Gravity Through Gravitational-Wave Astronomy

PHYSICAL REVIEW D 98, 104055 (2018)

New observational constraints on fðTÞ gravity through gravitational-wave astronomy

† ‡ Rafael C. Nunes,1,* Supriya Pan,2, and Emmanuel N. Saridakis3,4, 1Divisão de Astrofísica, Instituto Nacional de Pesquisas Espaciais, Avenida dos Astronautas 1758, São Jos´e dos Campos, 12227-010, SP, Brazil 2Department of Mathematics, Presidency University, 86/1 College Street, Kolkata 700073, India 3Department of Physics, National Technical University of Athens, Zografou Campus GR 157 73, Athens, Greece 4CASPER, Physics Department, Baylor University, Waco, Texas 76798-7310, USA

(Received 10 October 2018; published 28 November 2018)

We investigate the new observational constraints on fðTÞ gravity that arise from the effects of primordial gravitational waves (GWs) on the cosmic microwave background (CMB) anisotropies and the BB spectrum. We first show that on the GWs propagation in fðTÞ gravity we obtain only an amplitude modification and not a phase one, comparing to the case of general relativity in the background of ΛCDM cosmology. Concerning primordial GWs we find that the more the model departs from general relativity the larger is the GW amplitude decay, and thus a possible future detection would bring the viable fðTÞ gravity models five orders of magnitude closer to ΛCDM cosmology comparing to standard cosmological constraints. Additionally, we use the CLASS code and both data from the Planck probe, as well as forecasts from the near-future CORE collaboration, and we show that possible nontrivial constraints on the tensor-to- scalar ratio would offer a clear signature of fðTÞ gravity. Finally, we discuss on the possibility to use the properties of the GWs that arise from neutron stars mergers in order to extract additional constrains on the theory.

DOI: 10.1103/PhysRevD.98.104055

I. INTRODUCTION evaluated on some pivot scale. The Planck collaboration within ΛCDM þ r has reported r<0.10 at 95% confidence The LIGO and Virgo collaborations reported the first level (CL) by combining temperature, low-polarization, and direct detection of gravitational waves (GWs) through the 0 002 −1 GW150914 event [1], that was produced from the merger lensing, at the pivot scale k ¼ . Mpc [9]. of a pair of black holes with masses around 36 and 29 solar Concerning the research on the fundamental gravita- masses and the subsequent ringdown of a single black hole, tional interaction, the GWs detection can be extremely while similar GWs event were later reported in [2–6]. helpful since the underlying gravitational theory deter- Recently, the detection of a binary neutron star merger with mines both the properties of the GWs themselves (speed, a GW (GW170817 event [7]) and an electromagnetic polarization modes etc) as well as the properties of the counterpart (GRB 170817A event [8]) opened the window background on which they propagate (the expanding of multimessenger GW astronomy. Certainly, the detection universe). Thus, one can use the GWs measured properties of GWs is a potential indication for a new era in modern and extract valuable information for the structure of the astrophysics and cosmology, offering a new spectrum of fundamental gravitational interaction. For instance, from possibilities to investigate nature at the fundamental level. the recent GW170817 and GRB 170817A events, which On the other hand, another important aspect in cosmology showed that GWs propagate practically with the light is to investigate the cosmic origin through a possible speed, one may impose strong constraints or exclude detection of primordial gravitational waves and their effect various modified gravity theories of the literature [10–13] on the cosmic microwave background (CMB) in the very (see [14] for a latest review). low frequency band, which is expected to bring new One class of theories of modified gravity that have astrophysical and cosmological information. The primordial recently attracted the interest of the research is the torsional GWs can be quantified through the tensor-to-scalar ratio r gravitational modification, such as the fðTÞ gravity. In this theory one describes the gravitational interaction through *[email protected] the torsion instead of the curvature tensor, and thus the † [email protected] Lagrangian is a function of the torsion scalar T (see [15] for ‡ [email protected] a review). Hence, fðTÞ gravity corresponds to a novel

