Search for sub-solar mass ultracompact binaries in Advanced LIGO's second observing run

TL;DR

This paper reports a search for sub-solar mass ultracompact binaries using Advanced LIGO's second observing run data, finding no candidates and setting upper limits on their merger rates and dark matter contributions.

Contribution

The study extends previous searches by including component spin effects and provides new constraints on the merger rates and dark matter fraction of sub-solar mass black hole binaries.

Findings

No viable sub-solar mass binary candidates detected.

Upper limits on merger rates: <3.7 x 10^5 Gpc^-3 yr^-1 for 0.2 solar masses.

Constraints on dark matter fraction: less than 16% for 0.2 solar mass black holes.

Abstract

We present an Advanced LIGO and Advanced Virgo search for sub-solar mass ultracompact objects in data obtained during Advanced LIGO's second observing run. In contrast to a previous search of Advanced LIGO data from the first observing run, this search includes the effects of component spin on the gravitational waveform. We identify no viable gravitational wave candidates consistent with sub-solar mass ultracompact binaries with at least one component between 0.2 - 1.0 solar masses. We use the null result to constrain the binary merger rate of (0.2 solar mass, 0.2 solar mass) binaries to be less than 3.7 x 10^5 Gpc^-3 yr^-1 and the binary merger rate of (1.0 solar mass, 1.0 solar mass) binaries to be less than 5.2 x 10^3 Gpc^-3 yr^-1. Sub-solar mass ultracompact objects are not expected to form via known stellar evolution channels, though it has been suggested that primordial density fluctuations or particle dark matter with cooling mechanisms and/or nuclear interactions could form black holes with sub-solar masses. Assuming a particular primordial black hole formation model, we constrain a population of merging 0.2 solar mass black holes to account for less than 16% of the dark matter density and a population of merging 1.0 solar mass black holes to account for less than 2% of the dark matter density. We discuss how constraints on the merger rate and dark matter fraction may be extended to arbitrary black hole population models that predict sub-solar mass binaries.

Figures

Equations

$$\mathrm{FAR}(\log {\mathcal{L}}^{\ast })=\frac{N}{T}P\left(\log \mathcal{L}\ge \log {\mathcal{L}}^{\ast }|\mathrm{noise}\right),$$

$${\mathcal{R}}_i=\frac{2.3}{⟨VT{⟩}_i}$$

$$D_{\mathrm{horizon}}\propto {\mathcal{M}}^{5/6}\sqrt{{\int }_{f_{\min }}^{f_{\max }}\frac{f^{-7/3}}{S_n(f)}df}$$

$$\mathcal{R}({\mathcal{M}}_1,{\mathcal{M}}_2)={\int }_{{\mathcal{M}}_1}^{{\mathcal{M}}_2}\mathcal{R}(\mathcal{M})\times \psi (\mathcal{M})\mathrm{d}\mathcal{M}$$

$$f_{\mathrm{CO}}=\frac{{\rho }_{\lim }}{{\rho }_{\mathrm{CDM}}}\times \frac{1}{f_{\mathrm{obs}}}=\frac{\mathcal{R}(M)T_{\mathrm{obs}}M}{{\rho }_{\mathrm{CDM}}}\times \frac{1}{f_{\mathrm{obs}}}.$$

Full text

I Introduction

Gravitational wave and multi-messenger astronomy progressed remarkably in Advanced LIGO TheLIGOScientific:2014jea and Advanced Virgo’s TheVirgo:2014hva second observing run, which included the first observation of gravitational waves from a binary neutron star merger TheLIGOScientific:2017qsa and seven of the ten observed binary black hole mergers Abbott:2017vtc ; Abbott:2017gyy ; Abbott:2017oio ; o1o2catalogpaper . These detections, as well as the candidates presented in the gravitational wave transient catalog (GWTC-1) o1o2catalogpaper , have led to a better understanding of the populations of compact binaries detectable by ground based interferometers LIGOScientific:2018jsj . These observations, however, represent just a portion of the parameter space that Advanced LIGO and Advanced Virgo currently search DalCanton:2017ala ; 2018arXiv181205121M and are sensitive to Magee:2018opb . We report on an extension of the searched parameter space in data obtained during O2 to binaries with component masses $<1\thinspace M_{\odot}$. This is the second search for sub-solar mass ultracompact objects in Advanced LIGO data and the fourth since initial LIGO Abbott:2005pf ; Abbott:2007xi ; Abbott:2018oah , as well as the first search to incorporate spin effects in the modeling of the gravitational wave emission.

