aspects of cosmology and astroparticle physics in modified ... · rdepochinr2 gravity [e. arbuzova....

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!).


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!


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.


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.


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!


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).


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


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.


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)


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.


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


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, %)


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


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

)


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.


“Ankle” in UHECR Spectrum


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.


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?


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


aspects of cosmology and astroparticle physics in modified ... · rdepochinr2 gravity [e. arbuzova....

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