Foundations and special cases for this work here: Schwarzschild Black Hole in GR.
ORIGINAL ARTICLE: Sergi Sirera Lahoz, Testing the Speed of Gravity with Black Hole Ringdown, arXiv: 2301.10272.
DOI: https://doi.org/10.48550/arXiv.2301.10272
GITHUB
ORIGINAL ARTICLE: Sergi Sirera Lahoz, Testing the Speed of Gravity with Black Hole Ringdown, arXiv: 2301.10272.
DOI: https://doi.org/10.48550/arXiv.2301.10272
GITHUB
Article Abstract: We investigate how the speed of gravitational waves, , can be tested by upcoming black hole ringdown observations. We do so in the context of hairy black hole solutions, where the hair is associated with a new scalar degree of freedom, forecasting that LISA and TianQin will be able to constrain deviations of from the speed of light at the level from a single supermassive black hole merger. We discuss how these constraints depend on the nature of the scalar hair, what different aspects of the underlying physics they are sensitive to in comparison with constraints derived from gravitational wave propagation effects, which observable systems will place the most stringent bounds, and that constraints are expected to improve by up to two orders of magnitude with multiple observations. This is especially interesting for dark energy-related theories, where existing bounds from GW170817 need not apply at lower frequencies and where upcoming bounds from lower-frequency missions will therefore be especially powerful. As such, we also forecast analogous bounds for the intermediate-frequency AEDGE and DECIGO missions. Finally, we discuss and forecast analogous black hole ringdown constraints at higher frequencies (so from LVK, the Einstein Telescope and Cosmic Explorer) and in what circumstances they can yield new information on top of existing constraints on . All calculations performed in this paper are reproducible via a companion Mathematica notebook.
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This is the companion notebook to the paper https://arxiv.org/abs/2301.10272, where we perform all the calculations appearing there. Starting with the action for scalar-tensor theories, we find the modified Regge-Wheeler equation governing the evolution of odd perturbations in a Schwarzschild black hole with scalar hair. We find the corresponding quasinormal modes and apply a Fisher forecast to constrain the speed of gravitational waves. For details on the motivation, results and references we refer to the paper.
This notebook has hugely benefitted from resources provided by: - xAct ( http://www.xact.es/ ) - Johannes Noller - Reggie Bernardo’s public github repository ( https://github.com/reggiebernardo/notebooks/tree/main/thesis_notebooks ). - Public notebooks by Vitor Cardoso et al. ( https://centra.tecnico.ulisboa.pt/network/grit/files/ringdown/ ). - Discussions with Emanuele Berti on ways of calculating the Fisher matrix.If you use any of the results or techniques in this notebook please cite the repository and [2301.10272] ( https://arxiv.org/abs/2301.10272 ).If you have any comments or feedback please let me know. Thanks!Sergi Sirera LahozInstitute of Cosmology and Gravitation - University of Portsmouth
0. Setup and definitions
0. Setup and definitions
1. Quadratic action
1. Quadratic action
We start with the Horndeski action. In this (more pedagogical) notebook, for the sake of computation time, we perform the calculations for the subset of Horndeski defined below, which nonetheless does not affect the final results. It should also be noted that the contributions excluded here have been checked in separate notebooks with the same format (available upon request).
We use geometric units:
We use geometric units:
G=c=1
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L2=G2[phi[],X[]];L3=-G3[phi[],X[]]CD[b]@CD[-b]@phi[];L4=G4[phi[],X[]]RicciScalarCD[]+G4X[phi[],X[]](-CD[c]@CD[d]@phi[]CD[-c]@CD[-d]@phi[]);L5=G5[phi[],X[]]EinsteinCD[-b,-c]CD[b]@CD[c]@phi[]-G5X[phi[],X[]](-3CD[e]@CD[c]@phi[]CD[-e]@CD[-c]@phi[]CD[d]@CD[-d]@phi[]+2CD[-f]@CD[-k]@phi[]CD[f]@CD[j]@phi[]CD[k]@CD[-j]@phi[]);LT=Sqrt[-Detg[]](L2+L3+L4+L5)LTreduced=LT/.{G2[phi[],X[]]->0,G3[phi[],X[]]->0,G4[phi[],X[]]->G4[X[]],G4X[phi[],X[]]->G4X[X[]],G5[phi[],X[]]->0,G5X[phi[],X[]]->0}//Simplify
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We also define here the metric perturbations. We do not include scalar perturbations because we restrict to the odd sector.
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DefMetricPerturbation[g,h,ϵ];Unprotect[IndexForm];IndexForm[LI[x_]]:=ColorString[ToString[x],RGBColor["#00a898"]];Protect[IndexForm];IndexForm[LI[n]];
The following rules will take care of selecting contributions only from first order in the perturbations, applying differentiation by parts, specifying Ricci flat spacetimes , parametrizing the scalar hair and rewriting Horndeski G-functions.