There is no known mechanism for the formation of compact objects with masses well below a solar mass within the standard model of particle physics and the standard $\Lambda$CDM model of cosmology. Black holes must be heavier than the Chandrasekhar limit ($\sim 1.4\thinspace M_{\odot}$, set by the mass of the proton Chandrasekhar:1931ih ) while neutron stars are expected to have masses greater than about $0.9\thinspace M_{\odot}$ kapermeasuringNS ; strobel2001minimum . A detection of a sub-solar mass object in a merger would therefore be a clear signal of new physics. Indeed, there are several proposals that link sub-solar mass compact objects to proposals for the nature of dark matter, which makes up nearly 85% of the matter in the Universe. One possibility is that black holes with masses accessible to ground based interferometers could have formed deep in the radiation era from the prompt collapse of large primordial over-densities on the scale of the early time Hubble volume Carr:1975qj ; nakamura1997gravitational . The size and abundance of any such primordial black holes depends on the spectrum of primordial perturbations and on the equation of state of the early universe Jedamzik:1996mr ; Widerin:1998my ; Georg:2017mqk ; Byrnes:2018clq . An alternative inflationary mechanism proposes that vacuum bubbles nucleated during inflation may result in black holes (with masses that can be around a solar mass) after inflation ends Deng:2017uwc .

A different class of possibilities, explored more recently, is motivated by ideas for the particle nature of dark matter. For example, dark matter may have a sufficiently complex particle spectrum to support cooling mechanisms that allow dense regions to collapse into black holes at late times, in processes analogous to known astrophysical processes Shandera:2018xkn . Alternatively, dark matter may have interactions with nuclear matter that allow it to collect inside of neutron stars and trigger their collapse to black holes Kouvaris:2010jy ; deLavallaz:2010wp ; PhysRevD.40.3221 ; Bramante:2015dfa ; Bramante:2014zca ; Bramante:2017ulk ; Kouvaris:2018wnh . The details of when dark matter can collapse a neutron star to form a black hole or other exotic compact object are still under investigation Gresham:2018rqo , but the postulated black holes will have masses comparable to the progenitor neutron star mass, or perhaps smaller if some matter can be expelled by rapid rotation of the star during collapse.

A detection of a sub-solar mass black hole would have far-reaching implications. In the primordial black hole scenario, the mass and abundance of the black holes would constrain a combination of the spectrum of initial density perturbations on very small scales and the equation of state of the Universe at a time when the typical mass inside a Hubble volume was of the order of the black hole mass. For particle dark matter scenarios, the abundance of sub-solar mass black holes would provide a direct estimate of the cooling rate for dark matter. The black hole mass would constrain the masses of cosmologically abundant dark matter particles through, for example, the Chandrasekhar relation for fermions Shandera:2018xkn or analogous relations for non-interacting bosons 1984PhLB..140..329B ; UrenaLopez:2002gx . In the case that all black holes are observed to be near but not below the mass of neutron stars, the abundance of such objects would constrain the dark matter-nucleon interaction strength, as well as the dark matter self-interaction strength and mass(es) Kouvaris:2018wnh .

This letter reports the results of a search for gravitational waves from sub-solar mass ultracompact binaries using data from Advanced LIGO’s second observing run. No significant candidates consistent with a sub-solar mass binary were identified. The null result places the tightest constraints to date on the merger rate and abundance of sub-solar mass ultracompact binaries. We describe an extension of our merger rate constraints to arbitrary populations and models under the assumption that the horizon distance controls the sensitivity of the search. We once more consider the merger rate constraints in the context of merging primordial black hole populations contributing to the dark matter Abbott:2018oah . We describe how to extend the dark matter fraction parameterization to other models by separating LIGO observables from model dependent quantities. Finally, we conclude with a discussion of the implications of this search.