2470-0010=2018=98(10)=104055(9) 104055-1 © 2018 American Physical Society NUNES, PAN, and SARIDAKIS PHYS. REV. D 98, 104055 (2018) R gravitational modification, with no known curvature equiv- and the corresponding action reads S 1 d4xeT, where ffiffiffiffiffiffi ¼ 16πG A p alence, and it proves to lead to interesting cosmology at e ¼ detðeμ Þ¼ −g and G is the Newton’s gravitational – both early and late times [16 48] and be in agreement with constant (we set the speed of light to c ¼ 1). – observations [49 54]. If we use TEGR as the starting point for torsional Some preliminary investigations concerning the GWs modified gravity, the simplest such modification is fðTÞ properties in fðTÞ gravity have been performed in [55,56], gravity, with action where it was shown that their speed is equal to the light Z speed and thus the constraints from GW170817 are trivially 1 S ¼ d4xefðTÞ: ð3Þ satisfied (see also [57,58] for the GWs properties in 16πG other modified teleparallel gravities). In this work we are interested in studying the new observational constraints Variation of (3) with respect to the tetrads leads to the field that can be imposed on fðTÞ gravity from the advancing equations multimessenger GW astronomy. Moreover, we examine the −1∂ ρ μν ρ μν∂ effect which primordial GWs have on the cosmic micro- e μðeeASρ ÞfT þ eASρ μðTÞfTT wave background (CMB) anisotropies and we examine the λ ρ νμ 1 ν ρ ν − f e T μλSρ e f T 4πGe Θρ ; 4 corresponding constraints. Finally, we discuss on the T A þ 4 A ð Þ¼ A ð Þ possibility that GWs from neutron star mergers could lead ∂ ∂ ∂2 ∂ 2 Θ ν to additional constraints on the theory. where fT ¼ f= T, fTT ¼ f= T , and ρ denotes the The manuscript is organized as follows: In Sec. II we energy-momentum tensor of the matter sector. In the above present the equations that determine the propagation of equation we have introduced for convenience the “super- ” μν ≡ 1 Kμν δμ αν − δν αμ GWs in fðTÞ gravity, and we quantify their modification potential Sρ 2 ð ρ þ ρT α ρT αÞ. comparing to the case of general relativity in a background Applying fðTÞ gravity in a cosmological framework, of ΛCDM cosmology. In Sec. III we study the observa- namely imposing the homogeneous and isotropic geometry A tional constrains that arise from primordial GWs, and in eμ ¼ diagð1;a;a;aÞ, which corresponds to the spatially Sec. IV we use the CLASS code and data from the Planck flat Friedmann-Robertson-Walker (FRW) metric probe and forecasts from the near future CORE collabo- 2 2 2 i j ration in order to quantify the effects of primordial GWs on ds ¼ dt − a ðtÞδijdx dx ; ð5Þ the CMB anisotropies and the BB spectrum. In Sec. V we discuss on the possibility to extract constraints from GWs with aðtÞ the scale factor, and inserting it into the general that arise from neutron star mergers. Finally, in Sec. VI we field equations (4), we extract the Friedmann equations present the conclusions. 8π 2 G ρ − f TfT H ¼ 3 m 6 þ 3 ð6Þ II. GRAVITATIONAL WAVES IN fðTÞ GRAVITY In this section we describe the equations of fðTÞ gravity, 4πGðρ þ p Þ H_ ¼ − m m : ð7Þ as well as the formulation of the tensor modes propagation 1 þ fT þ 2TfTT in this cosmological context. In the framework of torsional gravity ones uses the tetrad In the above equations H ≡ a=a_ is the Hubble function, μ fields eA, which form an orthonormal base in the tangent with dots denoting derivatives with respect to t, and ρm, pm space of the underlying manifold ðM;gμνÞ, where gμν ¼ are respectively the energy density and pressure for the A B ηABeμ eν is the metric tensor defined on the manifold M. matter perfect fluid. Throughout this work we use the Greek indices to denote the Let us now study the perturbations of fðTÞ around a coordinate space and the Latin indices for the tangent space. FRW cosmological background, focusing on the gravita- Furthermore, unlike general relativity which uses the tor- tional wave part. We follow [55] and we perturb the sionless Levi-Civita connection, here we use the curvature- tetrads as wλ λ A less Weitzenböck connection Γνμ ≡ e ∂μeν [59]. Thus, the A A A A eμ ðxÞ¼e¯μ ðxÞþχμ ðxÞ; ð8Þ gravitational field is described by the torsion tensor ¯A λ ≡ ρ ∂ A − ∂ A where eμ represents the part of the tetrad corresponding to T μν eAð μeν νeμ Þ: ð1Þ metric components, which satisfies the equation gμνðxÞ¼ η A B η ¯A ¯B χA The Lagrangian of the teleparallel equivalent of general ABeμ eν ¼ ABeμ eν (the part μ that represents the degrees relativity (TEGR) is the torsion scalar T, constructed as [59] of freedom released from the local Lorentz transformation is not going to play any role in the analysis of this work). A 1 ρμν 1 ρμν ρ νμ The part e¯μ around a spatially flat FRW geometry writes as T ≡ T Tρμν þ T Tνμρ − Tρμ T ν; ð2Þ 4 2 [19,55,60]

