Screened Scalars and Black Holes
Ruth Gregory
Centre for Particle Theory
Durham UK
1402.4737 & coming soon..
With Anne Davis, Rahul
Jha, Jessie Muir
Image Credit: photojournal.jpl.nasa
Outline
  Black holes and no hair
  Screened scalars
  Chameleon hair
  Observational prospects
Black Hole Theorems
Black holes in 4D obey a set of theorems: We know they are
spherical, that they obey laws of thermodynamics, and that
they are characterized by relatively few “numbers” – or
“Black Holes Have No Hair”.
i.e. electrovac solutions are
uniquely specified by 3
parameters: M, Q, and J
No Scalar Hair
The essence of “no hair” is that the scalar field must have
finite energy, and fall off at infinity. Integrating the equation of
motion gives a simple relation, only satisfied for ϕ = ϕʹ′ ≡ 0
ϕ, ϕʹ′
finite
ϕ, ϕʹ′ →0
€
essence of their argument is to take the scalar equation of motion (shown here for
√
the Schwarzschild background), to multiply by V,φ (φ) g and integrate:
€
� ∞
�
�
€
d
∞
r2 V,φ2 + r(r − 2GM )φ� (V,φ ) = [V,φ r(r − 2GM )φ� ]2GM = 0.
dr
2GM
(2.2)
Sotiriou, Faraoni, 1109
Clearly, if V,φφ ≥ 0, as is the case with a wide range
of physically relevant potentials,
then the only possibility is that the integrand on the LHS is identically zero, i.e.,
φ� ≡ 0, φ = φmin . Although at first sight the potentials we are considering appear to
No-No Hair!
But this is highly idealised:
•  Static
•  Vacuum
•  Potential must be convex
And no hair has come to mean the much stronger “no field
profiles”.
2.1.1
2.1.2
Analytic analysis
Numerical Results
5
8
Screened Scalars
e interpreted as a ‘fifth force’ on matter due to its interac
1. Introduction
The screening arises because the scalar couples to matter
For a non-relativistic
particle,
this
takes
the
form
via a metric term – typically due to a conformal rescaling.
These mechanisms can be modelled generically with the Einstein frame action
� 2
�
�
�
�
Mp
√
1 µν
4
2
S = d x −g
R − g ∂µ φ∂ν φ − V (φ) + Sm Ψi , A (φ)gµν .
(1.1)
2
2
∂ ln A
ẍ = −
∇φ .
∂φ
A(φ) = e
φβ(φ)/Mp
β(φ)
≈1+φ
Mp
∂
∂Veff (φ, ρ)
As we
are always
the (φ)
low energy
regime, we
�φ
= in [V
+ (A(φ)
− take
1)ρ]φ �≡Mp.
.
The modified
∂φ‘Einstein’ equation is then written as
∂φ
�
1 � µν
µν
µν taken
upling function
is
therefore
to
be
G = a 2ρ dependent
Tm + Tφ
When calculating the e.o.m.
term appears.
M
p
where
T µν = 2
A(φ)
=e
,
δSm
βφ/Mp
.
(1.2)
(1.3)
∂
∂Veff (φ, ρ)
∂ ln A
�φ =
[V (φ) + (A(φ) − 1)ρ] ≡
. −
ẍ =
∇φ . (1.8)
∂φ
∂φ
∂φ
unction is therefore taken to be
∂
∂Veff (φ
Chameleon
�φ. =
[V (φ) + (A(φ) − 1)ρ] (1.9)
≡
A(φ) = eβφ/M
p
∂φ
The effective potential for the chameleon is a sum of two
The
function
is therefore
taken to be
and looks different
at different
densities
n potentialterms
is coupling
∂φ
A(φ) = eβφ/Mp .(1.10)
V (φ) = M 4+n φ−n = V0 φ−n ,
4+n
n integer
of
order
one,
and
we
define
V
≡
M
to simplify
10
10
A typical chameleon potential is 0
only the 8 leading order term from the 8coupling function, we see
6
V (φ)
= M 4+n φ−n = V0 φ−n ,
otential is6
V�Φ�
V�Φ�
4
4
V0 ρβφ
2 order one, and we
where
n (φ,
≥ ρ)1 ≈is an
integer
of
define V
2
+
,
(1.11)
Veff
n
φ
Mp
0
0
notation. Keeping
only the leading order term from the co
Φ
Khoury, Weltman 0309
that the effective potential is
Φ
at
Chameleon
Useful to sum up bounds in terms of Compton wavelengths:
So for a stellar size black hole with an accretion disc, λc
small, but for a SMBH in ambient galactic density λc more
comparable.
nsity, ρ, jumps from being roughly zero to the ambient galactic or local accretion
c value. Thus, although r2 V,φ2 is positive definite, the derivative of V,φ with respect
r contains a delta function, coming from the derivative of ρ. Combining this with
e intuition that φ, if nontrivial, will tend to roll towards large values near the
ack hole horizon, we see that the second term in the integrand can potentially be
ry large and negative, thus ruling out a simple “no-hair” proof, and opening the
