1.
Cape
Town,
11.03.2013
African
Ins7tute
for
Mathema7cal
Sciences
D-branes and the disformal dark sector
Danielle Wills and Tomi Koivisto
Institute for Theoretical Astrophysics
University of Oslo
Centre for Particle Theory
Durham University

2. gµ⌫ = gµ⌫ +
˜ ,µ ,⌫ (5)
1 + 2X ) = 1
C(
D( ) = D0 e ( 0)
On the
C( ) = 1 physical and relationV0 e 
V( ) =
between
gravitational geometry
D( ) = D0 e ( ) 0

V ( ) =Z V0 e 
p R p
S= d4 x g + g L (matter, gµ⌫ )
¯ ¯
Z  16⇡G
p R p
• For
d4 x
S =simplicity,
let
us
take
the
rela3on
to
be
given
by
a
scalar
Φ
g + g L (matter, gµ⌫ )
¯ ¯ (6)
16⇡G
• It
can
be
argued
that
the
most
general
consistent
rela3on
then
has
the
form
gµ⌫ = C( , X)gµ⌫ + D( , X) ,µ ,⌫
¯ [Bekenstein, Phys.Rev.
D48
(1993)
]
(7)
• C
≠
1
is
very
well
known
and
extensively
studied.
We’ll
focus
on
D
≠
0.
The
outline:
-­‐
What
is
it
good
for?
1)
Mo7va7ons,
Phenomenology:
the
screening
-­‐
Where
does
it
come
from?
2)
The
DBI
string
scenario
-­‐
How
to
detect
it?
3)
Cosmology:
background
expansion,
Large-­‐scale
structure
-­‐
So
what?
An
outlook
and
conclusion

3. Institute for Theoretical Astrophysics, University of Osloother
between scalar degree of freedom and
ter, which in turn could help to explain
arXiv:1205.3167v2
Frames of gravity becomes dynamically important at the p
There are myriad variations of such mod
of them the coupling can be effectively d
y as field-dependent mass of the dark matter p
Yukawa-type couplings can be motivated b
gµ⌫ =
˜ gµ⌫
relation to scalar-tensor theories, which inc
f (R) class of modified gravity [2].
However, for any other type of gravity
the relation between the matter and gravit
Brans-­‐Dicke
theory,
e.g.
f(R):
will be non-conformal. This can also be m
type 1
in a DBI ~ ~scenario where matter is all
2
L = φ R + V (φ ) L = R − (∂ϕ ) + U (ϕ ) + Lm (ϕ )
2
the additional dimensions [3]. When give
field φ, the disformal relation can be param
• The
generalisa3on
of
conformal
mapping
gµν = C(φ)gµν + D(φ)φ,µ φ,ν ,
¯
• Is
contained
in
any
modified
gravity*
commas denote partial derivatives.
where beyond
f(R)
• and
in
any
scalar-­‐tensor
theory
the most general physical
case, Bekenstein [
beyond
Brans-­‐Dicke
both functions C and D may also depend
*
The
generic
ghost
problem
of
higher
deriva3ve
theories
may
be
avoided
in
nonlocal
gravity
that
may
further
be
simpler case here.
but we will focus on the asympto(cally
free!
[Biswas,
TK,
Mazumdar:
PRL
(2012)].
plications of such a relation to cosmology

4. further relations between scalar- 4 
V( ) =  V0 e
Considerand itsfderivatives which gives rise to second or-
tensor
an (R) theory as C( ) = 1
der equations of motion in four space-time dimensions. Z
The addition of a scalar degree of freedom provides =
An example of both
S a d x

