Standard Sirens

Constraining the non-flat ΛCDM model

We investigate the capability of the Einstein Telescope to constrain the cosmological parameters of the non-flat ΛCDM cosmological model. Two types of mock datasets are considered depending on whether or not a short Gamma-Ray Burst is detected and associated with the gravitational wave event using the THESEUS satellite. Depending on the mock dataset, different statistical estimators are applied: one assumes that the redshift is known, and another one marginalizes over it assuming a specific prior distribution. We demonstrate that (i) using mock catalogs collecting gravitational wave events to which a short Gamma-Ray Burst has been associated, Einstein Telescope may achieve accuracy on the cosmological parameters of σH0 ≈ 0.40 km/s/Mpc, σΩk,0 ≈ 0.09, and σΩΛ,0 ≈ 0.07; while (ii) using mock catalogs collecting also gravitational wave events without a detected electromagnetic counterpart, Einstein Telescope may achieve accuracy on the cosmological parameters of σH0 ≈ 0.04 km/s/Mpc, σΩk,0 ≈ 0.01, and σΩΛ,0 ≈ 0.01. These results show an improvement of a factor 2-75 with respect to earlier results using complementary datasets [1].

In the framework of the ΛCDM cosmological model, the first Friedman equation can be recast as

which is required to compute the comoving distance

Consequently, the luminosity distance is

To build the mock catalogs of cosmological distances, we set as a fiducial cosmological model the ΛCDM with the following observational constraints:

A schematic description of the main ingredients needed to build the mock catalogs is the following one:

(i) the first step is to define the probability distribution p(z) of an event happening at redshift z. This probability distribution will depend on the astrophysics, i.e. the merger rate;

(ii) the merger rate of BNS defines the rate density per unit of redshift in the observer frame, and it depends on the Star Formation Rate (SFR);

(iii) one needs to define the mass distribution of the NS. We set it to be uniform in the interval [1, 2.5] M⊙;

(iv) one needs to set the spatial distribution of the merger event of BNS. We set it to be isotropic on the sky angles θ and φ, and uniform on the orientation angle cos i and the polarization ψ. Once the previous ingredients have been defined, and using the fiducial cosmological model, we can extract the fiducial redshift from the probability distribution p(z). Then, we can predict the SNR, ρ, for Einstein Telescope (ET) using the expected one-sided noise power spectral density. Finally, we select events having SNR above a fixed threshold. Specifically, we will create three mock catalogs with SNR thresholds equal to [9, 12, 15] for each observational period fixed in one year, and five and ten years. In total, we have nine mock catalogs containing all events that ET will be capable of detecting. From each one, we extract two sub-catalogs (realistic and optimistic cases) listing only the events of BNS merger that have a detected electromagnetic counterpart.

Finally, we have a total of 27 mock catalogs.

Figure: Distribution of the expected rate of detection of GW events with ET. Left, central, and right panels illustrate the detection rate for SNR> [9, 12, 15], respectively.

Figure: Distribution of the expected rate of detection of GW events with an electromagnetic counterpart for both the optimistic and the realistic cases in upper and lower row, respectively. Left, central, and right panels illustrate the detection rate for SNR> [9, 12, 15], respectively.

We will examine three different cases: (I) including the redshift information from the electromagnetic signal; (II) including selection effects due to the cut in SNR and flux with respect to (I); and, finally, (III) using all the GWs events detected by the ET, even those without electromagnetic counterpart, the so called dark sirens. The results shown below refer to the mock catalog that collects all the events with SNR$>9$ detected throughout ten years of observations (all results are shown in [1]).

Figure: The figures illustrate the 68%, 95% and 99.7% of confidence level obtained from the posterior distribution of the parameters of our baseline model from the realistic analyses, carried out on the events detected after ten years of observations and using all the events with SNR> 9. The panel above shows resuls without selection effect, while the panle below include them. The vertical red line in the histograms and a red point in the contour plot indicate the true values of the corresponding cosmological parameter. While, the vertical dashed line indicates the median value and the shaded band indicates the 1σ confidence interval.