ssibility of a nontrivial scalar profile.
Turning now to the case of the black hole, our setup is motivated by a physical
The “no-hair” theorem does not apply because dV/dr is not
cture of an astrophysical black hole, typically located within some larger distribumonotonic
– theblack
density
may jump.
n of matter. Although
astrophysical
holes profile
will be rotating,
for the purpose
establishing whether or not a nontrivial scalar profile is possible, it will suffice to
nsider a purely monopole
spherically
symmetric
set-up,the
in which
is
Demonstrate
explicitly
within
rulestheofblack
the hole
no-hair
game:
scibed by the Schwarzschild
metric:
static, spherically
symmetric black hole, but with density
�
� 2 �
�
2
Rs
Rs −1
profile
ds = − 1 − r dt + 1 − r
dr2 + r2 dΩ2 ,
(2.3)
Chameleon Hair
enoting Rs = 2GM for clarity), and the density profile by
�
0 Rs < r < R0 (Region I)
ρ(r) ≡
ρ� r > R0 (Region II)
(2.4)
he motivation for this profile is that the larger distribution of matter in which the
ack hole sits will be characterised by a density ρ� (taken to be constant in (2.4)),
hich is assumed to vary slowly on length scales comparable to the size of the black
le. Very close to the black hole however, we expect an approximately empty inner
gion, motivated by the fact that all black holes have an innermost stable circular
ρ� to be the density of an accretion disk, m� R0 will always be large (� 1). If ρ� is the
density of a galactic halo the chameleon will have m� R0 � 1 for stellar mass black
holes, and m� R0 � 1 for supermassive black holes. Thus, both limits will potentially
be relevant for chameleons. We then present numerical solutions for a range of
chameleon model data, comparing them to the derived analytic approximations.
Analytic Approximations
2.1.1 Analytic
analysis
Always
important to develop analytic understanding where
possible.
regions I, II with
small
and large
n order to both
exploreAnalyse
analyticinapproximations
and
perform
numerical integrations,
Compton
wavelength
we rewrite the
chameleon
equation in
of probe
motionlimit.
in terms of dimensionless variables:
2
φ
r
ρ
βR
∗
s
φ̂ =
, x=
, m̂2 = m2∗ Rs2 = (n + 1)
φ�
Rs
Mp φ ∗
giving
2
�
2x − 1 �
x
m̂
1
φ̂ +
φ̂ =
Θ[x − x0 ] −
x(x − 1)
(x − 1) (n + 1)
φ̂n+1
��
(2.6)
�
(2.7)
The physical set-up is that we have a dense extended region (II) in which the
chameleon will be held essentially constant at φ∗ . Nearer to the black hole, we have a
region of vacuum (region I) in which the chameleon is allowed to roll freely and is only
φ̂ = φ̂h −
m̂2
6(n +
1)φ̂n+1
h
[x2 + 2x + 2 ln x − 3] .
(2.10)
Matching the solutions at x0 with the asymptotic form (2.8) gives
�
�
2
m̂2
4x
+
4x
+
4
0
0
φ̂h = 1 +
x20 + 2x0 + 2 ln x0 − 3 +
n+1
2m̂x0 + 2 + m̂
6(n + 1)φ̂h
Large Compton
1+m̂/2
m̂2
C=
(x20 + x0 + 1)
.
m̂x
+
1
+
m̂/2
0
wavelength expect
x0
(2.11)
3(n + 1)φ̂n+1
For long Compton
a small perturbation to
h
the Writing
scalar,φ̂ =so
perturbatively.
gives:
1 +model
δ φ̂, and expanding
to leading order
φ̂ � 1 +
�
m̂2
6(n+1)
m̂2
3(n+1)
�
3x20 − x2 + 4x0 − 2x + 2 + 2 ln xx0
(x20 + x0 + 1) xx0 e−m̂(x−x0 )
�
x < x0
x > x0
(2.12)
1.020 interested in the horizon value of the chameleon field, and how
We are mainly
this differs from the asymptotic value, as this indicates the impact of the black hole
on the local scalar
profile, and we can read this off to leading
� �0.1order in m̂x0 as:
1.015
m
or,
m̂2 x20
φ̂h ≈ 1 +
,
2(n + 1)
Φ 1.010
Φ�
Here we see that δφh increases with the coupling function β, and
the density of the local environment, as well as the range in w
6 – (R ).
can –roll
0
1.005
1.000
1.
2
ρ
βR
�
0
δφ(2.13)
.
h ≈
2Mp
2.
3.
• m̂x0 � 1
4.
5.
6.
7.
8.
9.
Inr�Rthe
case
that
the
chameleon
is short range, we can u
s
approximation presented above, although without expanding
dominated by the potential:
which is solved by
�
2
� 1 �
� n+2
2 (n+2)2 n+2

