Ghost Free, Singularity Free Theory of Gravity and observational - PowerPoint PPT Presentation

d 4 x ⇥� g [ RF 1 ( ⇤ ) R + RF 2 ⇤ Ghost Free, Singularity Free Theory of Gravity and observational hints R λσ F 5 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ λ R µ ν + Anupam Mazumdar Lancaster University R ρ λ F 8 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ ρ R µ νλσ Warren Siegel, Tirthabir Biswas � Alex Kholosev, Sergei Vernov, Erik Gerwick, R µ νλσ F 10 ( ⇤ ) R µ νλσ + R ρ Tomi Koivisto, Aindriu Conroy, Spyridon Talaganis µ νλ F Phys. Rev. Lett. (2012), JCAP (2012, 2011), JCAP (2006) CQG (2013), gr-qc/1408.6205 R ν 1 ρ 1 σ 1 F 13 ( ⇤ ) ⇤ ρ 1 ⇤ σ 1 ⇤ ν 1 ⇤ Einstein’s GR is well behaved in IR, but UV is µ Pathetic; Aim is to address the UV aspects of Gravity

Tests of 1/r gravity: 10 − 4 cm − 10 26 cm 1109.6571 hep-ph/0611184 V = − Gm 1 m 2 ⇣ 1 + α e − r/ λ ⌘ r There is NO departure from inverse square law gravity

d 4 x ⇥� g [ RF 1 ( ⇤ ) R + RF 2 ⇤ Classical Singularities UV is Pathological, IR Part is Safe R λσ F 5 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ λ R µ ν + Z √− gd 4 x ✓ ◆ R S = 16 π G + · · · R ρ λ F 8 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ ρ R µ νλσ What terms shall we add such that gravity behaves better at small distances and R µ νλσ F 10 ( ⇤ ) R µ νλσ + R ρ at early times ? µ νλ F � � While keeping the General Covariance Z √− gd 4 x ✓ ◆ R R ν 1 ρ 1 σ 1 S = F 13 ( ⇤ ) ⇤ ρ 1 ⇤ σ 1 ⇤ ν 1 ⇤ 16 π G µ

d 4 x ⇥� g [ RF 1 ( ⇤ ) R + RF ⇤ Motivations Resolution to Blackhole Singularity � R λσ F 5 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ λ R µ ν Resolution for Quantum Mechanics & Gravity Blackhole Information Loss Paradox R ρ λ F 8 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ ρ R µ νλ � Resolution to Cosmological Big Bang R µ νλσ F 10 ( ⇤ ) R µ νλσ + R ρ Singularity Geodesically complete Inflationary Trajectory µ νλ While Keeping IR Property of GR Intact R ν 1 ρ 1 σ 1 F 13 ( ⇤ ) ⇤ ρ 1 ⇤ σ 1 ⇤ ν 1 µ

d 4 x ⇥� g [ RF 1 ( ⇤ ) R + RF Bottom-up approach ⇤ Higher derivative gravity & ghosts R λσ F 5 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ λ R µ ν Covariant extension of higher derivative ghost-free gravity Singularity free theory of gravity R ρ λ F 8 ( ⇤ ) ⇤ µ ⇤ σ ⇤ ν ⇤ ρ R µ νλ Background independent action of UV gravity R µ νλσ F 10 ( ⇤ ) R µ νλσ + R ρ 4d picture of Gravity µ νλ Corrections in EFT is a good M M p UV becomes R ν 1 ρ 1 σ 1 F 13 ( ⇤ ) ⇤ ρ 1 ⇤ σ 1 ⇤ ν 1 approximation in IR important µ

As a smoking gun example… String Theory Introduces 2 Parameters κ 2 ≈ g 2 s ( α � ) 12 � Fundamental Strings are Non-Local DBI action ameliorates the Point like Singularity of Coulomb Solution Z p d p +1 ζ S = − T p − det( γ ab + 2 πα 0 F ab ) � + DBI Action Provides a Description of Open Strings to All Orders in at One-Loop α 0 + Challenge for String Theorists: To Construct a similar Action for Closed Strings with All Orders in α 0 +

4th Derivative Gravity & Power Counting renormalizability � � 3 ( b + a ) R 2 � d 4 x √ g λ 0 + k R + a R µ ν R µ ν − 1 I = ✓ 1 ◆ k 4 + Ak 2 = 1 1 1 D ∝ k 2 − k 2 + A A Massive Spin-0 & Massive Spin-2 ( Ghost ) Stelle (1977) Utiyama, De Witt (1961), Stelle (1977) Modification of Einstein’s GR Extra propagating Modification degree of freedom of Graviton Propagator Challenge: to get rid of the extra dof

Ghosts Higher Order Derivative Theory Generically Carry Ghosts ( -ve Risidue ) with real “m”( No- Tachyon) Propagator with first order poles Ghosts cannot be cured order by order, finite terms in perturbative expansion will always lead to Ghosts !!