p
4 D( ) = D0 e R p ( 0)
g µ⌫ =+ gµ⌫ L (matter, g
g˜ g
¯ ¯
e generous extension of the possibilities. The most general
e derivatives which gives rise to second theory was first
s gravitational sector for a scalar-tensor or- V ( ) 16⇡G 0 e 
= V
y of motionmodified
g[14] and has ith
in
the
1st
order
formalism
[1]:
• A
by four space-time dimensions. considerable
derived in Horndeski ravity
w received
,
on of a scalarrecently [15–21]. It is provides the Horndeski gµ⌫g C( =X)gµ⌫ ++ Dv v ,µ ,⌫
attention degree of freedom given by a
tnsion of the possibilities. The most general Z ¯ =µ⌫ , Cgµ⌫ D( , X) ⌫ ˜ µ
Lagrangian
s sector for a scalar-tensor theory was first S = 4 p R p
d x g✓ + ◆ g L (matt
¯
l
[14] and has received5considerable
orndeski X d 16⇡G X
l LH = Li . (1)
ently
è It iss
the
Levi-­‐Civita
connec3on
of
gµ⌫ dX f,R gµ⌫ + f> 0 µ⌫
[15–21]. Γ
i given by the Horndeski i=2
˜ = C + XD ,Q R
-
Up to total derivative terms that do not contribute to the µ⌫ = C( , X)gµ⌫ + D( , X) ,µ
g
¯
- equations ofX 5
motion, the di↵erent pieces can be written gµ⌫ = gµ⌫ + ,µ ,⌫
˜
- • [19]
H
=
L
i
.Horndeski
scalar-­‐tensor
theory,
“covariant
galileon”:
as
L
(1)
h
- L 2 = G2
i=2(X, )
,
(2)
✓ ◆p
erivative terms that do not contribute to the G2 = G3 = G5 =X, G4 = > 0 2X
d 0 1+
w
ymotion, L
the
=
G
3
(X,pieces
,
can
be
written
(3)
3
di↵erent
)2
dX C + XD
⇥ matter slow down: ⇤
n
-
L
4
=
G4
(X,
made
G
4,X
(2
2
;µ⌫
;µ⌫
,
(4)
is
the
E-­‐H
theory
for
[2]
Clocks
)R
+ of
dark )
G2 (X,
)
,
;µ⌫
g
00
1 + D ˙ 2 ! 0 = 1 + D0 e (
a
L5 = G5 (X, )Gµ⌫
1 G5,X h(2 )3 (2) ¯
=
- 6 gµ⌫ = gµ⌫ + ,µ ,⌫
˜
n G3 (X, )2 , (3) ;µ i
;µ⌫ ;⌫ ;
- 3(2 ) ;µ⌫
⇥ + 2 ;µ ⇤ ;⌫ ; . (5)
G4 (X, )R + G4,X (2 ) 2 ;µ⌫
, (4) V ⇠e 
m
1.
[TK:
PRD
(2007)]
2.
[Zumalacarregui,
TK,
Mota:
PRD
(2013,
to
appear)]
;µ⌫
- Here R, Gµ⌫ are the Ricci scalar and the Einstein tensor, = G = G = 0 , G = (1 + 2X h G

5. ated e.g. The coupling will then generically involve second deriva- 0
✓ ◆
e of dark matter slow down: the distortion of causal structure.
to enter tives, which ) = 1
C( entail d X V ( ) = V0 e 
>0 (8)
a scalar g00 )= = 1 D D ˙ 2 ( 0 0 ) 1 + account
¯ + e ! taking + XD ) ˙2
For a point particle, and dX = Cinto D0 e ( the 0cor- (11)
D(
Interacting matter
ed as 0
rect weight of the delta function, we have Z 
√ ( ¯) = V0 e
V  p R p
(1) V g⇠µ=νg + ,µ ,⌫ S =
˜µ⌫ e µ⌫
(4)
−¯Lm = −Σm −¯µν x x δ (x − x(λ)) . (4)
g g ˙ ˙ d4 x g + ¯ (12) (9)
g L (matter, gµ
¯
16⇡G
p
nsidering ZFrom the point of view of the = G5 = 0frame, the proper
 G2 = G3 physical , G4 = 1 + 2X
p p D⇠e  (13) (10)
ued that time the particleR
4 experiences is dilated by the conformal
ks made ,of dark matter slow down:+ the g L (matter, gµ⌫D gives ga = C( , X)gµ⌫ + D( , X)
n (∂φ)2 = factor C. In 16⇡G
S d x g addition, ¯disformal factor )¯ ¯µ⌫ (6)
,µ ,⌫
vious ap- ⌦ ⇠ 2
direction-dependent effect proportional to theD0 e (
g00 = 1 + D ˙ 2 ! 0 = 1 + projection 0 ) ˙ 2
¯
(14) (11)
de vary- The
pthe four-velocity along the gradient ofrom
GR:
• of hysical
proper
distances
differ
f the field:
✓ ◆
k energy ↵2⇠ erm 2
V e
tensions ˙ 2 ¯ x ef ν
x ≡ gµνGµ xf = C1 2 + D(x · ∂φ) G
¯ ˙ ˙ x+
˙ ˙ . (5) (15) (12)
2
n of cou- The
equivalence
principle
is
vthe particle along its path
• Extremising the proper time of iolated:
D⇠e  (13)
-up that
in shows that xµ follows the x = µ geodesics: forces
on (1)
¨ = ¯ µ x↵ disformal x↵ x + 5th
it
↵ ˙ ˙ ↵ ˙ ˙
2
(16)
¯ ˙B xβ ⌦ 0⇠
• The
conformal
prototype,
xµ + Γµ xαrans-­‐Dicke
theory,
C(Φ)=exp(-­‐ακ(Φ-­‐Φ0)),
D=0:
¨ ˙ = , (6)
(14)
αβ
✓ ◆
Effec3ve
ravita3onal
coupling
g
↵2 rm Newton’s
force
+
extra
5th
force
between
mager
par3cles
Gef f = 1+ e G mediated
by
scalar
par3cles
(15)
ress: tomi.koivisto@fys.uio.no
2
How
to
reconcile
with
observa7ons?
1.
Make
the
field
very
massive
:
no
DE
2.
Make
α
very
small
:
uninteres7ng
nic address: tomi.koivisto@fys.uio.no
3.
Make
them
species-­‐dependent
:
coupled
DE
4.
Make
them
density-­‐dependent
:
chameleon