m̂
2

m̂2
1
x0 − x + (n+2)
��
2n(n+1)
φ̂ � −
(2.15) m̂
φ̂
�
1
+
1
�
�
(n + 1) φ̂n+1
n+2 � x �1+m̂/2

2
0

e−m̂(x−x0 )
n(n+1)
x
x<
x>
Small
Compton
This profile captures
a rapid transition to large near horizon
�
�
1/(n+2)
2
2
m̂
(n + not
2)2 (xsolve
1 − x)
it
does
the
equations
horizon, as it has
φ̂ � 1 +
. of motion at the
(2.16)
2n(n +
1)near x0the
For low Compton wavelength
expect
toasbe
the variation
. Inpotential
spite of this,
we more
will see from the n
does
indeed
captureapproximation
the essential features
of the field profile (see
important,
but to
trickier
analytic
more
Meanwhile,
matching
(2.8) atto
x0 get
gives:
both this expression, and the simpler geometry dominate
than indicative � 2 Indeed,
1 �
2
�
� n+2
2 n+2

m̂
(n+2)
data
(2.11)
give
the
same
dependence
of the horizon value on x0
2

x
−
x
+
x
<
x
0
0
2n(n+1)
(n+2)m̂
in
x
φ̂ � 1 + �
(2.17)
1
0
�order
�
�
1
�
� n+2
n+2
1+
m̂/2

x0
2
−m̂(x−x0 )

