Black Holes beyond General Relativity - LPSClpsc.in2p3.fr/barrau/aurelien/talk_montpellier_2005.pdf · 2005-11-25 · J. Grain , A. Barrau - LPSC Black Holes evaporate • Radiation

J. Grain , A. J. Grain , A. BarrauBarrau -- LPSCLPSC

Black Holes beyond General RelativityJulien GRAIN, Aurelien BARRAU,

(Laboratoire de Physique Subatomique et de Cosmologie - Grenoble)


What can micro-black holes “say” about new physics ?

• A reminder of the black-hole evaporation process in one slide

• Structure of the Lovelock gravity (Gauss-Bonnet term)

• Exact calculation of the Hawking radiation spectrum in this new framework

• Black holes in ADD braneworld: colliders and astrophysics


Black Holes evaporate• Radiation spectrum

• Hawking evaporation law

kGMhcTπ16

3

=

2

)(M

Mdt

dM α−=stGeVTgM

stGeVTgM11010101010

49

21116

=→=→=

=→=→= −

⎟⎠⎞⎜

⎝⎛ −−

Γ=sTBk

Q

eh

MsdQdt

Nd2

2

)1(

),,(ω

The possible existence (or absence) of primordial black holes formed by densityfluctuations in the early Universe has many cosmological consequences : previoustalks here on those topics by D. Polarski, A. Barrau, M. Lemoine, K. Jedamzik, etc.

I won’t go again through such points and focus on the « new BH physics » itself


Why do people search for extension of Einstein’s theory ?

• First of all, to build a quantum theory of gravity• String theory, loop quantum gravity and many others

• Secondly, to explain two cosmological observations• Accelerated expansion of the Universe (which can be explained

with a cosmological constant Λ)• Rotation curves of galaxies which can be explained by modifying

Newton’s law (although dark matter is the most straightforwardintepretation)

Gravity as a Taylor expansion in scalar curvature


Lovelock gravity• This extension is free of ghosts and keep second

order field equations

• If the Taylor expansion is limited to the second orderin scalar curvature: Gauss-Bonnet gravity

nn

nnnlove RLLcL ∝=∑ with

)4(2 2μνρσ

μνρσμν

μνα RRRRRRLGB +−++Λ−=


Lovelock gravity• This extension is free of ghosts and keeps second


• Stop the Taylor expansion to second order in scalarcurvature: Gauss-Bonnet gravity

∑=i

iilove LcL

)4(2 2μνρσ

μνρσμν


General Relativity





∑=i

iilove LcL

)4(2 2μνρσ

μνρσμν


General Relativity

Cosmological constant





∑=i

iilove LcL

)4(2 2μνρσ

μνρσμν


General Relativity

Cosmological constantGauss-Bonnet term





∑=i

iilove LcL

)4(2 2μνρσ

μνρσμν


Gauss-Bonnet gravity is also obtained as the lowenergy expansion of some STRING THEORIES


Large Extra-Dimensions • Hierarchy problem in standard physics:

• One of the solutions:Large extra dimensions

• Standard model fields confined into the brane. Only gravitons and scalar fields with no standard model charges allowed to propagate into the bulk

Arkani-Hamed, Dimopoulos, Dvali Phys. Lett. B 429, 257 (1998)


Calculation of the HawkingRadiation Spectrum

P. Kanti, J. Grain, A. BarrauPhys. Rev. D 71 (2005) 104002

J. Grain, A. Barrau, P. KantiPhys. Rev. D 72, 104016 (2005)

Derivation of the greybody factors(i.e. accurate coupling of black holes and quantum fields in

this new framework)


Scalar field confined on the brane

• D-dimensional Schwarzschild-like metric

• Induced 4-dimensional one keep informations of the D-dimensional one via h(r)

22

22

22

)()( −Ω−−= Ddr

rhdrdtrhds

( )22222

22 )(sin)(

)( ϕθθ ddrrh

drdtrhds +−−=

The D-dimensional metric function is involved in the 4-dimensional Klein-Gordon equation


Calculation of Greybody factors (1)• A potential barrier appears in the equation

of motion of fields around a black hole:

• Black holes radiation spectrum isdecomposed into three part:

0)1()(12

222 =⎟

⎠⎞

⎜⎝⎛ +−+ R

rrh

dydRr

dyd

rllω

kde

dtdN

HT

3)(1

1 ××±

≡ ∑l

l ωσω

Potential barrierTortoise coordinate

Break vacuum fluctuations

Cross the potentialbarrier

Phase space term

Black hole’s horizon


Calculation of Greybody factors (2)

)(

)(

)(

)(

1 ∞

∞

∞ −==in

out

in

hin

FF

FFAl

22

12)( ll

l Aω

ωσ +∝

Analytical calculations Numerical calculations

)(∞inF

)(∞outF

)(hinF

Equation of motion analyticallysolved but gives results only for

low and high energy regime

Equation of motion numericallysolved to obtain exact results in

the whole energy range


Analytical investigations• High energy regime: last

stable orbit for a test particles

• Solving the KG equationnear the event horizon (hypergeometricfunctions) and at spatial infinity (spherical Bessel functions) and finallymatch the two solutions in an intermediateregime

22

2)(11

rrh

bddr

r−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ϕ

Classical accessible regime

))(/min( rhrb <

Optical, high energy limit2min)( bg ×=∞→ πωσ


Numerical investigations (1)

• The Klein-Gordon equation is numerically solved from the eventhorizon until spatial infinity. The no out-going waves boundarycondition at the event horizon is applied.

