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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