2018/02/19-02/23 YKIS@YITP

Constraining modified gravity
with galaxy bispectrum
DY, S.Yokoyama, H.Tashiro, PRD96, 123516(2017)

Daisuke Yamauchi
Kanagawa University

Modified gravity theories➢General relativity : massless spin-2➢Modified gravity : A new d.o.f. is introduced toachieve the accelerating expansion
f(R), DGP, Galileons, (beyond-)Horndeski, massive gravity, …

✓Scalar-tensor theories• Gravity is mediated by gμν and φ• Can capture essential modification of gravity

Why modified gravity?
➢ Discovery of cosmic acceleration: Our understanding of the Universe
is incomplete!
➢ Need better understanding of gravity:-- Dark energy or modified gravity ?
• Small scale : consistent withexperiments in the solar-systemand on the Earth• Large scale : can be tested though
the cosmological observations[Planck2013]

Constraining modified gravity with LSS➢ Gravitational growth index γ :f(a) = Ωm(a)γ• a powerful tool to test the modified gravity
responsible for the present cosmic acceleration

ΛCDM
γ=0.545[ FastSound (Okumura+) 2015 ]Current constraint on growth index γ
γ = 0.566 +- 0.056 (68%CL) [Mueller+(2016) : SDSS-III]
γ = 0.609 +- 0.079 (95%CL) [Sanchez+(2017) : SDSS-III]
γ = 0.54 +- 0.11 (68%CL) [Grieb+(2016) : SDSS-III]
γ = 0.580 +- 0.082 (95%CL) [Zhao+(2018) : eBOSS DR14]
γ = 0.54 +- 0.19 (95%CL) [Gil-Marin(2018) : eBOSS DR14]

Is γ enough?
➢ Need new possible parameterizations to capture
the essence of the modified gravity theory!
--Quasi-nonlinearity of the growth
as a way to provide new insight• Late-time nonlinear gravitational evolution havenew information, which would not be imprinted
on the growth index

Second-order index

Second-order index ξf(a) = Ωm(a)γ➢ Gravitational growth index γ :-- In order to compare the observational data and theoreticalpredictions efficiently, introducing γ should be useful.

However,-- There are many models in which the expansion history andthe growth rate are same in the fiducial model.

-- The new parameter is needed to distinguish and hopefully
exclude these model !

➢ Second-order index ξ:λ(a) = Ωm(a)ξ

Evolution of density perturbation➢ The governing equations for matter fluctuations:δm + (1/a)∇・[(1+δm)v] = 0v + H v + (1/a)(v・∇) v = -(1/a)∇Φ
The effect of the gravity theory on the evolution ofthe matter perturbations δm appears only throughthe gravitational potential Φ.

Bispectrum due to nonlinear growthB(k1,k2,k3;z) = 2D4(z) Z1(k1;z)Z1(k2;z)Z2(k1,k2;z)P(k1)P(k2) +cycwith Z1(k;z) = b1+fμ2

Z2(k1,k2;z) = b2/2 + b1 F2(k1,k2;z) + fμ122 G2(k1,k2;z) +…
F2, G2 have information of nonlinear growth➢ Second-order perturbative kernel

F2(k1,k2;z) = αs(k1,k2) – (2/7) λ(z) γs(k1,k2)
G2(k1,k2;z) = αs(k1,k2) – (4/7) λθ(z) γs(k1,k2)[ λ=λθ=1 in EdS , λ,λθ≒1 in ΛCDM [Scoccimarro+Couchiman (2001)] ][Horndeski (1974),Deffayet+ (2011),Kobayashi+ (2011)]

Horndeski theory: The most general scalar-tensor theory withsecond-order field equations.

✓have 4 arbitrary functions of φ and X=-1/2(∇φ)2.
✓is quite useful for a comprehensive study ofmodified gravity.

EFT Parametrization
Instead of considering the Horndeski functions Ga(φ,X), the EFTparameters is used to specified the cosmological information :➢ Linear perturbation

M : effective Planck massαM : Planck mass run rate αM=dlogM2/dloga
αT : GW (tensor) speed excess cT2=1-αT
αB : braiding [Bellini+Sawicki (2014), Gleyzes+(2013), Bloomfield (2013),
αK : kineticity (see also Creminelli+(2009), Gubitosi+(2013), Bloomfield(2012))]➢ To include the 2nd order perturbations, two other functions
are required

αV1 , αV2 : Vainshtein screening amplitude[(see also Bellini+(2015))]

[Takushina+Terukina+Yamamoto (2014), Kimura+Kobayashi+Yamamoto(2012)]

Small-scale effective Lagrangian• Impose quasi-static approximation• Neglect the higher order metric perturbations Ψ & Φ• Keep all terms with second-derivative of Q = H δφ: ∇2Q can be large, leading to self-screening in the vicinity of source➢ Linear order
Leff(2) = aM2 [ 2Ψ∇2Φ – (1+αT)Ψ∇2Ψ – (αT+…)Q∇2Q - 2αBΦ∇2Q + 2(αM-αT)Ψ∇2Q ]➢ Nonlinear order
Leff(NL) = M2/aH2 [ -(αV1+…) L3Gal + (αV1+αV2)ΦE3Gal – (1/2)(αV2+…)ΨE3Gal +… ][see Kobayashi+Watanabe+DY(2014),Hirano+Kobayashi+Tashiro+Yokoyama(2018) for GLPV]Second-order index ξ in Horndeski➢ Second-order index ξ :λ(a) = Ωm(a)ξ
Large ξ model canbe realized inHorndeski !
ξΛCDM+GR = 3/572~0.00524

Simple model➢ Assume that ΛCDM as the background expansion, namely w(0)=-1.
➢ Taking the small braiding limit cB→0, the situation is drasticallysimplified :• γ and ξ depend only on cM and cV1.
• We can realize the large ξ model in the case which can not
be distinguished with the standard ΛCDM with GR up to thelinear growth of density fluctuations!

Fisher analysis

SurveyΔwDEΔγΔξ

SKA1MID

0.135

0.067

0.060

SKA2

0.00850.00870.0094

Euclid

0.016

0.021

0.018

Summary➢ Measuring the galaxy bispectrtum induced by the late-timenonlinear gravitational evolution of the density fluctuations
can be used to test the gravity theory through the linear
growth rate and the second-order kernel.
✓ Second-order index ξ :λ(a) = Ωm(a)ξ

Large ξ model➢ Set the form of Horndeski functions :➢ Search tracker solution, which is characterized by the condition➢ Solving the background EoMs, the EFT parameters are given by➢ Consider the small braiding limit and the large hierarchy |p/q|<<1:

Scalar-tensor theory after GW170817
New constraint brought by the discovery of a GW event
and its optical counterpart✓ The speed of gravitational waves (“cGW”) and thatof light (“c”) coincide to a very high precision :-3×10-15 < αT < 6×10-16✓ Several possibilities : (Extreme) fine-tuning This observation is a strong indication that“cGW” and “c” strictly coincide : cGW=c