Newtonian Potential and Geodesic Completeness in Infinite Derivative Gravity
NNewtonian Potential and Geodesic Completeness in Infinite Derivative Gravity
Aindriú Conroy and James Edholm Consortium for Fundamental Physics,Physics Avenue, Lancaster University,Lancaster, LA1 4YB, United Kingdom.
Recent study has shown that a non-singular oscillating potential – a feature of Infinite DerivativeGravity (IDG) theories – matches current experimental data better than the standard GR potential.In this work we show that this non-singular oscillating potential can be given by a wider class of the-ories which allows the defocusing of null rays, and therefore geodesic completeness. We consolidatethe conditions whereby null geodesic congruences may be made past-complete, via the RaychaudhuriEquation, with the requirement of a non-singular Newtonian potential in an IDG theory. In doingso, we examine a class of Newtonian potentials characterised by an additional degree of freedom inthe scalar propagator, which returns the familiar potential of General Relativity at large distances.
I. INTRODUCTION
In the century since Albert Einstein ushered in a newparadigm for modern physics by formulating a gravita-tional theory that is described by the curvature of space-time, General Relativity (GR) has withstood numerousexperimental tests [1]. It accurately describes our uni-verse all the way down to short distances ( . × − mfrom a source [2]) and has been found to be in agree-ment with experimental tests on gravitational redshiftand the equivalence principle [3]. More recently, the de-tection of gravitational waves lends more weight to thisalready colossal theory [4]. However, that is not to saythat GR does not have any shortcomings. At the classi-cal level, the theory breaks down at short distances in itsdescription of black holes, for instance. GR also cannotbe made geodesically past-complete, when an appropri-ate energy condition is met, indicating the presence of aninitial singularity in the theory [6–9]. In a geodesically-incomplete spacetime, causal geodesic congruences con-verge to a point in a finite ‘time’ (affine parameter). Insuch a scenario a freely falling particle or photon willsimply cease to exist in a finite ‘time’, which suggests aserious physical malady in the theory [10]. These short-comings allow us to consider GR to be a first approxima-tion to a broader theory.This notion of extending GR, via additional curva-ture terms in the gravitational action, forms the basisfor many modified theories of gravity. Significant exam-ples include local theories such as f ( R ) -gravity, which re-places the curvature scalar in the Einstein-Hilbert actionwith an arbitrary function f ( R ) , or Stelle’s 4th deriva-tive gravity, which can be seen as a generalisation of theGauss-Bonnet term in that it includes tensorial as wellas scalar modifications to the Einstein-Hilbert action.However, these finite higher-derivative models still comehand-in-hand with a number of weaknesses. Stelle’s the-ory, while perturbatively renormalizable [11], suffers from Dark energy notwithstanding the introduction of ghosts - physical excitations, charac-terised by negative kinetic energy [12]. f ( R ) -gravity, incomparison, may avoid the introduction of ghosts [3] butbreaks down at short distances, meaning that the the-ory can not be said to be UV-complete [13]. In contrast,Infinite Derivative Gravity (IDG) offers a means of ‘com-pleting’ GR in the UV regime (short distances). IDG ischaracterised by an action containing an infinite series ofd’Alembertian operators ( (cid:3) = g µν ∇ µ ∇ ν ) acting on thecurvature along with the mass scale of the theory, M .The IDG action was first derived in [14], where theform of the modified propagator was also calculatedaround a Minkowski background, see also [15, 16]. Thefull non-linear equations of