aspects of cosmology and astroparticle physics in modified ... · rdepochinr2 gravity [e. arbuzova....
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![Page 1: Aspects of Cosmology and Astroparticle Physics in Modified ... · RDEpochinR2 Gravity [E. Arbuzova. A. Dolgov, L. Reverberi, JCAP 02, 049 (2012)] f(R) = R+ R2 6m2 Quadratictermsarisefromone-loopcorrectionstoT](https://reader033.vdocuments.mx/reader033/viewer/2022060915/60a8f3016b16f841e4535842/html5/thumbnails/1.jpg)
Aspects of Cosmology and Astroparticle Physicsin Modified Gravity
Lorenzo ReverberiUniversità degli Studi di Ferrara and INFN, Sezione di Ferrara - Italy
SUPERVISORS Prof. A.D. DolgovDott. P. Natoli
REFEREES Prof. S. Capozziello – Università degli Studi di Napoli e INFN NapoliProf. S. Matarrese – Università degli Studi di Padova e INFN Padova
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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The Dark Energy Puzzle
Einstein Equations with Λ
Gµν −Λ gµν = Tµν
Smallness Problem
%V ∼ 10−123m4Pl
∼ 10−44Λ4QCD
Coincidence Problem
ΩV ∼ ΩmdΩV /dN ∼ max
MODIFIED GRAVITY: Gµν → G′µν
∗ f(R) Gravity∗ Scalar-Tensor Gravity∗ Gauss-Bonnet Gravity∗ Braneworld Models∗ . . .
MODIFIED MATTER: Tµν → T ′µν
∗ Quintessence∗ k-essence∗ Phantoms∗ Chameleons∗ . . .
f(R) Gravity
Gµν + (f,R − 1)Rµν −f −R
2gµν + (gµν−∇µ∇ν)f,R = Tµν
Additional dynamics = 1 scalar degree of freedom (scalaron), more solutions thanGR (and more complicated!).
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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RD Epoch in R2 Gravity [E. Arbuzova. A. Dolgov, L. Reverberi, JCAP 02, 049 (2012)]
f(R) = R+R2
6m2
Quadratic terms arise from one-loop corrections to Tµν in curved spacetime(Starobinsky 1980). Model originally proposed as a purely gravitationalmechanism of inflation.“GR” limit recovered for m→∞During the Radiation-Dominated epoch, induced oscillations of R:
R+ 3HR+m2 (R+ T ) = 0
%+ 3H(%+ P ) = 0
⇒
R+ 3HR+m2R = 0
%+ 4H% = 0
Solutions oscillate around the GR solution R = −T = 0 with frequency m.Initially, estimate amplitude analytically in linearised regime
H '1
2t+CH
t3/4sinmt R ' 0 +
CR
t3/4sinmt
In non-linear regime, use semi-classical approach (high frequency)
H 'α
2t+CH
tsinmt R ' 0 +
CR
tsinmt
One additional condition gives α > 1 in the presence of oscillations⇒ expansion is faster than in GR!
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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Gravitational Particle Production [E. Arbuzova. A. Dolgov, L. Reverberi, JCAP 02, 049 (2012)]
R-oscillations → gravitational particle production: e.g. massless,minimally-coupled scalar field
R+ 3HR+m2R =8π
m2Pl
gµν∂µφ∂νφ = · · · ' −m2
12πm2Pl
∫ t
t0
dt′R(t′)
t− t′
(∂2t −∆)φ+1
6a2Rφ = 0
Particle production and back-reaction on evolution of R:
%(R→ φφ) 'm(∆R)2
1152π
R→ R exp(−t/τR) with τR =48m2
Pl
m3
Eventually oscillations stop and the Universe expansion is the same as in GR, butit must happen before BBN!
τR . 1 s ⇒ m & 105 GeV
Relic Energy Density
%R
%therm∼β2Neff
κ
(1−
tin
τR
)κ arbitrary, ∼ O(1)β small . O(10−1)
Implications for Dark Matter (e.g. φ =LSP), CMB distorsion, etc.