II Search

We analyze data obtained from November 30, 2016 to August 25, 2017 during Advanced LIGO’s second observing run (O2).111Data from Advanced LIGO’s second observing run is available from the Gravitational Wave Open Science Center with and without noise sources linearly subtracted: https://www.gw-openscience.org Noise artifacts are linearly subtracted from the data; this includes strong sinusoidal features in both detectors due to injected calibration frequencies and the AC power grid, as well as laser beam jitter in the LIGO-Hanford detector data DavisMassinger2018 . 117.53 days of coincident data remain after the application of data quality cuts TheLIGOScientific:2017lwt ; Abbott:2016trn ; abbott2016binary ; Nuttall:2018 ; BBerger:2018 . The Advanced Virgo interferometer completed commissioning and joined Advanced LIGO in August 2017 for $15$ days of triple coincident observation o1o2catalogpaper , however, we only report on the analysis of data obtained by the LIGO Hanford and LIGO Livingston interferometers.

The search was conducted using publicly available gravitational-wave analysis software Cannon:2011vi ; Messick:2016aqy ; Sachdev:2019vvd ; gstlal ; lalsuite ; Ajith:2012mn ; Capano:2016dsf . The initial stage of the search performed a matched-filter analysis using a discrete bank of template waveforms generated using the TaylorF2 frequency-domain, post-Newtonian inspiral approximant Buonanno:2009 . The template bank used for this search was designed to recover binaries with component masses of $0.19-2.0\thinspace M_{\odot}$ and total masses of $0.4-4.0\thinspace M_{\odot}$ in the detector frame with 97% fidelity, as in Abbott:2018oah . The search presented here, however, additionally includes spin effects in the modeling of the gravitational waveform. The bank is constructed to recover gravitational waves originating from binaries with component spins purely aligned or anti-aligned with the orbital angular momentum, and with dimensionless spin magnitudes of $0.1$ or less. The inclusion of spin effects required denser placement of the waveforms in the template bank; the resulting bank had $992\thinspace 461$ templates, which is nearly twice as large as the non-spinning bank used in Abbott:2018oah .

In order to reduce the computational burden, matched filtering was only performed for a subset of Advanced LIGO’s full sensitive band Magee:2018opb . The choice to only analyze the 45–1024  Hz band led to a detector averaged signal-to-noise ratio (SNR) loss of 8% when compared to the full $\sim 10-2048\mathrm{Hz}$ frequency band. This estimated SNR loss is a property of Advanced LIGO’s noise curves and is independent of the templates used in the search; the discrete nature of the template bank causes an additional $\lesssim 3\%$ loss in SNR.

Gravitational-wave candidates that were found coincident in both the Hanford and Livingston detectors were ranked using the logarithm of the likelihood-ratio , $\mathcal{L}$ Cannon:2011vi ; Messick:2016aqy ; Sachdev:2019vvd . For a candidate with a likelihood-ratio of $\mathcal{L}^{*}$, we assign a false-alarm-rate of

[TABLE]

where $N$ is the number of observed candidates, $T$ is the total live time of the experiment, and $P\left(\log\mathcal{L}\geq\log\mathcal{L}^{*}|\mathrm{noise}\right)$ describes the probability that noise produces a candidate with a ranking statistic at least as high as the candidate’s.

The search recovered the previously detected signal GW170817 TheLIGOScientific:2017qsa , which was observed along with an electromagnetic counterpart GBM:2017lvd . This signal is consistent with a binary neutron star. No other viable gravitational wave candidates were identified. The next loudest candidate was consistent with noise and had a FAR of 3.25 per year.

III Constraint on binary merger rate

As in Abbott:2018oah , we consider nine populations of equal mass, non-spinning binaries that are delta-function distributed in mass, i.e. $m_{i}\in\{0.2,0.3,\dots,1.0\}$. We injected $913\thinspace 931$ fake signals into our data; the injections were randomly oriented and spaced uniformly in distance and isotropically across the sky. The recovered signals provide an estimate of the pipeline’s detection efficiency as a function of source distance for each equal mass population. This in turn allows us to estimate the sensitive volume-time accumulated for each mass bin. We once more use the loudest event statistic formalism Biswas:2007ni to estimate the upper limit on the binary merger rate to 90% confidence,

[TABLE]

These upper limits are shown for equal mass binaries and as a function of chirp mass in Fig. 1. Although our template bank includes systems with total mass up to $4\thinspace M_{\odot}$, we only place bounds on the merger rate of systems where both components are $\leq 1\thinspace M_{\odot}$. We estimate that detector calibration uncertainties o1o2catalogpaper ; 2017PhRvD..96j2001C ; Viets:2017yvy and Monte Carlo errors lead to an uncertainty in our rate constraint of no more than $20\%$.