Higher Derivative Action around Minkowski S = S E + S q Z .... R .... + R .... O .... .... R .... + R .... O .... .... R .... + · · · ] d 4 x √− g [ R .... O .... S q = .... R .... O .... .... R .... O .... .... R .... O .... R ∼ O ( h ) g µ ν = η µ ν + h µ ν � d 4 x √− gR µ 1 ν 1 λ 1 σ 1 O µ 1 ν 1 λ 1 σ 1 µ 2 ν 2 λ 2 σ 2 R µ 2 ν 2 λ 2 σ 2 S q = Unknown Infinite Covariant derivatives Functions of Derivatives

RF 2 ( � ) ∇ µ ∇ ν R µ ν + R µ ν F 3 ( � ∞ − → ∞ d 4 x √− g [ RF 1 ( � ) R + RF 2 ( � ) ∇ µ ∇ ν R µ ν + R µ ν F 3 ( � ) R µ ν + R ν � µ F 4 ( � ) ∇ ν ∇ λ R µ λ S q = + R λσ F 5 ( � ) ∇ µ ∇ σ ∇ ν ∇ λ R µ ν + RF 6 ( � ) ∇ µ ∇ ν ∇ λ ∇ σ R µ νλσ + R µ λ F 7 ( � ) ∇ ν ∇ σ R µ νλσ + RF 6 ( � ) ∇ µ ∇ ν ∇ λ ∇ σ R µ νλ λ F 8 ( � ) ∇ µ ∇ σ ∇ ν ∇ ρ R µ νλσ + R µ 1 ν 1 F 9 ( � ) ∇ µ 1 ∇ ν 1 ∇ µ ∇ ν ∇ λ ∇ σ R µ νλσ + R ρ + R µ νλσ F 10 ( � ) R µ νλσ + R ρ µ νλ F 11 ( � ) ∇ ρ ∇ σ R µ νλσ + R µ ρ 1 νσ 1 F 12 ( � ) ∇ ρ 1 ∇ σ 1 ∇ ρ ∇ σ R µ ρνσ ⇧ F 13 ( � ) ∇ ρ 1 ∇ σ 1 ∇ ν 1 ∇ ν ∇ ρ ∇ σ R µ νλσ + R µ 1 ν 1 ρ 1 σ 1 F 14 ( � ) ∇ ρ 1 ∇ σ 1 ∇ ν 1 ∇ µ 1 ∇ µ ∇ ν ∇ ρ ∇ σ R µ νλσ + R ν 1 ρ 1 σ 1 µ + R µ 1 ν 1 F 9 ( � ) ∇ µ 1 ∇ ν 1 ∇ µ f i,n ⇤ n . F i ( ⇤ ) = ⇤ n ≥ 0 d the most general no F 11 ( � ) ∇ ρ ∇ σ R µ νλσ + R µ ρ What Have We Gained ? Fundamental Theory Must ∇ ν ∇ ρ ∇ σ R µ νλσ + R µ 1 ν 1 ρ 1 σ 1 have Finite Parameters

Redundancies d 4 x √− g [ RF 1 ( � ) R + RF 2 ( � ) ∇ µ ∇ ν R µ ν + R µ ν F 3 ( � ) R µ ν + R ν � µ F 4 ( � ) ∇ ν ∇ λ R µ λ S q = + R λσ F 5 ( � ) ∇ µ ∇ σ ∇ ν ∇ λ R µ ν + RF 6 ( � ) ∇ µ ∇ ν ∇ λ ∇ σ R µ νλσ + R µ λ F 7 ( � ) ∇ ν ∇ σ R µ νλσ λ F 8 ( � ) ∇ µ ∇ σ ∇ ν ∇ ρ R µ νλσ + R µ 1 ν 1 F 9 ( � ) ∇ µ 1 ∇ ν 1 ∇ µ ∇ ν ∇ λ ∇ σ R µ νλσ + R ρ + R µ νλσ F 10 ( � ) R µ νλσ + R ρ µ νλ F 11 ( � ) ∇ ρ ∇ σ R µ νλσ + R µ ρ 1 νσ 1 F 12 ( � ) ∇ ρ 1 ∇ σ 1 ∇ ρ ∇ σ R µ ρνσ F 13 ( � ) ∇ ρ 1 ∇ σ 1 ∇ ν 1 ∇ ν ∇ ρ ∇ σ R µ νλσ + R µ 1 ν 1 ρ 1 σ 1 F 14 ( � ) ∇ ρ 1 ∇ σ 1 ∇ ν 1 ∇ µ 1 ∇ µ ∇ ν ∇ ρ ∇ σ R µ νλσ + R ν 1 ρ 1 σ 1 µ Z R + R F 1 ( ⇤ ) R + R µ ν F 2 ( ⇤ ) R µ ν + R µ ναβ F 3 ( ⇤ ) R µ ναβ ⇤ d 4 x √− g ⇥ = ∆ L = √− g ( α R 2 + β R 2 µ ν + γ R 2 αβ µ ν ) Gauss-Bonet ⇤ d 4 x √− g ( R 2 − 4 R 2 Gravity µ ν + R 2 µ ναβ ) ,

