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,-"+%(".%/#01)%+-)%2"+)3+04)%('540(%"(()2)#"+0'.67(('#/0.8%+'%4'/)#.%('54'2'89:%".9+-0.8%(".;+%<)%)=>2"0.)/%<9%+-)%('.1).+0'."2%>"#"/084:%0+%4?5+%<)2'.8%+' @

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*'/0B0)/%C#"10+9%<)9'./%D0.5+)0.

Chapter 1: Dark energy

• In the sky

• In the mind

Chapter 1: Dark energy

• In the sky

• In the mind

Cosmic Pie

!"#$%&'"($)*+,&$%#$

!"#$%&'()*#+*

Concordance Model

Our Universe can be precisely described by the above six cosmological parameters.H0: the Hubble expansion rate of the present UniverseΩb: the fraction of baryonic matter in the critical density of the present UniverseΩc: the fraction of cold dark matter in the critical density of the present Universe

Then, assuming a spatially flat Universe, which btw is consistent with observationsΩΛ = 1 - Ωb – Ωc

It is the fraction of dark energy in the critical density of the present Universe

- With different parameters, the Universe would have experienced different histories

- The nature of matters is to make the Universe clumping- Before 1998, we talked about the deceleration parameter q, because we

thought the Universe was clumping!

The Great Discovery

!"#$%&'$()*+$'(,%-./*

Observational Windows!"#$%%"&'%#%(#&"$)*+"#%,"#-"."/"+$%0(1#'$+$&"%"+#!2#$1-#0%#%*+1)#(*%#1"3$%04"56*+#7104"+)"#0)#$.."/"+$%013! $1#"8'".%$%0(1#(9#-$+:#"1"+3;#

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The Expansion History

!"##"$%&'()*+&,*$-&.$/)'

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6*(/"2&7'&8"%/(*&9::;

The Latest Status

!"#$%&''()&*(+,&-./..012345

Chapter 1: Dark energy

• In the sky

• In the mindMainly based on CYF, Saridakis, Setare, Xia, Phys.Rept. 2010

What we know about DE?Background equation:

3/1/ 03 -<=<+ rr pwp

Basic features: • Negative pressure• Violating strong energy condition• Almost not clustered at cosmological scales

Acceleration requires:

Theoretical implications: • Dynamical nature• Microscopic interpretation• Possible applications

Dark Energy 70%

Dark Matter 25%

Atoms 5%

Cosmological ConstantOriginally introduced by Einstein to realize a static universe by “mistake”, it has

become the simplest candidate of dark energy.

Dynamics: w=-1

Microscopic interpretation: vacuum energy

- The most severe fine-tuning problem in physics: one needs to cut off vacuum

energy by a factor of 1,000,000,000,000, 000,000,000,000, 000,000,000,000,

000,000,000,000, 000,000,000,000, 000,000,000,000, 000,000,000,000,

000,000,000,000, 000,000,000,000, 000,000,000,000.

- Coincidence problem: one cannot explain why such a tiny energy density

happens to be comparable with that of matter at today’s value, otherwise the

universe would look totally different!

Weinberg, 1989

Dynamical FieldsQuintessence:

Phantom:

Applications: • For certain potential forms, scalar fields may present tracking behavior to

alleviate the coincidence problem • They may seed cosmological perturbations to affect the large-scale structure of

the UniverseIssues: • No clues for microscopic origins• Classical and quantum instabilities

Caldwell, astro-ph/9908168

Ratra, Peebles, 1988

Categories of Dynamics!"#$%&'()*+,$-.$%(/0$#'#/1&$+/2+*(33&$/#3&$-'$4"#$#52(4*-'6-.6,4(4#$7(/()#4#/8$