motion were computed in [17]and the boundary terms found in [18]. The gravitationalentropy for the IDG action was investigated around an(A)dS metric in [19], while the form of the (A)dS prop-agator was given in [20]. Constraints were put on themass scale M of IDG, either by using data on the tensor-scalar ratio and spectral tilt of the Cosmic MicrowaveBackground [21], or by looking at the deflection of lightaround the Sun [22].In [23, 24], it was shown using a toy model of IDG thatit is possible to curtail the divergences of 1-loop diagramsand show that 2-loop diagrams are finite, while in [25] theUV finiteness of IDG theories were investigated. Furtherwork has focused on the Newtonian potential around aflat background [14, 26, 27], formulating the Hamiltonianof the IDG action [28] and avoiding cosmological singular-ities both through using the ansatz (cid:3) R = c R + c [29, 30]and via the Raychaudhuri equation [31–33].In this paper we investigate the Newtonian potential Φ( r ) , describing the gravitational field of a small staticspherically symmetric test mass in a flat space back-ground. In General Relativity, this diverges accordingto Φ( r ) ∼ − /r , becoming singular at the origin. IDGoffers a means of resolving this divergence.It was shown in [17] that the equations of motion forIDG can be formulated in terms of two arbitrary func-tions of the d’Alembertian operator a ( (cid:3) ) and c ( (cid:3) ) , whichalso characterise the modification to the gravitationalpropagator. In the case where these functions are equal, a r X i v : . [ g r- q c ] M a y i.e. a ( (cid:3) ) = c ( (cid:3) ) , no additional degrees of freedom otherthan the massless graviton enter the system. Previouswork [14, 26] has shown that, in this case, a non-singularNewtonian potential can be derived, where the potentialtakes the form of an error function Φ ∼ − Erf ( M r/ /r ,for the simplest choice of a ( (cid:3) ) = e − (cid:3) /M . If we gener-alise this further by taking a ( (cid:3) ) to be an exponential ofa higher order polynomial, the potential is modified byan oscillating function at these higher orders. Analysisby Perivolaropoulos [2, 34] has shown that this oscillat-ing function provides a better fit to experimental data onthe force of gravity at small distances than the standardGR theory. Although further analysis is needed, thereare tantalising hints that modified gravity could providethe correct description of the strength of gravity at smalldistances.However, work on the avoidance of singularities in IDGtheories [31–33, 35] has shown that we require a depar-ture from the simple choice of a ( (cid:3) ) = c ( (cid:3) ) in order forcausal geodesic congruences to be made past-complete.The aim of the present work is to consolidate the re-quirements of a non-singular theory of gravity, known asthe defocusing conditions, with the aforementioned con-verging Newtonian potential. To this end, we examinea wider class of Newtonian potentials, characterised bythe condition a ( (cid:3) ) (cid:54) = c ( (cid:3) ) , in tandem with the defocus-ing conditions derived in [32] around a Minkowski back-ground. This would allow us to avoid singularities bypermitting the defocusing of null rays in a theory with awell-defined Newtonian potential at short distances.In Section II, we give an overview of Infinite Deriva-tive Gravity (IDG), and show how the theory offers ameans of rendering null rays geodesically-complete. InSection III, we derive the Newtonian potential for IDG,and specifically look at the case which allows defocusing.Finally in Section IV we plot the Newtonian potentialand interpret the results. II. INFINITE DERIVATIVE GRAVITY