6. D( ) = 4 D0 e ( 0)
V ( ) =  4 V0 e 
Chameleonic screening Z 
pR p
S= d4 x g+ g L (matter,
¯
16⇡G
[Khoury and Weltman, PRL
(2004)
]
gµ⌫ = C( , X)gµ⌫ + D( , X) ,µ ,⌫
¯
• Spherically symmetric NR point matter source
Compton wavelength ✓ ◆
• The scalar potential decays beyond the Compton wavelenght 1/m> 0
d X
Effective potential
• High effective Suppose highscalar has a mass
 mass in the density regions
dX C + XD
 Consider non-relativistic matter 2
r gµ⌫ = g + ,µ ,⌫
(r)
Comptonµ⌫wavelength
˜
   m   4 G
2
p
G = G3 = G5 = 0 , G4 = 1a mas
2 Suppose the scalar has + 2X
Sphericallymade of dark solutions down:
Clocks symmetric matter slow 2  m2  4 G
g00 =
¯ 1 + D ˙2 ! 0 = 1 + D0 e (
d 2
2 d  Spherically symmetric solutions
 2   m2    4 G
 dr r dr 

V ⇠e
d 2
2 d 
 2   m2    4 G
GM  dr Dr⇠ e  
m 1
dr r
 (r)   exp  mr 
r GM
 (r) ⌦  ⇠
 exp  mr 
2
r
The scalar potential decays exponentially above✓the Compton ◆
The scalar potential rm
↵2 decays expo
wavelength m 1 Gef f = 1 +
wavelength 1 2
e G
m

7. µν µν
ensures that Einstein field equations have ◆ usual form
✓ the
ch models, but in all µν µν d X
s thatdescribed by a equations have the usual formcovariant conservation of
tively µν Einstein field G = 8πGT . However, the >0
Disformal screening
8πGT . However, energy momentum does not hold for
matter particle. Those the covariant conservation of
dX C +the coupled compo-
nents separately. Instead, we obtain that
momentum does not hold for the coupled compo-
XD
vated by a conformal
[TK, Mota & Zumalacarregui PRL
(2012)
]
separately. Instead, we obtain that µ Tm ≡ −Qφµ⌫ ,= gµ⌫ + ,µ ,⌫
hich includes also the µν g ,ν
˜ (3)
C D D
Tm ≡ −Qφ ,ν , Q =
µν
gravity modification,
µ where Tm − λ (3) ,µ Tm +
φ µλ
φ,µ φ,ν Tm µν
= G3 =the effects , disformal couplings 1
2C G2Addressing C 5 = 0of G4 = (1 + 2X)therefore
G 2C
d gravitational metric
C D D requires studying the field dynamics in high density, non- w
µλ µν fi
Q Clocks Tm −of dark matter m down: relativistic Tm involve second deriva-
lso be= motivated e.g. λ The coupling will then µ φ,ν environments. This regime can be explored
φ,µ T slow + φ, generically
er is Spherically symmetric, static NRof causal structure.a station-
• allowed made
2C to enter C 2Cusing the general configuration: e
tives, which entail the distortion scalar field equation (7) for t
For involve tog00neglecting D ˙ =into + D0 ecurvature [31]. ˙
hen given by thenterm proportionalparticle, densitythe remaining ρ(x)account the→cor-The
ary
oupling • will a scalar 1 distribution 1 in the limit ρ 0∞, 2
¯D = deriva- 2
Each generically a point second and+takingidentically!
vanishes spacetime
( ) and r
in
be parametrized as distortion weight of the delta result follows from taking the limit ρ C/D, φ2 in
which entail the rect of causal structure.
same function, we have ˙ v
• High density Dρ>>1 limit: √ (8): t
)φ point and g
−¯L account ¨ −¯ 2˙ V˙ δ φ2 
a ,µ φ,ν , particle,(1) taking into¯m = −Σmthe gµν˙xµ xν⇠ (4) (x 1− x(λ))β. ˙2 (4)
cor-
D
e
˙ t
eight of the delta function, we equation:
• The Klein-Gordon have φ≈− φ +C − =− φ , (15)
2D C 2D 2Mp s
vatives. Considering From the point of view of the physicalgeneral andthe second applies
frame, proper b
nstein¯[4] arguedfield−¯µν xµ xν δ (4) (x − x(λ)) .example model. The by , expression
• The that indeed slows downour (4) dilated
(e.g. if β>0)
√ where the first equality is 
D ⇠ e abovethe the departs sub-
−¯Lm = −Σm
g g time the particle experiences is
˙ ˙ to conformal m
c
• The evolution is independent ρof ∞from theill-defined. Spatialcoupling, for which
depend upon (∂φ)2 , the → the density
factor C. In addition, the disformal conformal D gives a
stantially
limit is
simple factor
derivatives become t
thehere. Previous ap- the physical frame, the proper are suppressed by projection
se point of view of direction-dependent irrelevant, as they
effect proportional to the a p/ρ factor w.r.t. p
he • The experiencesof the four-velocity along independent of importantly, the density, making
particle 5
th force just isn’t there the gradient of the field: equation t
mology include vary- is dilated by the time derivatives. More the local energy above
conformal
becomes fo
ation [6], dark energy
C. In addition,muchdisformalx2 ≡ of thefield evolution insensitive∂φ)2 D s
• Pretty the regardless gµν xthexνdetails+of Vhomogeneously, spatial gradi-
[9, 10] and extensions ¯ factor ˙D ˙ gives xafield D(x ·and .
˙ ¯ bodies.= C ˙ 2 rolls˙
µ
As the
to the presence of massive
(5) t
on-dependent effect proportional to the projection
⇤ Electronic ents between separate objects, which would give rise to a
generalization ofaddress: tomi.koivisto@fys.uio.no scalar force,thenot form. along its path
cou- fo
four-velocity along theExtremisingof the field: purely disformal case with exponential D, equa-
s a simple set-up that gradient the proper time of do particle
the
In the in
shows that it follows the disformal geodesics:
tion (15) can be integrated directly