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0 0 i 00 0 2 e¯μ ¼ δμð1 þ ψÞþaδμðG þ ∂ FÞ; h þ 2Hð1 − β Þh þ k h ¼ 0; ð14Þ i i ij T ij ij a a e¯μ ¼ a δμð1 − ϕÞ with primes denoting derivative with respect to the con- formal time, and where we have introduced the dimension- 1 δi δaj ∂ ∂ ∂ ∂ less parameter [55] þ μ 2 hij þ i jB þ jCi þ iCj ; 1 f_ ¯μ δμ 1 − ψ − δμi ∂ β − T e0 ¼ 0ð Þ ðGi þ iFÞ; T ¼ 3 : ð15Þ a HfT μ 1 μ e¯ ¼ δ ð1 þ ϕÞ a a a Therefore, we straightforwardly deduce that the speed of GWs is equal to one, i.e., equal to the speed of light, and 1 μi j thus the experimental constraint of GW170817 is trivially − δ δa hij þ ∂i∂jB þ ∂iCj þ ∂jCi ; ð9Þ 2 satisfied in fðTÞ gravity. However, as it was mentioned in [55], the correction term β reflects the effect on the ϕ ψ T where , , B and F are the scalar modes, Ci and Gi the gravitational waves due to the change that fðTÞ gravity transverse vector modes, and hij the transverse traceless brings on the background they propagate on. tensor mode, which lead to the standard perturbed metric In order to quantify the above effect we follow the analysis of [13,61]. In most of gravitational modifications −1 − 2ψ g00 ¼ ; one can bring the GW propagation equation in the form gi0 ¼ −a½∂iF þ Gi; 00 2 ν H 0 2 2 2μ2 Π 2 hij þð þ Þ hij þðcTk þ a Þhij ¼ ij; ð16Þ gij ¼ a ½ð1 − 2ϕÞδij þ hij þ ∂i∂jB þ ∂jCi þ ∂iCj: ð10Þ μ In the rest of the manuscript we set the scalar and vector where cT is the GW propagation speed, is the effective perturbations to zero, since we are interested in studying the graviton mass, ν is related to the effectively running Planck Π gravitational-wave sector. mass, and ij is a source term arising from possible addi- Inserting (9) into (1) we acquire tional fields. Hence, one can describe the deviation of GW propagation at cosmological scales comparing to general 1 i δ _ relativity through T 0j ¼ H ij þ 2 hij 1 −D −ikΔT i ∂ − ∂ h ¼ e e hGR; ð17Þ T jk ¼ 2 ð jhik khijÞ; ð11Þ and consequently the torsion scalar (2) is perturbed as where Z 0 2 2 2 1 T ¼ Tð Þ þ Oðh Þ¼6H þ Oðh Þ; ð12Þ D νH τ0 ¼ 2 d ð18Þ with Tð0Þ the zeroth-order quantity. Thus, we deduce that at linear order the torsion scalar remains unaffected, which quantifies the amplitude modification, i.e., the damping lies behind the fact that in fðTÞ gravity the gravitational factor, while waves do not have extra polarization modes [57]. Z Moreover, the perturbed super-potential can be written as a2μ2 ΔT ¼ 1 − c − dτ0 ð19Þ 0j δ − 1 _ jk 1 ∂ − ∂ T 2 2 Si ¼ H ij 4 hij and Si ¼ 4a2 ð jhik khijÞ. Inserting k the perturbed quantities into the field equations (4), and neglecting the matter sector, we obtain [55] quantifies the phase modification, i.e., the time delay. 1 In the case of fðTÞ gravity Eq. (14) implies that cT ¼ , 1 ∇2 μ ¼ 0, Π ¼ 0 and ν ¼ −2β . Therefore, we deduce that in _ 2 ̈ _ ij T 4fT ðH þ 3H Þδij þ −hij þ hij − 3Hhij Δ 0 4 a2 fðTÞ gravity T ¼ , and thus we do not obtain any phase _ modification in comparison to general relativity. However, h ν þ 4f_ Hδ − ij − fδ ¼ 0; ð13Þ the quantity , which carries the information of the fðTÞ T ij 4 ij modification, will lead to a nonzero D, and thus to an amplitude modification of the gravitational waves compar- ð0Þ where the derivative fT is calculated at T ¼ T . Hence, ing to general relativity. Hence, this modification can in the perturbation part of the above equation leads to the principle be measured in GW observations, offering equation of motion for the gravitational waves in fðTÞ an observational signature of this class of modified gravity. cosmology, namely In the following sections we quantify this behavior.

