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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
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
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
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.
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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.
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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!
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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).
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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.
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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)
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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.
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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
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, %)
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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
)
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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
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.
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
“Ankle” in UHECR Spectrum
L. Reverberi Cosmology and Astroparticle Physics in Modified Gravity
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
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
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