**Gravitational Waves in Scalar-Vector-Tensor Gravity**

## Abstract

The general relativity (GR) and gravitational waves have been found for over 100 years, To explain different novel phenomena, modified gravity theories came into being. Now, in the great era of gravitational waves detection, shall we wonder: what's the main differences between gravitational waves in GR and in other theories of gravity? Can we test these models with gravitational waves data and judge who is the better? Firstly, this thesis reviews the basic theory of gravitational waves in GR, talks about the background and history of modified gravity and specifically scalar-vector-tensor gravity, introduces the detection methods and 3 runs of LIGO/Virgo's observation and the outcomes so far. Then, the radiation and waveforms of gravitational waves emitted in the inspiral phase of binary black hole coalescence with linearized gravity theory in MOdified Gravity (MOG) scalar-vector-tensor gravity has been derived. Besides, the thesis also compares the black hole masses inferred by MOG and GR before and after the merger phase. In the ringdown regime of binary black hole coalescence which is well described by the Quasi-Normal-Mode model in the black hole perturbation theory, this thesis fits the numerical relativity shear modes data with both MOG and GR with different numbers of overtones included. Then, the upper constraint \(\left(\alpha \leq 0.141_{-0.043}^{+0.042}\right)\) of parameter \(\alpha\) in MOG has been given by the AIC, ALCc and BIC estimators of the fitting. Finally, this thesis concludes: based on nowadays' gravitational waves observations, GR is better than MOG as a gravity theory.

##### Key Words: Gravitational Waves; Modified Gravity; Quasi-Normal-Mode

See also the full **Chinese version** in .pdf here.

And the thesis oral denfense presentation in .pdf here and in .pptx here.

Below are my notes about the theoretical correlation between gravitational waves and black hole shear modes.

## Shear modes

Define \(\mathcal S\) a smooth closed-2-dimensional spacelike surface. It has 2 future-directed null-normal denoted \(\ell^a\) and \(n^a\) , one is outgoing and another is ingoing ( \(\ell^a\) corresponds to the esterior and \(n^a\) the interior when project on \(\mathcal S\) ). Also, \(\ell \cdot n=-1\). We have a Riemannian metric restricted to the tangent space of \(\mathcal S\) : \[ \widetilde q_{ab}=g_{ab}+\ell_an_b+n_a\ell_b, \] Also, we can get \(\sigma_{a b}^{(\ell)}\) from: \[ \widetilde{q}_{a}^{c} \widetilde{q}_{b}^{d} \nabla_{c} \ell_{d}=\frac{1}{2} \Theta_{(\ell)} \widetilde{q}_{a b}+\sigma_{a b}^{(\ell)}+\omega_{a b}^{(\ell)}. \] where \(\Theta_{(\ell)}\) is the expansion of \(\ell^a\). In the cases where we discuss marginally outer trapped surface (MOTS), the requirement from its definition is \(\Theta_{(\ell)}\) (generally has no requirement on \(\Theta_{(n)}\)). This is because we are referring to the trapped surfaces bound on the null future, so the surface cannot expand in its future (hence the term trapped), results in the null-normal \(\ell^a\) has vanishing expansion: \[ \Theta_{(\ell)}:=q^{ab}\nabla_a\ell_b=0 \] here \(q^{ab}\) denotes the intrinsic metric on \(\mathcal S\).

Now, we can also introduce vectors tangent to \(\mathcal S\) denoted \(m^a\) which satisfies \(m\cdot \overline m=1\) ( \(\overline m\) is the complex conjugate of \(m\) ), that can be used to reduce the tracefree tensor \(\sigma_{ab}\) on \(\mathcal S\) to be a single complex field: \[ \sigma :=\sigma_{ab}m^am^b=m^am^b\nabla_a\ell_b. \]

## Gravitational waves

The ingoing and outgoing gravitational radiation are described by the perturbation of Weyl tensor (traceless part of Riemann tensor): \[ \Psi_{0}=C_{a b c d} l^{a} {m}^{b} l^{c} {m}^{d}, \quad\Psi_{4}=C_{a b c d} n^{a} \bar{m}^{b} n^{c} \bar{m}^{d} \] \(\Psi_4\) can be expanded in spin-weighted harmonics \(_{-2}Y_{\ell,m}\) of spin-weight -2.

The correlation between gravitational waves and shears near the event horizon can be express in 2 ways.