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Viable f(R) Models of Dark Energy
Cosmological viability
f,R > 0 (graviton 6= ghost)
f,RR < 0 (scalaron 6= tachyon)
Among the many models capable of generating the observed accelerated cosmologicalexpansion, at least three survive several important local (Solar system, Eötvös) andcosmological (BBN, CMB) tests:
F (R) = f(R)−R = Rc ln[e−λ + (1− e−λ)e−R/Rc
]Appleby-Battye 2007
F (R) = f(R)−R = −λRc
1 + (R/Rc)−2nHu-Sawicki 2007
F (R) = f(R)−R = λRc
[(1 +
R2
R2c
)−n− 1
]Starobinsky 2007
As |R| decreases in the history of the Universe, solutions tend to de-Sitter, withconstant curvature:
R ∼ λRcCosmologically, the possibilities of distinguishing these models from ΛCDM are small,but additional constraints may come from astrophysics, astroparticle physics, etc.
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Framework and Basic Equations [L. Reverberi, Phys. Rev. D 87, 084005 (2013)]
Let us consider an astronomical cloud under the following assumptions:
“high” density %m %c ∼ 10−29 g cm−3 (R/Rc 1)
low gravity |gµν − ηµν | 1 (derivatives might be large though!)
spherical symmetry + homogeneity ⇒ neglect spatial derivatives
pressureless dust: Tµµ ∝ % (not necessary but reasonable)
We define a new (scalaron) field ξ ∼ F,R:
ξ ≡ −3F,R = 6nλ
(Rc
R
)2n+1
This field fulfills the very simple equation of motion:
ξ +R+ T = 0 ⇔ ξ +∂U(ξ, t)
∂ξ= 0
SINGULARITY
R→∞ for ξ = 0
Along the GR solution we have ξ ∝ T−(2n+1) 6= 0 but oscillations may allow ξ = 0!
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Scalaron Potential [L. Reverberi, Phys. Rev. D 87, 084005 (2013)]
ξ ∼ R−(2n+1)
0Ξ
Ubottom corresponds to the GR solutionR = −Tnot symmetric around the position of thebottom
for increasing T , the bottom rises:U0 = −3nλRc |Rc/T |2n
potential is finite for ξ = 0 ⇔ R→∞ξ oscillates with frequency ω; the potentialchanges on a timescale tcontr:
U(ξ) = T ξ −A(n,Rc) ξ2n
2n+1 ω0 tcontr ' 0.5%n+129 t10√
(2n+ 1)nλ
“Slow-Roll” Regime: ω tcontr 1
Oscillations are slow w.r.t. changes of the potential, so the motion of ξ is mainlydriven by changes of U (and initial conditions if ξ0 6= 0)
“Fast-Roll” Regime: ω tcontr 1
Oscillations are fast, so they are practically adiabatic. Near a given time t, ξ oscillatesbetween two values ξmin and ξmax with roughly U(ξmin) = U(ξmax).
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Slow-Roll Regime [L. Reverberi, Phys. Rev. D 87, 084005 (2013)]
Let us first consider the slow-roll regime, that is ω0 tcontr 1. The initial “velocity”of the field dominates over the acceleration due to the potential, so in firstapproximation
ξ(t) = ξ0 + ξ0t
This can also be understood as follows:
ξ ∼ξ
t2contr, R+ T =
∂U
∂ξ∼ ω2ξ ⇒
ξ
R+ T∼
1
ω2 t2contr 1
Therefore the trace equation reduces to
ξ +R+ T ' ξ = 0
0.05 0.10 0.15 0.20 0.25 0.30ttcontr
0.2
0.4
0.6
0.8
1.0
ΞΞ0
0.00 0.05 0.10 0.15 0.20 0.25 0.30ttcontr
1.5
2.0
2.5
3.0
RR0
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Slow-Roll Regime – Singularity [L. Reverberi, Phys. Rev. D 87, 084005 (2013)]
Initial Conditions
R0 = −T0 R0 = −(1− δ)T0
The singularity appears when ξ = 0, that is at
Singularity – Critical T and t
tsing
tcontr= −
ξ0
ξ0'
1
(2n+ 1)|1− δ|Tsing
T0= 1 +
1
(2n+ 1)|1− δ|
n = 3%29 = 30
t10 = 10−5
⇒ω0 tcontr
2π' 0.2
æ
æ
æ
æ
æ
æ
æ
æ æ æ ææææææææææ
0.10 1.000.500.20 2.000.300.15 1.500.70È1-∆È
0.001
0.005
0.010
0.050
0.100
0.500
Dtsingtsing
“Cusp” due to change in sign of ∆t/t.Precision is outstanding, given therelatively large ω0 tcontr ' 1. Takingn = 3
%29 = 100
t10 = 10−9
δ = 0
⇒ ω0 tcontr ' 10−2
yields ∆tsing/tsing ' 10−7.
