Will Gravitational Wave Sirens Determine the Hubble Constant?
MMNRAS , 1–7 (2018) Preprint 20 December 2018 Compiled using MNRAS L A TEX style file v3.0
Will Gravitational Wave Sirens Determine the HubbleConstant?
Arman Shafieloo, , (cid:63) Ryan E. Keeley, † Eric V. Linder , , ‡ Korea Astronomy and Space Science Institute, Daejeon 34055, Korea University of Science and Technology, Yuseong-gu 217 Gajeong-ro, Daejeon 34113, Korea Berkeley Center for Cosmological Physics & Berkeley Lab, University of California, Berkeley, CA 94720, USA Energetic Cosmos Laboratory, Nazarbayev University, Astana, Kazakhstan 010000
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Lack of knowledge about the background expansion history of the Universe from inde-pendent observations makes it problematic to obtain a precise and accurate estimationof the Hubble constant H from gravitational wave standard sirens, even with electro-magnetic counterpart redshifts. Simply fitting simultaneously for the matter densityin a flat Λ CDM model can reduce the precision on H from 1% to 5%, while notknowing the actual background expansion model of the universe (e.g. form of darkenergy) can introduce substantial bias in estimation of the Hubble constant. Whenthe statistical precision is at the level of 1% uncertainty on H , biases in non- Λ CDMcosmologies that are consistent with current data could reach the 3 σ level. To avoidmodel-dependent biases, statistical techniques that are appropriately agnostic aboutmodel assumptions need to be employed. Key words: distance scale – gravitational waves – cosmological parameters – darkenergy
Gravitational waves emitted from inspiral and coalescenceof binary compact objects can be used to measure a dimen-sional quantity – the time or frequency associated with thewave form. By modeling the expected wave form within gen-eral relativity these events can be standard sirens, measur-ing dimensional cosmic distances. Since most cosmic mea-surements involve dimensionless quantities (often ratios ofdistances), this makes standard siren distances potentiallyuseful in a distinct way. In particular, they have been pro-posed to measure the absolute distance scale of the universe,or Hubble constant H (Schutz 1986; Holz & Hughes 2005;Dalal et al. 2006). This is an exciting prospect.Locally, at very low redshifts z (cid:46) . , the source dis-tance is related linearly to the distance through the Hubblelaw, d = H − z . This means that the redshift to the sourcemust also be determined, but it is not uniquely providedby the gravitational wave (GW) observations. The moststraightforward way to obtain the redshift is to use GWsystems with electromagnetic (EM) counterpart events (e.g.X-ray or optical flashes associated with the merger), where (cid:63) Email: shafi[email protected] † E-mail: [email protected] ‡ E-mail: [email protected] the redshift comes from the EM measurement. (Crosscorre-lation with redshift surveys is an alternative area of inves-tigation, e.g. Zhang (2018); see Abbott et al. (2018) for ageneral review of GW detectors).However, low redshift means a small volume in whichevents can occur, hence small numbers of observed GW+EMsystems and poor precision on H . Upcoming gravitationalwave detectors – more sensitive runs of LIGO-Hanford andLIGO-Livingston (Aasi et al. 2015), Virgo (Acernese et al.2015), and new interferometers in India and Japan (Iyeret al. 2011; Aso et al. 2013) – will be able to detect GWevents expected to have EM counterparts (e.g. binary neu-tron stars) out to higher redshift, z ≈ . , with further gener-ations including space based detectors such as LISA (Amaro-Seoane et al. 2013, 2017) reaching z ≈ or beyond. Thesewill detect significantly more events, and some papers haveheld them out as the means to measure the Hubble constantto 1% or better precision (see, e.g., Chen et al. (2018)).Obtaining a 1% measurement of H from low redshiftobservables can be important for cosmology because of thediscordance between the inference of H from observations ofthe cosmic microwave background (CMB) (Planck Collabo-ration et al. 2018) and the value observed locally from cal-ibrating the supernova (SN) distance ladder with Cepheids(Riess et al. 2016). An independent measurement of H would offer keen insight into this tension (e.g. Feeney et al. © a r X i v : . [ a s t r o - ph . C O ] D ec Shafieloo et al. (2018)). If H from GW preferred the Planck value, thenthat might indicate an unaccounted for systematic withCepheid calibrations of SN distances. If it agrees with theCepheid+SN measurement, then that might lend more cre-dence to the idea this discordance is explained by newphysics.At redshifts z (cid:38) . , though, the distance does not de-pend solely on H but on an integral over the expansion his-tory H ( z ) , with all components of energy density in the uni-verse – in particular matter and dark energy – contributing.Without knowledge of these components one cannot cleanlyseparate out the Hubble constant.Astrophysical systematics such as binary orbit in-clination effects, peculiar velocities, and signal to noise(Malmquist-like) biases can also affect the use of GW sirensto constrain H (see, e.g., Mortlock et al. (2018)). For localsirens this could include coherent velocity flows (cf. Hui &Greene (2006); Cooray & Caldwell (2006)).Our focus here is two-fold: investigation of the precisionwith which H can be constrained in a background more gen-eral than Λ CDM – and indeed whose form may be unknown,and investigation of the accuracy, i.e. the bias suffered whenthe background is not accounted for correctly. Chen et al.