Advanced LIGO and Virgo’s horizon distance scales as:

[TABLE]

where $S_{n}(f)$ is the noise spectra of the detector and $f_{\mathrm{min}}$ and $f_{\mathrm{max}}$ are $45$ Hz and $1024$ Hz, respectively.222The waveform model used to generate our template bank, TaylorF2, truncates the waveform at an upper frequency $f_{\mathrm{ISCO}}$, which corresponds to radiation from the innermost stable circular orbit of a black hole binary with mass $M_{\mathrm{total}}$. This frequency is above $f_{\mathrm{max}}$ for all non-spinning waveforms in our template bank and so does not impact $D_{\mathrm{horizon}}$. For a null result, we therefore expect $\mathcal{R}(\mathcal{M})\propto\mathcal{M}^{-15/6}$ provided the horizon distance controls the sensitivity of the search. The observed power law dependence of the rate constraint on the chirp mass is within $\sim 4\%$ of the expected $\mathcal{M}^{-15/6}$ dependence; this is well within the error bound on the rate upper limit and is strong evidence that the chirp mass is the primary parameter that dictates the sensitivity of the search. Therefore our upper limits from equal mass systems also apply to unequal mass systems within the range of mass ratios we have searched over. For verification, we performed a small injection campaign over five days of coincident data with injected component masses distributed between $0.19\thinspace M_{\odot}$ and $2.0\thinspace M_{\odot}$ with at least one component $<1.0\thinspace M_{\odot}$. The search sensitivity remained a function of the chirp mass; this implies that the rate constraints found from the equal mass injection sets can therefore be applied to systems with arbitrary mass ratios provided that both component masses lie within $0.20\thinspace M_{\odot}$ and $1.0\thinspace M_{\odot}$ where our injection sets were performed.

The Advanced LIGO and Virgo rate upper limit can be expanded as:

[TABLE]

where $\mathcal{R}$ is the rate density as a function of chirp mass and $\psi(\mathcal{M})$ denotes the black hole population distribution in chirp mass. We ignore the effects of redshift due to the small detector range for sub-solar mass binaries. Setting $\psi(\mathcal{M})=\delta(\mathcal{M})$ then reveals the form of the LIGO constraining rate density, $\mathcal{R}(\mathcal{M})$, which is shown in Figure 1. For a given model, $\psi(\mathcal{M})$, $\mathcal{R}(\mathcal{M}_{1},\mathcal{M}_{2})$ provides the LIGO rate constraint on that model for chirp masses between $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$. The resulting rate constraints allow direct comparison of sub-solar mass ultracompact object models with LIGO observations.

IV General constraints on sub-solar mass black hole dark matter

We convert our limits on the merger rate of sub-solar mass ultracompact objects into a constraint on the abundance of primordial black holes using our fiducial formation model Sasaki:2016jop first developed in nakamura1997gravitational ; Ioka1998 and used previously in LIGO analyses Abbott:2005pf ; Abbott:2018oah . We consider a population of equal mass primordial black holes that is created deep in the radiation era. We model the binary formation via three-body interactions, though others have considered the full field of tidal interactions Ali-Haimoud:2017rtz . By equating the model’s predicted merger rate with the merger rate upper limit provided by Advanced LIGO and Virgo, we can numerically solve for the upper limit on the PBH abundance. These constraints are shown in Figure 2.333The normalization of the PBH distribution used in our fiducial model Sasaki:2016jop differs by a factor of 2 from the normalization in (nakamura1997gravitational, ). As such, our fiducial model (used here and in (Abbott:2018oah, )) predicts a more conservative PBH merger rate and leads to less constraining limits on $f_{\mathrm{PBH}}$ than would be attained using the model of (nakamura1997gravitational, ).

This interpretation is highly model dependent; the mass distribution, binary fraction, and binary formation mechanisms all have a large effect on the expected present day merger rate and consequently the bounds on the primordial black hole composition of the dark matter. The Advanced LIGO and Virgo observables can be separated from the model dependent terms:

[TABLE]

Here we use $f_{\mathrm{CO}}$ to refer to the dark matter fraction in ultracompact objects instead of $f_{\mathrm{PBH}}$ to emphasize that this is generally applicable to other compact object models that could contribute to the dark matter Shandera:2018xkn , and not just PBHs. The first term, $\rho_{\mathrm{lim}}/\rho_{\mathrm{CDM}}$, represents the upper limit on the fraction of the dark matter contained in presently merging sub-solar mass ultracompact binaries. In the second term, $f_{\mathrm{obs}}$ describes the fraction of sub-solar mass ultracompact objects that are observable by Advanced LIGO and Virgo for a particular model. This is set by the binary fraction and the probability density of binaries merging at present day. Note that the merger rate density must be converted from a function of chirp mass to total mass; this can be done by mapping to total mass for each mass ratio on an equal chirp mass curve.

