest constraints on PBHs (M ∼
∗ This
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10 − 100M ) coincide with the masses of black holes detected by the Laser Interferometer Gravitational Observatory (LIGO) [7–9]. This intriguing possibility lead to a revival of PBH models [10–12] (see [13] for pre-detection work) that could simultaneously provide the right dark matter abundance, explain the high merger rate and progenitor masses inferred by the first LIGO detections while being compatible with other bounds. Unfortunately, uncertainties in the small-scale distribution of PBHs remain an obstacle to constrain their abundance on the basis of current gravitational wave (GW) observations (although see [14]). Other methods are needed to reliably test the PBH-DM hypothesis. Given the dark nature of PBHs, a promising technique is to probe their gravitational influence on the propagation of light. Microlensing observations, based on monitoring a field of stars and search for magnification caused by compact objects moving near the line of sight, yields one of the most powerful constraints in the range of mases M . M , right below the LIGO band. The characteristic √timescale for microlensing-induced variations grows as M and this technique becomes ineffective for PBH over M & 10M [15]. In this paper we will derive lensing constraints on the PBH abundance using Supernovae as standard candles of known luminosity. This measurement is different from traditional microlensing of stars in a number of ways. Instead of comparing the same star at different times, this method compares different type Ia supernovae, some of which will be highly magnified by a PBH. The main advantage of this method is that it does not rely on the movement of the PBHs, making SNe lensing sensitive to larger PBH masses than stellar microlensing. Instead,
2
Eridanos II
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FIG. 1: Bounds on the abundance of PBHs as a function of the mass (95 % confidence level). The analysis of SNe lensing using the JLA (solid) and Union 2.1 compilations (dashed) constrain the PBH fraction in the range M & 0.01M . This range includes the masses of black hole events observed by LIGO (gray), only weakly constrained by previous data including micro-lensing (EROS [16]), the stability of stellar compact systems (Eridanus II [17, 18]) and CMB [19]. The CMB excluded regions correspond to Planck-TT (solid), Planck-full (dotted) for the limiting cases of collisional (red) and photo-ionization (orange) (see [19] for details).
−0.1
the effectiveness of this technique is limited by the finitesize of SNe if the PBHs are sufficiently light. Finally, SNe probe the very deep universe, as opposed to specific nearby regions of the sky. Our results provide stringent constraints, ruling out the DM-PBH model in the mass range detected by LIGO at high significance. Our work improves on previous analyses [20] and is complementary to other techniques based on lensing such as caustic crossing [21, 22], as well as bounds derived from the CMB [19, 23–26] (see also [27] for earlier work). Section II describes the effect of PBHs on the magnification of distance sources. In Section III we describe the likelihood and how we model the SNe, including systematic effects. In Section IV we present the bounds derived from our analysis. We conclude in Section V.
SNE LENSING BY COMPACT OBJECTS
Gravitational lensing of small sources like SNe is sensitive to the abundance of compact objects. This section presents the statistical predictions of lensing magnification, including the effects of a variable PBH fraction and the large scale structure (LSS) of the universe. We will then consider how the constraints are affected by assumptions on the PBH mass distribution.
0.0
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FIG. 2: Effects of the PBH fraction on the magnification probability density function (equation 6), including compact objects and cosmological large scale structure. Compact objects produce 1) a displacement of the maximum of the PDF towards a demagnified universe and 2) a larger probability of large magnifications. The cases shown correspond to no PBH (solid) and all of the dark matter (but not baryons) in PBH (dashed) at z = 1. Also shown is the empty beam (vertical dotted line). We see that LSS never reaches empty beam values: all LSS lines of sight pass through matter for z = 1.
A.
II.