(2018) have put forth the exciting prospect of 1-2% measure-ment of H from GW but within a calculation assuming notonly Λ CDM but a perfectly known matter density. While DiValentino et al. (2018) have looked beyond a Λ CDM back-ground, this was only for non-GW probes, implementing theGW data as purely a H prior from Chen et al. (2018). Duet al. (2018) have included a full dynamical dark energybackground, but for far future GW data sets of 1000 sirensout to z = and in combination with other probes alreadygiving strong cosmology constraints. Our approach is to ex-amine the role of the background expansion model on boththe precision and accuracy of H determination from midand moderately long term GW experiments.In Sec. 2 we present the framework for the analysis, in-cluding the GW+EM datasets corresponding to next, andnext next, generation GW experiments and our simulationmethodology. Precision on H is treated in Sec. 3, where weexamine how it degrades with greater freedom for the ex-pansion history: first including just the matter density anda cosmological constant, then allowing for dynamical darkenergy with assumption of the standard w – w a time depen-dence. Accuracy is the focus in Sec. 4, where we show howassuming a Λ CDM cosmology can significantly bias the re-sults in the regime of 1% precision. In Sec. 5 we concludeand discuss the statistical techniques needed to infer H ( z ) without bias from high precision datasets. As stated in the Introduction, GW standard sirens do notdirectly measure H but rather measure luminosity distances D L throughout the cosmic volume to which the detectors aresensitive. Each event has a cosmic redshift associated with it,which must be obtained from EM counterparts. The distanceis then related to the cosmic expansion rate through D L ( z ) = ( + z ) cH ∫ z dz (cid:48) h ( z (cid:48) ) , (1) where h ( z ) is the Hubble rate scaled to the present value, H ( z )/ H , and a spatially flat universe is assumed. The as-sumptions about the background expansion model h ( z ) havea direct impact on extraction of H . Even under the assump-tion of flat Λ CDM the matter density must also be known: h ( z ) = Ω m ( + z ) + − Ω m .To be concrete about how uncertainties in the expan-sion history can affect model dependent inferences usingGWs, we generate mock luminosity distance datasets froma given background cosmology. We consider two alterna-tive possibilities: 1) a flat Λ CDM model with Ω m = . and h ≡ h ( z = ) = . , and 2) time varying dark energy withassumption of w ( z ) = w + w a z /( + z ) which provides moreflexibility to the expansion history.For the latter we choose two models consistent with cur-rent data, specifically lying on the 68% confidence contour ofthe Pantheon supernovae plus Planck CMB plus SDSS BAOplus HST H combined data fit of Scolnic et al. (2018).To generate a realistic mock dataset of GW luminositydistances, we sample the event redshift distribution basedon the assumption that the GW events have a constant rateper comoving volume. That is, N ( z ) = dNdz = dNdV c dV c dz , (2)where we assume dN / dV c is constant and calculate dV c / dz from our fiducial input cosmology.We perform this sampling for two cases. One is a nextgeneration case corresponding roughly to the ∼ z = . and draw 120 events from thisdistribution, with distance errors normally distributed witha σ precision of 13%, roughly following Chen et al. (2018).The other is a next next generation case where we takethe maximum redshift to be z = . and draw 600 eventsfrom the distribution, with 7% distance precision. One real-ization each of these samplings can be seen in Fig. 1.Again we emphasize that any projection of the distanceinformation from these redshifts to H = H ( z = ) requiresthe assumption of an uncertain background model. Testingwhether or not this model is true is of course one aim of anyanalysis of cosmological datasets, along with H . Combining multiple cosmological probes within a givenmodel can give tight constraints on parameters, including H . As a rough rule of thumb, note that the CMB alreadytightly constrains the combination Ω m h , to about 0.3%(Planck Collaboration et al. 2018), fairly independently oflate time physics (though still power-law form of the primor-dial power spectrum is assumed). Thus δ h / h ≈ ( / ) δ Ω m / Ω m so a prior of 0.03 on Ω m from large scale structure probesgives a 3% constraint on h . Adding other probes such as su-pernovae would tighten this further. Thus we want another,individual probe at the level of ∼ on h . Let us exploreunder what conditions GW sirens can provide this in them-selves.Table 1 shows the constraints on H from next gener-ation GW data, either alone or with external priors. For MNRAS , 1–7 (2018) ravitational Waves Determine Hubble Constant? z N ( z ) z N ( z ) Next Next Gen.Next Gen.