8. Potential signatures?
Our assumptions are violated if we have:
• Matter velocity flows
- Suppressed by v/c. Binary pulsars?
• Pressure
- Potential instability if p>C/D-X. Astrophysics?
• Strong gravitational fields
- Gravity coupling not suppressed by Dρ. Black holes?
• Spatial field gradients
- Potential remnants of LSS formation. Even Solar system?
Systematic study requires developing the PPN formalism
[Work under progress with Kari Enqvist and Hannu Nyrhinen]

9. Disformal couplings from DBI:
Flux compactifications in Type IIB string theory
• In
flux
compac3fica3ons
of
Type
IIB
string
theory,
warping
can
arise
from
the
backreac3on
of
fluxes/objects
onto
the
compact
space
→
warped
throats
• Single
Dp-­‐branes
can
move
as
probes
in
this
geometry,
with
a
DBI
ac3on
Warped throat
CY3
D3,
h(r) wrapped D5…
D7
• The
disformal
coupling
arises
generically
from
this
set-­‐up,
as
we
will
now
see….

10. 0

V ( ) = V0 e
Disformal couplings from DBI:
 Z
p R p
FluxS compactificationsLin Type )IIB string theory
= d4 x g
16⇡G
+ g (matter, gµ⌫
¯ ¯
• Recall:
gµ⌫ = C( , X)gµ⌫ + D( , X) ,µ ,⌫
¯
– The
disformal
metric:
this
arises
from
the
pull-­‐back
of
the
10
dimensional
metric
on
the
worldvolume
of
a
moving
D-­‐brane
– The
scalar
field:
the
changing
posi3on
coordinate
of
the
brane
is
a
scalar
field
from
the
four-­‐dimensional
point
of
view
(we
consider
mo3on
in
one
transverse
direc3on
only)
– The
func7ons
C
and
D:
both
are
given
by
the
warp
factor,
1/C=D=√h
– The
disformally
coupled
maVer:
whatever
stuff
resides
on
the
moving
brane

11. Disformal couplings from DBI:
Example – D3-brane in AdS5
• Consider
a
probe
D3-­‐brane
moving
in
the
radial
direc3on
of
an
AdS5-­‐type
geometry
induced
by
a
stack
of
D3-­‐branes
– The
disformal
metric:
– The
scalar
field
ac7on:
+
poten3al
+
charge
D-brane probe in AdS5
geometry from stack of D3-
branes

