002884544 001__ 2884544
002884544 005__ 20240719043536.0
002884544 0248_ $$aoai:cds.cern.ch:2884544$$pcerncds:FULLTEXT$$pcerncds:CERN:FULLTEXT$$pcerncds:CERN
002884544 037__ $$9arXiv$$aarXiv:2312.04632$$castro-ph.CO
002884544 035__ $$9arXiv$$aoai:arXiv.org:2312.04632
002884544 035__ $$9Inspire$$aoai:inspirehep.net:2734193$$d2024-07-18T04:10:28Z$$h2024-07-19T02:00:03Z$$mmarcxml$$ttrue$$uhttps://inspirehep.net/api/oai2d
002884544 035__ $$9Inspire$$a2734193
002884544 041__ $$aeng
002884544 100__ $$aMangiagli, Alberto$$mmangiagli@apc.in2p3.fr$$uAPC, Paris$$vUniversité Paris Cité, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France
002884544 245__ $$9arXiv$$aMassive black hole binaries in LISA: constraining cosmological parameters at high redshifts
002884544 269__ $$c2023-12-07
002884544 300__ $$a33 p
002884544 500__ $$9arXiv$$a33 pages, 21 figures. Submitted to PRD. Improved text quality and
added clarifications
002884544 520__ $$9arXiv$$aOne of the scientific objectives of the Laser Interferometer Space Antenna (LISA) is to probe the expansion of the Universe using gravitational wave observations. Indeed, as gravitational waves from the coalescence of a massive black hole binary (MBHB) carry direct information of its luminosity distance, an accompanying electromagnetic (EM) counterpart can be used to determine its redshift. This method of $bright$$sirens$, when applied to LISA, enables one to build a gravitational Hubble diagram to high redshift. In this work, we forecast the ability of LISA-detected MBHB bright sirens to constrain cosmological models. The expected EM emission from MBHBs can be detected up to redshift $z\sim 7$ with future astronomical facilities, and the distribution of MBHBs with detectable counterpart peaks at $z\sim 2-3$. Therefore, we propose several methods to leverage the ability of LISA to constrain the expansion of the Universe at $z\sim 2-3$, a poorly charted epoch in cosmography. We find that the most promising method consists in using a model-independent approach based on a spline interpolation of the luminosity distance-redshift relation: in this case, LISA can constrain the Hubble parameter at $z\sim2-3$ with a relative precision of at least $10\%$.
002884544 540__ $$3preprint$$aCC BY 4.0$$uhttp://creativecommons.org/licenses/by/4.0/
002884544 595__ $$cHAL
002884544 595_D $$aV$$d2023-12-14$$sabs
002884544 595_D $$aV$$d2023-12-18$$sprinted
002884544 65017 $$2arXiv$$agr-qc
002884544 65017 $$2SzGeCERN$$aGeneral Relativity and Cosmology
002884544 65017 $$2arXiv$$aastro-ph.CO
002884544 65017 $$2SzGeCERN$$aAstrophysics and Astronomy
002884544 690C_ $$aCERN
002884544 690C_ $$aPREPRINT
002884544 700__ $$aCaprini, Chiara$$uU. Geneva (main)$$uCERN$$vUniversité de Genève, Département de Physique Théorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland$$vCERN, Theoretical Physics Department, 1 Esplanade des Particules, CH-1211 Genéve 23, Switzerland$$vInfinis -Toulouse (L2IT-IN2P3), Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France
002884544 700__ $$aMarsat, Sylvain$$uL2IT, Toulouse$$vLaboratoire des
002884544 700__ $$aSperi, Lorenzo$$uPotsdam, Max Planck Inst.$$vMax Planck Institute for Gravitational Physics (Albert Einstein Institute) Am Mühlenberg 1, 14476 Potsdam, Germany
002884544 700__ $$aCaldwell, Robert R.$$uDartmouth Coll.$$vDepartment of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755
002884544 700__ $$aTamanini, Nicola$$uL2IT, Toulouse$$vLaboratoire des
002884544 8564_ $$82501752$$s77251$$uhttp://cds.cern.ch/record/2884544/files/error_bars_H0.png$$y00022 Uncertainty on $h$ for all the realisations in the analysis for $(h, \Omega_m)$. Error bars correspond to 90th confidence interval. \am{add that the x-axis is irrelevant and the order is irrelevant} \am{the point is that this plot give the information about the fact the median is consistent while Fig.~\ref{fig:H0_and_omegam} tells us information only about the error}
002884544 8564_ $$82501753$$s91759$$uhttp://cds.cern.ch/record/2884544/files/comparison_js_divergence.png$$y00018 Comparison between JS divergence values from different priors, as stated in the legend. On the x-axis we report the JS divergence adopted in the main text, i.e. with uniform prior in [0.1,50].