But: short contraction timescales⇒ (maybe) unphysical.
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Fast-Roll Regime [L. Reverberi, Phys. Rev. D 87, 084005 (2013)]
Harmonic Regime
Oscillations of ξ are initially small, so we can expand it as
ξ = ξGR + ξ1 = ξGR +α sin
∫∫∫ tω dt′
with α/ξGR 1. At first order we find
ξ1 + ω2ξ1 ' 0 ⇒ α ∼ ω−1/2
Naively, one may suppose that the singularity condition is α = ξGR.
Anaharmonic Regime
As α→ ξGR, the asymmetry of the potential becomes more evident and oscillationsare no longer harmonic. Energy conservation for ξ:
1
2ξ2 + U(ξ, t)−
∫∫∫ tdt′
∂T
∂t′ξ(t′) = const.
The singularity condition becomes
U(ξ) +1
2ξ2∣∣∣∣max
= U(ξsing)
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Fast-Roll Regime – Singularity [L. Reverberi, Phys. Rev. D 87, 084005 (2013)]
Singularity – Critical T and t
Tsing
T0=
[T 2n+20 t2contr
6n2(2n+ 1)2 δ2|Rc|2n+1
] 13n+1
'[
0.28%2n+229 t210
n2(2n+ 1)2 δ2
] 13n+1
tsing
tcontr=Tsing
T0− 1
æ
æ
æ
æ
æ
æ
ææ
ææ
æ æ
æ æææ æ æ
1 2 5 10 20 50106tcontr
0.001
0.002
0.005
0.010
0.020
0.050
0.100
DTsingTsing
n = 3%29 = 102
δ = 0.5
⇒tsing
tcontr∼ O(1)
Relative errors tend to constant value∼ 2 · 10−3 (maybe numerical feature?).“Cusp” due to change in sign of ∆T/T .Precision is nevertheless satisfactory.Computational time proportional to totalnumber of oscillations:
Nosc ∼∫ tsing
ω dt ∝(%n+129 t10
) 5n+53n+1
Large %29 = expect better agreement, butdifficult to test numerically.
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Avoiding the Singularity [E. Arbuzova. A. Dolgov, L. Reverberi, Phys. Rev. D 88, 024035 (2013)]
Gravitational Particle Production
Oscillations of R induce gravitational particle production, resulting in an exponentialdamping
R→ R exp(−t/τR)
not effective in SLOW-ROLL regime!
enhanced by high-curvature terms (see below)
High-Curvature Corrections
As |R| → ∞, high-curvature terms become important, e.g. terms ∼ R2.The full model to be considered is
f(R) = R−R2
6m2+ FΛ(R)
This “new” model combines ultraviolet (QFT in curved spacetime) and infrared (DarkEnergy) extensions to GR. Now R→∞ gives ξ →∞, and U(ξ →∞) =∞ so thesingularity is inaccessible. The effects of R2 are parametrised by
g(n,m,Rc, %0) =(8πGN %0)2n+2
6n|Rc|2n+1m2' 10−94
[%0
%c
]2n+2 [105 GeVm
]2 1
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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Potential and Regimes [E. Arbuzova. A. Dolgov, L. Reverberi, Phys. Rev. D 88, 024035 (2013)]
-0.1 0.1 0.2 0.3 0.4 0.5 0.6Ξ
-0.05
0.05
0.10
0.15
0.20
UΞ
U(ξ) 'Tξ − α(n, %0, Rc,m) ξ
2n2n+1 (ξ > 0)
Tξ + β(n, %0, Rc,m) ξ2 (ξ < 0)
ξ(R) = 6nλ
(Rc
R
)2n+1
− g(R
Rc
)
Harmonic Regime
When oscillations of ξ are small, the potential is always ∼ harmonic andoscillations are ∼ adiabatic.
