Study of Weak Gravitational Waves in the Field Theory of Gravitation

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Shifts of interference fringes corresponding to an increase in length of one arm and a decrease in the other would indicate the passage of gravitational waves. Two black holes about 1. The black holes were 36 and 29 times the mass of the Sun and formed a new black hole 62 times the mass of the Sun. In the merger, three solar masses were converted to energy in gravitational waves; the amount of power radiated was 50 times more than that of all the stars shining in the universe in that moment.

As of , LIGO has made 10 detections of gravitational radiation. Nine were from mergers of a black hole binary, and one was the merger of a neutron star binary. As stated above, studies of gravity allow the masses and densities of celestial bodies to be estimated and thereby make it possible to investigate the physical constitutions of stars and planets. Because gravitation is a very weak force , however, its distinctive effects appear only when masses are extremely large. The idea that light might be attracted gravitationally had been suggested by Michell and examined by the French mathematician and astronomer Pierre-Simon Laplace.

Predictions by classical physics and general relativity that light passing close to the Sun might be deflected are described above. There are two further consequences for astronomy. Light from a distant object may pass close to objects other than the Sun and be deflected by them.

In particular, they may be deflected by a massive galaxy. If some object is behind a massive galaxy, as seen from Earth, deflected light may reach Earth by more than one path. Operating like a lens that focuses light along different paths, the gravity of the galaxy may make the object appear multiple; examples of such apparently double objects have been found. Both Michell and Laplace pointed out that the attraction of a very dense object upon light might be so great that the light could never escape from the object, rendering it invisible.

Such a phenomenon is a black hole. The relativistic theory of black holes has been thoroughly developed in recent years, and astronomers have conducted extensive observations of them. One possible class of black holes comprises very large stars that have used up all of their nuclear energy so that they are no longer held up by radiation pressure and have collapsed into black holes less-massive stars may collapse into neutron stars. Supermassive black holes with masses millions to billions times that of the Sun are thought to exist at the centres of most galaxies.

Black holes, from which no radiation is able to escape, cannot be seen by their own light, but there are observable secondary effects. Present GW observations place severe limits on deviations from GR. Among the different constraints, the most stringent ones are the propagation speed equal to the speed of light and the absence of emission of additional polarizations. The key question is then. Of course, we do not have a complete answer to this question. In the following, we survey different proposals of viable theories and highlight some lessons we have learned in light of current bounds.

Before considering which theories are compatible with present constraint on the speed of GWs, it is important to discuss how far reaching this new measurement is. The first thing to note is that due to the closeness of the BNS, the constraint only applies basically to present time. However, one should be careful about this argument for several reasons. This is because although the cosmological evolution might lead to the precise cancellation of the dangerous terms, there will be deviations from the cosmological background along the path of the GWs toward the detector, for instance, when they cross the Milky Way.

A second remark is that constraints in the dispersion relation only apply to the characteristic wave numbers of the compact binary systems detected so far. As a consequence, in a phenomenological approach, one could envision modifications of the dispersion relation only arising at cosmological scales Battye et al. This could, in principle, lead to modified gravity effects at large scales not affecting present GW constraints.

However, in practice, only theories with non-local couplings or higher derivative interactions with ghost degrees of freedom are known to have this dispersion relation. It would be interesting to study in depth the theoretical framework allowing for this modified propagation. Theories with a Lorentz-invariant ultra-violet UV completion are presumed to have luminal GW propagation.

Therefore, one would expect higher dimensional operators to erase any anomalous speed beyond the cutoff scale, which in this case might already happen in the LIGO band. This might give us valuable information about the cutoff scale of the effective theory of DE. Another lesson from GW, as it was discussed in section 5. In other words, the GW-cone and the light-cone are the same. Satisfying this constraint, concrete DE models have been proposed Kase and Tsujikawa, If they are not, one would need to tune the GW speed order by order in perturbation theory.

Using the results of Pirtskhalava et al. Thus, the tree-level condition is not modified see also Santoni et al. Within the scalar-tensor theories compatible with the constraint on the speed of GWs, there have been extensive efforts to explore interesting phenomenology. One immediate question is whether the survival theories can provide accelerated expansions at late times without a cosmological constant as covariant Galileons were providing. It was found that indeed there are scaling solutions with a late time de Sitter behavior.

Still, a full comparison with present cosmological observations is missing due to the lack of appropriate Boltzmann codes for these higher-derivative theories. Another attractive feature of Horndeski gravity was the possibility to have self-tuning solutions Charmousis et al. This was an attempt to solve the cosmological constant problem by counterbalancing the large bare vacuum energy with the energy momentum of the scalar field. However, Fab Four models realizing this behavior predict an anomalous GW speed.