002884544 8564_ $$82501754$$s72549$$uhttp://cds.cern.ch/record/2884544/files/dl_errors_vs_z_alter.png$$y00002 Scatter plot of the luminosity distance uncertainty at 1$\sigma$ from LISA parameter estimation, as a function of redshift. The blue points correspond to all the MBHBs we have simulated within the Q3d model, i.e.~the 90 years of data. \textcolor{\colorref}{We convolve the luminosity distance uncertainty from the GW detection with the lensing and peculiar velocity errors as described in Sec.~\ref{sec:cat_construction}, to get the luminosity distance uncertainties adopted in the cosmological analysis}. The green solid line represents the errors from peculiar velocities as in Eq.~\ref{eq:pv_error}. The red (yellow) dashed line corresponds to the lensing error as in Eq.~\ref{eq:lensing_Q3} without (with) the delensing correction. \textcolor{\colorref}{The grey dotted-dashed line corresponds to the arbitrary cut-off we impose on the $1\sigma$ error on $d_L$ in order to choose which systems ro rerun, assuming their sky position is known, i.e. $\sigma_{68,d_L} > 0.5 \, \rm \sigma_{\rm delens}$ or $0.5\sigma_{\rm pv}$. The grey points above the grey dotted-dashed correspond to the subset of systems for which we rerun the parameter estimation assuming perfect localisation, leading to the corresponding blue points (connected with a vertical thin dashed grey line)}. For points below the grey dotted-dashed line, the error on $d_L$ is already dominated by lensing or peculiar velocities.
002884544 8564_ $$82501755$$s31442$$uhttp://cds.cern.ch/record/2884544/files/number_of_info_real_vs_threshold.png$$y00019 Number of informative realisations as a function of the JS divergence for the three astrophysical models, as in the legend. The JS divergence is computed for $h(z_p=3)$ with the prior adopted in the main text.
002884544 8564_ $$82501756$$s24834$$uhttp://cds.cern.ch/record/2884544/files/test_gauss_dl.png$$y00016 Upper left panel: ratio between the true value of the luminosity distance and the median value from the $d_L$ posterior distributions. Upper right panel: ratio between the $1\sigma$ uncertainty from fisher analysis and from $d_L$ posterior distribution. Lower left panel: skewness of the $d_L$ posteriors. Lower right panel: same as the left one but for the kurtosis. Aquamarine (crimson) lines correspond to the distribution for the entire MBHBs catalogues (the subset of EMcps). This plot is only for the Q3d model. Overall, the $d_L$ posterior distributions for the EMcps can be considered as Gaussian distributions.
002884544 8564_ $$82501757$$s115570$$uhttp://cds.cern.ch/record/2884544/files/uncer_bin_approach.png$$y00013 Same as Fig.~\ref{fig:H0_and_omegam} but for $h(z_p)$ and $d_C(z_p)$ from the \hyperref[itm:bins]{\color{mycolormodel}redshift bins} approach at different pivot redshifts, according to the x-axis labels. Colors represent different astrophysical models as reported in the legend. The cumulative distributions reach the fraction of \emph{informative} realisations (see Sec.~\ref{subsec:bins}): for example at $z_p=3.5$, only $\sim75\%$ of the Q3d and Q3nd realisations provide useful constraints.
