Post on 11-Aug-2020
Challenges for the Matter Bounce Scenario
Jerome QuintinMcGill University
ITC, CfA, CambridgeSeptember 7, 2018
Based on work with Robert Brandenberger (McGill U.),Yi-Fu Cai (USTC, Hefei, China), and other collaborators
PRD 92, 063532 (2015) [arXiv:1508.04141]JCAP 1611, 029 (2016) [arXiv:1609.02556]JCAP 1703, 031 (2017) [arXiv:1612.02036]JCAP 1801, 011 (2018) [arXiv:1711.10472]
Introduction and Motivation• We search for a causal mechanism that can explain the observed
large scale structures of our universe from primordial fluctuations• The standard picture is ‘horizon exit’ and ‘horizon re-entry’• E.g., inflation: a(t) ∝ eHt, H ' const., H−1 = Hubble radius
tBig Bang
tinflation begins
treheating xp
t H−1λ ∝ k−1
horizon
inflation
post inflation
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 2 / 28
Successes of Inflation
Inflation explains:
• the formation of structure problem
• the horizon problem
• the flatness problem
• the monopole problem
Also, it gives (in general):
• nearly scale-invariant power spectra of curvature and tensorperturbations
• small non-Gaussianities
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 3 / 28
Problems with Inflation• Trans-Planckian problem Brandenberger & Martin [hep-th/0005209, hep-th/0410223]
• Singularity problem Borde & Vilenkin [gr-qc/9612036]; Borde et al. [gr-qc/0110012]; Yoshida & JQ [1803.07085]
• Unpredictability, unlikeliness, initial conditions problem, measureproblem Ijjas, Steinhardt, & Loeb [1304.2785, 1402.6980]
• and more conceptual issues Brandenberger [1203.6698]
tBig Bang
tinflation begins
treheating xp
t H−1λ ∝ k−1
horizon
super-Planckian density
`Pl
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 4 / 28
Aside: Inextendibility of Inflation
• de Sitter spacetime (a(t) = a0eHt in FRW) has a past boundary at
finite affine length (when a→ 0)• Yet, the full spacetime is geodesically complete. It can be extended in
the appropriate coordinate basis (global coordinates, not FRW)
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 5 / 28
Aside: Inextendibility of Inflation
• Similarly, any spacetime is extendible iff Ricci is finite in theappropriate basis (not FRW!)
• Result: extendible iff H/a2 is finite Yoshida & JQ [1803.07085]
• Consequence: if leading energy component is quasi-de Sitter(p ' −ρ), then the spacetime is extendible only if sub-leadingcomponent severely violates the Null Energy Condition (p ≤ −(5/3)ρ)
• E.g. 1: Starobinsky inflation is inextendible→ paralarallelypropagated curvature singularity
• E.g. 2: Small field inflation is C0 extendible, but unstable againstinitial condition fluctuations Brandenberger [1601.01918]
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 6 / 28
Alternatives to Inflation
• There are a few alternative scenarios for the very early universe• E.g., nonsingular bouncing cosmology (Ekpyrotic scenario, matter
bounce scenario, pre-Big Bang scenario)
t
a(t)
t
H(t)
→ Need theory beyond Einstein gravity to avoid singularity
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 7 / 28
Nonsingular Bouncing Cosmology• Solves the problems of standard Big Bang cosmology• Free of the trans-Planckian problem• Can avoid the initial Big Bang singularity
τB−
τB
τB+
xc
τ
|H|−1λ ∝ k−1
`Pl
What about the connection to observations?Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 8 / 28
Matter Bounce Scenario• Perturbations exit the Hubble radius in a matter-dominated
contracting phase, when a(τ) ∝ τ2 (p = 0)• With an initial quantum vacuum, curvature perturbations have a
scale-invariant primordial power spectrum Wands [gr-qc/9809062]; Finelli & Brandenberger[hep-th/0112249]
v′′k +
(k2 − 2
τ2
)vk = 0 , ′ ≡ d
dτ
vk = aRk√
2ε =√
3aRk , ε ≡ − H
H2=
3
2
(1 +
p
ρ
)=
3
2
• Same for tensor modes:
u′′k +
(k2 − 2
τ2
)uk = 0 , uk =
1
2ahk
• Advantage: implementable with a single canonical scalar field; onlyadiabatic perturbations
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 9 / 28
Power Spectra for the Matter Bounce
• Power spectra:
PR(k) ≡ k3
2π2|Rk|2 =
1
48π2H2B−
M2Pl
Pt(k) ≡ 2× k3
2π2|hk|2 =
1
2π2H2B−
M2Pl
• Tensor-to-scalar ratio:r ≡ PtPR
= 24
• Observations: r < 0.07 (2σ) BICEP2 [1510.09217]
−→ Ruled out!