α=0 α = 0.85
Magnification by compact objects
Gravitational lensing by compact objects has two distinct effects, summarized in Fig. 2: • Most objects appear dimmer than the average, as most light beams do not pass near any lens. The characteristic demagnification corresponds to the empty-beam distance. • Few objects undergo significant magnification, as their light beams pass very close to a lens. These bright outliers are far less likely in a microscopic DM scenario. Note that both effects are balanced so that the mean magnification remains the same as in a homogeneous universe. In PBHs-dominated universe, the line of sight to most sources will not lay near any compact object. Those sources will be demagnified and appear fainter, affecting its perceived angular-diameter distance D(µ, z) = √
¯ DE (z) D(z) =√ . 1+µ 1 + ∆µ
(1)
In the first equality we have defined the magnification ∆µ relative to filled-beam distance, i.e. the angular di¯ ameter distance of the homogeneous cosmology D(z) = R dz0 1 . The second equality defines µ relative to the 0 1+z H(z ) empty-beam distance [28, 29] Z z 1 DE (z) = dz 0 . (2) 0 )2 H(z 0 ) (1 + z 0
3 Some sources will appear highly magnified by a compact object near the line of sight. The lensing probability distribution function (PDF) of a universe filled by a uniform comoving density of PBHs only depends on the mean magnification µ ¯. Numerical simulations [30] have found that that the PDF is well approximated by PC (µ, µ ¯) = A
1 − e−µ/δ (µ + 1)2 − 1
3/2 if µ > 0 ,
(3)
and 0 otherwise. The parameters A, δ depend on µ ¯ and are chosenRto normalize the distribution and enforce the mean µ ¯ ≡ dµ µPC (µ, δ). In the high-magnification limit the PDF decays as PC (µ) ∝ µ−3 , as has been shown in the limit of a single lens [20] and by detailed numerical studies with a distribution of point lenses [30, 31]. We note that our analysis is in the regime of low optical depth (low average convergence and shear): we will use data up to z ∼ 1 where µ < 0.14 and the PDF for point lenses only depends on the total magnification [20, 30]. In this regime caustics are isolated and from individual lenses only and magnification is well below the threshold where collective effects (caustic networks) become important. In this limit the PDF (equation 3) is independent of the PBH mass as long as lenses and sources can be considered point-like, with the finiteness of sources requiring that MPBH & 10−2 M (see Sec. II C). In this case, the statistical distribution of the lensing images in fact independent of both the mass spectrum and the clustering properties of the point masses, provided that the clustering is spherical [31]. In the specific case of a PBH-only universe the mean magnification in the PDF (equation 3) has to ensure that the mean distance corresponds to the homogeneous cosmology (∆µ = 0 in equation 1). This corresponds to µ ¯ = µF where the full-beam magnification 2 ¯ µF ≡ DE (z)/D(z) − 1.
(4)
Keeping the dependence on µ ¯ will allow us to include the effects of LSS clustering in the next section. B.
PBH fraction and Large Scale Structure
We need to generalize the simplified model of the previous section to a realistic universe in which a fraction of compact objects α≡
ρPBH ρM = fPBH , ρM ρDM
(5)
traces the underlying LSS distribution. Note we are distinguishing the fraction over the total matter density including baryons α from the fraction of DM fPBH . We will work with the former, as it is more convenient to incorporate the effects of LSS. Lensing by compact objects (equation 3) and LSS will each contribute with a weight depending on the PBH
fraction α. For a given line of sight, the mean magnification is that associated with the LSS distribution, regardless of α. In the absence of compact objects this LSS PDF is shown in figure 2: one can see that PDF samples µ0 values peaked around 0 (mean beam), with a tail of rare events towards high magnifications caused by matter in centers of halos (galaxies and clusters). In the presence of compact objects a fraction α of LSS magnification µ0 along each line of sight is spread out further with its own PBH PDF: this is constructed such that it conserves the mean magnification αµ0 of that line of sight (determined by LSS PDF), but its distribution is more peaked at empty beam, with a tail towards high magnifications. The total magnification µ is then obtained by adding a contribution (1 − α)µ0 to the contribution from compact objects, where the compact objects PDF has a mean magnification αµ0 [32]. This approach yields a combined lensing PDF Z PL (µ; z, α) = 0