• Once a suffiently high value of r is reached, analytical solutions at spatial infinity are fitted on the numerical solutions to extractthe amplitude of ingoing and outgoing modes

yiineAr ω

ωψ −=)(,l

reB

reBr

ri

out

ri

in

ωω

ωψ +=−

)(,l



• Allows the determination of the tunnel probability for each multipole and at a given energy…

• …and consequently, to determine the greybodyfactors

22 1

in

out

BBA −=l

∑ +=l

l

l 22

12)( Aω

ωσ



• There are three numerical uncertainties for such a calculation, in addition to numerical integration:

• but ε small enough to induce a small error

• The numerical integration is stopped before spatial infinity,the induced error was checked to subdominant

• The summation over multipoles cannot be carried out untilinfinity. However, it can be safely truncated as the effective potential increases with the angular momentum quantum number. The high multipole contributions to thegreybody factors are irrelevant.

ε+= hini rr


What about non-vanishing spin particles…

• A master equation of motion is obtained using theNewman-Penrose formalism

• With asymptotic solutions

2

2221

)( )1()1(with

0)(

2)(

rrhssjj

Pdrdh

rhrisris

rhr

drdP

drd

ssss

=Δ−−+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−++⎟

⎠⎞

⎜⎝⎛ ΔΔ −

λ

λωωω

yisout

yiinhs eAeArrP ωω Δ+=→ −)(

reB

reBrP

ri

outs

ri

ins

ωω

+=∞→ −

−

21)(


… and particles emitted in the bulk• In ADD braneworld, only scalar fields with vanishing

standard model charges are allowed to propagate in the bulk

• With asymptotic solutions

0)3()()()(02

2022 =⎟

⎠⎞

⎜⎝⎛ +−−+⎟

⎠⎞

⎜⎝⎛ −

− PDrrh

drdPrrh

drd

rrh D

D lω

yiinhs eArrP ω−=→ )(

22)(

−−

−

+=∞→D

ri

outD

ri

insreB

reBrP

ωω


Let’s add a second order term in scalar curvature

Gauss-Bonnet Gravity


Absorption Cross-Section-results for scalar on the brane-

d=6 210−=α

Increase of the high energy limit induced by theGauss-Bonnet term

J. Grain, A. Barrau, P. KantiPhys. Rev. D 72, 104016 (2005)


Absorption Cross-Section-results for fermions and gauge-bosons on the

brane-

Increase of the high energy limit induced by the Gauss-Bonnetterm

GB coupling affects the greybody factors in the opposite way ofthe number of dimensions does

fermions Gauge-bosons


Absorption Cross-Section-results for scalar in the bulk-

Increase of the high energy limit induced by the Gauss-Bonnet term

Quantitatively, GB coupling has more influence for fiedsin the bulk


Radiation spectra-influence of the black-hole mass-

∗= MMBH410

Low masses: increase of GB coupling leads to a decrease of the flux

High masses: increase of GB coupling leads to an increase of the flux

∗= MMBH 10


Bulk-to-brane emission rates ratios-influence of the black-hole mass-

∗= MMBH410

Low masses:BHs mainly decay in the brane

High masses: important contirubution of the bulk channel

∗= MMBH 10


Let’s add a cosmological constant


(A)dS Universe

Presence of an event horizon at

μνμνμνμν πTgRgR 821 =Λ+−

Λ−−=

2)2)(1( ddRdS

De Sitter (dS) Universe

Positive Λ

Anti-De Sitter (AdS) Universe

Negative Λ

Cosmological constant


Black Holes in such a space-time

• Two event horizons and

22

22

)2)(1(2

222

)2)(1(22

)1()1(

1

1 −−−

Λ−−Λ Ω−

−−−−−=

−

− dddr

ddrdr

rdrdtrds

d

d μμ

Metric function h(r)

De Sitter (dS) Universe

HR dSR


Calculation of Greybody factors

Analytical calculations Numerical calculations

)(∞inF

)(∞outF

)(hinF

Equation of motion analyticallysolved at the black hole’s and the

de Sitter horizon

Equation of motion numericallysolved from black hole’s horizon

to the de Sitter one

De Sitter horizon


The problem of non-asymptotical flatness

• For the temperature • For the greybody factors, usingthe optical theorem requiresasymptotically free states

asymptotically flat

dS UniverseHrdr

rdhrh

T )(41

)(1

0 π=Λ

Hrflat dr

rdhT )(41π

=

)()())(,( 22

2

yUyUrhVdyd ω=⎥

⎦

⎤⎢⎣

⎡+− l

⎩⎨⎧

∞→→

→r

rrryV H when 0))((

⎩⎨⎧

→→

→dS

H

rrrr

ryV when 0))((


Absorption Cross-Section-results for scalar in dS universe-

d=4 210−=Λ

The divergence comes from thepresence of two horizons

P. Kanti, J. Grain, A. BarrauPhys. Rev. D 71 (2005) 104002


Fluxes and bulk-to-brane ratios-results for scalar in dS universe-

BHs mainly decay via the branechannel

Emssion of ultra-soft particles due to the infrared divergence of σ


Micro Black holes at the LHC

A. Barrau, J. Grain & S. Alexeev

Phys. Lett. B 584 (2004) 114

We will see…Let’s hope!!!