Figure 1.
One realization of the redshift distribution of GW+EMevents for the next generation case of 120 total events observedout to maximum redshift z = . (upper panel) and for the nextnext generation case of 600 total events observed out to maximumredshift of z = . (lower panel). The next generation events areoverplotted on the lower panel for comparison.Background Prior σ ( h ) σ ( h )/ h Λ CDM none 0.036 5% Λ CDM σ ( Ω m ) = . Λ CDM fix Ω m w – w a ∗ none 0.039 6% w – w a ∗ σ ( Ω m ) = . w – w a ∗ fix Ω m Table 1.
Constraints on H from the next generation GW+EMset are given under various backgrounds and priors. An asteriskdenotes a broad prior of 1 on both w and w a , since the Fisherinformation approach is inaccurate for extended degeneracies. Wesee that GW siren constraints are sensitive to the input model. this table alone, the numbers come from Fisher informa-tion computation; all other numbers in the article are fromMCMC. They are in good agreement where they overlap.When the background is fixed to Λ CDM, and furthermorethe matter density is perfectly known, then the uncertaintyis σ ( h ) = . or approximately 1.2%. When external in-formation is used at the level of a prior on matter density of σ ( Ω m ) = . then the H precision has a modest increase to1.4%. However, such combination of probes can be done aswell between non-GW probes (e.g. supernovae, strong lenses,large scale structure, CMB), so we should look at what GW sirens themselves deliver. For GW alone, even with fixingthe background to Λ CDM, the uncertainty is σ ( h ) = . or 5%. This will not allow them to make a statistically sig-nificant statement on the tension between the Planck value(which used the same assumption of Λ CDM) and the localdistance ladder value.To illustrate the effect of the matter density covariancewith H within the Λ CDM model, we perform a Monte Carlosimulation of next generation GW data. That is, we generaterealizations of the data in a particular Λ CDM case with Ω m = . and h = . , and fit for these two parametersunder the assumption we know the background is Λ CDM.The 1D and 2D joint confidence contours appear in Fig. 2.The covariance between Ω m and h is clear, showing that– just as with other distance probes – external data to breakparameter degeneracies is necessary. Only if Ω m is well con-strained does the uncertainty on h reduce to the 1-2% level(still under the assumption of Λ CDM). Of course, if the ex-ternal data has systematics that shift the value of Ω m , thenthe value of h derived from the GW plus this data will bebiased.Another, intuitive way of seeing the difficulty in GWsirens (or any cosmic distance) in determining cleanly H isprovided in Fig. 3. We plot realizations of the ratio of GWluminosity distances for Λ CDM cosmologies with parame-ters drawn from the posterior relative to the input cosmol-ogy. The size of the scatter at different redshifts indicates atwhich redshifts the distance is better constrained. Both theparameter covariances and the number of events at a givenredshift enter into the scatter. Unfortunately, redshift zeroand hence D L ( z (cid:28) ) = H − z has large uncertainty. Thus wedo not expect H to be constrained near the 1% level whenwe are appropriately agnostic about the expansion history.If we allow for standard w - w a dynamical dark energyfreedom in the background we see an even bleaker picture.Now, even with the matter density perfectly known, GWsirens deliver only σ ( h ) = . , i.e. 6%. With Ω m as a fitparameter the covariance increases the MCMC uncertaintyon h , even in the next next generation case. The MCMCcontours in the Ω m – h plane are shown in Fig. 4. Thus thecosmological model assumed plays a critical role in the con-straints from GW sirens at cosmic distances. Apart from the issue of precision on H , if insufficient free-dom is given to the background cosmology fit then the re-sulting value of H (and Ω m ) will be biased. We explore themagnitude of this effect by choosing two alternative w – w a model points on the 68% confidence contour of the currentjoint probe analysis in Scolnic et al. (2018) and generatingGW data sets in these cosmologies. If these are then an-alyzed within the Λ CDM model, parameter biases ensue.The two cosmologies used are ( w , w a )=( − . , − . ) and( − . ,0.35), with Ω m = . , h = . , and by constructionare consistent with current combined data sets.After generating our mock luminosity distance datasetfrom GW+EM observations, we then use MCMC samplingto infer the 1D parameter fits and 2D joint confidence con-tours for a Λ CDM model. This allows us to assess the biasinduced by the incorrect background model assumption, and
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Figure 2.