002884544 8564_ $$82501758$$s38968$$uhttp://cds.cern.ch/record/2884544/files/H0_and_omega_and_beta_uncer_all_mission_time.png$$y00011 Same as Fig.~\ref{fig:H0_and_omegam} but for $h$ and $\beta$ for the \hyperref[itm:lcdm_beta]{\color{mycolormodel}$(h, \Omega_m, \beta)$} model. The parameter $\beta$ models a possible deviation from zero in the matter equation of state, i.e. $\omega_m = \beta$. We forecast a relative accuracy on $\beta$ of $<20\%$ with 4 yrs of LISA observations and $\lesssim 10\%$ with 10 yrs for at least 50\% of the realisations.
002884544 8564_ $$82501759$$s26092$$uhttp://cds.cern.ch/record/2884544/files/ratio_dl_with_and_without_skypos.png$$y00003 Ratio of the luminosity distance error fixing the sky position in the parameter inference over the one obtained in the full analysis, i.e. including as parameters the two sky position coordinates. Both errors are at 90\% level. Colors and line styles correspond to the three astrophysical models, as reported in the legend.
002884544 8564_ $$82501760$$s96783$$uhttp://cds.cern.ch/record/2884544/files/js_divergence_for_different_prior.png$$y00017 Scatter plot of JS divergence computed between the posterior distribution of $h(z_p=3)$ and different priors (as specified in the legend) versus the inferred median value of $h(z_p=3)$. The dashed black line represents the value of $h(z_p=3)\sim3.06$, according to $\Lambda$CDM.
002884544 8564_ $$82501761$$s46342$$uhttp://cds.cern.ch/record/2884544/files/cpl_uncer_all_mission_time.png$$y00009 Same as Fig.~\ref{fig:H0_and_omegam} but with $h, \Omega_m$ and $\omega_0$ from the \hyperref[itm:cpl]{\color{mycolormodel}$(h, \Omega_m, \omega_0, \omega_a)$} model. $\omega_a$ is unconstrained and not reported. The addition of two parameters worsen the estimates on $(h, \Omega_m)$.
002884544 8564_ $$82501762$$s25266$$uhttp://cds.cern.ch/record/2884544/files/corner_plot.png$$y00007 Corner plot for $(h, \Omega_m)$ for the \lcdm model in 4 yrs of LISA mission, for the average, best and worst realisations of the Q3d MBHBs formation scenario, i.e.~those corresponding to the median, 5th and 95th percentile of the distribution pf the realisations (more details in the text). Colors according to legend.
002884544 8564_ $$82501763$$s61579$$uhttp://cds.cern.ch/record/2884544/files/Hz_and_dlz_uncer_splines_90perc_paper.png$$y00014 Relative errors at 90\% for $d_L(z)$ (upper panels) and $h(z)$ (lower panels) as a function of redshift from the \hyperref[itm:splines]{\color{mycolormodel}spline interpolation} model. Error bars correspond to the $90\%$ uncertainty on the relative errors from the distribution of realisations. Light blue boxes highlight the uncertainties at the knot redshifts. Splines can constrain the luminosity distance to $<10\%$ in 4 yrs at $1<z<4$ and up to $z\sim6$ in 10 yrs of observations.
002884544 8564_ $$82501764$$s41217$$uhttp://cds.cern.ch/record/2884544/files/Xi0_omega0_uncer_belgacem19_all_mission_time.png$$y00010 Same as Fig.~\ref{fig:H0_and_omegam} but for $\omega_0$ and $\Xi_0$ for the \hyperref[itm:belgacem19]{\color{mycolormodel}$(h, \Omega_m, \omega_0, \Xi_0)$} model. Note that we report the absolute $1\sigma$ uncertainties for comparison with Tab.2 in \cite{Belgacem19}. The uncertainties on $h$ and $\Omega_m$ coincide with the CMB priors. We can constrain $\Xi_0$ to $<10\%$ using only MBHBs bright sirens.
