ABSTRACT (original article): We investigate black hole solutions with time-dependent (scalar) hair in scalar-tensor theories. Known exact solutions exist for such theories at the background level, where the metric takes on a standard GR form (e.g. Schwarzschild-de Sitter), but these solutions are generically plagued by instabilities. Recently, a new such solution was identified in arXiv:2310.11919, in which the time-dependent scalar background profile is qualitatively different from previous known exact solutions - specifically, the canonical kinetic term for the background scalar X is not constant in this solution. We investigate the stability of this new solution by analysing odd parity perturbations, identifying a bound placed by stability and the resulting surviving parameter space. We extract the quasinormal mode spectrum predicted by the theory, finding a generic positive shift of quasinormal mode frequencies and damping times compared to GR. We forecast constraints on these shifts (and the single effective parameter

β
controlling them) from current and future gravitational wave experiments, finding constraints at up to the
Ο(
-2
10
)
and
Ο(
-6
10
)
level for LVK and LISA/TianQin, respectively. All calculations performed in this paper are reproducible via a companion Mathematica notebook. CITATION (original article): Sergi Sirera, Johannes Noller, Stability and quasinormal modes for black holes with time-dependent scalar hair, arXiv:2408.01720. https://doi.org/10.48550/arXiv.2408.01720
This is the companion notebook to the paper https://arxiv.org/abs/2408.01720, where we perform all the calculations appearing there.
Starting with the action for our theory, we study a black hole solution with a scalar that is linearly dependent on time. We first show that this solution indeed solves the equations of motion of the theory, and show an interesting relation between the covariant metric and scalar equations. We then study the stability of odd parity perturbations and show a region in parameter space safe from instabilities. We then obtain the modified Regge-Wheeler equation and the effective potential. We find the corresponding quasinormal modes via the WKB method and apply a Fisher forecast analysis to constrain the parameter of the theory.
This notebook has hugely benefitted from resources provided by: - xAct ( http://www.xact.es/ ) - Johannes Noller - Kazufumi Takahashi - WKB package ( http://arxiv.org/pdf/1904.10333.pdf )
If you use any of the results or techniques in this notebook please cite the repository and [2408.01720] ( https://arxiv.org/abs/2408.01720 ), [2301.10272] ( https://arxiv.org/abs/2301.10272 ).
If you have any comments or feedback please let me know. Thanks!
Sergi Sirera Lahoz​​Institute of Cosmology and Gravitation - University of Portsmouth​​
​​August 2024

0. Setup and definitions


1. Theory and background

Action

In[]:=
L2=G2[phi[],X[]];​​L3=-G3[phi[],X[]]CD[b]@CD[-b]@phi[];​​L4=G4[phi[],X[]]RicciScalarCD[]+G4X[phi[],X[]](
2
(Scalar[CD[b]@CD[-b]@phi[]])
-CD[c]@CD[d]@phi[]CD[-c]@CD[-d]@phi[]);​​L5=G5[phi[],X[]]EinsteinCD[-b,-c]CD[b]@CD[c]@phi[]-
1
6
G5X[phi[],X[]](
3
(CD[b]@CD[-b]@phi[])
-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)
Out[]=
-g
G
2
[ϕ,X]+
G
4
[ϕ,X]R-
G
3
[ϕ,X]
b
∇
∇
b
ϕ+
G
b
c
G
5
[ϕ,X]
b
∇
c
∇
ϕ+
G
4 X
[ϕ,X]
2

b
∇
∇
b
ϕ
-
∇
c
∇
d
ϕ
c
∇
d
∇
ϕ-
1
6
G
5 X
[ϕ,X]
3

b
∇
∇
b
ϕ
-3
d
∇
∇
d
ϕ
∇
e
∇
c
ϕ
e
∇
c
∇
ϕ+2
∇
f
∇
k
ϕ
f
∇
j
∇
ϕ
k
∇
∇
j
ϕ
In[]:=
(*thisrulespecifiesourtheoryofinterestinnaturalunits*)​​ncXtheory=G2[phi[],X[]]->-2Λ+2
2
Mpl
η
X[]
,G3[phi[],X[]]->0,G4[phi[],X[]]->
2
Mpl
+λ
X[]
,G4X[phi[],X[]]->D
2
Mpl
+λ
X[]
,X[],G5[phi[],X[]]->0,G5X[phi[],X[]]->0​​(*thefollowingtworulescanbeappliedatanypoint*)​​(*1.thisrulesetsgeometricunits*)​​ToGeomUn={Mpl->1,lam2->1}​​(*2.thisrulebringsthetheoryfrom2-parameterformto1-parameterform*)​​ToBeta={η->1,λ->2
2
β
}
Out[]=

G
2
[ϕ,X]-2Λ+2
2
M
Pl
η
X
,
G
3
[ϕ,X]0,
G
4
[ϕ,X]
2
M
Pl
+λ
X
,
G
4 X
[ϕ,X]
λ
2
X
,
G
5
[ϕ,X]0,
G
5 X
[ϕ,X]0
Out[]=
{
M
Pl
1,
Λ
2
1}
Out[]=
{η1,λ2
2
β
}
Our Lagrangian is then
In[]:=
L=LT/.ncXtheory​​(*one-parameterversion*)​​%/.ToBeta​​(*ingeoemetricunits*)​​%/.ToGeomUn
Out[]=
-g
-2Λ+R
2
M
Pl
+λ
X
+2
2
M
Pl
η
X
+
λ
2

b
∇
∇
b
ϕ
-
∇
c
∇
d
ϕ
c
∇
d
∇
ϕ
2
X
Out[]=
-g
-2Λ+R
2
M
Pl
+2
2
β
X
+2
2
M
Pl
X
+
2
β