Figure: The figures illustrate the results obtainated from the optimistic analyses. The panel above shows resuls without selection effect, while the panle below include them.

Figure: The figures illustrate the 68%, 95% and 99.7% of confidence level obtained from the posterior distribution of the parameters of our baseline model from the dark sirens analyses, carried out on the events detected after ten years of observations and using all the events with SNR> 9. The vertical red line in the histograms and a red point in the contour plot indicate the true values of the corresponding cosmological parameter. While, the vertical dashed line indicates the median value and the shaded band indicates the 1σ confidence interval.

These results show the huge potential of ET to strongly improve current constraints on the cosmological parameters of non-flat ΛCDM cosmology and, hopefully, solve the current tensions involving both the Hubble constant and the curvature density parameter

Investigating some solutions to the Hubble tension

We probe four cosmological models which, potentially, can solve the Hubble tension according to the dark energy equation of state. In this context, we demonstrate that the Einstein Telescope is capable of achieving a relative accuracy below 1% on the Hubble constant independently of the specific dark energy model [2]. We use mock catalogs containing gravitational wave events for one, five, and ten years of observations, and above signal-to-noise ratio equal to nine built to investigate the non-flat ΛCDM cosmological model in [1]. The four model we took into account are the following ones:

Non-flat ωCDM: the modification to the expansion rate appears as

Interacting Dark Energy: in such a model the expansion rate is given by

Emergent Dark Energy: the simplest parameterization yelds to

Time-Varying Gravitational Constant: the modification to the expansion rate is given as

Figure: the panels report the predicted luminosity distance as a function of redshift for the non-flat ωCDM (upper left panel), IDE model (upper right panel), emergent DE model (lower left panel), and time-varying gravitational constant model (lower right panel). In each panel, we depict our fiducial model (blue solid line), and the DE models where [H0, Ωm,0, Ωk,0, ΩΛ,0] are set to their fiducial values, and the extra-parameters are varied. For each model, we also show the residuals with respect to the ΛCDM model.

Results: Non-flat ωCDM

Results: Interacting Dark Energy

Results: Emergent Dark Energy

Results: Time-Varying Gravitational Constant

We foresee that the Hubble constant is always constrained with less than 1% uncertainty, thereby offering a potential solution to the Hubble tension. The accuracy on the other cosmological parameters is at most comparable with the one currently obtained using multiple probes, except for the emergent dark energy model for which the Einstein Telescope alone will be able to improve the current limits by more than one order of magnitude.

Probing Interacting Dark Sector

We have probed the capability of third-generation Gravitational Waves (GW) interferometers, such as the Einstein Telescope and Cosmic Explorer, to constrain a cosmological model with an interacting dark sector. We focused on GW events with a detected electromagnetic counterpart, which included the Gamma-Ray Burst and Kilonova emission. We assume the first one to be detected by the THESEUS satellite, while the second one is to be detected by the Vera Rubin Observatory. We probed three different interaction kernels :

CASE I: When the interaction is proportional to the DE energy density, the Hubble function becomes:

CASE II: When the interaction is proportional to the DM energy density

CASE III: When the interaction is related to the total dark energy densities (DM + DE)

We found that the posterior estimation of the cosmological parameters is biased due to the existing degeneracies between the dark and matter sectors. We also found that introducing an external prior on the matter density parameter breaks the degeneracy, removes the bias, and improves the accuracy of the dark sector parameters.

Bibliography:

M. Califano, I. de Martino, D. Vernieri, S. Capozziello, "Constraining ΛCDM cosmological parameters with Einstein Telescope mock data", 2022, MNRAS, 518, 3372-3385.
M. Califano, I. de Martino, D. Vernieri, S. Capozziello, "Exploiting Einstein telescope to solve the Hubble tension", 2023, Physical Review D, 107, 123519.
M. Califano, I. de Martino, R. Lazkoz, "Probing Interacting Dark Sector with the next generation of gravitational-wave detectors", 2024, Phys. Rev. D 110, 083519

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