002884544 8564_ $$82501765$$s24102$$uhttp://cds.cern.ch/record/2884544/files/comparison_posterior_hzp.png$$y00020 Comparison between the posterior distributions on $h(z_p)$ and the corresponding priors, according to legend. Here we select a particular event realisation of Q3d \am{for future note: I selected the realisation 1}
002884544 8564_ $$82501766$$s63662$$uhttp://cds.cern.ch/record/2884544/files/Hz_and_dlz_uncer_90.png$$y00008 Relative errors at 90\% for $h(z)$ (upper panels) and $d_L(z)$ (lower panels) as a function of redshift, derived using~\cref{eq:hubble_rate_h0_omegam} and \cref{eq:dl} from the posterior samples of the \hyperref[itm:lcdm]{\color{mycolormodel}$(h, \Omega_m)$} model, shown in \cref{fig:H0_and_omegam}. Error bars correspond to the $90\%$ uncertainty coming from the distribution over realisations. The three colors correspond to different astrophysical models, according to the legend. To avoid null values in the lower panels, the first point for $d_L$ is at $z=0.1$.
002884544 8564_ $$82501767$$s48841$$uhttp://cds.cern.ch/record/2884544/files/matter_only_uncer_all_mission_time.png$$y00012 Same as Fig.~\ref{fig:H0_and_omegam} but for $h(z_p)$ and $d_C(z_p)$ from the \hyperref[itm:matteronly]{\color{mycolormodel}matter-only} approximation. Left panels: $z_p=2$. Right panels: $z_p=3$. In 10yr of observation, we expect constraints on $h(z_p=2)$ at $3-5\%$ and on $h(z_p=3)$ at $\lesssim 10\%$.
002884544 8564_ $$82501768$$s42105$$uhttp://cds.cern.ch/record/2884544/files/H0_and_omega_uncer_all_mission_time.png$$y00006 Cumulative distributions over realisations of the relative uncertainties for $h$ (left panel) and $\Omega_m$ (right panel) in the \hyperref[itm:lcdm]{\color{mycolormodel}$(h, \Omega_m)$} model, namely $\Lambda$CDM. Solid (dashed) lines correspond to 10 (4) yrs of observations with LISA. Colors represent different astrophysical models as described in the legend and the grey area represents uncertainties larger than $100\%$. We expect relative errors on $h$ of $\lesssim 5\%$ in 4 yrs and $\lesssim 2\%$ in 10 yrs for at least $50\%$ of the realisations. The relative errors on $\Omega_m$ descend below $\lesssim 10\%$ for at least $50\%$ of the realisations only in the 10 yrs LISA scenario.
002884544 8564_ $$82501769$$s27983$$uhttp://cds.cern.ch/record/2884544/files/convergence_error.png$$y00015 Median relative uncertainty on $h$ as a function of the number of realisations. Errors bars represent the 90 percentile. For all astrophysical models, 100 realisations are sufficient to construct a statistically representative sample.
002884544 8564_ $$82501770$$s38539$$uhttp://cds.cern.ch/record/2884544/files/dl_and_z_bins_paper.png$$y00001 Representation of the \hyperref[itm:bins]{\color{mycolormodel}redshift bins} and \hyperref[itm:matteronly]{\color{mycolormodel}matter-only} approximation models, according to the legend. The blue line corresponds to the comoving distance in the standard \lcdm Universe; the yellow dotted-dashed line represents the matter-only approximation from Eq.~\ref{eq:matter-only} with $z_p=3$ and the two orange dashed lines the redshift bins approach in Eq.~\ref{eq:linear_fit} for two redshift bins with $z_p=1.5$ and $z_p=3$ (black stars). Green points correspond to the MBHBs from a random realisation of Q3d with the corresponding errors on redshift and comoving distance, accounting also for lensing and peculiar velocities errors as described in Sec.~\ref{sec:cat_construction}. For low-redshift events, the errors are smaller than the size of the dot.