As mentioned in the introduction, ghosts are physi-cal excitations bearing negative kinetic energy. Theseexcitations are represented by a negative residue in thegravitational propagator. When interactions in such asystem take place, the vacuum decays into both positiveand negative energy states. This is known as the Ostro-gradsky instability [36].Previous attempts to resolve singularities by modify-ing gravity, such as Stelle’s 4th derivative gravity [11],resulted in the introduction of ghosts, where the Hamil-tonian of the theory was unbounded from below due tothe Ostrogradsky instability [36]. By adding an infinitenumber of derivatives to the theory this instability maybe avoided through an appropriate choice of the functions a ( (cid:3) ) and c ( (cid:3) ) . In [28] it was shown that the Hamiltonianof IDG is indeed bounded from below.The IDG action, which is the most general, torsion-free and parity invariant action of gravity, that is quadraticin curvature was first derived in [14, 35] S = 12 (cid:90) d x √− g (cid:18) M p R + RF ( (cid:3) ) R + R µν F ( (cid:3) ) R µν + C µνλσ F ( (cid:3) ) C µνλσ (cid:19) , (1)where R is the Ricci curvature scalar, R µν is the Riccitensor, C µνλσ is the Weyl tensor and M p is the Planckmass. Each F i ( (cid:3) ) = (cid:80) ∞ n =0 f i n (cid:3) n /M n is a function ofthe d’Alembertian operator (cid:3) ≡ g µν ∇ µ ∇ ν . M is thescale of modification of our theory and the f i n are thedimensionless coefficients of the series.The equations of motion for IDG around a Minkowskibackground are given by [32] κT µν = a ( (cid:3) ) R µν − η µν c ( (cid:3) ) R − f ( (cid:3) ) ∇ µ ∇ ν R, (2)where we have defined a ( (cid:3) ) = 1 + M − P (cid:18) F ( (cid:3) ) + 2 F ( (cid:3) ) (cid:19) (cid:3) ,c ( (cid:3) ) = 1 + M − P (cid:18) − F ( (cid:3) ) − F ( (cid:3) ) + 23 F ( (cid:3) ) (cid:19) (cid:3) ,f ( (cid:3) ) = M − P (cid:18) F ( (cid:3) ) + 2 F ( (cid:3) ) + 43 F ( (cid:3) ) (cid:19) , (3)which abide by the constraint a ( (cid:3) ) − c ( (cid:3) ) = f ( (cid:3) ) (cid:3) .From (2) we can derive the propagator around a flat back-ground [14, 15] Π( k ) = P (2) k a ( − k ) − P (0) s k ( a ( − k ) − c ( − k )) . (4)The simplest choice is to set a ( (cid:3) ) = c ( (cid:3) ) equal to the ex-ponential of an entire function, which by definition doesnot have any roots, and therefore the propagator willnot receive any additional degrees of freedom other thanthe massless graviton. It was shown in [16, 33, 35] thatthe scalar sector of the propagator can have at most oneadditional pole compared, without the introduction ofghost-like degrees of freedom. This allows us to write ( a ( (cid:3) ) − c ( (cid:3) )) R = 2 (cid:0) (cid:3) /m − (cid:1) ¯ a ( (cid:3) ) R, (5)where ¯ a ( (cid:3) ) is the exponential of an entire function and so a ( − k ) − c ( − k ) has a single pole in the scalar sector at (cid:3) → − k = m , which produces an extra spin-0 particleof mass m . By expanding to first order in (cid:3) , we find that m ≡ M P / (6 f − f − M p /M ) [32]. We know that m > so that the particle has real mass and thereforeno tachyons are introduced. We denote the linearised curvatures around a Minkowski back-ground as R , R µν , R µνρσ . A. Defocusing conditions
The Raychaudhuri equation is a model-independentidentity which relates the geometry of spacetime to thecontribution of gravity via the curvature. It says thatfor a null tangent vector k µ , satisfying k µ k µ = 0 , theexpansion parameter θ = ∇ µ k µ is described by dθdλ + 12 θ ≤ − R µν k µ k ν , (6)where λ is the affine parameter [10]. In order to havean expansion parameter which is positive and increas-ing, and therefore allow the defocusing of the null rays,we require R µν k µ k ν < . In GR, using the Einsteinequation, G µν = κT µν , we cannot fulfil this conditionbecause the Null Energy Condition (NEC) requires that T µν k µ k ν ≥ . By the Hawking-Penrose Singularity The-orem, this inability to defocus will always lead to a singu-larity. To be precise, a spacetime cannot be null geodesi-cally complete in the past direction