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III. PRIMORDIAL GRAVITATIONAL WAVES predictions for fðTÞ gravity and ΛCDM are practically identical. Thus, for small angular scales no deviations are In this section we investigate the deviation of GW expected compared to standard ΛCDM cosmology. On the propagation in fðTÞ gravity, comparing to general rela- other hand, we can see significant deviations at large tivity, focusing on the primordial gravitational waves. As a angular scales. specific model, and without loss of generality, we will Additionally, the theoretical CMB BB spectrum should consider the power-law one, which is the most viable one, also present a peak at l ≃ 5, still to be detected by future nevertheless our analysis can be performed for every fðTÞ experiments, due to the effects of tensor modes on the form. The power-law scenario corresponds to [17]: scattering during the reionization epoch. At low-l, the large fðTÞ¼T þ αð−TÞb: ð20Þ angular is dominated by modes that have not entered the horizon at recombination and therefore are approximately with α and b the two parameters. Inserting this into (6) at constant. Thus, the effects due to fðTÞ modification can be present time we obtain quantified on the reionization peak (which should be located at l ≃ 5), where we can note different predictions 1 − Ω 0 − Ω 0 for a range of values of model parameter b compared to the α ¼ð6H2Þ1−b m r ; ð21Þ 0 2b − 1 reference ΛCDM scenario. Interestingly, future measurements of the reionization peak by CMB spectrum could be with Ωm0, Ωr0 respectively the current values of the matter used to discriminate between fðTÞ gravity and standard and radiation density parameters, and H0 the Hubble cosmology scenario. parameter at present, and thus the only free model param- In Fig. 2, following the approach of [61], we depict the eter is b. The value b ¼ 0 corresponds to recovery of propagation of GWs as the function of the scale factor, for general relativity and of ΛCDM cosmology (in this case the α parameter is related to the cosmological constant). ΛCDM 0.015 We insert the above power-law fðTÞ form into the b = 0.01 -3 modified propagation equations of the previous section, b = 10 -4 and we use the publicly available CLASS (Cosmic Linear 0.01 b = 10 -5 Anisotropy Solving System) code [62] in order to inves- b = 10 -6 tigate the effects of the gravitational waves on the CMB BB 0.005 b = 10 h -7 anisotropy and also to calculate the propagation of the b = 10 gravitational waves through cosmic time. 0 In Fig. 1 we show the CMB BB spectrum for the fðTÞ gravity and the ΛCDM cosmological model as the reference model, for tensor modes only. In drawing the graphs -0.005 we have considered various values of the free model 0.015625 0.03125 0.0625 0.125 0.25 0.5 1 parameter b. All other parameters are fixed in the same a way for both models, based on Planck 2015 results [63]. As we observe, for the angular scale l>20, the theoretical 0.0005 ΛCDM -6 b = 10 Λ -7 CDM b = 10 b = 0.01 0.00025 -3 b = 10 -4 b = 10 0.1 h 0 -5 b = 10 BB