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 10 / 28
Possible Resolution #1• What if cs � 1, e.g. with a k-essence scalar field?• Curvature perturbations are amplified:
v′′k +
(c2sk
2 − 2
τ2
)vk = 0 =⇒ PR(k) =
1
48π2cs
H2B−
M2Pl
=⇒ r = 24cs
• r < 0.07 ⇐⇒ cs . 0.003• But cs � 1 =⇒ strong coupling Baumann et al. [1101.3320]
• So the scalar three-point function is also amplified Li, JQ et al. [1612.02036]
f localNL ' −165
16+
65
8c2s• Cannot simultaneously satisfy observational bound on r andf localNL = 0.8± 5.0 (1σ) Planck [1502.01592]
• Also, cs � 1 with a fluid =⇒ Jeans (gravitational) instability =⇒black hole formation (see ITC Luncheon talk!) JQ & Brandenberger [1609.02556]
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 11 / 28
Small sound speed: non-Gaussianities
• Non-Gaussianity dimensionless shape function for cs = 0.2 (left plot)and cs = 0.87 (right plot)
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 12 / 28
Small sound speed: black hole formation
−10 −8 −6 −4 −2 0log10(|H|/MPl)
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
log10(α
)
cs = 10−4
−12.0
−10.5
−9.0
−7.5
−6.0
−4.5
−3.0
−1.5
0.0
log
10(p
robab
ility)
−10 −8 −6 −4 −2 0log10(|H|/MPl)
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
log10(α
)
cs = 10−3
−12.0
−10.5
−9.0
−7.5
−6.0
−4.5
−3.0
−1.5
0.0
log
10(p
robab
ility)
α ≡ RBH/|H|−1
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 13 / 28
Possible Resolution #2
• What if R grows during the nonsingular bounce phase?
rbefore bounce
robs=
∣∣∣∣1 +∆R
Rbefore bounce
∣∣∣∣2• Creates large non-Gaussianities JQ et al. [1508.04141]
fNL ∝(
∆RRbefore bounce
)#
• =⇒ cannot simultaneously satisfy observational constraints on rand fNL
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 14 / 28
Matter Bounce No-Go Theorem
• A lower bound on the amplification of curvature perturbations R⇐⇒ an upper bound on the tensor-to-scalar ratio r⇐⇒ a lower bound on primordial non-Gaussianities fNL
• With Einstein gravity + a single (not necessarily canonical) scalarfield:
satisfying the current observational upper bound on r cannot bedone without contradicting the current observational constraints on
fNL (and vice versa) JQ et al. [1508.04141]; Li, JQ et al. [1612.02036]
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 15 / 28
Evading the No-Go Theorem
• Multiple fields, e.g., matter bounce curvaton scenario (entropy modessourcing curvature perturbations) Cai et al. [1101.0822]
• Or with a single field, go beyond Einstein gravity−→ need to modify tensor modes
• Add a nontrivial mass mg to the graviton:
L(2)tensor ⊃ a2[(h′ij)2 − (∇hij)2
]→ a2
[(h′ij)2 − (∇hij)2 −m2
ga2h2ij
]=⇒ u′′k +
(k2 +m2
ga2 − a′′
a
)uk = 0
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 16 / 28
Tensor Power Spectrum with Massive Gravity
• mg � |H∗| at horizon exit:
Pt(k) ' 1
2π2
(k
aMPl
)nt∣∣∣∣HB−MPl
∣∣∣∣2−nt
, nt '8m2
g
3H2∗� 1
• mg � |H∗| at horizon exit:
Pt(k) ' 2
π2
(k
a
)3 1
M2Plmg
• In both cases, r(0.05 Mpc−1)� 0.07 Lin, JQ & Brandenberger [1711.10472]
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 17 / 28
Tensor Power Spectrum with Massive Gravity• Note that 0.05 Mpc−1 ≈ 10−58MPl
10-53 10-43 10-33 10-23 10-13
10-13
10-12
10-11
10-10
10-9
10-8
kphysical/MPl
Ptensor
0.00 0.05 0.10 0.15 0.20
0.00
0.02
0.04
0.06
0.08
0.10
mg/|H*|
nt
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 18 / 28
Tensor Power Spectrum with Massive Gravity
0.00 0.05 0.10 0.15 0.20
10-4
0.001
0.010
0.100
1
10
mg/|H*|
r 0.0
5
0.00 0.02 0.04 0.06 0.08 0.10
10-4
0.001
0.010
0.100
1
10
nt
r 0.0
5
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 19 / 28
Tensor Power Spectrum with Massive Gravity
• If mg � |H(t)|, r ≈ 0
10-53 10-43 10-33 10-23 10-13
10-170
10-130
10-90
10-50
10-10
kphysical/MPl
Ptensor