µ 1−α
dµ0 PLSS (µ0 , z)PC [µ − µ0 (1 − α), αµ0 ]
(6) where PLSS (µ, z) is the PDF associated to LSS. We see that the result is a convolution of the two PDFs. We take PLSS from turboGL [33, 34] for a Planck cosmology [35]. This code computes the PDF of LSS using the halo approach, which should be more accurate than simulations [32] due to the high dynamic range of the halo mass profile resolution needed. We note however that around the peak there is an excellent agreement between the different LSS PDFs in the literature. Still, there are effects that are model dependent in the centers of the halos, such as the stellar and baryonic contribution, which affect the rare event tail of LSS PDF. So far there have been only a handful of SN detected that have been strongly lens magnified, and all of these events are consistent with the lensing effect generated by LSS in the centers of galaxies or clusters [36–38]. These events are often a result of a targeted search towards special objects such as clusters, and their probability density cannot easily be translated onto our plot. Here we simply remove SN with large magnification where the SN has passed through a center of a halo, since the modeling uncertainties are too large to use these events to distinguish between the LSS lensing and compact object lensing. Note that the effect of the cosmological parameters on the lensing PDF is very weak [39] and variations can be neglected in the range allowed by Planck. The combined PDF preserves both the maximum at the empty-beam distance and the high magnification tail (see Fig. 3).
C.
Finite sources and PBH mass distribution
The magnification PDF relies on the assumption that sources and lenses are point-like. To quantify whether a given SNe is an effective point-like source for a BH of a given mass M we take the ratio of the angular size and
4 Cumulative NSN e
Probability density function 3.5
z = 1.2
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FIG. 3: Redshift dependence of lensing probabilities: a cosmologically sizeable PBH population displaces the maximum of the PDF towards the empty-beam distance. This displacement is compensated with a tail of higher probability of finding strong magnifications. Left: normalized lensing PDF and predictions for no PBH (solid) and full PBH (dashed) universes. Right: total cumulative distribution using a logaritmic scale to highlight the high-magnification tail of the PDF. Horizontal lines correspond to 1,2 events, vertical lines indicate where 3 and 5σ outliers are expected. The residuals have been normalized to a fiducial error σ = 0.15 magnitudes to facilitate visualization.
the (angular) Einstein radius: s r θS RS M DL ≈ 0.034 · , θE 1.5 · 109 km M D DLS
(7)
S
where D are angular diameter distances in Mpc, L, S refer to the lens and source and RS is the radius of the source (the factor in parenthesis is of order 1 for type Ia SNe). We demand the ratio (equation 7) to be small in order for the approximation to be valid. Because the Schwarzschild radius is smaller than the Einstein radius for all but the most extreme configurations, the point-like lens assumption is satisfied for PBHs. We use a simple model to account for the mass dependence: an BH contributes to alpha only if θSN e /θE < .
Hereafter we take = 0.05 as a very conservative assumption to ensure the validity of the approximations in the magnification PDF.One can then derive the effective BH fraction α ˜ (zS , M ) = α · fL (zS , M ) ,
(8)
(now a function of the source redshift and the PBH mass) in terms of the effective lenses fraction R zS ρPBH (z)Θ (θS /θE − ) dz R zS fL (zS , M ) = 0 . (9) ρPBH (z)dz 0 Where Θ(x) = 1 if x > 0 and 0 otherwise (a smooth function can be used, but the results depend very weakly on
5 A.
fL(zS) (effective lenses fraction)
1.0
The effect of compact objects is to change the apparent distance of SNe
0.8 PBH mass 10 2M 10M 1M 10 3M 10 1M 10 4M
0.6 0.4 0.2 0.0 0.2
0.4
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0.8 zS
1.0
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1.4
FIG. 4: Effective lenses fraction as a function of the source redshift for a monochromatic BH distrubution with different masses, cf. (equation 9). The reduction in effective lenses fraction leads to the degrading of the constraints for low masses in Fig. 1. Note that the 10 and 1M curves are practically indistinguishable.