12. Disformal couplings from DBI:
Coupling to Matter
• Now
lets
couple
the
scalar
to
mager:
– Open
string
endpoints
→
U(1)
vector
fields
on
the
world-­‐volume
– These
can
acquire
masses
via
Stückelberg
couplings
to
bulk
2-­‐forms
→
The
massive
vectors
(or
their
decay
products)
are
-­‐
dark
to
our
standard
model
-­‐
disformally
coupled
to
our
metric
g
D-brane probe in AdS5
geometry from stack of D3-
branes
• Finally
lets
summarise
the
geometric
picture:
– Transverse
open
string
oscilla7ons
→
scalar
field
→
dark
energy
in
cosmology?
– Parallel
open
string
oscilla7ons
→
vector
field
→
dark
maVer
in
cosmology?
[Work under progress with Ivonne Zavala]

13. ν ν C−2DX
der, partial differential equation. Its hyperbolic char- presence ofG2 = G3 energy 5 ede 0 , 3(1 G4 = 21[1
early dark = G Ω= = + w)/γ
density keepsthe signaturemodifiedthe time derivative term
the correct sign of gravity [2]. The new features nus unity asymptotically. Th˙
acter depends on class of of the tensor M , which
f (R) µν
appear when the disformal factor Dφ
arXiv:12
involves the However, a large pressure cantensor.
if D > 0. coupled matter energy-momentum flip the sign of the order avoid a singularitythat the eff
grows towards to one. Then the clocks in tick fo
However, for any introducingit, ofinstability.modification,, slow wherea de Sitt
For a perfect fluid, in coordinates comoving
spatial µ derivatives coefficient, type gravity ¯ universe ˙ enters1intoand make=
other with an dark matter, g00 = −1 + Dφ2 down Q th
Toy model: ΦDDM
D
Mµν = δνthe relation between the matter and gravitational state for dark matter approach m
The present analysis focuses onderivative term effective equationsistance + 1 + 2X ,µ ,⌫also
non-relativistic environ- gµ⌫ of metric pathology was
˜ = gµ⌫ to
− C−2DX diag(−ρ, p, p, p). Positive energy
density keeps the correct sign of the time
Dp nus unity asymptotically. The field also begins to freez
ments, However, a large pressure can flipwill sign of can also be a motivated the effectiveThe coupling
if D > 0. will be non-conformal.theThisfurther to avoid singularity in e.g. scenariogµν , and th
and hence C−2DX 1 be the assumed. self-coupling metric ¯ [7, 8].
spatial derivativesDBIaddress the effects of pressure, includ-
Future work will type scenario where matter is allowed to de Sitter stage. Thiswhich re
in a coefficient,on non-relativistic environ- universe enters into Thus, the disformal coupl
introducing an instability. a enter tives, natural e
The present analysis focuses under which the stability sistance to pathology was also observed in the disforma
ing the circumstances condition C( ) = 1
The dark ingredients:
ments, andthe additional will be further assumed.When given scenario [7, 8].
Dp
hence C−2DX 1 dimensions [3]. 11 self-coupling bythat triggers the For a point
can breakwill address the effects of[12].
down dynamically pressure, includ-
a scalar
4
transition to
Future work
field φ,cosmological model. condition consider an disformal as  D0 steeper mechanism
the disformal relation can beThus, the D( ) relatively rect weight of
parametrized = e (
The coupling provides a the slop0)
• A canonical quintessence field Φ
ing the circumstances under which the stability Let us
An example
can break down where the [12]. acts as quintessence and the the transition to an V0 e the √
- VI. COSMOLOGY
application dynamically field that triggers
V ( ) the slopehigher  ratio β/
is, i.e. the 4 accelerated expansion
=  the
n
• DDM living in
disformal coupling is used gµν to trigger µν + D(φ)φ,µrelatively steeper (1)asofseendisformal−¯Lm
An example cosmological model. =Let us consider an
¯ The φ ,
C(φ)g cosmic acceleration. happens,
,ν g¯
functio
in FIG.1. Th
ΦDDM cosmology
application where the viability quintessence and the is, i.e. the higher a short “bump” in the equatio