Z R + R F 1 ( ⇤ ) R + R µ ν F 2 ( ⇤ ) R µ ν + R µ ναβ F 3 ( ⇤ ) R µ ναβ ⇤ d 4 x √− g ⇥ = g µ ν = η µ ν + h µ ν ⇤ ⌅ 1 2 h µ ν a ( ⇤ ) ⇤ h µ ν + h σ µ b ( ⇤ ) ∂ σ ∂ ν h µ ν d 4 x S q = − (3) + hc ( ⇤ ) ∂ µ ∂ ν h µ ν + 1 2 hd ( ⇤ ) ⇤ h + h λσ f ( ⇤ ) ∂ σ ∂ λ ∂ µ ∂ ν h µ ν ⇧ . ⇤ a ( � ) = 1 − 1 1 2 F 2 ( � ) � − 2 F 3 ( � ) � = 2( ∂ [ λ ∂ ν h µ σ ] − ∂ [ λ ∂ µ h νσ ] ) R µ νλσ b ( � ) = − 1 + 1 1 2 F 2 ( � ) � + 2 F 3 ( � ) � 2( ∂ σ ∂ ( ν h σ = µ ) − ∂ ν ∂ µ h − ⇤ h µ ν ) R µ ν c ( � ) = 1 + 2 F 1 ( � ) � + 1 ∂ ν ∂ µ h µ ν − ⇤ h = 2 F 2 ( � ) � R d ( � ) = − 1 − 2 F 1 ( � ) � − 1 a + b = 0 2 F 2 ( � ) � f ( � ) = − 2 F 1 ( � ) � − F 2 ( � ) � − 2 F 3 ( � ) � . c + d = 0 b + c + f = 0 F 3 ( ⇤ ) is redundant around Minkowski

Graviton Propagator µ ) + c ( ⇤ )( η µ ν ∂ ρ ∂ σ h ρσ + ∂ µ ∂ ν h ) b ( ⇤ ) ∂ σ ∂ ( ν h σ a ( ⇤ ) ⇤ h µ ν + 1 4 f ( ⇤ ) ⇤ − 1 ∂ σ ∂ λ ∂ µ ∂ ν h λσ = � κτ µ ν + η µ ν d ( ⇤ ) ⇤ h + = 0 = 0 = 0 ν ,µ + ( b + c + f ) h αβ ν = 0 = ( c + d ) ⇤ ∂ ν h + ( a + b ) ⇤ h µ � κτ ⇧ µ τ µ , αβν a + b = 0 Bianchi Identity c + d = 0 b + c + f = 0 λσ h λσ = ⇤⇧ µ ν Π − 1 h = h T T + h L + h T µ ν Π = P 2 P 0 P 0 ak 2 + ( a − 3 c ) k 2 + s w ( c − a + f ) k 2

Covariant Modification of a Graviton Propagator : Only 1 Entire Function UV Π = P 2 P 0  P 2 k 2 − P 0 � s = 1 ak 2 + s ( a � 3 c ) k 2 2 k 2 a a ( k 2 ) = c ( k 2 ) Demand: k 2 → 0 Π µ ν λσ = ( P 2 /k 2 ) � ( P 0 s / 2 k 2 ) Recovers GR lim IR a (0) = c (0) = � b (0) = � d (0) = 1 ONLY 1 Non-Singular, Analytic functions at k=0, is required to Ameliorate the UV property of GR � ‘a’ should be an Entire Function & cannot contain non-local operators, such as a ( ⇤ ) ∼ 1 / ⇤

Ghost Free Gravity a ( � ) = c ( � ) ⇒ 2 F 1 ( � ) + F 2 ( � ) + 2 F 3 ( � ) = 0 Entire Function  P 2 k 2 − P 0 Π = P 2 P 0 � = 1 s s ak 2 + 2 k 2 a ( a � 3 c ) k 2 a ( ⇤ ) = c ( ⇤ ) = e − ⇤ /M 2 Some function of k which falls faster than1 /k 2 M 2 and F 3 = 0 ⇒ F 1 ( � ) = e − � M 2 − 1 = − F 2 ( � ) a ( � ) = e − � 2 