this choice). We will neglect PBH accretion and energy loss via GW emission and assume ρPBH (z) ∝ (1 + z)3 . Realistic PBH models will be characterized by a mass distribution that accounts for their initial generation and subsequent evolution. An extended mass function P (M ) can be easily incorporated in the framework of the effective lenses fraction, generalizing the effective PBH fraction to Z α ¯ (zS , P ) = α · P (M )fL (zs , M )dM , (10) with P (M ) is normalized. One can straightforwardly generalize the effective PBH fraction to a redshiftdependent mass function P (M, z) accounting for evolution of the PBH population in equations (9,10). We note that extended mass functions tend to be better constrained by the data (e.g. [40–42]). Moreover, in the limit of point sources the lensing PDF is insensitive to the masses and small-scale clustering properties of the PBH population [31]. For these reasons we will assume a monochromatic mass function (equation 8) in what follows, as is both realistic enough and represents the most conservative case. III.
Magnification and global likelihood
LENSING LIKELIHOOD FOR IA SNE
The PBHs fraction α affects the observed luminosity of type Ia supernovae via gravitational lensing. In this section we will present the likelihood used to constrain the model, as it will be applied latter to the Joint Lightcurve Analisis (JLA) [43] and Union 2.1 [44] datasets. The discussion includes the effects of lensing, a general SNe intrinsic luminosity distribution and standardization, as well as potential sources of systematic errors such as outliers, correlated noise and selection bias.
¯ L (z) D DL (z, ∆µ) = √ 1 + ∆µ
(11)
¯ L (z) = (1 + z)2 D(z) ¯ where D is the average/full beam luminosity distanc and the magnification ∆µ is now defined ¯ The distance with respect to the average distance D. modulus of observed SNe then reads ¯ DL (z) + 25 − 2.5 log10 (1 + ∆µ) . (12) mth = 5 log10 Mpc The probability of a given magnification ∆µ is given by PL (µ, z, α) (equation 6) evaluated at µ = ∆µ+µF (equation 4). Note that for small magnifications the last term 2.5 can be expanded as 2.5 log10 (1 + δµ) ∼ log(10) δµ. We will adopt a form of Bayesian hierarchical modeling approach for statistical analysis. We assume the priors on all parameters are flat, so that the posterior is proportional to the likelihood only. We introduce for each SNe a latent variable which is the true distance modulus, which we do not observe, and instead we observe its noisy version constructed from absolute magnitude, color and stretch. The regression parameters that correct for stretch and color are assumed to be the same across all SN. This can be generalized in the hierarchical models to allow each SN to have its own value for magnitude, color, and stretch, each controlled by a prior with its own hyper-parameters that can be determined from the analysis itself [45]. However, it has been argued this approach is prone to selection bias effects [46] and we do not pursue it here. There are two general approaches one can follow to solve this: first, one can simultaneously derive the posteriors of all the latent variables and the parameters of the model, which requires an analysis of posteriors in a high dimensional space, for example using Hamiltonian Monte Carlo sampling [47]. The second approach is to analytically marginalize over the latent variables. This requires computing the convolution integrals of the true lensing probability distribution with the noise probability distribution. These integrals cannot be performed analytically, and so need to be done numerically, separately for each model, so it needs to be varied over α, z, and intrinsic parameters of the SNe PDF, as well as for each noise level. In this paper we adopt this analytic marginalization approach, which requires us to numerically compute a lot of convolution integrals and interpolate between them. Note that this is a second convolution on top of the one between LSS and PBH discussed above. However, as a consequence of these analytic marginalizations over latent variables we can work with a handful of variables only. The likelihood for each SNe measurement is then a convolution of the total lensing PDF with the intrinsic
6 error associated to a SNe Z ~ α) = dµPL (µ; , zi , α)PSN e (mi , σi , zi , µ, θ) ~ , (13) Li (θ, where mi is the observed distance modulus (equation 12) and σi the corresponding error. The vector θ~ collectively denotes parameters describing the cosmology, as well as the standardization and statistics of the SNe population (Sec. III B). We will assume that the SNe Q are independent and hence the total likelihood L = i Li is the product of the individual likelihood for each SNe. This is a very reasonable assumption for lensing, as correlations induced by compact objects occur on very small angular scales ∼ θE , and the SNe are observed in random points in the sky. Observational covariances in the SNe datasets due to common modeling and systematics are important. The likelihood (equation 13) is non-gaussian and thus the covariance matrix for the samples can not be straightforwardly included. We will discuss how to model the covariance in Sec. III D.