Equation (5) applies to any dark matter model that predicts the formation of dark compact objects. The abundance of those dark compact objects can then be expressed as a fraction of the dark matter density.

V Conclusion

We have presented the second Advanced LIGO and Advanced Virgo search for sub-solar mass ultracompact objects. No unambiguous sub-solar mass gravitational wave candidates were identified. The null result allows us to place tight constraints on the abundance of sub-solar mass ultracompact binaries.

This work represents an expansion of previous initial and Advanced LIGO and Advanced Virgo sub-solar mass searches. First, we have broadened the searched parameter space to increase sensitivity to systems with non-negligible component spins. Second, we have presented a method to extend our constraints on the binary merger rate to arbitrarily distributed populations that contain sub-solar mass ultracompact objects. Combined with the existing rate limits, this may already be enough to begin constraining collapsed particulate dark matter models Shandera:2018xkn or the cross section of nuclear interactions Kouvaris:2010jy ; deLavallaz:2010wp ; PhysRevD.40.3221 ; Bramante:2015dfa ; Bramante:2014zca ; Kouvaris:2018wnh . Finally, we have provided a method to separate Advanced LIGO and Advanced Virgo observables from model dependent terms in our interpretation of the limits on primordial black hole dark matter.

Ground based interferometer searches for sub-solar mass ultracompact objects will continue to inform cosmological and particle physics scenarios. Advanced LIGO and Advanced Virgo have begun a year long observing run in early 2019, with improved sensitivities Aasi:2013wya . Advanced Virgo will have more coincident time with the Advanced LIGO detectors over its next observing run, which will improve network sensitivity and aid in further constraining the above scenarios.

VI Acknowledgments

The authors gratefully acknowledge the support of the United States National Science Foundation (NSF) for the construction and operation of the LIGO Laboratory and Advanced LIGO as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. The authors gratefully acknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French Centre National de la Recherche Scientifique (CNRS) and the Foundation for Fundamental Research on Matter supported by the Netherlands Organisation for Scientific Research, for the construction and operation of the Virgo detector and the creation and support of the EGO consortium. The authors also gratefully acknowledge research support from these agencies as well as by the Council of Scientific and Industrial Research of India, the Department of Science and Technology, India, the Science & Engineering Research Board (SERB), India, the Ministry of Human Resource Development, India, the Spanish Agencia Estatal de Investigación, the Vicepresidència i Conselleria d’Innovació, Recerca i Turisme and the Conselleria d’Educació i Universitat del Govern de les Illes Balears, the Conselleria d’Educació, Investigació, Cultura i Esport de la Generalitat Valenciana, the National Science Centre of Poland, the Swiss National Science Foundation (SNSF), the Russian Foundation for Basic Research, the Russian Science Foundation, the European Commission, the European Regional Development Funds (ERDF), the Royal Society, the Scottish Funding Council, the Scottish Universities Physics Alliance, the Hungarian Scientific Research Fund (OTKA), the Lyon Institute of Origins (LIO), the Paris Île-de-France Region, the National Research, Development and Innovation Office Hungary (NKFI), the National Research Foundation of Korea, Industry Canada and the Province of Ontario through the Ministry of Economic Development and Innovation, the Natural Science and Engineering Research Council Canada, the Canadian Institute for Advanced Research, the Brazilian Ministry of Science, Technology, Innovations, and Communications, the International Center for Theoretical Physics South American Institute for Fundamental Research (ICTP-SAIFR), the Research Grants Council of Hong Kong, the National Natural Science Foundation of China (NSFC), the Leverhulme Trust, the Research Corporation, the Ministry of Science and Technology (MOST), Taiwan and the Kavli Foundation. The authors gratefully acknowledge the support of the NSF, STFC, MPS, INFN, CNRS and the State of Niedersachsen/Germany for provision of computational resources. Computing resources and personnel for this project were provided by the Pennsylvania State University. This article has been assigned the document number LIGO-P1900037.