if R µν k µ k ν > [9].However, in IDG this is not the case and we will seethat it is possible to have defocusing. By contracting theequations of motion (2) with k µ k ν , we can see that thecontribution of gravity to the Raychaudhuri equation is R µν k µ k ν = 1 a ( (cid:3) ) (cid:20) κT µν k µ k ν + k µ k ν f ( (cid:3) ) ∇ µ ∇ ν R (cid:21) , (7)which was studied in a cosmological setting [32]. Whenconsidering a static spherically symmetric perturbationaround a flat background, then the defocusing conditionis R µν k µ k ν = 1 a ( (cid:3) ) (cid:20) κT µν k µ k ν + ( k r ) f ( (cid:3) ) (cid:3) R ( r ) (cid:21) < , (8)where (cid:3) R ( r ) = r ∂ r ( r ∂ r ) R ( r ) . Note that the function a ( (cid:3) ) acting on the curvature cannot be negative, as thiswould lead to the Weyl ghost. This is because a ( − k ) isthe modification to the spin-2 part of the propagator, so itmust be positive to avoid negative residues [15]. There-fore if the NEC holds true, we arrive at the minimumdefocusing condition a ( (cid:3) ) − c ( (cid:3) ) a ( (cid:3) ) R ( r ) < . (9)The first observation to make is that defocusing cannotoccur in the case of a ( (cid:3) ) = c ( (cid:3) ) . However, if we allow anextra scalar propagating mode, from (5) we can say thatthe relationship between a ( (cid:3) ) and c ( (cid:3) ) is given by [32] c ( ¯ (cid:3) ) = a ( ¯ (cid:3) )3 (cid:2) (cid:0) − (cid:3) /m (cid:1) ˜ a ( ¯ (cid:3) ) (cid:3) , (10) We have taken the simplest case of the Raychaudhuri equationhere by making two simplifications. Firstly, we take the congru-ence of null rays to be orthogonal to the hypersurface, so that thetwist tensors vanish. Secondly, the shear tensor gives a positivecontribution to the right-hand side and so we can neglect it forour purposes. where ˜ a ≡ ¯ a ( (cid:3) ) a ( (cid:3) ) is an exponent of an entire function.Hence the propagator is given by [32] Π( k ) = 1 a ( − k ) (cid:34) P (2) k + P (0) S k (1 + k /m )˜ a ( − k ) (cid:35) , (11)while the minimum condition for null rays to defocus is(9) becomes (1 − (cid:3) /m )˜ a ( (cid:3) ) R ( r ) > R ( r ) . (12) III. NEWTONIAN POTENTIAL
When we take the metric generated by a small staticspherically symmetric test mass added to a flat spacebackground, following the method of [14, 37], ds = − (1 + 2Φ( r )) dt + (1 − r )) η ij dx i dx j , (13)this is akin to perturbing the flat space metric η µν as g µν = η µν + h µν , where h = h = − and h ij = h ij = − η ij . (14)As a result, the scalar curvature and component of theRicci curvature tensor around the flat Minkowski back-ground are given by R = 4∆Ψ( r ) − r ) , R = ∆Φ( r ) , (15)where ∆ ≡ η ij ∂ i ∂ j is the Laplace operator. Then fromthe equations of motion (2), the trace and componentequations of motion are − κρ = κT = 12 ( a ( (cid:3) ) − c ( (cid:3) )) R,κρ = κT = a ( (cid:3) ) R + 12 c ( (cid:3) ) R. (16)Here we have taken the weak field approximation so that ρ (cid:29) p , where ρ is the density and p is the pressure of thetest mass. Therefore T = ( − ρ + 3 p ) ≈ − ρ and T ≈ ρ .Combining this with (15) leads to ∆Ψ( r ) = − c ( (cid:3) )( a ( (cid:3) ) − c ( (cid:3) )) ∆Φ( r ) , ∆Φ( r ) = a ( (cid:3) ) − c ( (cid:3) ) a ( (cid:3) ) ( a ( (cid:3) ) − c ( (cid:3) )) κρ. (17)As our point source is of mass µ , its density is approx-imated by a 3-dimensional Dirac-delta function: ρ = µ δ ( r ) . Next we use a Fourier transform to calculate Φ( r ) by using the same method as for calculating theCoulomb potential [38, 39]. When the Dirac-delta func-tion is Fourier transformed, it becomes [14] ρ = µ δ ( r ) = µ (cid:90) d k (2 π ) e i k · r . (18) We have noted that the time derivatives of R ( r ) and R ( r ) vanish As we go into momentum space, we take (cid:3) → − k , giving Φ( r ) = − κµ (2 π ) (cid:90) ∞−∞ d k a − ca ( a − c ) e i k · r k = − κµ π r (cid:90) ∞−∞ dk ( a − c ) a ( a − c ) sin( kr ) k . (19)Trivially, we can use (17) to see that Ψ( r ) = κµ π r (cid:90) ∞−∞ dk ca ( a − c ) sin( kr ) k . (20)Note that if we set a ( − k ) = c ( − k ) , Φ( r ) = Ψ( r ) = − κµ π r (cid:90) ∞−∞ dk a sin( kr ) k . (21)This case was discussed in [14, 26]. For the simplest case,where a ( (cid:3) ) = e − (cid:3) /M , it results in the /r fall of the po-tential seen in GR being modified by the error function,i.e. Φ( r ) ∼ Erf ( r ) r . For large r , Erf ( r ) ≈ but at shortdistances, Erf ( r ) ∼ r , so Φ( r ) ∼ . Therefore the /r behaviour is retained at large distances but at short dis-tances, the modification means that the potential simplytails off to a constant and the potential is no longer sin-gular. A. Potential for IDG with defocusing
In this paper we will extend the calculation of the IDGpotential to the case where a (cid:54) = c , which allows us toavoid singularities. We describe the relationship between a ( (cid:3) ) and c ( (cid:3) ) using (10), and by inserting (10) into (19)we find that Φ( r ) = − κµ π r × (cid:90) ∞−∞ dk (cid:20) − m ˜ a ( − k )( m + k ) (cid:21) sin( kr ) k a ( − k ) . (22)The calculation we need to perform is f ( r ) = (cid:90) ∞−∞ dk (cid:20) − m ˜ a ( − k )( m + k ) (cid:21) sin( kr ) k a ( − k ) , (23)where Φ( r ) ∼ − f ( r ) r . We can write ˜ a ( − k ) = e τ ( − k ) and a ( − k ) = e γ ( − k ) , and this results in f ( r ) = (cid:90) ∞−∞ dk (cid:34) − m e − τ ( − k ) m + k (cid:35) e − γ ( − k ) sin( kr ) k . (24)This is our main result - we have shown that we can havedefocusing as well as a non-singular Newtonian potential.This potential returns to the GR value in the infraredlimit, i.e for large values of r . Note that in the limit M → ∞ we return to a non-local theory,which is fourth-order gravity [40]. B. Conditions on a ( − k ) and ˜ a ( − k ) Next we investigate the conditions that must be placedon a ( − k ) and ˜ a ( − k ) , and therefore what we can sayabout τ ( − k ) and γ ( − k ) .First we look at the spin-0 part of the propagator.From (11), we have Π( k ) = 1 e γ ( − k ) (cid:20) P (2) k + P (0) k (1 + k /m ) e τ ( − k ) (cid:21) , (25)where P (2) is the spin-2 projection operator and P (0) is the spin-0 projection operator. We require that in theUV, the propagator is exponentially suppressed, allowingus the possibility of curtailing divergences [23]. From thespin-0 sector of (25), we observe the condition γ ( − k ) + τ ( − k ) > . Fortunately, this condition ensures that theintegral (24) converges. Note that a priori, it is possiblethat there are coefficients in front of the exponentials e γ ( − k ) and e τ ( − k ) , but by requiring that we return tothe GR propagator in the infrared limit k (cid:28) m, M , weare obliged to set the coefficients to 1. Also, by requiringthe spin-2 part of the propagator (25) to be exponentiallysuppressed we have γ ( − k ) > .From looking at (12) in momentum space, the de-focusing condition imposes a constraint: τ ( − k ) ≥ .Thus, we have the functions a ( − k ) = e γ ( − k ) and ˜ a ( − k ) = e τ ( − k ) where γ ( − k ) is positive function and τ ( − k ) is a non-negative function. In fact, the result ofthese constraints is that the integral (24) converges.We therefore have two free mass scales M and m and two unspecified functions τ ( − k ) and γ ( − k ) . Wewill henceforth choose both the mass scales to be thePlanck mass M P and choose the simplest possible ver-sion of τ ( − k ) , which is τ = 0 . This choice means thatall of the freedom in the model is tied up in the func-tion γ ( − k ) , which we will choose to be the monomial γ ( k ) = ( Ck /M P ) D . We therefore have only two freeparameters, C and D . IV. PLOTTING THE RESULTSA. Choosing a form for a ( − k ) and ˜ a ( − k ) For the simplest choice τ ( − k ) = 0 and γ ( k ) =( Ck /M P ) D , this gives f ( r ) = (cid:90) ∞−∞ dk (cid:20) − M P M P + k (cid:21) e − ( Ck /M P ) D sin( kr ) k . (26)In Fig. 1, we take D = 1 and plot f ( r ) for differentchoices of C using (26). Clearly increasing the value of C Note that for large r , then Ψ( r ) = Φ( r ) , i.e. the Eddingtonparameter − Φ / Ψ is equal to one, as expected from experimentalbounds using data from the Cassini probe [41]. FIG. 1: We plot f ( r ) vs r for different C and D = 1 ,where γ ( k ) = ( Ck /M P ) D . See (26). For illustrativepurposes, we have taken M p = 1 m − . We compare ourresults to the a ( (cid:3) ) = c ( (cid:3) ) case seen in (21). Here, wesee that as C increases, the effect of IDG can be seenincreasingly further away from the origin.FIG. 2: We plot f ( r ) vs r for different D and C = 1 where γ ( k ) = ( Ck /M P ) D . See (26). The plot showsthat for D > the potential oscillates. As the value of D increases, so too does the magnitude of theseoscillations. For illustrative purposes, we have taken M p = 1 m − and again we find that our results reduce tothat of GR at large distances.moves the point at which the non-locality kicks in furtheraway from the origin. Next in Fig. 2 we look at how thepotential varies for the same C but with different valuesof D , which is the power of k in (26). As we increase thevalue of D , our potential begins to oscillate, as was foundin [26]. These oscillations grow in size as D increases untilabout D = 10 .This is because lim D →∞ e − ( Ck /M P ) D = rect ( Ck /M P ) , (27)where rect ( x ) is the rectangle function, which is defined FIG. 3: We plot f ( r ) vs r for different D greater than10 with C = 1 , where γ ( k ) = ( Ck /M P ) D . See (26).We can see that for D > , increasing D does notaffect the potential. For illustrative purposes, we havetaken M p = 1 m − . We can parameterise these curves as α r for r < and α cos( θr + θ ) /r for r > , where α , α , θ and θ are constants, as in [34].by rect ( x ) = 1 for | x | < and rect ( x ) = 0 for | x | > .For D > , (27) is a very good approximation and soincreasing the value of D does not change the potential.Our next task is to investigate the choice of large pow-ers of k which has been shown to fit recent experimentaldata [34] at small distances for the a ( (cid:3) ) = c ( (cid:3) ) case. Wewill see whether we can still obtain oscillating solutionswith conditions necessary to realise defocusing. In Fig. 3we plot (26) for different choices of D > and note thatit can still be parameterised accurately as f ( r ) = (cid:40) α r for < r < , α θr + θ ) r for < r. In other words, the oscillating solution which washinted at by experimental data can be produced by amodified gravity solution which also allows geodesic com-pleteness.
V. CONCLUSION
We have found the Newtonian potential for a widerclass of Infinite Derivative Gravity (IDG) theories thanwere previously investigated and analysed various casesof the theory. Analysis of data from experimental testshas hinted that, at small distances, an oscillating non-local potential provides the best fit to experimental data.We have shown that an IDG theory constrained to allowthe defocusing of null rays, and therefore geodesic com-pleteness, still produces a non-singular potential whichreturns to the standard GR result at large distances. Thisresult can still be parameterised as the oscillating func-tion which provides a good fit to the data.By allowing defocusing, it is necessary to introduce ex-tra parameters into the model, although we can reducethe freedom in these parameters by making appropriatechoices about the mass scales and the form of the func-tions. Our results can be tested experimentally, whichwill allow us to put constraints on our parameters. Fu-ture research could look at moving away from the min-imal model and reintroducing the choice of parameters which we reduced, or finding the potential around a deSitter background rather than a flat space background.