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;$,*)73#$7(/()#4/*<(4*-'=

>(4#1-/*#,=$

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! B2*'4-)=$9$+/-,,#,$6A

! G-%*.*#%$1/(H*4&=$,##$>"(7$I

J4(42,=$

! ? .*4,$9#33$

! K&'()*+(3$)-%#3,$(/#$)(/1*'(33&$.(H-/#%L#'18$M('18$N"('18$IOOPQ$R24#/#/8$>--/(&8$IOOPQS*(8$#4$(3T8$IOOUQ$N"(-8$#4$(3T8$IOAI8$IOAV

No-Go Theorem!"#$" %&'"(')*+

,"(+%&'"(-'.+"/+01(2+'3'(45+-3+%&'+678+0-)'3.-"319+,:;+<3-='(.'+0'.>(-?'0+?5+1+

.-349'+@'(/'>%+/9<-0+"(+1+.-349'+.>191(+/-'90+A-%&+1+4'3'(->+B#'..'3>'+C14(134-13D+

A&->&+)-3-)1995+>"<@9'.+%"+$:D+-%.+E"F @1(1)'%'(+>133"%+>("..+"='(+%&'+

>".)"9"4->19+>"3.%13%+?"<301(5G@&13%")+0-=-0'H+

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!"#D+'%+19HD+K&5.H:'@%H+LM8M,'34D+'%+19HD+LMMNO+P-2)13D+LMMNO

Q<D+LMMNO+R-1D+!"#D+'%+19HD+LMMS

Model BuildingsThe Key:

To realize the dynamics of w=-1 crossing over, one ought to break at least one condition presented in the No-Go theorem for dark energy.

Models: • Gauss-Bonnet Modified gravity – Cai, Zhang, Wang, CTP 2005• Yang-Mills model – Zhao, Zhang, CQG 2006• DGP brane-world – Zhang, Zhu, PRD 2007• Interacting DE – Wang, et al., PLB 2005; RPP 2016• Effective Lagrangian – CYF, et al., PLB 2007; CQG 2008• Horndeski DE – Matsumoto, PRD 2018• ……

ApplicationsExtended perturbation theory:

The parameter space would be enlarged by involving perturbations

Applications to the primordial Universe: • Non-singular bouncing cosmologies

- CYF, et al., JHEP 2017; JCAP 2008; SCPMA 2014• Emergent Universe paradigms

- CYF, et al., PLB 2012; PLB 2014

More applications?

Chapter 2: Modified gravity

What we know about gravity

10-3 cm 1 AU 1 kpc 1 Mpc 1 Gpc

Extra dimensions MOND Cosmological MG

Einstein’s GR has been precisely probed here

Why we modify gravity

Theoretical perspective: Quantum gravity, such as string theory, LQG, SUGRA, generally predicts

a modification to GR. Namely, – the scalar-vector-tensor theory

Historical perspective: • A modification to GR was initiated to explain the anomalous rotation

curves of galaxies – MOdified Newtonian Dynamics by Milgrom• The first and so far most successful inflation model is based on modified

gravity – R2 model by Starobinsky

Phenomenological perspective:There is no reason that gravity theory can’t be altered at cosmological

scales so that it can drive cosmic acceleration – F(R) theory

How many MGs

How many MGs

Chapter 2: Modified gravity

• Effective field theory of f(T) and beyond

Based on 1801.05827 and 1803.09818, by Li Chunlong, Cai Yong, XueLingqin, Emmanuel Saridakis & CYF

See also CYF, Capozziello, De Laurentis, Saridakis, RPP 2016 for warm-up (121 pages)

Introduction to teleparallel gravity

Metric:

Tangent space descriptions:

Associate a tangent space to each spacetime point and work in terms of that tangent space.

The dynamical variable can be regarded as tetrad

!"# Tetrad (vierbein): $"%

eμA, μ, ν, ρ . . . = 0, 1, 2, 3, A, B, C = 0, 1, 2, 3.

which is an orthonormal basis for the tangent space at each point of the manifold.