#### (i) News funtion

The \((\ell,m)\) component of the News function \(\mathcal{N}^{(\ell, m)}\) is defined as: \[ \mathcal{N}^{(\ell, m)}(u)=\int_{-\infty}^{u} \Psi_{4}^{(\ell, m)} d u \] The outgoing energy flux is related to the integral of \(|\mathcal N|^2\) over all angles, while \(|\sigma^2|\) appears in the energy flux falling into the black hole (though there are also other contributions).

#### (ii) Balance law

In general relativity, usually the notion of energy is tied to a choice of a vector field. Here we consider vector fields \(\xi^{a}=N \ell^{a}\) where the lapse \(N\) is tied to radial coordinates such that \(D_{a} r=N_{r} \hat{r}_{a}\). (Since \(\xi^{a}=N \ell^{a}\), as usual the term "lapse" refers to space-time evolution, not to "evolution" along \(\left.\hat{r}^{a} .\right)\) Thus each \(r\) determines a \(N_{r}\).

Note that the \(\mathcal S\) we have given above is actually leaf of a smooth, three-dimensional, spacelike submanifold \(H\) (dynamical horizon) foliation. By calculating the flux of energy associated with \(\xi_{(r)}^{a}\) for any radial coordinate \(r\) with an appropriate combination of Cauchy \(\left(q_{a b}, K_{a b}\right)\) scalar and vector constraints: \[ \begin{gathered} H_{S}:=\mathcal{R}+K^{2}-K^{a b} K_{a b}=16 \pi G T_{a b} \hat{\tau}^{a} \hat{\tau}^{b} \\ H_{V}^{a}:=D_{b}\left(K^{a b}-K q^{a b}\right)=8 \pi G T^{b c} \hat{\tau}_{c} q_{b}^{a} \end{gathered} \] we can obtain the following: \[ \mathcal{F}_{m}^{(r)}=\frac{1}{16 \pi G} \int_{\Delta H} N_{r}\left\{H_{S}+2 \hat{r}_{a} H_{V}^{a}\right\} d^{3} V \] Since \(H\) is foliated by 2 -spheres, we can perform a \(2+1\) split of the various quantities on \(H\). Using the GaussCodazzi relation we rewrite \(\mathcal{R}\) in terms of quantities on \(S\) : \[ \mathcal{R}=\tilde{\mathcal{R}}+\tilde{K}^{2}-\tilde{K}_{a b} \tilde{K}^{a b}+2 D_{a} \alpha^{a} \] where \(\alpha^{a}=\hat{r}^{b} D_{b} \hat{r}^{a}-\hat{r}^{a} D_{b} \hat{r}^{b}\). Next, the fact that the expansion \(\theta_{(\ell)}\) of \(\ell^{a}\) vanishes leads to the relation \[ K+\tilde{K}=K_{a b} \hat{r}^{a} \hat{r}^{b} \]

\[ \begin{aligned} \int_{\Delta H} N_{r} \widetilde{\mathcal{R}} d^{3} V=& 16 \pi G \int_{\Delta H} T_{a b} \hat{\tau}^{a} \xi_{(r)}^{b} d^{3} V \\ &+\int_{\Delta H} N_{r}\left\{|\sigma|^{2}+2|\zeta|^{2}\right\} d^{3} V, \end{aligned} \]

Let us now interpret the various terms appearing in this equation. The first integral on the right side of this equation is the flux \(\mathcal{F}_{m}^{(r)}\) of matter energy associated with the vector field \(\xi_{(r)}^{a}\). It is natural to interpret the second term as the flux \(\mathcal{F}_{g}^{(r)}\) of \(\xi_{(r)}^{a}\) energy in the gravitational radiation: \[ \mathcal{F}_{g}^{(r)}:=\frac{1}{16 \pi G} \int_{\Delta H} N_{r}\left\{|\sigma|^{2}+2|\zeta|^{2}\right\} d^{3} V \] This expression shares four desirable features with the Bondi-Sachs energy flux at null infinity. First, it does not refer to any coordinates or tetrads; it refers only to the given dynamical horizon \(H\) and the evolution vector field \(\xi_{(r)}^{a}\). Second, the energy flux is manifestly non-negative. Third, all fields used in it are local; we did not have to perform, e.g., a radial integration to define any of them. Finally, the expression vanishes in the spherically symmetric case: if the Cauchy data \(\left(q_{a b}, K_{a b}\right)\) and the foliation on \(H\) is spherically symmetric, \(\sigma_{a b}=0\) and \(\zeta^{a}=0\).