In this region, one can easily estimate the behaviour of ξ and hence R as % varieswith time (e.g. semiclassical approx.)
Anharmonic Regime
As ξ crosses 0 (previously: singularity), R exhibits narrow spikes(g very small ⇒ small variations in ξ corresponds to a huge variation of R)
In this regime, we need to use the energy conservation to find R(t, %)
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Curvature Evolution [E. Arbuzova. A. Dolgov, L. Reverberi, Phys. Rev. D 88, 024035 (2013)]
R(t)−RGR(t)
R0'
βharm(t) sin
[∫ t
dt′ ω(t′)
](harmonic region)
βspikes(t)∞∑k=1
exp
[−
(t− k δt)2
2σ2
](spike region)
βharm ∼1
tcontr
[2n+ 1
(%/%0)2n+2+ g
]−3/4
βspikes ∼(
1
g
[2n+ 1
(%/%0)2n+2+ g
]−1/2
−1
n(%/%0)2n
+
(%
%0
)2)1/2
0.1 0.2 0.3 0.4ttcontr
1.1
1.2
1.3
1.4
RR0
0.2 0.4 0.6 0.8 1.0 1.2 1.4ΚΤ
2
4
6
8
10
y
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Curvature Evolution [E. Arbuzova. A. Dolgov, L. Reverberi, Phys. Rev. D 88, 024035 (2013)]
R(t)−RGR(t)
R0'
βharm(t) sin
[∫ t
dt′ ω(t′)
](harmonic region)
βspikes(t)∞∑k=1
exp
[−
(t− k δt)2
2σ2
](spike region)
βharm ∼1
tcontr
[2n+ 1
(%/%0)2n+2+ g
]−3/4
βspikes ∼(
1
g
[2n+ 1
(%/%0)2n+2+ g
]−1/2
−1
n(%/%0)2n
+
(%
%0
)2)1/2
Particle production by general oscillating R
%pp '1
576π2 ∆t
∫dω ω
∣∣∣R(ω)∣∣∣2
∣∣∣Rharm(ω)∣∣∣2
∆t∼ δ(ω ± ω0)
∣∣∣Rspikes(ω)∣∣∣2
∆t∼ exp(−ω2
σ2)∑j
δ
(ω −
2πj
δt
)
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Particle Production [E. Arbuzova. A. Dolgov, L. Reverberi, Phys. Rev. D 88, 024035 (2013)]
Harmonic Region
%pp
GeV s−1m−3' 3.6× 10−141 C1(n)
[%
%0
]4n+4 [%0%c
]1−n [1010 ystcontr
]2(small %)
' 2.5× 1047 C2(n)[ m
105 GeV
]4 [ %c%0
]5n+3 [1010 ys
tcontr
]2(large %)
The back-reaction on R is the usual exponential damping
%→ % exp
[−2
∫ t
t0
dt′ Γ
]
Spike Region
%pp
GeV s−1m−3' 3.0× 10−47 C3(n)
[ m
105 GeV
]2 [ %%0
]2n+2 [ %c%0
]3n+1 [1010 ystcontr
]2The back-reaction on R leads to a more complicated condition for the limit % at whichoscillations effectively stop[
%max
%0
]3n+4
' 6× 10123 C4(n)
[%c
%0
]3n+3 [1010 ystcontr
]L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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Cosmic Ray Flux [E. Arbuzova. A. Dolgov, L. Reverberi, Phys. Rev. D 88, 024035 (2013)]
Considering a physical contracting source having mass M , we can estimate the totalluminosity due to gravitational particle production:
Harmonic Region
L
GeV s−1' 7.3× 10−74 C1(n)Ns
[M
1011M
] [%
%0
]4n+3 [ %c%0
]n [1010 ystcontr
]2Even for high densities and short contraction times, this value is practically alwaysnegligible. However, produced particles ∼ monochromatic, perhaps detectable signalin a certain range of parameters.