Having addressed the field acts asof the theory inthe the ratio β/γ, the faster the transitio
The Friedmann used to trigger cosmic usual form happens, as seen in FIG.1. This transition also produce
let us equations have the Z 
g • 2 extra parameters wrt ΛCDM, everything at Planck scale
Solardisformal coupling is commas denote acceleration.
System, where consider its cosmological implica-
tions. Using the Einstein Framethe usual form the Fried- S =
The Friedmann equations have description,
From the poin
partial derivatives.4 Considering of state, which may hav
a short “bump” in the equation observational cons
interesting
p R p
e
mann equations have the general˙ 2
the most usual form 8πG physical case, Bekenstein d observational16⇡G + time L (matte
˙
φ2 interesting x full g consequences. Wethe analy
background g performed
[4] argued that with MCMC part ¯
The Friedmann equations:
2
Canonical field + DDM:
, H +K = 8πG φ (ρ + + V ), full background MCMC analysis a modified versio
- ● H 2 + K functions 3 + V ) , D may alsoofdepend upon CMBEasy [14] using the U
both 8⇡G = (ρ + C and 2 of
(∂φ) ,2
CMBEasy [14] using the Union2factor C. comp
Supernovae In
k 3 ˙2
2
Φenergy [18]. Th
lation [15], WiggleZdata [16], cos
H + 2˙ we will(⇢ + 4πG V the simpler case here. WiggleZ baryon acousticdirection-depe
2 baryon ac
d but = H 3 4πG − 2 + (ρ + , ˙ 2 (57) lation [15], Previous ap-
˙ aH + 2 focus on
), scale
H + H 2 = − = (ρ + 2φ2 − 2V ) 2φ − 2V ) , mic microwave background angular scale [17] and bound
˙
s
plications of
3 3 gµ⌫ = C( ,microwaveof theX) ,µ
¯dark energy vary- + background an
mic X)g
µ⌫
4⇡G such a relation to cosmology include[18]. The obtained constraints ar
on early
D( , four-ve
f ˙
H +H 2
= (⇢ + 2 ˙2 2V ) , (58) on early dark
but the conservation equations for matter and the scalar
but the ing speed 3 from (3), theories [5], and the scalar darkWe see in FIG.2. We see th
conservation equations for matter inflation in FIG.2. shown that for steep slopes γ an
field have to be computedof light (7):
shown
[6], energy
e The (non)conservation equations:
field have to equations for from and the
β, the background evolution becomes ◆
but the conservationbe computedmatter (3), (7):scalar ilar to ΛCDM. At this ✓ background higher bound
[7, 8], ˙ gravitational alternatives to [9, 10] and extensions X β, the there are no evolution ˙2
increasingly sim
x
¯
¨ ˙
field have + 3Hρ computed from3H φ + V = −Q0 ,
ρ to be = Q0 φ , φ + (43, 45):
d level
˙ (8) ΛCDM. At this0
>
-
of [11] dark, matter. 3H φ + V = −Q0 , χ2 (8) 538.79ofto C χ2 XD 538.91 level t
ilar t
on these parameters, and the model is completely viabl
- ρ + 3Hρ = Q0 φ
˙ ˙ φ¨ + The disformal with disf =
˙ generalization versus + Extremising fi
dX cou- ΛCDM = (best
on these parameters, and the
g ●
-
(Non)conservation equations:
⇢ + 3H⇢ = Q0 ˙ ,
˙
were the background order coupling factor reads
¨ + 3H ˙ ˙+ V 0 = C Q0 ,
(59) WMAP7 parameters). However, the model is essentiall
pled quintessence here introduced is a simple set-up that = 538.79 versus χ
background order φ2 ) + D φ features different relation
(60)
withasχ2quite obvious when one look
were the C useful to+studycoupling2 factor reads the from ΛCDM, (1) disfparameters). Howevis shows that it
is − 2D(3H φ V + C ˙ ˙
generic ρ , (9) at the effective dark matterin
of WMAP7equation of state in FIG.1.
, Q0 = gµ⌫ = g +
˜
- different 2D(3H φ + V + matter + D φ2 Cosmological Perturbations.µ⌫A more,µ ,⌫ asdescrip
scenarios.
were ⇢ is the energy density of the φ2 )
˙
˙
2 C + D(ρ − coupled C
˙ 2
com-
˙ different from ΛCDM, is qu
realistic
s
C −
ponent and the background coupling factor reads ) C φ at the cosmological perturbations. e
effective dark matter I
Q0 =
after solving away the higher derivatives. In the2
ρ , (9)
tion requires considering
- 0 ˙ 2 C +2D(ρ −˙ 2 )
0 0
A ˙ 0 φ ˙ following the Newtonian gauge, the linearized field equation is