Forecast posterior for mock data generated from a Λ CDM cosmology for next generation (top) and next next gener-ation (bottom) sensitivities. The 2D posterior for h and Ω m showsthe 68.3%, 95.4%, 99.7% confidence regions in increasingly lightershades of blue. The 1D posteriors show the 68.3% confidence re-gions in dashed blue. The input values for Ω m and h are indicatedwith solid black lines. its significance relative to the statistical precision. The dataquality will be particularly important for this last question:as the precision improves a given bias becomes more im-portant. Therefore we study both next generation and nextnext generation GW data sets.We present the results for the next generation case inFig. 5. The precision obtained for the two input models iscomparable, with ∼
6% uncertainty on H and ∼
50% on Ω m . Figure 3.
The distance uncertainty is shown as a scatter plotfrom a posterior predictive distribution sampling of the distances(relative to the input cosmology), for next generation GW data.
Clearly next generation GW alone will not give the desiredconstraint. The bias, due to misassuming Λ CDM, shifts thefit contours so that the true input values are at the edge of68% confidence contour in each case.Figure 6 repeats the analysis for the next next gener-ation data case. The parameter precision is now stronglyimproved, to 1.1% on H and 8% on Ω m . However, thebias is much more severe, with the true values lying outsidethe 99.7% joint confidence contour, i.e. roughly σ bias (al-though the 1D values do not accurately show this tension).Note that knowing the actual value of matter density herefrom other observations could increase the bias in estima-tion of H due to our wrong assumption of the backgroundexpansion model.The biases are evident even in a simple constant w ex-tension to Λ CDM. If we take w = − . from the 68% confi-dence contour of the current joint data constraint of Scolnicet al. (2018), Fig. 7 shows the input values of ( h , Ω m ) havebeen biased to outside the 99.7% joint confidence contour inthe Λ CDM analysis (indeed to more than the equivalent of σ ).From Eq. 1 and Fig. 3 we can see that the bias mustexist if the cosmological framework used in the fit does notcover the true cosmological model. For GW data to constrainprimarily H , then h ( z ) should be indistinguishable from 1at the level of the statistical precision. Similarly, if we go be-yond H to include Ω m , then to fit these parameters withoutbias the h ( z ) for the Λ CDM cosmology fit should be indistin-guishable from the true (potentially non- Λ CDM) cosmology,again at the level of the statistical precision.The direction and magnitude of the parameter biasesis a combination of the data properties (e.g. redshift distri-bution), cosmological parameter covariances, and parametervalues. We have verified the MCMC results through the an-alytic Fisher bias formalism (Knox et al. 1998; Linder 2006),which makes these dependencies more explicit. The bias on
MNRAS , 1–7 (2018) ravitational Waves Determine Hubble Constant? Figure 4.
Forecast posterior for mock data generated from a Λ CDM cosmology, but marginalizing over w and w a in theMCMC fit, for next generation (top) and next next generation(bottom) sensitivities. a parameter (such as h or Ω m ) is given by δ p i = (cid:16) F − (cid:17) ij (cid:213) k ∂ O k ∂ p j σ k ∆ O k (3) = (cid:16) F − (cid:17) ij (cid:2) ( + w ) F jw + w a F jw a (cid:3) , (4)where ∆ O k is the difference between the true distance andin the assumed model. Here the second line holds when theassumed cosmological model is a subset of a cosmologicalmodel with additional parameters, as Λ CDM is a subset of
Figure 5.