Spike Region
L
GeV s−1' 6.0× 10
20C2(n)Ns
[M
1011M
] [m
105GeV
]2 [ %%0
]2n+1 [ %c%0
]3n+2[
1010ystcontr
]2
Potentially large luminosity, particles produced at energies from MeV to scalaron massm > 105 GeV, potentially up to 1019 − 1020 eV, hence with possible implications forthe UHECR “ankle” problem.
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“Ankle” in UHECR Spectrum
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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Spherical Symmetric Solutions [E. Arbuzova, A. Dolgov, L. Reverberi, Astrop. Phys. 54, 44 (2014)]
and Gravitational Repulsion
A spherically-symmetric metric can always be cast in the simple form
ds2 = A(t, r) dt2 −B(t, r)dr2 − r2dΩ
Solving the modified gravity equations assuming A,B 1 and gives:
B ' 1 +B(GR)
A ' 1 +A(GR) +Rr2
6
The dynamics of a test particle in a gravitational field is governed by
r = −A′
2= −
1
2
[R(t)r
3+
2GNMr
r3m
]
Gravitational repulsion if
|R| & 8πGN %
strange time-dependent repulsive behaviour of gravity in contracting systems
maybe: creation of this shells separated by vacuum
Cosmic voids: ISW Effect, etc.
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
![Page 20: Aspects of Cosmology and Astroparticle Physics in Modified ... · RDEpochinR2 Gravity [E. Arbuzova. A. Dolgov, L. Reverberi, JCAP 02, 049 (2012)] f(R) = R+ R2 6m2 Quadratictermsarisefromone-loopcorrectionstoT](https://reader033.vdocuments.mx/reader033/viewer/2022060915/60a8f3016b16f841e4535842/html5/thumbnails/20.jpg)
Conclusions and Future Work
Results
curvature oscillations in RD epoch in R2 gravity and consequent particleproduction → relic density (Dark Matter?)
curvature singularities in contracting systemscuring singularities with addition of high-curvature terms and particle production
general results of gravitational particle production by oscillating curvatureproduction and possible detection of UHECR
spherically symmetric solutions in modified gravity: repulsive behaviour
In Progress
(modified) Jeans analysis of structure formation: deviations from GR andimplications for Baryon Acoustic Oscillations, mass spectrum, maybe Plancklow-multipole riddle?
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
![Page 21: Aspects of Cosmology and Astroparticle Physics in Modified ... · RDEpochinR2 Gravity [E. Arbuzova. A. Dolgov, L. Reverberi, JCAP 02, 049 (2012)] f(R) = R+ R2 6m2 Quadratictermsarisefromone-loopcorrectionstoT](https://reader033.vdocuments.mx/reader033/viewer/2022060915/60a8f3016b16f841e4535842/html5/thumbnails/21.jpg)
Work done under the supervision of Prof. A.D. Dolgov and in collaboration with E.V.Arbuzova (Novosibirsk State University and Dubna University, Russia)
PAPERS
Peer-Reviewed
E.V. Arbuzova, A.D. Dolgov and L. Reverberi, Astrop. Phys. 54, 44 (2014)
E.V. Arbuzova, A.D. Dolgov and L. Reverberi, Phys. Rev. D 88, 024035 (2013)
L. Reverberi, Phys. Rev. D 87, 084005 (2013)
E.V. Arbuzova, A.D. Dolgov and L. Reverberi, Eur. Phys. J. C 72, 2247 (2012)
E.V. Arbuzova, A.D. Dolgov and L. Reverberi, JCAP 02, 049 (2012)
Proceedings
L. Reverberi, J. Phys. Conf. Ser. 442, 012036 (2013)[doi:10.1088/1742-6596/442/1/012036].
CONFERENCESSW7 - Cargèse (France) - May 2013
2D IDAPP - Ferrara (Italy) - October 2012
DICE 2012 - Castiglioncello (Italy) - September 2012
SW6 - Cargèse (France) - May 2012
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity