particles of dark energy observable e ects -...

IntroductionFifth forces and screening

Particles of dark energy

MotivationObservable effects

Outline

1 IntroductionMotivationObservable effects

2 Fifth forces and screeningScreening in relativistic starsTorsion pendulum testsCosmic expansion vs. growth

3 Particles of dark energyScalar-photon oscillationGammeV-CHASE

Amol Upadhye Tests of Modified Gravity




What if w(z) ≈ −1?

Many models other than Λ give w ≈ −1

V(φ)

minimum of V (φ)

V(φ)

slow roll

V(φ)

flat potential

What sorts of models are consistent with observations?

How can these models be distinguished using the data?

One class of models: modified gravity and scalar dark energy





Modified gravity and scalar fields

Since our universe looks 4-dimensional (at least since BBN), theremust be an effective 4-D description of modified gravity. Thesimplest models reduce to 4-D matter-coupled scalar field theories.

Modified gravity

f (R) gravity:action S =∫ d4x

√−g16πGN

f (R)

DGP, etc.:non-compactextra dimension

Kaluza-Klein,etc.: compactextra dimension

Effective scalar

Conformaltransformation⇒ chameleon

Decoupling limit(weak gravity)⇒ Galileon

Small extradimension limit⇒ radion

New physics

matter coupling,self-interactionV (φ)

matter coupling,non-canonicalkinetic term

matter coupling,photon (gaugefield) coupling





Effects of modified gravity

These new scalars can lead to:

fifth forces between masses;

equivalence principle violations;

variations in fundamental constants;

new particles.

Since gravity looks like General Relativity locally, fifth forces mustbe screened.

chameleon screening: large effective mass locally

Vainshtein screening: effectively weak coupling at high density

symmetron mechanism: field decouples at high density assymmetry is restored





Chameleon mechanism

V(φ)





Chameleon mechanism

Vint = βmat ρmat φ / MPl

V(φ)





Chameleon mechanism

φmin(ρlow)

(meff2 = V’’ is small)

V(φ)Veff(φ,ρlow)





Chameleon mechanism

φmin(ρlow)

(meff2 = V’’ is small)

φmin(ρhigh)

(meff2 is large)

V(φ)Veff(φ,ρlow)

Veff(φ,ρhigh)





Thin-shell screening

Chameleon field equation of motion: φ = V ′(φ)− βm

MPlTµµ

Linear regime: V ′ negligible

Static: ∇2φ = − βm

MPlTµµ

Nonrelativistic: Tµµ ≈ −ρ

e.o.m. ≈ Poisson equation∇2Ψ = 4πGρ = 1

2βmMPl∇2φ

φ = 2βmMPlΨ + constant(scalar follows thegravitational potential)

Transition regime: Ψ ∼ χscr

χscr = 12βmMPl

∆φ(max)

Nonlinear regime: φ negligible

Nonrelativistic limit:V ′(φ) = βm

MPlρ

⇒ φ→ φbulk(ρ) (constant)

-1.5

-1

-0.5

0

0.1 1 10 100r / rs

φ/φ∞-1Ψ/|Ψs|





At which scale should we probe each model?


V (φ) ∝ φn + const. ⇒ meff ∝ ρn−2

2n−2 (use lab for n . −12 , n > 2)

1e-50

1e-40

1e-30

1e-20

1e-10

1

1e-30 1e-20 1e-10 1

mef

f ∝ ρ

(n-2

)/(2n

-2) [e

V]

density ρ [g/cm3]

V(φ) ∝ φ n

n=-1n=1/2n=2/3

n=4

labo

rato

ry

cosm

olog

y



Screening in relativistic starsTorsion pendulum testsCosmic expansion vs. growth

Screening in nonrelativistic stars (χscr = 0.0014)


0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 1e+10 2e+10 3e+10 4e+10

φ / M

Pl (

solid

) and

2βΨ

(das

hed)

distance from center, r [m]

Ψs = 0.0006






0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 1e+10 2e+10 3e+10 4e+10

φ / M

Pl (

solid

) and

2βΨ

(das

hed)


Ψs = 0.0006Ψs = 0.0009






0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 1e+10 2e+10 3e+10 4e+10

φ / M

Pl (

solid

) and

2βΨ

(das

hed)


Ψs = 0.0006Ψs = 0.0009Ψs = 0.0012






0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 1e+10 2e+10 3e+10 4e+10

φ / M

Pl (

solid

) and

2βΨ

(das

hed)


Ψs = 0.0006Ψs = 0.0009Ψs = 0.0012Ψs = 0.0017






0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 1e+10 2e+10 3e+10 4e+10

φ / M

Pl (

solid

) and

2βΨ

(das

hed)


Ψs = 0.0006Ψs = 0.0009Ψs = 0.0012Ψs = 0.0017Ψs = 0.0022




Equations of motion: field and metric

metric: ds2 = −N(r)dt2 + dr2

B(r) + r 2(dθ2 + sin2θ dϕ2)

hydrostatic equilibrium: P ′(r) = − N′2N (ρ+ P)

equation of state: ρ(r) = constant (1g/cm3)

modified Einstein eq. (trace, tt, rr), fR = dfdR , φ = −MPl

2βmlog fR :[

f ′′R +

(2

r+

N ′

2N+

B ′

2B

)f ′R

]B =

dV

dfR− 8πG

3(ρ− 3P)

(−1 + B + rB ′)fRr 2

+

[f ′′R +

(2

r+

B ′

2B

)f ′R

]B = −8πGρ+

f − RfR2

(−1 + B + rBN ′/N)fRr 2

+

(2

r+

N ′

2N

)f ′RB = 8πGP +

f − RfR2





φ(r) in a relativistic star (χscr = 0.1)


1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1e+09 1e+10 1e+11 1e+12 1e+13

φ / M

Pl

r [m]




Thin-shell screening in relativistic stars

chameleon screening suppresses growth of φ:

1e-05

0.0001

0.001

0.01

0.1

1

1e-05 0.0001 0.001 0.01 0.1 1

(φ0

- φ∞

) / M

Pl

GM∗ /r∗

∆φ(max)=0.1

∆φ(max)=0.001

GM ∗/r ∗

relativisticnon-relativistic

AU and W. Hu (2009)Amol Upadhye Tests of Modified Gravity




Fifth-force constraints from a torsion pendulum

Eot-Wash Experiment

http://www.npl.washington.edu/eotwash





φ4 chameleon field in Eot-Wash pendulum


-6 -4 -2 0 2 4 6

-0.6-0.4

-0.2 0

0.2 0.4

0.6 0.8 1

1.2

0

5

10

15

20

25

φ [mm-1]

20 15 10 5

x [mm]

z [mm]

φ [mm-1]




φ4 chameleon field in Eot-Wash pendulum

AU, S. Gubser, J. Khoury (2006)





Eot-Wash constraints

E. Adelberger, et. al. (2007); D. Kapner, et. al. (2007)

V (φ) = λ4!φ

4

β

λ

βm





Chameleons with small quantum corrections


∆V1−loop(φ) = meff(φ)4

64π2 log(meff(φ)2

µ2

)⇒ meff , φbulk change

Eot-W

ash φ

4

0.01 0.1 1 10 100 1000matter coupling ξ

0.0001

0.001

0.01

0.1

mas

s m

φ(ρ l

ab) [

eV]

excluded by Eot-Wash

large quantum corrections

Eot-W

ash φ

4

0.01 0.1 1 10 100 1000matter coupling ξ

0.0001

0.001

0.01

0.1

mas

s m

φ(ρ l

ab) [

eV]

excluded by Eot-Wash

large quantum corrections

AU, W. Hu, J. Khoury (2012)




Self-accelerated DGP: Ωm sets expansion and growth


23

1e−05 0

0.6 21.5 22

2000

0.7

22.5

4000

0.8

23.5

0.0001

24

6000

0.9

24.5 0.4

1 0.6 0.2 0.8

0.001

1 0.4 1.2

10

1.4 0.6 0.8

100

1

100

Gro

wth

δ/a

a

m−

M

z

1000 1000

l(l+1

)Cl /(2

π )l(l+

1)P

l

κ /

(2

) π

l

c) d)

b)

ΛCDM:w =−1,w =01

1DGP: =0.27DGP: =0.20Ωm

Ωm

0

a)

SUGRA:w =−0.8,w =0.30

M. Ishak, AU, D. Spergel (2006)




Combined data exclude self-accelerated DGP

choose Ωm to fit expansion (SNe) ⇒ large CTT` at low `

ΩK helps fit expansion but makes low-` power larger

suppressing initial large-scale power ruins low-` fit to CEE`

⇒ self-accelerated DGP ruled out to 4.8σ (w.r.t. ΛCDM)

10 100 1000

0

2000

4000

6000

10 100 1000

0

1000

2000

3000

4000

5000

6000

7000

WMAP 5 k

min=0

kmin

=8 x 10-4 Mpc-1

(+1

)CTT

/2(

K2 )

10 100 1000

10-3

10-2

10-1

100

101

10 100 1000

10-3

10-2

10-1

100

101

kmin

=0 k

min=8 x 10-4 Mpc-1

(+1

)CE

E /2(

K2 )

(W. Fang, et. al, 2008)





f (R) model with V (φ) ∝ φ1/2

f (R) gravity “looks like” dark energy with w ≈ −1f (R)− R ∝ 1/R + const.⇒ V (φ) ∝ φ1/2 + const. withχscr > 10−4 has unscreened fifth forces, hence an abundanceof large clusters which is inconsistent with observations.

effective w(z)

0.01

0.01

0.03

0.03

0.1= Xscr

0.1= Xscr

z

wef

fw

eff

2

-0.95

-0.9

-1

-1.1

-1

-1.05

4 6 8

(b) n=4

(a) n=1

power spectrum 8

with the halo model better reveals the internal consis-tency of our simulations. The agreement between thesmallest box and the larger boxes with coarser resolu-tion and smaller particle number is ! 20% in case of the128Mpc/h boxes, and ! 40% for the 256Mpc/h boxes. Inthe following, we show results from the 128Mpc/h boxesfor the largest mass halos, in order to increase halo statis-tics, and from the 64 Mpc/h boxes for all other masses.

Fig. 8, top panel, shows the stacked halo profiles forthree mass bins, for !CDM and full f(R) simulationswith |fR0| = 10!4. The lower panel of Fig. 8 shows therelative deviation between !CDM and f(R) halo pro-files. When scaled to the same overdensity radius, halosin !CDM and f(R) apparently have very similar profiles,especially in the inner part of the halo. Although a pre-cise measurement of the NFW scale radius is not possiblewith our limited resolution, it is apparent that there areno dramatic e"ects of modified gravity on the halo con-centration c300 ! r300/rs. Moreover the deviations areconsistent with zero well within r300. The same holds forthe no-chameleon f(R) simulations.

For the intermediate and larger halo masses, there isan enhancement of the halo profile at r/r300 " few, i.e.in the transition region between one-halo and two-halocontributions. The smallness of the enhancement of !hm

can be explained by a partial cancellation between theincreased linear power spectrum and reduced linear biasin f(R) (§ B and § III B). However, a quantitative un-derstanding of the behavior of the halo-mass correlationat these radii is not possible with the simple halo modeladopted here, as it fails in the transition region betweenone and two-halo terms (see Fig. 7). In the small fieldsimulations, the deviations in the halo profiles are toosmall to be measured with our current suite of simula-tions.

Given the relative smallness of the modified gravitye"ects on halo profiles, the main e"ect of enhanced forcesin the large field simulations is to change the mass andhence the abundance and bias of halos.

D. Halo Model Power Spectrum

We can now put the halo properties together and dis-cuss statistics that can be interpreted under the halomodel paradigm outlined in §B. The matter power spec-trum Pmm is especially interesting in that the enhance-ment in the large field f(R) simulations found in [12] wasnot well described by standard linear to nonlinear scal-ing relations [22]. Without an adequate description ofthe large field limit, robust upper limits on |fR|, whichshould be available from current observations, are di#-cult to obtain.

The halo model provides a somewhat more physicallymotivated scaling relation between the linear and non-linear power spectra [23]. Specifically we use the samerange of ST predictions for the mass function and linearbias discussed in the previous sections in Eq. (B8). In ad-

FIG. 9: Power spectrum enhancement relative to !CDM for fulland no-chameleon simulation and di"erent fR0 field strengths. Theshaded band shows the predictions from the halo model using pa-rameters derived from spherical collapse (see text).

dition, we vary the concentration parameter of the halos,using either an unmodified cv(Mv) relation [Eq. (B6)], oran unmodified c300(M300) ! r300/rs. The latter relationis motivated by our finding that the inner parts of haloprofiles are unmodified in f(R) when referred to the sameoverdensity radius (§ III C). Converting c300 to the virialconcentration, we obtain a "10% higher cv, which in-creases the power spectrum enhancement at k " 1h/Mpcthrough the 1-halo term [Eq. (B8)].

The range of halo model predictions is shown in Fig. 9for di"erent values of fR0, together with the simulationresults from [12]. The upper boundary of each shadedband corresponds to unmodified spherical collapse pa-rameters and unchanged c300, while the lower boundaryis using the modified spherical collapse parameters, as-suming enhanced forces throughout in the f(R) predic-tion, and unchanged cv.

The halo model provides a reasonable approximationto the relative deviations in the large field limit out tothe k " 1 # 3 h/Mpc scales that can be resolved by thesimulations. The modified collapse provide a somewhatbetter and more conservative approximation for the pur-poses of establishing upper limits for |fR0| " 10!4.

The halo model still fails to capture the chameleon sup-pression in the small field limit. Its failure is apparenteven at |fR0| = 10!5 for 0.1 ! k(h/Mpc) ! 1 and is rel-atively larger than the error in the mass function, linearbias and halo profiles themselves. This range also corre-sponds to the regime where the one halo and two haloterms are comparable, i.e. where our simple prescription

cluster counts 4

FIG. 1: The halo mass function as a function of M300 measuredin !CDM simulations with bootstrap errors on the mean. Theupper panel combines di"erent box sizes from 64 to 400Mpc/h andcompares results with the Sheth-Tormen prediction rescaled fromMv to M300 as described in the text. The lower panel shows therelative deviations from this prediction separately for di"erent boxsizes.

III. RESULTS

In this section we present the results obtained fromN-body simulations of the f(R) models for the halomass function (§III A), halo bias (§III B), density profiles(§III C) and matter power spectrum (§III D). In all cases,we compare the simulation results with predictions usingscaling relations based on spherical collapse calculations,the Press-Schechter prescription and findings from simu-lations of !CDM. These calculations are detailed in theAppendices.

Since spherical collapse predictions depend on thegravitational force modification, we give a range of pre-dictions in each case. The extremes are given by collapsewith standard gravity and with enhanced forces through-out. The former follows the !CDM expectation of a lin-ear density extrapolated to collapse of !c = 1.673 and avirial overdensity of "v = 390; the latter modifies theseparameters to !c = 1.692 and "v = 309 as detailed inAppendix A.

Neither assumption for the nonlinear collapse is com-pletely valid given the evolving Compton wavelength andthe chameleon mechanism. Moreover, the evolution oflinear density perturbations used as the reference for thescaling relations in Eqs. (B1), (B4), (B8), and (B10) as-sumes in both cases the full linear growth of the f(R)

FIG. 2: Relative deviations of the f(R) halo mass functions from!CDM, with |fR0| = 10!4 (top panel), 10!5 (middle panel), and10!6 (lower panel). In each case, blue squares denote the fullsimulations, while red triangles (displaced horizontally for visibil-ity) denote the no chameleon simulations. The shaded band showsthe range of enhancement expected from spherical collapse rescaledfrom Mv to M300.

model through "(M), including the e#ects of the evolvingbackground Compton wavelength but not the chameleonmechanism. Thus unmodified spherical collapse parame-ters do not equate to unmodified spherical collapse pre-dictions.

A. Mass Function

In Fig. 1, we show the halo mass function measuredfrom our suite of !CDM simulations along with the boot-strap errors described in §II C. For reference, we comparethe simulations to the Sheth-Tormen (ST) mass functionof Eq. (B1). The ST formula gives the mass function interms of the virial mass and we rescale it to M300 as-suming an NFW profile (see Appendix B). Our !CDMsimulations are consistent with the 10-20% level of accu-racy expected of the ST formula and internally betweenboxes of di#ering resolution.

Next, we compare the f(R) and !CDM simulations.Our measurement of the halo mass function itself is lim-ited by statistics and to a lesser extent, resolution (seeFig. 1). However, we can reduce the impact of both ef-fects by considering the relative di#erence between thehalo mass functions measured in f(R) and !CDM simu-lations with the same initial conditions and resolution.

(Hu Sawicki 2007; Schmidt Lima Oyaizu Hu 2009; Schmidt Vikhlinin Hu 2009)




Scalar-photon oscillationGammeV-CHASE

Photons coupled to chameleon dark energy

Equations of motion (βφ MPl):

∂µ

(βγφMPl

Fµν)

= 0

φ = −V ′(φ)− βmMPl

ρmat − βγ4MPl

FµνFµν

Plane wave perturbations about background φ0 and ~B0 = B0x(Raffelt and Stodolsky 1988; AU, Steffen, and Weltman 2010):(

− ∂2

∂t2 − ~k2)ψφ = m2

effψφ +βγkB0

MPlx · ~ψγ(

− ∂2

∂t2 − ~k2)~ψγ = ω2

P~ψγ +

βγkB0

MPlk × (x × k)ψφ

φ→ γ oscillation in relativistic case:

Pγ↔φ = ~ψ∗γ · ~ψγ =4k2β2

γB20

(∆m2)2M2Pl

sin2(

∆m2L4k

) ∣∣∣k × (x × k)∣∣∣2

low-mass, ~k ⊥ ~B0: Pγ↔φ ≈ β2γB

20L

2

4M2Pl





Window as a quantum measurement device

0 5 10 15 20x [meff

-1]

density ρbackground φ0φ0 + δφphoton





A simple afterglow experiment

(a) Production phase: photons streamed through ~B0 region; someoscillate into chameleons

(b) Afterglow phase: chameleons slowly oscillate back intophotons, escaping chamber





GammeV-CHASE apparatus

1 Multiple magnetic field runs

2 Partitioning of magnetic field region

3 Modulation of detector

4 Vacuum maintained by ion pump





Expected afterglow signal


0.01

1

100

10000

1e+06

1e+08

1e+10

1e+12

-1500 -1000 -500 0 500 1000 1500 2000 2500 3000

afte

rglo

w ra

te [s

ec-1

]

time [sec]

0.05 T0.09 T

0.2 T

0.45 T

1.0 T

2.2 T5.0 T

observation period βγ=3e13




Constraints on dark energy


V (φ) = M4Λ exp(φN/MN

Λ ) ≈ M4Λ + M4−N

Λ φN

matter coupling βm

phot

on c

oupl

ing

β γ

Collider constraints

g γ =

βγ /

MP

l [G

eV-1

]

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

10000 1e+08 1e+12 1e+16 1e+20 1e+24 1e+10

1e+11

1e+12

1e+13

1e+14

1e+15

1e+16

1e+17

N=-1N=-2N=-4N=4, λ=10-2

N=4, λ=10-4

matter coupling βm

phot

on c

oupl

ing

β γ

Collider constraints

g γ =

βγ /

MP

l [G

eV-1

]

1e-8

1e-7

1e-6

1e-5

1e-4

1e-3

1e-2

10000 1e+08 1e+12 1e+16 1e+20 1e+24 1e+10

1e+11

1e+12

1e+13

1e+14

1e+15

1e+16

1e+17

N=-1N=-2N=-4N=4, λ=10-2

N=4, λ=10-4

(J. Steffen, AU, et. al, 2010; AU, J. Steffen, A. Chou 2012)




Conclusions

Many models “look like” Λ in that w(z) ≈ −1;

f (R) modified gravity (chameleon);

DGP/cascading brane world models (Galileon);

compact extra dimensions (radion).

Modifications to gravity lead to 5th forces testable at many scales:

laboratory: torsion pendulum experiments (Eot-Wash);

stellar systems: Kepler’s law, relativistic stars;

cosmology: expansion H(z) vs. growth G (z).

Modified gravity models reduce to 4-D scalar theories coupled tomatter, and, possibly, to gauge fields.

production and detection through scalar-photon oscillation





The End.


particles of dark energy observable e ects -...

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