002884544 8564_ $$82501771$$s38396$$uhttp://cds.cern.ch/record/2884544/files/js_divergence_vs_hvalue.png$$y00004 Example of JS divergence computed between the posterior distribution of $h(z_p=3)$ and an uniform prior between $[0.1,50]$ versus the inferred median value of $h(z_p=3)$. Each point corresponds to a single realisation for each astrophysical model, according to legend. The horizontal grey dotted-dashed line corresponds to the arbitrary cut-off of 0.5 on the value of the JS divergence (more details in the text). The blacked dashed line represents the true value of $h(z_p=3)\sim3.06$, according to $\Lambda$CDM. Uninformative realisations show inferred median values at $\sim25$, corresponding to the midpoint of the prior range, and JS divergence closer to 0, i.e. the posterior is similar to the prior.
002884544 8564_ $$82501772$$s64794$$uhttp://cds.cern.ch/record/2884544/files/pp_plot_H0_and_omega.png$$y00021 PP-plot for $h$ and $\Omega_m$
002884544 8564_ $$82501773$$s95897$$uhttp://cds.cern.ch/record/2884544/files/plot_probes_final_paper.png$$y00000 : width=1\textwidth
002884544 8564_ $$82501774$$s2011744$$uhttp://cds.cern.ch/record/2884544/files/2312.04632.pdf$$yFulltext
002884544 8564_ $$82501775$$s93269$$uhttp://cds.cern.ch/record/2884544/files/ppplot_for_hz3.png$$y00005 Cumulative distribution of events with p-value smaller or equal to the abscissa. Colors and line styles according to legend. The dotted lines represent the 90\% uncertainty region expected from the finite number of realisations. The dashed black line represents the expected value. Grey dashed line: same as the orange one for Pop3 but including all the realisations. It is clear that selecting only the informative realisations provide results consistent with the expected values.
002884544 8564_ $$82545404$$s122393$$uhttp://cds.cern.ch/record/2884544/files/uncer_bin_approach_referee_paper.png$$y00013 Same as Fig.~\ref{fig:H0_and_omegam} but for $h(z_p)$ and $d_C(z_p)$ from the \hyperref[itm:bins]{\color{mycolormodel}redshift bins} approach at different pivot redshift, as specified in the x-axis labels. Colors represent different astrophysical models as reported in the legend. \textcolor{\colorref}{The vertical dashed black line represents the model's accuracy.} The cumulative distributions reach the fraction of \emph{informative} realisations (see Sec.~\ref{subsec:bins}): for example at $z_p=3.5$, only $\sim75\%$ of the Q3d and Q3nd realisations provide useful constraints.
002884544 8564_ $$82545405$$s51946$$uhttp://cds.cern.ch/record/2884544/files/js_divergence_vs_hvalue_referee.png$$y00004 \textcolor{\colorref}{The JS divergence as a function of the median of the posterior on the parameter $h(z_p)$, for all the EMcps catalogues simulated for the three MBHB formation scenarios, and taking as example the redshift bin centred on $z_p=3$. The JS divergence is computed between the posterior distribution of $h(z_p=3)$ and its uniform prior, which is extended to the interval $[0.1,50]$ to distinguish the \emph{uninformative} realisations. Each point corresponds to a single realisation for each astrophysical model, according to the legend. The horizontal grey dotted-dashed line corresponds to the arbitrary cut-off of 0.5 which we impose on the value of the JS divergence to select the informative realisations (more details in the text). The black dashed line represents the \lcdm true value, $h(z_p=3)= 3.06$. Uninformative realisations show inferred median values far from the \lcdm one and, specifically, decaying towards $ h(z_p=3)\sim25$, corresponding to the midpoint of the prior range. Furthermore, the JS divergence decays towards 0, meaning that the posterior is similar to the prior. Informative realisations, on the other hand, cluster around the \lcdm value, and have $\mathrm{JS}\gtrsim 0.5$.}
002884544 8564_ $$82545406$$s104937$$uhttp://cds.cern.ch/record/2884544/files/ppplot_for_hz3_referee.png$$y00005 Cumulative distribution of events with p-value smaller or equal to the abscissa. The solid, dashed and dot-dashed lines represent the \emph{informative} realisations only, for the different MBHB formation scenarios according to the legend. The dotted lines represent the 90\% uncertainty region expected from the finite number of realisations. The dashed black line represents the expected value for unbiased results. Grey dashed line: same as the orange one for Pop3, but including all the realisations, even the uninformative ones (according to the JS criterion (see the meain text). It is clear that selecting only the informative realisations provide results consistent with the expected values.