l -0.00025 D 0.01 -0.0005

0.4 0.5 0.6 0.7 0.8 0.9 1 a

0.001 FIG. 2. Propagation of the gravitational waves at k ¼ 10 100 −1 l 0.01 Mpc ,inΛCDM scenario and in the fðTÞ gravity power-law model (20) for various values of the free model BB 1 BB 1 −1 FIG. 1. The CMB BB power spectrum, Dl ¼ lðl þ ÞCl = parameter b, as a function of the scale factor a ¼ð þ zÞ . 2πμK, for the ΛCDM cosmology and the fðTÞ gravity power-law Upper graph: full cosmological history. Lower graph: late-time model (20), for various values of the free model parameter b. cosmological history.

104055-4 NEW OBSERVATIONAL CONSTRAINTS ON fðTÞ … PHYS. REV. D 98, 104055 (2018) various values of the model parameter b of the power-law Concerning the Planck data we use the likelihood fake form (20), in order to investigate how b affects the planck realistic included in MONTEPYTHON code [67], amplitude damping. It is already known that the amplitude taking into account the temperature, polarization and CMB of GWs decays rapidly immediately after the tensor-modes lensing extraction, and we adopt noise spectra roughly entry into the horizon, while before the entrance to the matching those expected from the full Planck results. The horizon the amplitude is practically constant. The expan- baseline parameter space is given by sion of the Universe leads to damping of the tensor modes 0 P ≡ 100ω ω 1010 τ with the term proportional to h . We have noted that for f b; cdm; ln As;ns; reio;H0;r;bg; a<0.01, namely at early times, taking different ranges in orders of magnitude of b, only a tiny deviation from where the parameters in P from left to right are respectively ΛCDM paradigm appears. On the other hand, for a> the baryon density, the cold dark matter density, the 0.01 the damping factor becomes significant, and we can amplitude and slope of the primordial spectrum of metric see that already at the matter era the amplitude of GWs fluctuations, the scalar spectral index, the optical depth to becomes practically null (or extremely small) for b>10−5. reionization, the Hubble constant, the tensor-to-scalar ratio, For instance, for b ¼ 10−2 we have h ≃ 0 at a ≃ 0.04, while and the free parameter (i.e., b) of the power-law fðTÞ for b ¼ 10−5 we get that h ≃ 0 at a ≃ 0.4. Moreover, when model (20). The priors used for the model parameters are ω ∈ 0 005 0 1 ω ∈ 0 01 0 99 the Universe enters into the accelerated phase only for the the following: b ½ . ; . , cdm ½ . ; . , −6 −7 1010 ∈ 2 4 4 0 ∈ 0 5 1 5 τ ∈ 0 01 0 8 cases b ¼ 10 ; 10 the tensor-modes oscillations are ln As ½ . ; . , ns ½ . ; . , reio ½ . ; . , ∈ 50 90 ∈ 0 1 0 ∈ 0 0 1 non-null, however in the case b ¼ 10−6 the modes quickly H0 ½ ; , r ½ ; . , and b ½ ; . . −7 Concerning the CORE analysis we use the likelihood decay and only in the case b ¼ 10 they survive up to present time. Concerning the phase, we do not observe any CORE m5 also included in MONTEPYTHON. The exper- difference, since as we discussed there is no phase imental specifications for CORE data in our analysis are Δ 0 summarized in Table I. modification in fðTÞ gravity [ T ¼ in (17)]. 