cai deft 20190703 · 2019-08-07 · introduction to teleparallel gravity metric: tangent space...
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Chapter 1: Dark energy
• In the sky
• In the mind
Chapter 1: Dark energy
• In the sky
• In the mind
Cosmic Pie
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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
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Observational Windows!"#$%%"&'%#%(#&"$)*+"#%,"#-"."/"+$%0(1#'$+$&"%"+#!2#$1-#0%#%*+1)#(*%#1"3$%04"56*+#7104"+)"#0)#$.."/"+$%013! $1#"8'".%$%0(1#(9#-$+:#"1"+3;#
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The Expansion History
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The Latest Status
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Chapter 1: Dark energy
• In the sky
• In the mindMainly based on CYF, Saridakis, Setare, Xia, Phys.Rept. 2010
What we know about DE?Background equation:
3/1/ 03 -<=<+ rr pwp
Basic features: • Negative pressure• Violating strong energy condition• Almost not clustered at cosmological scales
Acceleration requires:
Theoretical implications: • Dynamical nature• Microscopic interpretation• Possible applications
Dark Energy 70%
Dark Matter 25%
Atoms 5%
Cosmological ConstantOriginally introduced by Einstein to realize a static universe by “mistake”, it has
become the simplest candidate of dark energy.
Dynamics: w=-1
Microscopic interpretation: vacuum energy
- The most severe fine-tuning problem in physics: one needs to cut off vacuum
energy by a factor of 1,000,000,000,000, 000,000,000,000, 000,000,000,000,
000,000,000,000, 000,000,000,000, 000,000,000,000, 000,000,000,000,
000,000,000,000, 000,000,000,000, 000,000,000,000.
- Coincidence problem: one cannot explain why such a tiny energy density
happens to be comparable with that of matter at today’s value, otherwise the
universe would look totally different!
Weinberg, 1989
Dynamical FieldsQuintessence:
Phantom:
Applications: • For certain potential forms, scalar fields may present tracking behavior to
alleviate the coincidence problem • They may seed cosmological perturbations to affect the large-scale structure of
the UniverseIssues: • No clues for microscopic origins• Classical and quantum instabilities
Caldwell, astro-ph/9908168
Ratra, Peebles, 1988
Categories of Dynamics!"#$%&'()*+,$-.$%(/0$#'#/1&$+/2+*(33&$/#3&$-'$4"#$#52(4*-'6-.6,4(4#$7(/()#4#/8$
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No-Go Theorem!"#$" %&'"(')*+
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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
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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
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A∂νeAμ = − eA
μ ∂νeλA
,(-.#(&/$"&.(-+ Tλμν ≡ Γ̂λ
νμ − Γ̂λμν = eλ
A(∂μeAν − ∂νeA
μ )
,(-.#(&/.)010-+ T = 14 TρμνTρμν + 1
2 TνμρTρμν − TμνμTαν
α
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R = − T − 2∇μTμ Tμ = gμνTλνλ
! ,;"/0)$#(&/(7/7<,=/$;"(-:+ S = ∫ d4x −gM2
P
2 f (T )
EFT of teleparallel gravity
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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 %,-
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<6-=$#5#-7.$(5$%/!28ρeffDE = M2
P
2 [T (0) − f (T (0)) + 2T (0)fT(T (0))] ,
peffDE = − M2P
2 [4 ·H[1 + fT(T (0)) + 2T (0)fTT(T (0))] − f (T (0)) + T (0) + 2T (0)fT(T (0))] .
Perturbation theory - scalar! !"#$#%&'(%)*%+#),-./
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k2 /a2 ≫ H2
k2ϕ = a2
2M2P fT
δρm . Effects of the additional scalar ! vanishes.
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Perturbation theory - scalar
Perturbation theory - tensor
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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
- & - ----- & -
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