Black Holes Creation

• Two partons with a center-of-mass energy moving in opposite direction

• A black hole of mass and horizon radius is formed if the impact parameter is lower than

s

From Giddings & al. (2002)

sM BH = hrhr


Precursor Works

• Computation of the black hole’s formation cross-section• Derivation of the number of black holes produced at the LHC• Determination of the dimensionnality of space using Hawking’s law

From Dimopoulos & al. 2001

Banks, Fischler hep-th/9906038

Giddings, Thomas Phys. Rev. D 65, 056010 (2002)

Dimopoulos, Landsberg Phys. Rev. Lett 87, 161602 (2001)


Gauss-Bonnet Black holes’ Thermodynamic

Non-monotonic behaviourtaking full benefit of evaporation process(integration over black hole’s lifetime)

Boulware, Deser Phys. Rev. Lett. 83, 3370 (1985)

Cai Phys. Rev. D 65, 084014 (2002)

Properties derived by:


The flux Computation

• Analytical results in the high energy limitThe grey-body factors are constant

• is the most convenient variable

Harris, Kanti JHEP 010, 14 (2003)


The Flux Computation (ATLAS detection)

• Planck scale = 1TeV• Number of Black Holes produced

at the LHC derived by Landsberg• Hard electrons, positrons and

photons sign the Black Hole decay spectrum

• ATLAS resolution

A. Barrau, J. Grain & S. AlexeevPhys. Lett. B 584 (2004) 114


The Results -measurement procedure-

• For different input values of (D,λ), particles emitted by the full evaporation process are generated

spectra are reconstructed for each mass bin

• A analysis is performed

•

2χ


The Results-discussion-

• For a planck scale of order a TeV, ATLAS can distinguish between the case with and the case without Gauss-Bonnet term.

Important progress in the construction of a full quantum theory of gravity

• The results can be refined by taking into account more carefully the endpoint of Hawking evaporation

• The statistical significance of the analysis should be taken with care

2χ


Astrophysical production of μBH

• Such BHs should decay Is there a conflict with observed CR spectrum? Antiprotons are a good probe as .

1<<pp

BHISMCR →+

)'()('

)(' '

,2

QQfdtdE

NdBoostedE

dEdN

dtdQdN

Egq

BHCRp

⎟⎟⎠

⎞⎜⎜⎝

⎛⊗⊗⊗≡ σ

Propagation of antiprotons through the galaxies using adiffusive two-zone model:

Maurin, Taillet, Donato, Salati, Barrau, Boudoul, review article published in “Research Signapost” (2002) [astro-ph/0212111]


Antiprotons fluxes

Much lower than the observed, secondary spectrum; thought it isharder. Early Universe production is also OK because of theirbrief lifetime. Relic production OK, DM < 10^12g / galaxy

A. Barrau, C. Féron & J. Grain Astrophys. J., 630 (2005) 1015


ConclusionBig black holes are fascinating…

But small black holes are far more fascinating!!!

Thanks to the people whomade the pictures


Primordial Black holes in our Galaxy

F.Donato, D. Maurin, P. Salati, A. Barrau, G. Boudoul, R.TailletAstrophy. J. (2001) 536, 172

A. Barrau, G. Boudoul et al., Astronom. Astrophys., 388, 767 (2002)

Astrophys. 398, 403 (2003)

Barrau, Blais, Boudoul, Polarski, Phys. Lett. B, 551, 218 (2003)


CosmologicalCosmological constrainconstrain usingusing PBHPBH

Small black Small black holesholes couldcould have have beenbeenformedformed in in thethe earlyearly universeuniverse

StringentStringent constrainsconstrains on on thethe amountamount ofofPBH in PBH in thethe galaxygalaxy::

TheThe antianti--proton flux proton flux emittedemitted by PBH by PBH isisevaluatingevaluating usingusing an an improvedimprovedpropagation propagation schemescheme for for cosmiccosmic raysrays

This This leadsleads to to constrainconstrain on on thethe PBH PBH fractionfraction

New New windowwindow ofof detectiondetection usingusing lowlowenergyenergy antianti--deuterondeuteron

9104 −×<ΩPBH

2710−≤PBHβ

Documents

Black Holes beyond General Relativity - LPSClpsc.in2p3.fr/barrau/aurelien/talk_montpellier_2005.pdf · 2005-11-25 · J. Grain , A. Barrau - LPSC Black Holes evaporate • Radiation