2 3000 In summary, the more b departs from its general relativity In our forecasts, we assume lmin ¼ , lmax ¼ , and f 0.70. We forecast the f T gravity with the set of value b ¼ 0 the larger is the GWamplitude decay comparing sky ¼ ð Þ P to ΛCDM scenario. This is because for larger b values the cosmological parameters shown in . tensorial modes enter in the cosmological horizon earlier, In the forecasting analysis we assume the fiducial values 2 22 0 119 3 07 0 962 0 05 compared to the standard prediction within ΛCDM cosmol- of the above parameters as: f . ; . ; . ; . ; . ; 68 0 0 1 0 005 ogy, and thus the GWamplitude goes rapidly to zero already . ; . ; . g. The details of the methodology used in in radiation and matter eras. Hence, we deduce that a the CORE likelihood can be seen in [65,66]. possible future detection of primordial gravitational waves In the upper graph of Fig. 3 we present the parametric − ≲ 10−7 space in the ns r plane, where for the analysis we have would imply that b (taking the specific scale −1 −1 assumed the pivot scale at k 0.05 Mpc , which is the k ¼ 0.01 Mpc ), bringing the viable fðTÞ gravity models ¼ Λ standard in the literature. As we observe, r<0.23 at five orders of magnitude closer to CDM cosmology 0 96 0 005 comparing to standard cosmological constraints based on 95% confidence level (CL) and ns ¼ . . at – 68% CL from the Planck data. These results are also in SN Ia, BAO, CMB, HðzÞ data [49 54]. This fact reveals the Λ capabilities of gravitational-wave astronomy, since such accordance with the Planck team assuming CDM [68], strong constraints were possible to be obtained only through where no evidence for a non-null r is also reported and n < 1 5σ ’ Solar-System data [64]. s up to CL. According to the authors knowledge, this is the first observational constraints on inflation parameters (plane ns − r) obtained directly in the context IV. PLANCK AND CORE CONSTRAINTS of fðTÞ gravity. In this section we present new observational constraints on fðTÞ gravity, arising from the CMB data from Planck TABLE I. Experimental specifications for CORE, with fre- collaboration [63], as well as from forecasts that are quency channels dedicated to cosmology, and beam width, expected from near-future probes such as CORE [65,66]. temperature sensitivity, and polarization sensitivity for each Similarly to the previous section we will use the CLASS channel, in units of μK arcmin. code [62]. We mention that in the present analysis we only Δ Δ consider the CMB data, neglecting the influences of the Channel [GHz] FWMH [arcmin] T P external data, since we are particularly interested in 130 8.51 3.9 5.5 quantifying and analyzing the effects of Eqs. (14)–(15), 145 7.68 3.6 5.1 i.e., the modified GWs propagation in fðTÞ gravity, as well 160 7.01 3.7 5.2 as its effects on CMB. The fðTÞ gravity has been recently 175 6.45 3.6 5.1 195 5.84 3.5 4.9 well constrained from the geometrical data [51,54], and 220 5.23 3.8 5.4 using the CMB data for the first time in [53].