Indeed, an infinite set of self-tuning models compatible with GW were found in Babichev et al. Again, a detailed cosmological analysis is left for future work. In the realm of Horndeski gravity, one could search for other definite predictions. This model, named no slip gravity Linder, , has the property of predicting that gravity should be weaker than GR in the late universe.

This could be tested with growth of structure observations in the next generation galaxy redshift surveys.

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Present observations severely constrain deviations from GR at small scales. Screening mechanisms are thus needed to surpass these bounds Chu and Trodden, ; de Rham et al. Modified gravity theories can display different types of screening mechanisms see reviews Brax, ; Joyce et al. For theories with derivative interactions this is achieved with the Vainshtein mechanism, which screens the fifth force when the local curvature is larger than a given threshold.

Such mechanism has been extensively studied for Horndeski theory Kimura et al. For theories beyond Horndeski of the GLPV class the screening works similarly outside the source, but there is a breaking of Vainshtein screening inside matter Kobayashi et al. This suggests using astrophysical systems, such as neutron stars, to test these theories Koyama and Sakstein, ; Babichev et al.

The question then is whether the viable scalar-tensor theories in light of GW tests can display a successful screening and if there are any observational signatures to test them. This was addressed by different groups soon after the announcement of GW Crisostomi and Koyama, b ; Dima and Vernizzi, ; Langlois et al. Still, these recent analyses show that within DHOST theories satisfying the constraint in the speed of GWs, screening is effective outside non-relativistic bodies, but there could be a breaking inside matter as well. This deviation from GR inside compact bodies is only predicted for theories beyond Horndeski.

Moreover, the emission of additional polarizations is highly constrained as well. Depending on the gravity theory, compact objects might emit extra radiation see Herdeiro and Radu, for a review on no-hair theorems. An interesting question is if cosmologically relevant theories compatible with the bound on the speed of GWs can exhibit scalar hair in the black-hole solutions.

In Tattersall et al. Analysis of black-hole solutions including screening effects have not been studied so far. Strong field effects are yet possible in theories not aimed at explaining cosmology, for instance spontaneous scalarization in neutron stars Damour and Esposito-Farese, or even in black-holes Antoniou et al. However, one should note that this kind of solutions may also induce an anomalous propagation speed due to the spatial scalar field profile Papallo and Reall, Possible constraints from this effect should be investigated further. Gravitational wave astronomy has opened a new window to test gravity and dark energy.

Multi-messenger probes are specially promising for this task. This has provided an independent, standard siren measurement of the Hubble constant H 0. Moreover, GW already constrains large classes of DE models. In particular, the bound on the speed of GWs was significantly strong. Other multi-messenger tests of DE are possible, such as probing the GW luminosity distance or searching for additional polarizations.

These tests will become more relevant in the future when more events will be available. Still, there remain important challenges in this GW program to probe DE. From the observational side, it will be crucial to achieve a global synergy in the quest of multi-messenger GW astronomy. On the one hand, GW detectors will have to improve their sensitivity and enlarge the network to detect more events and localize them better.

On the other hand, observatories around the world should be available to follow-up triggers. Moreover, improved galaxy catalogs might be necessary to maximize the chances of localization. Lastly, cross-correlations between GWs and other cosmological probes might be an interesting endeavor. From the theoretical side, the main challenge will be to analyze the GW propagation over non-cosmological backgrounds, understanding the possible interplay of additional polarizations.

This will be relevant for instance for GWs traveling through a screened region. Furthermore, degeneracies between modified gravity predictions and astrophysical properties should be studied in more detail. For example, possible signatures of phenomenology beyond GR in neutron stars could possibly be the same as modifications of the equation of state.

Altogether, the future of multi-messenger GW astronomy appears promising. In the coming years standard sirens will be routinely detected and we will be able to apply the different GW tests of gravity to a much higher precision. The new techniques brought by GW astronomy will bring us closer to unveil the nature of dark energy. All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We are grateful to E. Bellini, J. Bernal, D. Bettoni, D. Blas, L. Heisenberg, G. Horndeski, J. Smirnov, G. Tasinato, and F. Vernizzi for comments on the manuscript. We would like to thank also the authors of Max et al. LIGO is funded by the U. National Science Foundation. These elements can be viewed as either breaking the fundamental assumptions or including additional fields. This approach can be generalized to gravity theories with additional vector and tensor fields as well Ezquiaga et al. For the case of the triad, the symmetry group is SO 3. Aasi, J.

First low frequency all-sky search for continuous gravitational wave signals.

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