002884544 8564_ $$82545407$$s34521$$uhttp://cds.cern.ch/record/2884544/files/number_of_info_real_vs_threshold_referee.png$$y00019 Number of informative realisations as a function of the JS divergence for the three astrophysical models, as in the legend. The JS divergence is computed for $h(z_p=3)$ with the prior adopted in the main text.
002884544 8564_ $$82545408$$s117898$$uhttp://cds.cern.ch/record/2884544/files/comparison_js_divergence_referee.png$$y00018 Comparison between JS divergence values for $h(z_p=3)$ for different priors, as stated in the legend. On the x-axis we report the JS divergence adopted in the main text, i.e. with uniform prior in [0.1,50].
002884544 8564_ $$82545409$$s101456$$uhttp://cds.cern.ch/record/2884544/files/splines_uncertainties_referee_for_appendix.png$$y00020 \textcolor{\colorref}{Difference between the luminosity distance from spline interpolation $d_L^{spline}$ and the $\Lambda$CDM luminosity distance $d_L^{ \Lambda CDM}$, normalised over $d_L^{ \Lambda CDM}$ and for different choices of knot redshifts as reported in the legends. Upper and lower panels report the same quantity: we split the results and use solid and dashed lines only to facilitate the visual comparison. At the values of the knots, the spline is forced to pass through $\Lambda$CDM values and the difference goes to zero.}
002884544 8564_ $$82545410$$s40829$$uhttp://cds.cern.ch/record/2884544/files/dl_and_z_bins_paper_referee.png$$y00001 Representation of the \hyperref[itm:bins]{\color{mycolormodel}redshift bins} and \hyperref[itm:matteronly]{\color{mycolormodel}matter-only} models. The blue, solid line corresponds to the comoving distance in the standard \lcdm Universe; the yellow, dotted-dashed line represents the matter-only approximation from Eq.~\ref{eq:matter-only} with $z_p=3$; the two red, dashed lines denote the redshift bins approach in Eq.~\ref{eq:linear_fit} for two redshift bins with $z_p=1.5$ and $z_p=3$ (black stars) \textcolor{\colorref}{and the vertical dotted-dashed grey lines visualise the boundaries of the corresponding redshift bins}. The green points show the MBHBs from a random Universe realisation, \textcolor{\colorref}{based on the astrophysical model Q3d, assuming 10 yrs of data (and therefore correspondingly of LISA mission duration),} with the respective \textcolor{\colorref}{1$\sigma$} errors bars on the redshift and comoving distance, accounting also for lensing and peculiar velocities errors (as described in Sec.~\ref{sec:cat_construction}). For the low-redshift events, the errors are smaller than the size of the dot.
002884544 8564_ $$82545411$$s125207$$uhttp://cds.cern.ch/record/2884544/files/matter_only_uncer_all_mission_time_all_z.png$$y00012 Same as Fig.~\ref{fig:H0_and_omegam} but for $h(z_p)$ and $d_C(z_p)$ from the \hyperref[itm:matteronly]{\color{mycolormodel}matter-only} approximation at different pivot redshift, as specified in the x-axis labels. \textcolor{\colorref}{Colors represent different astrophysical models as reported in the legend. The vertical dashed black line represents the model's accuracy. In 10yrs of observation, we expect constraints on $h(z_p=2)$ at $3-5\%$. At higher redshift, estimates get slightly worse but we still have relative errors on $h(z_p)$ of $10\%$ at $z_p=7$.}
002884544 8564_ $$82545412$$s136023$$uhttp://cds.cern.ch/record/2884544/files/js_divergence_for_different_prior_referee.png$$y00017 Scatter plot of JS divergence computed between the posterior distribution of $h(z_p=3)$ and different priors (as specified in the legend) versus the inferred median value of $h(z_p=3)$. The dashed black line represents the value of $h(z_p=3)\sim3.06$, according to $\Lambda$CDM.
002884544 960__ $$a11
002884544 980__ $$aPREPRINT