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CMB, we can again see that H0 is fully compatible with local measurements, namely H0 ¼ 73.2 1.74 [69], even at 1σ CL. We proceed motivated by the future CMB experiments, and we apply the inherent modifications due to fðTÞ gravity aiming to impose bounds on the baseline parameters of the model. In the upper panel of Fig. 3 we additionally present the parametric space in the plane ns − r via CMB CORE forecasts. In this way we can see how much can CORE improve the constraints compared to the current Planck ones. We find r ¼ 0.1 0.020 and ns ¼ 0.96 0.0017 at 68% CL (as we mentioned above, we assume r ¼ 0.1 and ns ¼ 0.96 in the forecast data simulation). Let us define the improvement with respect to the constraints arising from the σ σ 1 Planck data as Ii ¼ Planck= CORE. Thus, for the parameters 11 5 2 94 of our interest, we find that Ir ¼ . and Ins ¼ . . Within the ΛCDM paradigm, the CORE collaboration [66] finds for instance r ¼ 0.0042 0.00028, assuming r ¼ 0.0042 in the forecast analysis. For other forecastings using CORE estimation see [65,66]. In the lower graph of Fig. 3 we depict the bounds that CORE results can impose directly on the free model parameter of the power-law fðTÞ model, and in particular we find that b ¼ 0.0050 0.00049. Here, we note an improvement of Ib ¼ 12.2. As we can see, future constraints could improve current constraints with respect to the Planck data, on the fðTÞ gravity free parameter, by a factor of 12. A precise measurement of this parameter, with that magnitude of improvement, can offer an opportunity to check deviations from the general theory of relativity. The value b ¼ 0.0050 is the best-fit value that arises from the Planck data too, thus it is reasonable to consider it for performing our forecast simulation. Hence, we may con- clude that the forecast errors on b may limit a non-null value on this parameter by taking future CMB missions. This will be a clear signature that a deviation from ΛCDM cosmology is favored.

FIG. 3. Upper graph: One-dimensional marginalized posterior V. GRAVITATIONAL WAVES FROM MERGERS distributions and 68%, 95% CL regions in the ns − r plane at k ¼ 0.05 Mpc−1, from the Planck data and CORE forecasts. In the previous section we investigated the effects of Lower graph: One-dimensional marginalized posterior distribu- fðTÞ gravity on the primordial GWs. In this section we tions and 68%, 95% CL regions in the H0 − n plane [where b is present a preliminary discussion on the possibility to use the free parameter of the fðTÞ gravity model (20)] from the the properties of the detected GWs that arise from mergers, Planck data and CORE forecasts. in order to impose constraints on fðTÞ gravity and on modified gravity in general. In this direction, the GWs that In the lower graph of Fig. 3 we present the parametric are accompanied by electromagnetic counterparts are proved to be extremely efficient. space in the H0 − b plane. As we can see, H0 ¼ 69 5.42 km=s=Mpc and b ¼ 0.005 0.006, both at 68% CL. In the case of detection of GWs that arise from black As already noted in [53] (within a different theoretical and holes mergers ones does not have any other information observational perspective), fðTÞ gravity can provide high apart from the properties of the GWs at the moment they reach earth, and the direction in space they came from (in H0 values, and therefore a possible solution to the H0 tension might be available in this context [53]. the case where three different detectors are used). Hence, In this work, incorporating the propagation modification of the tensor modes due to fðTÞ gravity and its effects on 1The label i runs over the parameters given in P.