In[]:=
(*keeponlyfirstorder*)onlyfirst={h[LI[2],x_,y_]:>0(*,ψ[LI[2]]->0*)}(*integrationbypartsrules*)IBPrule=XX_*h[LI[1],aa_,bb_]CD[cc_]@CD[dd_]@h[LI[1],ee_,ff_]->-CD[cc]@(XX*h[LI[1],aa,bb])*CD[dd]@h[LI[1],ee,ff](*Ricciflatspacetimes*)Ricciflat={RicciCD[-a_,-b_]0,RicciScalarCD[]0,EinsteinCD[-a_,-b_]0}(*scalarhair*)DefConstantSymbolhatphi,PrintAs->"";DefConstantSymbol[phic,PrintAs->""];DefConstantSymbol[eps,PrintAs->"ϵ"];DefConstantSymbol[mass,PrintAs->"M"];DefScalarFunction[δϕ];GoHair={phi[]hatphi+epsδϕ[r[]]}GoHairn=δϕ->phic(*scalarfieldkineticterm*)Xtoϕ=X[]-CD[-b][phi[]]CD[b][phi[]](*rewritingG4derivatives*)G4dervs={Derivative[1][G4][-1/2*Scalar[CD[-b][phi[]]*CD[b][phi[]]]]->G4X[-1/2*Scalar[CD[-b][phi[]]*CD[b][phi[]]]],Derivative[1][G4X][-1/2*Scalar[CD[-b][phi[]]*CD[b][phi[]]]]->G4XX[-1/2*Scalar[CD[-b][phi[]]*CD[b][phi[]]]],Derivative[2][G4][-1/2*Scalar[CD[-b][phi[]]*CD[b][phi[]]]]->G4XX[-1/2*Scalar[CD[-b][phi[]]*CD[b][phi[]]]]}
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Here we perturb the Lagrangian to second order.
We use rules defined above to keep only first order metric perturbations, integrate by parts and implement Ricci-flatness.
We use rules defined above to keep only first order metric perturbations, integrate by parts and implement Ricci-flatness.
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LTreduced/.XtoϕPerturbation[%,2]//ExpandPerturbation//NoScalar//Expand;%/.{Perturbation[phi[]]->0,Perturbation[phi[]]^2->0,Perturbation[phi[],2]->0};%/.onlyfirst/.Ricciflat//ContractMetric;%/.IBPrule//NoScalar//ToCanonical;δL=%/.G4dervs//ContractMetric//Simplification
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2. Quadratic action in components
2. Quadratic action in components
Components set up
Components set up
We define the chart (coordinate system) and metric components. We use MetricInBasis[...] to tell xCoba that these are the components of the abstract metric g. We also define the spherical harmonic functions and the odd perturbation metric components.
We set m=0, in our case is fully general. This entails that derivatives with respect to ϕ vanish, which simplifies the odd metric components (this is the coordinate ϕ, not the scalar).
We set m=0, in our case is fully general. This entails that derivatives with respect to ϕ vanish, which simplifies the odd metric components (this is the coordinate ϕ, not the scalar).
In[]:=
DefChart[BH,Md,{0,1,2,3},{t[],r[],θ[],ϕ[]}];DefScalarFunction/@{A,B};DefScalarFunction[CC,PrintAs->"C"];gB=DiagonalMatrix-A[r[]],,CC[r[]],CC[r[]];MetricInBasis[g,-BH,gB];MetricCompute[g,BH,All,CVSimplifyToCanonical];DefConstantSymbol/@{l,m,ω};DefScalarFunction[PL,PrintAs""];(*becauseofourbackground,wecansetm=0withoutlossofgenerality*)m0={m->0,D[XX_,ϕ[]]->0};(*associatedLegendrepolynomials*)(*weexploitthefactthatwecansetm=0withoutlosinggenerality*)PL''[θ_]:=-(l(l+1)-)PL[θ]-Cot[θ]PL'[θ]/.m0;PLrules=PL[θ[]]^2*Sin[θ[]]->,Sin[θ[]]*Derivative[1][PL][θ[]]^2->,Cos[θ][θ][θ]->-2,Cos[θ]Cot[θ][θ]->;(*sphericalharmonics*)YLM:=PL[θ[]]/.m0;(*oddparityh*)(*again,weusem=0*)DefScalarFunction[h0,PrintAs""];DefScalarFunction[h1,PrintAs""];hO=0,0,-h0[t[],r[]],Sin[θ[]]D[YLM,θ[]]h0[t[],r[]],0,0,-h1[t[],r[]],Sin[θ[]]D[YLM,θ[]]h1[t[],r[]],-h0[t[],r[]],-h1[t[],r[]],0,0,{Sin[θ[]]D[YLM,θ[]]h0[t[],r[]],Sin[θ[]]D[YLM,θ[]]h1[t[],r[]],0,0}/.m0;(*wedisplayhowthemetricperturbationslooklike*)hO//Flatten;Apply[PolynomialGCD,%];%*hO//MatrixForm
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In order to perform a variation with respect to the perturbation functions, we first need to convert them to ‘tensors’ in the xAct language. This is so that the variation can also take into account derivatives of the function, which would be otherwise excluded. This step has no deep significance but it is necessary so that xAct can treat perturbation functions as it should.