B.
SNe standardization, population and errors
We want to allow for a sufficiently general, nongaussian PSN e likelihood in (equation 13) that can accommodate the distribution of intrinsic luminosities of type Ia SNe. This distribution will be a function of the normalized deviation between the prediction (equation 12) and the observation: xi =
1 (mob,i − mth (zi , µ) − m) ¯ . σi
(14)
Here m ¯ controls the mean of the SNe intrinsic magnitude (the observed magnitude mob,i and corrected dispersion σi are discussed below). Since one signature of the signal we are searching for is the non-gaussian PDF we need to allow for the intrinsic scatter of SN luminosities to be non-gaussian as well. Even if this exactly mimicked the signal at one redshift this cannot be the case at all redshifts, hence the data can distinguish between these two models due to their different redshift dependencies. For the PDF there are a few possible options. A popular one is to use gaussian mixture model, but here we will instead use a non-gaussian PDF of the form k3 x 1 PSN e (x) = N 1 + erf √ exp − |x|2−k4 . 2 2 (15) Here the parameter k3 , k4 introduce a skewness and kurtosis, respectively, and N is a normalization constant. Since the lensing PDF is non-gaussian (cf. Fig. 3), the inclusion of extra parameters in the likelihood allows the exploration of possible degeneracies between them. We will show below that the data (in the absence of outliers
discussed separately) do not show much preference for the non-gaussian parameters, with a strong correlation between the two, so there is no need to explore more general PDFs given that even this parametrization leads to overfitting of the PDF. The observed SNe distance modulus corrects the bolometric magnitude m∗B for stretch X1 and color C mob,i = m?B,i − (MB − αX1,i + βCi ) ,
(16)
where α, β are nuisance parameters and MB is the a constant offset. In the Union 2.1 compilation the data provided has been already standardized, α, β, MB are fixed. The JLA compilation includes an offset correction depending on the host mass galaxy as MB = MB1 if Mstellar < 1010 M and MB = MB1 + ∆M otherwise. In the JLA case, the parameter vector θ~ also contains ∆M , α, β, which are sampled along with the other parameters. Note that m ¯ is degenerate with both MB and H0 , ~ fix MB , H0 ¯ ∝ 1/H0 . Therefore we will vary m as D ¯ in θ, to their fiducial values and remove the mean posterior of m, ¯ which is degenerate with both. The standard deviation is corrected in quadrature for intrinsic dispersion and gravitational lensing 2 2 σi2 = σob,i + k2 − σL (z)
(17)
2 The observational error σob,i is obtained from the diagonal of the covariance matrix, including systematics. For the JLA sample we construct the covariance matrix including the standardization parameters (equation 16) and their covariances, as described in Ref. [43]. SNe data includes an intrinsic magnitude dispersion to account for variability in luminosity after standarization (on top of the observational error). We correct the dispersion via k2 , which is allowed to have a negative sign to cancel the intrinsic scatter included in the error (and which may be model dependent). Finally, we remove the lensing 2 (z) to avoid double counting, as we are contribution σL including realistic LSS lensing in the likelihood (equation 13).
C.