ηAB = ηAB = diag(−1,1,1,1)gμν = ηABeA

μ eBν

Introduction to f(T) gravity

!"#$%"&'()* )(&&")$#(&+ Γ̂λμν ≡ eλ

A∂νeAμ = − eA

μ ∂νeλA

,(-.#(&/$"&.(-+ Tλμν ≡ Γ̂λ

νμ − Γ̂λμν = eλ

A(∂μeAν − ∂νeA

μ )

,(-.#(&/.)010-+ T = 14 TρμνTρμν + 1

2 TνμρTρμν − TμνμTαν

α

! ,"1"20-011"1/345#601"&$/(7/8"&"-01/9"10$#6#$:+

R = − T − 2∇μTμ Tμ = gμνTλνλ

! ,;"/0)$#(&/(7/7<,=/$;"(-:+ S = ∫ d4x −gM2

P

2 f (T )

EFT of teleparallel gravity

!"#$#%%#&'()#$%(#*+$'"#,-.$/01!2$+#3&-(4'(,5$,%$'#*#46-6**#*$7-6)('.8

S = ∫ d4x −g[ M2P

2 Ψ(t)R − Λ(t) − b(t)g00 + M2P

2 d(t)T0] + S(2) .

(T 0 = g0μTνμν) 9:6+-6'(& 6&'(,5 %,-

*(5#6- 4#-':-;6'(,531,-$%/!2$7-6)('.8

<6-=$#5#-7.$(5$%/!28ρeffDE = M2

P

2 [T (0) − f (T (0)) + 2T (0)fT(T (0))] ,

peffDE = − M2P

2 [4 ·H[1 + fT(T (0)) + 2T (0)fTT(T (0))] − f (T (0)) + T (0) + 2T (0)fT(T (0))] .

Perturbation theory - scalar! !"#$#%&'(%)*%+#),-./

0(1)-.,#.&2#*2(3

4-,//-.&(5*#),-.3 k2ϕ = a2

2M2P fT (1 − a2M2

M2P fT H2k2 ) δρm . M2 /M2

P ∼ H4

k2 /a2 ≫ H2

k2ϕ = a2

2M2P fT

δρm . Effects of the additional scalar ! vanishes.

4-/)60(1)-.,#.&'#%#7()(%3 γ = ψϕ

= 1 + a2M2

fT H2k2M2P − a2M2 .

k2 /a2 ≫ H2

γ = 1

,.)%-8*"(8&+9&):(&;,-$#),-.&-<&):(&=$-"#$&>-%(.)?&,.;#%,#."(&@A

Perturbation theory - scalar

Perturbation theory - tensor

! !"#$%& '"&()&*+(,%#$-

.,$'"&$,%#/&"0+(,%#$1,'-

Perturbation theory - tensor

!(#) = −# + (#) ( = (6+,-)(./))1 − Ω2,24 − 1

Summary I• Our understanding of late-time cosmic acceleration remains far far away

from the reality

• Dark energy physics: - The concordance model of ΛCDM can fit to data well- A dynamical model is phenomenologically interesting and marginally

indicated by observations- The precise measurement of the EoS parameter is crucial in

examining the nature of DE- A proof of theoretical No-Go makes the DE study become

phenomenologically fruitful

Summary II• Torsional based modified gravity:

- MG in terms of torsion can be applied to drive cosmic acceleration- Teleparallel gravity and extensions can be depicted by EFT dictionary- The EFT approach is powerful to help testing this type of MG- Different from f(R), there is no propagating dof in pure f(T) gravity- The effect of local Lorentz violating scalar dof would be manifest at

k2/a2 ~ H2

- Testing f(T) is possible via the measurement of dispersion relations of cosmological GWs in the era of GW astronomy

• Outlook: Accumulated high-precision data from multiple messengers are expected to probe the physics of late-time cosmic acceleration in the future

- & - ----- & -

! Review, Article, Letter, News & Views, Editor’s Focus

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