To implement this we use a simple method of substituting an r-derivative with a partial derivative contracted with a normalized vector in the r-direction. We do the same for t-derivatives.
Here we are defining these normalized vectors and the tensors for each of the functions.
To implement this we use a simple method of substituting an r-derivative with a partial derivative contracted with a normalized vector in the r-direction. We do the same for t-derivatives.
Here we are defining these normalized vectors and the tensors for each of the functions.
These rules allow us to convert the odd functions to ‘tensors’ (denoted with a tilde).
These rules allow us to express B[r] and its derivatives in terms of M and vice versa.
We also make use of the following two rules to make the conversion from abstract covariant tensors into their components faster.
The first one allows us to contract the metric ‘by hand’ when we have a combination of abstract indices in basis and a specific component index. ContractBasis is not able to do that but it reduces the amount of terms produced by TraceBasisDummy.
The second one sends to zero contributions which we have checked are zero when going to Basis (we show this below). This also reduces considerably the amount of terms that need to be computed.
The first one allows us to contract the metric ‘by hand’ when we have a combination of abstract indices in basis and a specific component index. ContractBasis is not able to do that but it reduces the amount of terms produced by TraceBasisDummy.
The second one sends to zero contributions which we have checked are zero when going to Basis (we show this below). This also reduces considerably the amount of terms that need to be computed.
3. One-variable action
3. One-variable action
This rule (of the same form as we used before) implements integration by parts on q.
The following rule helps to evaluate the derivatives of the a-coefficients.
We now use what we found above to write the quadratic Lagrangian in terms of the single degree of freedom q.
The following rule sets the specific expressions for α and β.
The expression for γ is not given in https://arxiv.org/abs/1710.07669. We have confirmed that α and β in that paper are the same as we obtain here, but we have nothing to compare our γ to. Hence, we provide the expression for γ here and carry on with the calculation. The fact that we later find the same potential as they do confirms that our γ’s match.
We proceed, as the paper ( https://arxiv.org/abs/1710.07669 ), by defining a new Q variable that will simplify our action
We will need another integration by parts rule for Q
We also introduce Tortoise coordinates
Comparing it to our result we confirm that they are the same. Note that we have to introduce a minus sign to match the signs. This is because V appears with a minus sign in the action (44) of https://arxiv.org/abs/1710.07669.
The rule below is just to write the potential as V in the Lagrangian in the next line (just aesthetics).
4. Modified Regge-Wheeler equation
4. Modified Regge-Wheeler equation
Having recovered the action above, we only need to vary it with respect to the only variable Q to obtain the modified Regge-Wheeler equation.
We define ω, which will be our quasinormal frequencies, and specify that G-functions should be evaluated on the background
And now we write the full modified RW equation
And we compare the two
This confirms the equation has the correct form.
5. Quasi-normal modes
5. Quasi-normal modes
We proceed to use the parametrized ringdown formalism ( https://arxiv.org/abs/1901.01265 ) to obtain the quasinormal modes. We first check that we can write the modified Regge-Wheeler equation in the following form.
with the following substitutions for the potentials
The prefactors and the minus sign in the definitions of the a-coefficients need to be there to match our expression to how they are defined. If confusing, this should be clearer in the paper.
Using the parametrized prescription, this means we can write the quasinormal mode frequencies as
Note that above there is a sum over the index i which is not explicit.
7. Plots
7. Plots
The plots in the paper are made in python (available upon request) and here we show a Mathematica version. We refer to the paper for further explanations.
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Testing the Speed of Gravity with Black Hole Ringdown
by Sergi Sirera Lahoz
Wolfram Community, STAFF PICKS, June 30, 2023
https://community.wolfram.com/groups/-/m/t/2948738
by Sergi Sirera Lahoz
Wolfram Community, STAFF PICKS, June 30, 2023
https://community.wolfram.com/groups/-/m/t/2948738