104055-6 NEW OBSERVATIONAL CONSTRAINTS ON fðTÞ … PHYS. REV. D 98, 104055 (2018) assuming that general relativity is the underlying gravita- universe). Hence, one can use the GW features in order to tional theory and that ΛCDM cosmology is the background impose constraints on the various gravitational theories and cosmological model one can calculate the distance of the offer a way to distinguish them. In the particular case of merger, the involved black hole masses, and the properties fðTÞ gravity (where it was recently shown that GWs of the GWs at the moment of emission [1–6]. Definitely, propagate with the light speed [55,56], without extra assuming a modified gravity and its implied cosmology as polarization modes [33]), we showed that one obtains only the underlying theory will lead to different calculations, and an amplitude modification and not a phase one on the GWs thus one faces degeneracies that do not allow for distin- propagation, comparing to the case of general relativity in guishing different theories. the background of ΛCDM cosmology. However, the situation changes radically in the case of Concerning primordial GWs, and focusing without loss detection of GWs that arise from neutron stars mergers, of generality on the power-law fðTÞ model, we showed that which are accompanied by an electromagnetic counterpart. the more the model departs from general relativity, i.e., the In this case apart from the properties of the GWs at the more the exponent b departs from its general relativity moment they reach earth and their origin direction, one can value b ¼ 0, the larger is the GW amplitude decay additionally calculate their propagation speed, the distance comparing to ΛCDM scenario. Hence, we deduced that of the merger, and the involved neutron stars masses, a possible future detection of primordial gravitational independently of the underlying gravitational theory and waves would imply that b ≲ 10−7 (analyzing on the scale cosmological scenario, just using the implied physics from k ¼ 0.01 Mpc−1), bringing the viable fðTÞ gravity models the electromagnetic information [7,8]. Therefore, without five orders of magnitude closer to ΛCDM cosmology assuming an underlying cosmology one can find the comparing to standard cosmological constraints based on properties of the GWs at the time of their emission too. SN Ia, BAO, CMB, HðzÞ data [49–54]. This fact reveals the Hence, knowing the GWs properties at both the emission capabilities of gravitational-wave astronomy, since such and detection time one can extract information for the strong constraints were possible to be obtained only background on which they propagated and thus impose through Solar-System data [64]. constraints on the gravitational theory that determines it. Additionally, we used the CLASS code in order to In the case of fðTÞ gravity, as we described in Sec. II, the quantify the primordial GWs effect on the CMB anisotro- effect of the fðTÞ modification on the underlying cosmol- pies and the BB spectrum. We used both the data from the ogy leads to an amplitude modification comparing to the Planck probe, as well as forecasts from the near-future Λ GWs propagating in CDM scenario under general rela- CORE collaboration. As we showed, possible nontrivial tivity, without a phase change. constraints on the tensor-to-scalar ratio would favor a In principle one can follow the above roadmap and model parameter b different from its general relativity impose new constraints on fðTÞ gravity. In practice value, offering a clear signature of fðTÞ gravity comparing however, the involved procedure is very complicated and to ΛCDM cosmology. one needs to perform a thorough investigation on the GWs Finally, we examined the possibility of constraining fðTÞ generation from neutron star mergers in the framework of gravity through the detection of GWs that arise from neutron fðTÞ gravity itself. In these lines the analysis can be based stars mergers, which are accompanied by an electromagnetic on the recent work [14], focusing on the strain amplitude of counterpart. In this case, apart from observing the properties the generated GWs. Since the necessary calculations are of the GWs at the time of detection one can use the features lengthy and complicated and lie beyond the scope of the of the electromagnetic observations and the deduced physics present work, which is mainly the investigation of cosmo- of the neutron stars to find the properties of the GWs at the logical GWs, it is left for a separate project. time of their emission, without any assumption on the underlying theory of gravity and the background cosmo- VI. CONCLUSIONS logical evolution. Hence, knowing the GWs properties at The advancing GWs multimessenger astronomy offers a both the emission and detection time it is possible to extract new window to observe nature, and amongst others to information for the background on which they propagated extract information about cosmology and gravity. In this and thus impose constraints on the gravitational theory that work we focused on the case of fðTÞ gravity, and we determines it. Since the basic scope of the present work is studied the new observational constraints that arise from the mainly the investigation of cosmological GWs, the above effect of primordial GWs on the cosmic microwave back- detailed investigation is left for a separate project. ground (CMB) anisotropies and the BB spectrum. In summary, we showed how one can use information The main feature of the investigation that lies at the basis from the advancing multi-messenger GW astronomy in of the analysis, is that the underlying gravitational theory order to extract new observational constraints on fðTÞ determines both the properties of the GWs themselves gravity. As we saw, fðTÞ gravity remains in agreement with (speed, polarization modes, etc) as well as the properties of observations and thus a good candidate for the description the background on which they propagate (the expanding of nature.

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ACKNOWLEDGMENTS This article is based upon work from CANTATA COST (European Cooperation in Science and Technology) The authors thank the referee for some specific clarifying action CA15117, EU Framework Programme Horizon points. The authors would also like to thank J. C. N. 2020. de Araujo and M. E. S. Alves for useful discussions.

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