VI. ACKNOWLEDGEMENTS
The authors would like to thank David Burton andAnupam Mazumdar for their invaluable help in preparingthis paper. [1] C. M. Will, Living Rev. Relativity, “The Confrontationbetween General Relativity and Experiment” 17, (2014),4[2] D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gund-lach, B. R. Heckel, C. D. Hoyle and H. E. Swanson, “Testsof the gravitational inverse-square law below the dark-energy length scale,” Phys. Rev. Lett. (2007) 021101doi:10.1103/PhysRevLett.98.021101 [hep-ph/0611184].[3] T. Clifton, P. G. Ferreira, A. Padilla and C. Sko-rdis, “Modified Gravity and Cosmology,” Phys.Rept. (2012) 1 doi:10.1016/j.physrep.2012.01.001[arXiv:1106.2476 [astro-ph.CO]].[4] B. P. Abbott et al. [LIGO Scientific and Virgo Collab-orations], “Observation of Gravitational Waves from aBinary Black Hole Merger,” Phys. Rev. Lett. (2016)no.6, 061102[5] R. P. Geroch, “What is a singularity in general rela-tivity?,” Annals Phys. (1968) 526. doi:10.1016/0003-4916(68)90144-9[6] S. Kar, “An introduction to the Raychaudhuri equa-tions,” Resonance J. Sci. Educ. (2008) 319.doi:10.1007/s12045-008-0013-1[7] A. Vilenkin and A. C. Wall, “Cosmological singu-larity theorems and black holes,” Phys. Rev. D (2014) no.6, 064035 doi:10.1103/PhysRevD.89.064035[arXiv:1312.3956 [gr-qc]].[8] T. Vachaspati and M. Trodden, Phys. Rev. D (1999) 023502 doi:10.1103/PhysRevD.61.023502 [gr-qc/9811037].[9] S. W. Hawking and G. F. R. Ellis. “The Large Scale Struc-ture of Space-Time" Cambridge Monographs on Mathe-matical Physics. Cambridge University Press, (2011)[10] R. M. Wald, “General Relativity", Chicago, Usa: Univ.Pr. (1984) 491p[11] K. S. Stelle, Phys. Rev. D 16, 953 (1977) “Renormaliza-tion of higher-derivative quantum gravity”[12] P. Van Nieuwenhuizen, “On ghost-free tensor lagrangiansand linearized gravitation,” Nucl. Phys. B (1973) 478.doi:10.1016/0550-3213(73)90194-6[13] D. Blas, “Aspects of Infrared Modifications of Gravity,”arXiv:0809.3744 [hep-th].[14] T. Biswas, E. Gerwick, T. Koivisto and A. Mazum-dar, “Towards singularity and ghost free theoriesof gravity,” Phys. Rev. Lett. , 031101 (2012)doi:10.1103/PhysRevLett.108.031101 [arXiv:1110.5249[gr-qc]].[15] T. Biswas, T. Koivisto and A. Mazumdar, “Nonlo-cal theories of gravity: the flat space propagator,”arXiv:1302.0532 [gr-qc]. [16] L. Buoninfante, “Ghost and singularity free theories ofgravity,” arXiv:1610.08744 [gr-qc].[17] T. Biswas, A. Conroy, A. S. Koshelev and A. Mazumdar,“Generalized ghost-free quadratic curvature gravity,”Class. Quant. Grav. (2014) 015022 Erratum: [Class.Quant. Grav. (2014) 159501] doi:10.1088/0264-9381/31/1/015022, 10.1088/0264-9381/31/15/159501[arXiv:1308.2319 [hep-th]].[18] A. Teimouri, S. Talaganis, J. Edholm and A. Mazum-dar, “Generalised Boundary Terms for Higher Deriva-tive Theories of Gravity,” JHEP (2016) 144doi:10.1007/JHEP08(2016)144 [arXiv:1606.01911 [gr-qc]].[19] A. Conroy, A. Mazumdar, S. Talaganis and A. Teimouri,“Nonlocal gravity in D dimensions: Propagators, en-tropy, and a bouncing cosmology,” Phys. Rev. D (2015) no.12, 124051 doi:10.1103/PhysRevD.92.124051[arXiv:1509.01247 [hep-th]].