Outliers
A critical prediction of the PBH model on SNe lensing is the existence of highly magnified events. If existing, these events would be removed from the sample by the outlier rejection and hence bias the result against the PBH model. The simplest approach to outliers is simply “clipping” data points whose residual over the prediction is above some threshold, i.e. those points that deviate beyond what is expected by lensing magnification (cf. section 2.1 of [48]). Other prescriptions are more sophisticated, including the fitting to two mixed SNe populations, one consisting of “real” type Ia events, and an outlier population with a large scatter [49]. In order to test the effect of outliers we consider the supernovae rejected in compiling the Union 2.1 dataset,
7
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46
↑ sub-luminous
44 42 40 ↓ super-luminous
38 36
Union 2.1 Outliers
34 7.5
±3σ
∆m/σm
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Here C¯ij is the simplified JLA covariance matrix (Table F2 in Ref. [43]) but setting the diagonal and next-todiagonal elements to zero (C¯ii = C¯i,i±1 = 0) as they represent the combined error at each redshift and the correlations induced by the data compression. For the sake of comparison we will also discuss the effect of using the diagonal-free (C¯i,i±1 6= 0) and the full covariance matrix (C¯ii 6= 0). The sum is over J = (27, 28, 29), and normalized by NJ = 3: this gives an average over the redshifts z = (0.679, 0.799, 0.940) on which the sensitivity to the PBH fraction is strongest, while discarding the last entries to avoid auto-correlations (accounted in the measurement errors) and next-neighbor correlations (induced by the specific procedure used to generate the matrix). In the model of equation (18) γ is a free parameter to be sampled with a Gaussian prior of width σγ = 1.
2.5 0.0
E.
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FIG. 5: Outliers in the Union 2.1 Sample. The outliers are predominantly sub-luminous, except at high redshift (where they would be too faint to be detectable). This is opposed to what is expected in the PBH model, which predicts only super-luminous outliers.
see Fig. 5, where a visual inspection shows that their distribution is not compatible with the expectations of the PBH model. The predictions in Fig. 3 show that the PBH model predicts super-luminous events, without any sub-luminous counterpart. In contrast, Union 2.1 outliers are predominantly sub-luminous, and have an asymetric distribution: sub-luminous SNe deviate as much as ∆m/σm ∼ 8, while super-luminous SNe only depart as much as ∆m/σm ∼ −4. We note that this is unlikely due to observational systematics, since super-luminous outliers would be much easier to detect.
D.
Correlated noise
Correlations in the noise model arise due to the correlations in the calibration errors, but are difficult to include in the non-gaussian likelihood (equation 13). We will model correlations by adopting a rank one (plus diagonal) covariance matrix approximation. We model the observed magnitude in equation (16) by adding to it a redshift dependent term that has been extracted from the rank one decomposition of the covariance matrix mc = γ · e(z) ,
with e(zi ) =
1 X C¯ij p . NJ Cjj j∈J
(18)
Selection bias
Another potential issue is the selection bias in PBH models: high-redshift supernovae may be selectively brighter. This selection (or Malmquist) bias is usually corrected through a frequentist procedure where the mean bias effect is simulated and corrected for, separately for each of the surveys (e.g. SDSS, SNLS...). The overall effect can be difficult to visualize, since the color and stretch parameters entering the standardization relation (16) are correlated with the intrinsic magnitude, all of which can play a role in selection bias effects. Despite the large size of the selection bias effect on color at the upper redshift range of any specific survey (figure 11 of [43]), the overall bias correction is small (less than 0.04 in µ, figure 5 of [43]). It is important to note that the different surveys give consistent results with each other after correction (figure 11 of [43]) despite the fact that they can have a large color selection bias. This gives confidence that the selection effects are properly corrected over the range of models we are considering here. We note that the changes in peak value of µ we are considering here are up to 0.1, which is a smaller change than the differences between the accelerating and non-accelerating universe models, so as long as the bias correction is valid for the standard cosmological models it should also be valid for our models.
IV.