2

b
∇
∇
b
ϕ
-
∇
c
∇
d
ϕ
c
∇
d
∇
ϕ
X
Out[]=
-g
-2Λ+R1+2
2
β
X
+2
X
+
2
β

2

b
∇
∇
b
ϕ
-
∇
c
∇
d
ϕ
c
∇
d
∇
ϕ
X

Perturbations

We also define here the metric perturbations. We do not include scalar perturbations because we restrict to the odd sector.
In[]:=
DefMetricPerturbation[g,h,ϵ];​​Unprotect[IndexForm];​​(*thisselectsacolourtodifferentiateindicesreferringtotheorderinperturbationsfromotherindices*)​​IndexForm[LI[x_]]:=ColorString[ToString[x],RGBColor["#00a898"]];​​Protect[IndexForm];​​IndexForm[LI[n]];​​DefTensorPerturbation[δϕ[LI[order]],phi[],Md];

Rules


Coordinate basis


2. Background equations of motion


3. Quadratic action


4. One-variable action


5. Stability conditions

Here we will define stability conditions and identify the parameter space satisfying them
In[]:=
PowerContract[expr_]:=expr//.{m_^q_n_^q_:>(mn)^q/;!IntegerQ[m]&&!IntegerQ[n],m_^q_n_^p_:>(m/n)^q/;q>=0&&p==-q&&!IntegerQ[m]&&!IntegerQ[n]}
In[]:=
b1t[r[]]/.Tob1t/.Tob//Simplify​​%/.ToFGHJ/.ncXtheory/.Xtoϕ/.CDphirules1/.CDphirules2/.psirules//Simplify;​​b1tcomp=%//PowerExpand//Simplify​​b2[r[]]/.Tob1t/.Tob//Simplify​​%/.ToFGHJ/.ncXtheory/.Xtoϕ/.CDphirules1/.CDphirules2/.psirules//Simplify;​​b2comp=%//PowerExpand//Simplify​​b4[r[]]/.Tob1t/.Tob//Simplify​​%/.ToFGHJ/.ncXtheory/.Xtoϕ/.CDphirules1/.CDphirules2/.psirules//Simplify;​​b4comp=%//Simplify
Out[]=
2
ℋ[r]
2
r
B[r][r]
Out[]=
4
4
M
Pl
2
r
2
2
M
Pl
B[r]+
2
q
λ
λ+
2
M
Pl
η
2
r
Out[]=
B[r][r]
2
ℋ[r]
2
r
ℱ[r][r]+
2
[r]
Out[]=
2
2
M
Pl
2
r
λ+
2
M
Pl
η
2
r
2
2
M
Pl
B[r]+
2
q
λ
λ+
2
M
Pl
η
2
r

2
2
M
Pl
λ+2
4
M
Pl
η
2
r
+
2
q
3/2
λ
λ+
2
M
Pl
η
2
r
Out[]=
ℋ[r]
Out[]=
2
2
M
Pl
In[]:=
b1t[r[]]/.Tob1t/.Tob/.ToFGHJ/.ncXtheory/.Xtoϕ/.CDphirules1/.CDphirules2/.psirules/.ToM/.ToGeomUn//Simplify;​​%/.r[]->
2mass
R
//Simplify;​​%*
2
R
8
2
mass
//Simplify​​b1tSimp=(%//PowerExpand//ExpandDenominator//Simplify//PowerContract)/.{Sqrt[xx_]*Sqrt[yy_]->Sqrt[xx*yy]}/.
3/2
(xx_)
yy_
->
xx
yy
xx
​​b2[r[]]/.Tob1t/.Tob/.ToFGHJ/.ncXtheory/.Xtoϕ/.CDphirules1/.CDphirules2/.psirules/.ToM/.ToGeomUn//Simplify;​​%/.r[]->
2mass
R
//Simplify;​​%*
2
R
8
2
mass
//Simplify​​b2Simp=(%//PowerExpand//ExpandDenominator//Simplify//PowerContract)/.{Sqrt[xx_]*Sqrt[yy_]->Sqrt[xx*yy]}/.
3/2
(xx_)
yy_
->
xx
yy
xx

We can then plot the parameter space where these conditions are satisfied.
And here we show several sections of this valid parameter space region.

6. Modified Regge-Wheeler equation

Here we will derive the modified Regge-Wheeler equation. We start by defining the tortoise coordinates.
This means we can write the modified Regge-Wheeler equation as
where
Let us now look at the form of these expressions specific for our theory
We now show that we can write this potential in a simplified form.
We work with the following redefinition of the beta parameter.

7. Quasinormal modes via the WKB method

Here we will obtain the complex quasinormal frequencies form the effective potential by applying the WKB formula. To do so, we use the package made available in https://arxiv.org/pdf/1904.10333.pdf, which you will need to add to your directory.

9. Large-r effects on QNMs

Scalar

CITE THIS NOTEBOOK

Stability and quasinormal modes for black holes with time-dependent scalar hair​
by Sergi Sirera and Johannes Noller
Wolfram Community, STAFF PICKS, August 16, 2024
​https://community.wolfram.com/groups/-/m/t/3249669