Forecast posteriors of the Λ CDM parameters for thenext generation case, where the distances to the GW+EM eventsare generated from currently viable ( w , w a ) cosmologies. The toppanel is for ( w , w a ) = (− . , − . ) and the bottom panel is for (− . , . ) . a w – w a cosmology. This parameter bias estimation is ingood agreement with the full Monte Carlo analysis we use. Gravitational wave sirens in conjunction with electromag-netic counterparts provide a new distance measure for theuniverse. New and improved detectors with better sensitiv-ities and further redshift reach are exciting developmentsthat will deliver abundant science from significant numbers
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Figure 6.
As Fig. 5 but for next next generation data. of events. While GW provide an absolute distance measure-ment they are not a panacea – covariance between the evo-lution of the expansion H ( z ) and the absolute scale today H still exists.We have quantified how assuming that the expansionhistory H ( z ) is purely of the Λ CDM form to infer the valueof H from GW luminosity distance data will yield inac-curate results should the true cosmology be different than Λ CDM. This holds even if the deviation from Λ CDM cos-mology is modest, within the 68% confidence level constraintfrom the current combination of data from several probes.Indeed we find that even a constant w model within the cur-rent 68% joint confidence contour can deliver almost a σ bias if inappropriately analyzed within Λ CDM. Fixing the
Figure 7.
As Fig. 6 but for distances generated from a constant w = − . cosmology. value of the matter density Ω m within Λ CDM, as is some-times adopted in predicting 1% precision on H from GW,is further problematic for accuracy.Next generation GW events will probe deeper into theuniverse, so all the freedom that enters into H ( z ) , from theimperfectly known matter density Ω m and dark energy prop-erties, will both dilute the precision and open up the poten-tial for bias as we try to project distances to the very lowredshift behavior involving only the Hubble constant H . Be-ing properly agnostic about the expansion history translatesinto uncertainties in H that are well above 1%. To quan-tify this, we carry out a Monte Carlo analysis simulating theGW+EM event distance data for next generation (roughly2026) and next next generation experiments.Next generation data reaches 1.2% precision on H onlyif both Λ CDM is assumed and Ω m is perfectly known, witha degradation to 1.4% if Λ CDM is assumed and an externalprior on Ω m is used. For GW themselves, the precision is5% when restricted to Λ CDM. Allowing for uncertainty inthe cosmological model by including dynamical dark energysuch as with w , w a dilutes the precision to 7% – barelymore constraining than the single local GW binary neutronstar event already measured (Abbott et al. 2017a,b). Thuscosmic GW data can clearly not be implemented as a pureprior on H .This is in no way a failing of GW data. Any cosmicdistance measurement has the same issues with covariances(and note strong lensing time delays involve H in a similarway to GW), and potential biases if unduly restricted to thewrong expansion model.The biases exist if fixing to the wrong Ω m , the wrongconstant w , or in general assuming a wrong model or formof dark energy. (Note that the w – w a form does fit the dis-tances out to z = to 0.1% in a wide variety of viable models(de Putter & Linder 2008).) In this work we show that for MNRAS , 1–7 (2018) ravitational Waves Determine Hubble Constant? the case of next next generation GW data we can have morethan σ bias in estimation of H while precision of the es-timation can be very tight at 1.1%. One safe approach toadvocate is to carry out the analysis with model indepen-dent reconstruction techniques.If one could achieve substantial samples of GW+EMevents at z (cid:46) . then most of the parameter covariancevanishes and one does get purer determination of H inde-pendent of background cosmological model (modulo issuesof peculiar velocities and coherent flows). This would be anexciting prospect, though event rates are currently too un-certain to obtain a clear estimate of the H leverage. ACKNOWLEDGEMENTS
We thank KIAS, where this project was discussed among theco-authors, and especially Stephen Appleby for hospitality.A.S. would like to acknowledge the support of the NationalResearch Foundation of Korea (NRF- 2016R1C1B2016478).EL is supported in part by the Energetic Cosmos Laboratoryand by the U.S. Department of Energy, Office of Science,Office of High Energy Physics, under Award DE-SC-0007867and contract no. DE-AC02- 05CH11231.
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