RESULTS: LIMITS ON PBH ABUNDANCE
We now describe the bounds on the PBH abundance α = ΩΩPBH , summarized by Fig. 1. Our baseline standard M analysis refers to the case of point-like SNe. We then discuss the effects of MPBH on α the effect outliers and correlations. We constrain the PBH fraction by sampling the total likelihood (equation 13) over the parameters representing cosmology (α, Ωm ), the SNe population (m, ¯ k2 , k3 , k4 ), and standardization (a, b, ∆M JLA sample only). We
8 JLA Union 2.1 Union 2.1 (outliers)
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0.0
0.8
k3
1.6
−0.25 0.00
0.25
k4
0.50
0.2
0.4
0.6
0.8
α (PBH)
FIG. 6: Marginalized constraints on the model parameters for SNe compilations: JLA (blue) Union 2.1 (green) and Union 2.1 including outliers (red) (see Sec. III C). Only the PBH fraction α and SNe population parameters used in Eqs. 6 are shown.
impose a Gaussian prior Ωm = 0.309 ± 0.006 consistent with CMB+BAO constraints within a flat ΛCDM model [50]. The likelihood (equation 13) was sampled using the emcee code [51] and the results analyzed with GetDist [52]. Results will be presented for the JLA [43] and Union [44] datasets, as described in Sec. III. The bounds on the PBH fraction are α < 0.346 (JLA) and α < 0.405 (Union) at 95% confidence, assuming that SNe are point sources MPBH 0.01M . The allowed re-
gions are presented in Fig. 6 and summarized on Table I. Both supernova samples produce consistent results, with slight variations in the parameters characterizing the SNe population: the Union sample is best fit by a non-zero skewness k3 = 0.46+0.29 −0.27 , while the JLA sample exhibits a significant degeneracy between k3 and the mean m. ¯ None of the parameters is degenerate with the PBH fraction α because 1) it produces redshift dependent effects and 2) low redshift supernovae are very effective at fix-
9 JLA Union +0.035 0.002+0.073 −0.001 −0.078 −0.034 +0.52 +0.45 0.01−0.45 −0.23−0.38 0.04+0.68 0.46+0.29 −0.65 −0.27 +0.20 0.13−0.22 0.19+0.16 −0.17 0.126+0.012 − −0.012 2.63+0.14 − −0.15 −0.047+0.022 − −0.023 +0.011 0.310−0.011 0.309+0.011 −0.012 < 0.346 < 0.405
TABLE I: 95% limits on the population, standarization and cosmological parameters obtained for the base analysis (MPBH 0.01M ), see Fig. 6. The degrading of the constraints seen in Fig. 1 affects only the PBH fraction α.
ing the population parameters. The constraints on Ωm are entirely dominated by the external prior, and will be therefore not shown. Our results for α are consistent with the forecasted sensitivities estimated in Ref. [32] given our supernova sample size. The finite SNe size degrades the constraints if the PBH mass is sufficiently low. We derive constraints for monochromatic mass distributions MPBH /M = 100, 10, 1, 0.1, 0.01, 0.003 and 0.001 using the prescription for the effective PBH fraction (equation 9). The results, shown in Fig. 1, indicate that the constraints degrade for MPBH . 0.01M but remain basically unaffected and consistent for higher masses. This behavior is expected from the dependence of the effective PBH fraction with the MPBH , Fig. 4. Changing the PBH mass does not affect the constraints on the remaining parameters. Including the outliers in the Union sample changes the constraints only slightly to α < 0.446 (95%). This is because the data are not compatible with the peak of the lensing PDF being at the empty beam distance (equation 2), see Fig. 3. In addition, the majority of outliers are sub-luminous, with only a few super-luminous cases, cf. Fig. 5. The main difference in the analysis with outliers is the preference towards a non-zero kurtosis k4 = 0.52+0.11 −0.12 , different from zero at ≈ 5σ significance, see Fig. 6. The interplay between the outliers and higher order SNe population parameters highlights the importance of including a sufficiently detailed model, e.g. equation (15). The results are robust against the inclusion of correlated noise, which weakens the constraints only slightly. The correlation model based on the JLA simplified covariance matrix (equation 18) changes the 95% confidence bounds to α < 0.363 when both the diagonal and elements next to the diagonal are removed and α < 0.372 when only the the diagonal is removed, see Fig. 7. Using the full correlation matrix (including the diagonal, which artificially increases the errors) degrades the bounds to only α < 0.400. None of these different prescriptions are sufficient to significantly weaken the constraints on the
Non-diagonal Non-diagonal +1
0.8
0.6
α (PBH)
Parameter m ¯ 100∆σ 2 k3 k4 a b ∆M ΩM α (PBH)
0.4
0.2
0.0 −3.0
−1.5
0.0
1.5
3.0
γ (SNe correlation)
FIG. 7: Effects of correlated noise on the PBH constraints following the model in Sec. III D (equation 18). Using the correlation matrix without the diagonal terms (orange) weakens the constraints more than the case without the diagonal and next-to-diagonal (dark cyan). The horizontal blue lines mark the 1,2 and 3 σ constraints in the standard analysis (γ = 0).