14. gµ⌫ = C( , X)gµ⌫ D( D( = X) D0 + ⌫
¯ + ) dX  Cµ e, XD
, , (7)
4 
V( ) =  V0 e
ΦDDM: the background story
✓ ◆
d Z X  gµ⌫ = gµ⌫ + ,µ ,⌫
˜
p >0
R p (8)
S = C 4 x XD g
dX d+ + g L (matter, gµ⌫ )
¯ ¯ (6)
16⇡G
1
G2 = G3 = G5 = 0 , G4 = (1 + 2X)
Converging to the µ⌫ = gµ⌫ + ,µ
,⌫ Radiation era, Matter era
Disformal “freezing”
gscaling attractor*
˜ (9) Sitter era
De
gµ⌫ = C( , X)gµ⌫ + D( , X) ,µ ,⌫
¯ (7)
matter Practically
slow down:
The quintessence
arbitrary initial0 , Gd = (1˙+ 2X)◆1
G2 = G3 = G5 =
✓
X= 1 + D e ( Acceleration
(10)
g00 = 1 + D
¯ 4 2
scaling solution0 ) ˙ 2
>0 0 (8)
conditions dX C + XD
down:
13
g00 =
¯ 1+D = V ⇠ e) ˙ 2 with exponential V
The coupling then triggers acceleration
The early evolution of 2 usual “self-tuning” scalar
˙ the (
1 + µ⌫ 0= gµ⌫ +
gD e
˜ 0
,µ ,⌫ (11) (9)
he * a saddle point if β>γ. Then as field
gravitational metric also avoids problems with 10 108
n gravity tests and the subtleties related to the  1
ce of di↵erent frames, hence e 3 , the the anal-
rolls down the G2 ⇠ G = G5 = 0 , G4 = (1 + 2X)