2 2
e
x
>
x
m̂
x
0
0
n(n+1)
x
φ̂h ≈ φ̂c
,
(n + 1)
This profile captures a rapid transition to large near horizon values, although
14
where
φ̂n+2
(n horizon,
+ 2)2 /(2n)
the been
potential
dominated
expression
c at=
it does not solve the equations of
motion
the
as for
it has
tailored
to
12
geometry
Re-expressing
in terms
of the dime
the variation near x0 . In spite of
this, asdominated
we will seeexpression.
from the numerical
work,
it
� �1000
10
m
gives
does indeed capture the essential features of the field profile (see next
� subsection).
� 1
2
2 n+2
Φ 8
φ
∝
(n
+
2)
V
R
.
h
0 0
Indeed, both this expression, and the simpler geometry dominated
approximation
Φ�
6 give the same dependence of the horizon value on x0 and m̂ to leading
data (2.11)
order in x40
1
� 2 2 � n+2
m̂
x
2
0
φ̂h ≈ φ̂c
,
(2.18)
(n
+
1)
–7–
0
1.
n+2
φ̂c =
2.
2
3.
s
forr�Rthe
4.
5.
where
(n + 2) /(2n)
potential dominated expression, and 1/6 for the
geometry dominated expression. Re-expressing in terms of the dimensionful variables
gives
Horizon Data : m
10
�
small m
1
0.1
Φh � Φ�
Φ�
�
large m
0.01
0.001
full analytic
approxn.
numerical data
10�4
0.01
0.1
1
�
m
10
100
1000
Horizon Data : n
10
1
� �100
m
0.1
Φh � Φ�
Φ�
0.01
0.001
� �0.01
m
10�4
10�5
0
5
10
n
15
20
This can be interpreted as a ‘fifth force’ on matter due to its interaction with the
scalar field. For a non-relativistic particle, this takes the form
Observation
∂ ln A
ẍ = −
∇φ
∂φform:
Scalar forces are roughly of the
�
βφ
β
aφ � −
�
Mp
Mp
�
φh − φ∗
∆R
2/(n+2)
∆R ∼ Min{R0 , R0
The
(1.7)
�
(1.8)
m−n/(n+2)
}
∗
(1.9)
where φ̂n+2
= (n + 2)∂2 /(2n) for the potential dominated expression,
andρ)
1/6 for the
∂Veff (φ,
c
Probe
limit
scalar
does
not
back-react
strongly
on
geometry
�φ = expression.
[V (φ) Re-expressing
+ (A(φ) −in 1)ρ]
≡the dimensionful.variables
geometry dominated
terms of
∂φ
∂φ
– also
gives means geometry dominates local motion. Instead
�
� 1
2
2 n+2
φ
∝
(n
+
2)
V
R
.
(2.19)
coupling
function
therefore
taken
h
0 to
0tobe
compare
energyisloss
from
scalar
gravitational
radiation.
�
�
� � 92 �
�
�
�
� Ė �
βφ/M
p
R
φ
−
φ
M
M
� φ �
0
h
�
BH
. BH3
.
�
� ∼ β(φ� ) A(φ) = e
� ĖGR �
Rs
∆R
Mp
mt
A typical chameleon potential is
�
�
4+n −n
−n
(2.20)
(1.10)
(1.11)
gives
where φ̂n+2
= (n + 2)2 /(2n) for the
dominated
expression, and 1/6 for the
� potential
� 1
c
2
2 n+2
φh ∝ Re-expressing
(n + 2) V0 R0in terms
. of the dimensionful variables
(2.19)
geometry dominated expression.
gives
�
�
� 92 � 2
� 1
� �
�
�
�
� Ė �
2 n+2
φh R∝0 (n
+φ2)
V0φR
. MBH
(2.19)
� φ �
h−
� 0 MBH
.
(2.20)
�
� ∼ β(φ� )
� ĖGR �
Rs
∆R
Mp3
mt
�
�
� � 92 �
�
�
�
� Ė �
R0
φh − φ� MBH MBH
� φ �
.
(2.20)
�
� ∼ β(φ� )
� ĖGR �
Rs
∆R
Mp3
mt
Estimates
�
�
n
�
� n+2
�
�
2
Best case scenario:
of
solar
mass
object
to SMBH
M
m
� Ėφ � infall
∗
(2.21)
� ∼ 10−28−60/(n+2) β∗ BH
�
� ĖGR �
Mp m t Mp
�
�
n
�
� n+2
�
� Ė �
2 �2
2(n+3)
M
m
M
�
�
φ
∗
BH
−23+
BH
−28−60/(n+2)
(n+1)(n+2) β
Short range:
(2.21)
� ∼1010
�
∼
≈ 10−11 − 10−5
(2.22)
∗
� ĖGR �
Mp⊙mt Mp
�
�2
2(n+3)
M
BH
∼ 10−23+ (n+1)(n+2)
≈ 10−11 − 10−5
(2.22)
M⊙
Long range:
�
�
� Ė �
(φh − φ� ) MBH
� φ �
�∼β
�
� ĖGR �
Mp
mt
�
�
�
�3
� Ė �
(φh2−ρφ�� ) MMBH
� φ �
BH
−42
� ∼10β β
�
∼
∼ 10−18 − 10−9
� ĖGR �
Mρpcos m
Mt ⊙
�
�3
MBH
−42 2 ρ�
∼ 10 β
∼ 10−18 − 10−9
ρcos M⊙
(2.23)
(2.23)
(2.24)
(2.24)
In Progress: Kerr Accretion Disc
In reality black holes
rotate, and the accretion
disc is localized near the
equatorial plane. Model
this by a uniform matter
shell extending from
RISCO to ~100GM, and
solve perturbatively for φ,
roughly massless away
from disc.
Summary
  No hair not applicable – screened scalars have nontrivial
profiles even in static spherically symmetric black hole
environments.
  Analytic tools are quite accurate predictors of actual profiles.
  General picture similar (nontrivial profiles sourced by varying
density) with more realistic accretion disc model, though profile
different.
  Unfortunately it seems these will not give observable effects.