fraction of primordial black holes. V.
CONCLUSIONS
We have presented constraints on the primordial black hole fraction via the lack of gravitational lensing signatures on type Ia supernovae observations. Our analysis includes the full lensing probability distribution for PBHs, the effect from LSS, the dependence on the PBH mass and a non-gaussian model for supernovae luminosities. The JLA and Union 2.1 datasets lead to consistent results in the PBH fraction, both by the lack of demagnification, as well as the absence of highly magnified, superluminous SNe outliers. Our results provide stringent bounds on the PBH abundance, α < 0.346 (JLA) and α < 0.405 (Union) at 95% c.l., rejecting the hypothesis of DM entirely formed by PBH at the level of 5.01σ (JLA) and 4.28σ (Union). The results are robust when considering data outliers: we find their inclusion results in a better fit by a nongaussian SNe distribution, but our constraints on the PBH abundance do not change. Including correlated noise in the SNe measurements weakens the constraints only slightly. The analysis applies to PBH masses MPBH & 10−2 M , including the mass range of black holes detected by LIGO, which was weakly constrained until recently. This closes a viability window for PBH-DM models and rules out the possibility of a connection between LIGO observations and macroscopic dark matter. The SNe constraints are complementary to other tests (see Fig. 1: they overlap with the microlensing bounds on the low mass end, which are more constraining for MPBH .
10 2.5M , while for MPBH & 60, other probes (CMB anisotropies, stability of stellar compact systems) become more stringent. Since SNe lensing constraints are independent of the mass spectrum and spatial distribution they offer a bound on the total PBH fraction above 0.01M . SNe also probe random directions and reach into cosmological distances, and are not limited to specific and nearby systems. Hence, our results may be extended to models of clustered PBH that avoid constraints from stellar micro-lensing [53], as long as the PBH clusters can be considered point-like lenses (cf. Sec II). As they probe the late universe, SNe also rule out scenarios where initially light PBHs merge into more massive ones, potentially weakening CMB constraints. The best strategy for future analyses is considering the PBH lensing model when building the SNe catalog, thus fitting the PBH fraction along with all other parameters. In doing so, issues such as outlier rejection, covariances, selection bias and other potential systematics would be addressed consistently at all stages. The best way to approach this is through hierarchical Bayesian modeling: in this paper we have performed convolution integrals to analytically marginalize over latent variables, but future analyses could instead perform numerical MC sampling marginalization. Beyond testing the PBH model, such a comprehensive analysis would address the degeneracies that exist between the PBH fraction and other cosmological parameters [19, 54]. Larger and more consistent SNe catalogues (e.g. [55, 56]) will significantly increase the
constraining power of this technique. Moreover, a variety of techniques involving gravitational waves [14, 57–60], lensing of fast radio bursts [61] and others hold considerable promise to rule out a significant fraction of macroscopic dark matter. Heavy, compact objects can not comprise the totality of the dark matter in the universe. Closing the LIGO band for macroscopic dark matter strengthens the case for particle physics explanation and weakens the type of early universe scenarios proposed to abundantly produce PBHs. Our results are supported by recent studies that provide further evidence against PBH models in the mass range probed by SNe lensing, including caustic crossing [21, 22], radio and X-ray emission [62] and revised estimates of LIGO BH merger rates [14] (although see [63]). With no LIGO MACHO allowed by observations the future of primordial black holes turns even darker.
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Acknowledgments
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