V = simplifying D ⇠ e (12)
Coupled Matter (10)
cosmologicalto matter:
begins observations.
rk matter slow down: 10 107
udy the dynamics within a particular example, Scalar Field
D⇠e 
s on a simple Disformally + D ˙ 2 ! 0 =Matter D e ( (13)
g00 = 1 Coupled Dark 0) ˙2
4
¯ 1+ 0 (11)
Ρ Mpc
M) model, constructed with the following prescrip- 10 106
Γ 10 all
• The field slows rolling V ⇠ e 
exponentially Β 5Γ
(12)
ark Matter disformally coupled to a canonical 10 105
Β 15 Γ
alar• field, following Eq. (59-61).
Eventually dark matter freezes too
i.koivisto@fys.uio.no Β 40 Γ
n exponential parametrization is never flipped!
• However, the sign for the ⇠ e 
D disformal 10 104
(13)
0.1 0.2 0.5 1.0 2.0 5.0 10.0
ys.uio.no the scalar field potential:
lation and a
( 0 )/Mp
B = B0 e , (73) 0.5
/Mp

15. µ
perfectD , X)g in D( p, p, p).
δa g− = C( fluid, µ⌫ +coordinates ,⌫ Positive energy G2 = G3 = G5equation = (1 φ2˙2X) 1down and m
ν ¯µ⌫ C−2DX diag(−ρ, , X) ,µ comoving with it,
˙
dark matter, g(7) =G4 1of D + ,2slow darkD e
effective ¯ 000 = −1 + state = 1 + matt
g ,
=
µ µ D
¯
00 + D for 0
eeps δν −correct sign of thep, p). derivative term effective equation of state for dark matter appr
ν = the C−2DX diag(−ρ, p, time Positive energy nus unity asymptotically. The field also
✓Clocks made ◆
dX a + XD
X
ΦDDM: constraints
However,da correct sign of of dark matter slowterm
>0
the time the sign down:
sity keeps the large pressure can flipderivative of the nus unity asymptotically. Thethe effective me
> 0. However, C large pressure can flip the sign of the
to avoid a singularity in field also begins
erivatives coefficient, introducing an instability. to universe 2enters + in the effectiveemetric gµν ,
tial derivativesfocuses on introducing an instability.
g00 = 1 +
¯
(8)
a ˙ = 1 into ( V ⇠) ˙ 2  ¯
avoid D singularity D e a de Sitter stage.
0
0
ent analysis coefficient, non-relativistic environ- universe enters pathologySitter also observed
sistance toMota & a de was stage. (2012)
]na
[TK,
into Zumalacarregui PRL
This
Dp focuses on non-relativistic environ-
e present analysisgµ⌫ + ,µ will be further assumed. sistance to pathology was also observed in the 3d
nd hence gC−2DX ˜µ⌫ =
Dp
1 ,⌫ self-coupling scenario [7, 8].
(9)
nts, and hence C−2DX effects of pressure, includ- self-coupling scenario [7, 8]. D ⇠ e 
work will address the 1 will be further assumed. V ⇠e 
We used the the effects data:
ircumstancesaddress following of 1
ure work will under which the stability condition
0.5 pressure, includ- Thus, the disformal coupling provid
G2 = G3 = G5 = 0 , G4 = (1 + 2X) Thus, the (10)
the circumstances under which the stability condition
k down dynamically [12]. that triggers the transition provides a me
disformal coupling to an accele
D⇠e 
wn: • down dynamically [12]. distance – redshift diagram The relatively steeper the slope of the ex
break Supernovae Ia luminosity Let us consider an that triggers the transition to an accelerated di
ample Baryon acousticmodel.
• cosmological oscillation Let The relatively steeper the slope ofβ/γ,disformal
An example ˙cosmological model. (scaleus2 consider an is, i.e. the higher the ratio the the fast
0.0
happens, ΦDDM cosmology
0) ˙
0 = • 1 + D the microwave background angular scale
on where !field actsD0 e 0 = 1 + as
2 Scalar quintessence and the
Field
licationCosmic the field acts as Matter
(11)
is, i.e. the higher the ratio β/γ, the faster the tr
w
where Coupled quintessence and the
l coupling is used 0.5 trigger cosmic acceleration. happens, as seenseen in FIG.1. transition also p
to
• BBN constraints onΓearly dark energy
as
in Φ:
FIG.1. This
This transiti
dmann priorsV ⇠Hubble Βthe usual form
10 all
a● Evolution of
ormal coupling on used to trigger cosmic acceleration. (BBN)short “bump” in the equation of state,
is
• +equationse have rate (HST) and baryon fraction a short “bump” in the equation of state, which m
⇤ 40 Γ
e Friedmann equations haveElectronic address: tomi.koivisto@fys.uio.no of self-tuningTRACKING"
the usual form (12) The "EXACT
Practically ARBITRARY ”Disformal freez
interesting observational consequences.
●
Β 15 Γ
scalar field
interesting observational consequences. We perf
V B
Β 5Γ initial conditions DE SITTER expa
8πG φ ˙ 2˙ 2 0.01 full background MCMC analysis with a
D ⇠ e10  + φ +tomi.koivisto@fys.uio.no full background MCMC analysis with a modified
1.0
2
H And+ = = 8πG (ρ + bounds on γ and β/γ:
of CMBEasy [14] using the Union2 Su (13)
4 0.001 0.1 1 10 100
⇤ Electronic address:
+ K obtained lower +V )),, a
H 2
K 3 (ρ V
3 22 of CMBEasy [14] using the Union2 Supernovae
4πG lation [15], WiggleZ baryon acoustic sca
˙• + Since 2 ⌦−⇠(blue) (ρ different2choices of field coupling slope β. Highmic [15], WiggleZbackground angular data [
2 4πG + 2φ ˙ 2−scaling(red) and coupled mat- FIG. 2: Marginalizedbaryon two-sigma regions obtained
FIG. 1: Equation of state for the
H H + H = − for(ρ + 2φ − 2V the
H = ter
˙
2
during ˙the 2V ) , era lation (14)
acoustic scale
microwave one and angular scale [17]scale scale
),
3 β/γ (solid, dashed)preferredfit to observations, microwave background (Green), CMB angular and
mic from Supernovae (Blue), BAO
high values for γ are not produce enough acceleration.on + early dark energy bounds (Orange), The obtained
values of 3 give a good
while low values (dotted) do on early dark energyincluded [18]. obtained the HST
early dark energy The and combined con-
[18]. a prior on H0 constr
the • conservation freezing for matter and the scalar shown in FIG.2. We Bang Nucleosynthesis from slope
conservation the equations is matter and the scalar
Since equations for then swifter, straints. All contours
shown inb HFIG.2. see that for steep stee
[21] and Ω 0 from Big We see that for
2
[22].
d have tocomputed from (3),β/γ are preferred equations β, β, the background evolution becomes i
high values for continuity and Euler
e to be be computed from (3),(7):
while the perturbed (7): for the background evolution becomes increasin
o.no
• The expansiondark mattera(t) then resembles ΛCDMto to ΛCDM.this level there are no higher
coupled history are ilar ilar ΛCDM. At At this level there are n
˙Q0 φ , φ + + 3H φ Q0 V = −Q0 ,0 ˙ (8)˙
˙ ¨ φ 3H θ˙ + V˙ = −Q 0 ,
¨ ˙ φ ˙ + CDM ˙are
3Hρ 3Hρ =φ ,though both Λ and φδ = 3Ψ + Qvery δQ φ , (11)on these parameters,the model model is c
ρ + = Q0
˙ (8) on
δ+ + δ φ + different! these parameters, andthe late time dependence dur-
For our example model, and the is complete
a ρ ρ ρ ing dark energy domination produces a large enhance-
Q0 ˙ Q0 with2 2 the matter versus efΛCDM 2 538.91
with mentχof = 538.79growth, δGχ2 /G ∼χ(γV /ρ)2 =
χdisf disf = 538.79 versus ΛCDM 1, f
=
e the background order+couplingfactor kreads ρ δφ .
background order coupling ρ factor reads
˙
θ θ H+ φ = 2 Φ+ (12)
WMAP7 parameters).to avoid However, early is es
WMAP7 parameters). the effects model dark
as γ 10 is required However, the of the mo
energy. Such behavior is in tension with large scale struc-
The general coupling perturbation2 δQ is a much moredifferent from ΛCDM, quite obvious when o
different observations, and also is as in conformally coupled
from ΛCDM, as occurs is quite obviou
˙ ˙ C C ˙ 22˙ ˙2 ture