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 20 / 28
Anisotropy problem• Consider
ds2 = −dt2 + a23∑i=1
e2θi(dxi)2 ,
3∑i=1
θi = 0
• Einstein gravity = Friedmann equations + anisotropies:
θi + 3Hθi = 0 =⇒ θi ∼ a−3 =⇒ ρθ ∼3∑i=1
θ2i ∼ a−6
• Analogous to a massless scalar field
Lθ = −1
2∂µθ∂
µθ =⇒ pθ = ρθ (w = 1)
• Anisotropies dominate at high energies:
H2 =1
3M2Pl
(ρ0ma3
+ρ0rada4
)− k
a2+
Λ
3+ρ0θa6
• Not a problem for Ekpyrosis which has w � 1 Garfinkle et al. [0808.0542]
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 21 / 28
Anisotropies with Massive Gravity• With a massive graviton Lin, JQ & Brandenberger [1711.10472]
Lθ = −1
2∂µθ∂
µθ − 1
2m2gθ
2
−→ θi + 3Hθi +m2gθi = 0 , ρθ ∼
3∑i=1
(θ2i +m2gθ
2i )
• mg � |H| =⇒ ρθ ∼ a−3 (i.e., pθ = 0)
• No anisotropy problem anymore:
H2 =1
3M2Pl
(ρ0ma3
+ρ0rada4
)− k
a2+
Λ
3+ρ0θa3
• Would need mg � |HB−|, but mg < 7.2× 10−23 eV (2σ) today−→ requires symmetry breaking or mg(t) −→ might be natural withchameleon coupling and solve other problems of massive gravity likethe Higuchi bound De Felice et al. [1711.04655]
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 22 / 28
Effective Massive Gravity Action
• ADM decomposition:
ds2 = −N2dt2 + γij(dxi +N idt)(dxj +N jdt)
• Effective massive gravity action:
S =
∫dt d3x
√γ
(N
(M2
Pl
2R− Λ
)−m2
gV [γij ]
)• The nonderivative potential V [γij ] is independent of the lapse N
−→ only 2 DoF propagate (2 polarization states of GWs), butdiffeomorphism invariance is broken Comelli et al. [1407.4991]
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 23 / 28
Effective Massive Gravity Action
• In the EFT language, take
V [γij ] ∼M2Plδγ
ij δγij
where δγij denotes the traceless part of the linear perturbations ofthe spatial metric Lin & Labun [1501.07160]
• =⇒ background unaffected (Friedmann equations unchanged)scalar and vector perturbations unaffectedtensor perturbations and anisotropies receive a mass term (mg)
• Can be implemented with Stückelberg scalar fields (additional slides)
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 24 / 28
Conclusions
• The matter bounce scenario is an alternative to inflation
• It naturally predicts a very large tensor-to-scalar ratio (r)
• r cannot be suppressed by enhancing curvature perturbations;otherwise fNL is too large
• Tensor modes can be suppressed with a (Lorentz-violating) massivegraviton
• Anisotropies are suppressed likewise, but need mg(t)
• In sum, the simple idea of the matter bounce doesn’t fit withobservations; needs nontrivial theory to work
• Motivates us to keep looking for and testing alternative scenarios forthe very early universe
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 25 / 28
Acknowledgments
Thank you for your attention!
I acknowledge support from the following agencies:
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 26 / 28
Additional Slides: Action with Stückelberg Scalar Fields
• Introduce 1 timelike (φ0) and 3 spacelike (φi) Stückelberg scalar fieldswith the following VEVs:
φ0 = f(t) , φi = xi
• Impose symmetries:
φi → Λijφj , φi → φi + Ξi(φ0)
Λij : SO(3) rotation operator; Ξi : generic function of φ0
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 27 / 28
Additional Slides: Action with Stückelberg Scalar Fields• The following quantities are invariant under the symmetries Dubovsky et
al. [hep-th/0411158]
X = gµν∂µφ0∂νφ
0
Zij = gµν∂µφi∂νφ
j − (gµν∂µφ0∂νφ
i)(gαβ∂αφ0∂βφ
j)
X
• The following operator is traceless Lin & Sasaki [1504.01373]
δZij =Zij
Z− 3
δk`ZikZj`
Z2, Z = δijZ
ij
• Construct quadratic operator graviton mass term:
Lmass ∼M2Plm
2gδikδj`(δZ
ij)(δZk`)
• Resulting theory has 2 gravitational DoF Lin, JQ & Brandenberger [1711.10472]
Jerome Quintin (McGill U.) Challenges for the Matter Bounce Scenario 28 / 28