[20] T. Biswas, A. S. Koshelev and A. Mazumdar, “Consis-tent higher derivative gravitational theories with stablede Sitter and anti–de Sitter backgrounds,” Phys. Rev. D (2017) no.4, 043533 doi:10.1103/PhysRevD.95.043533[arXiv:1606.01250 [gr-qc]].[21] J. Edholm, “UV completion of the Starobinskymodel, tensor-to-scalar ratio, and constraints on non-locality,” Phys. Rev. D (2017) no.4, 044004doi:10.1103/PhysRevD.95.044004 [arXiv:1611.05062 [gr-qc]].[22] L. Feng, “Light Bending in the Infinite Derivative Theo-ries of Gravity,” arXiv:1703.06535 [gr-qc].[23] S. Talaganis, T. Biswas and A. Mazumdar, “Towardsunderstanding the ultraviolet behavior of quantumloops in infinite-derivative theories of gravity,” Class.Quant. Grav. (2015) no.21, 215017 doi:10.1088/0264-9381/32/21/215017 [arXiv:1412.3467 [hep-th]].[24] S. Talaganis, “Quantum Loops in Non-Local Gravity,”PoS CORFU (2015) 162 [arXiv:1508.07410 [hep-th]].[25] S. Talaganis and A. Mazumdar, “Towards UV Finitenessof Infinite Derivative Theories of Gravity and Field The-ories,” arXiv:1704.08674 [hep-th].[26] J. Edholm, A. S. Koshelev and A. Mazumdar, “Be-havior of the Newtonian potential for ghost-free grav-ity and singularity-free gravity,” Phys. Rev. D (2016) no.10, 104033 doi:10.1103/PhysRevD.94.104033[arXiv:1604.01989 [gr-qc]].[27] V. P. Frolov and A. Zelnikov, “Head-on collisionof ultrarelativistic particles in ghost-free theoriesof gravity,” Phys. Rev. D (2016) no.6, 064048doi:10.1103/PhysRevD.93.064048 [arXiv:1509.03336 [hep-th]].[28] A. Mazumdar, S. Talaganis and A. Teimouri, “Hamilto-nian Analysis for Infinite Derivative Field Theories andGravity,” arXiv:1701.01009 [hep-th][29] T. Biswas, T. Koivisto and A. Mazumdar, JCAP (2010) 008 doi:10.1088/1475-7516/2010/11/008[arXiv:1005.0590 [hep-th]].[30] T. Biswas, A. S. Koshelev, A. Mazumdar andS. Y. Vernov, JCAP (2012) 024 doi:10.1088/1475-7516/2012/08/024 [arXiv:1206.6374 [astro-ph.CO]].[31] A. Conroy, A. S. Koshelev and A. Mazumdar, “Geodesiccompleteness and homogeneity condition for cosmicinflation,” Phys. Rev. D (2014) no.12, 123525doi:10.1103/PhysRevD.90.123525 [arXiv:1408.6205 [gr-qc]].[32] A. Conroy, A. S. Koshelev and A. Mazumdar, “Crite-ria for resolving the cosmological singularity in InfiniteDerivative Gravity,” arXiv:1605.02080 [gr-qc].[33] A. Conroy, “Infinite Derivative Gravity: A Ghost andSingularity-free Theory,” arXiv:1704.07211 [gr-qc].[34] L. Perivolaropoulos, “Sub-millimeter Spatial Oscillationsof Newton’s Constant: Theoretical Models and Labora-tory Tests,” arXiv:1611.07293 [gr-qc]. [35] T. Biswas, A. Mazumdar and W. Siegel, “Bouncing uni-verses in string-inspired gravity,” JCAP (2006) 009doi:10.1088/1475-7516/2006/03/009 [hep-th/0508194].[36] R. P. Woodard, “Ostrogradsky’s theorem on Hamilto-nian instability,” Scholarpedia (2015) no.8, 32243doi:10.4249/scholarpedia.32243 [arXiv:1506.02210 [hep-th]].[37] A. Conroy, T. Koivisto, A. Mazumdar and A. Teimouri,“Generalized quadratic curvature, non-local infraredmodifications of gravity and Newtonian poten-tials,” Class. Quant. Grav. (2015) no.1, 015024doi:10.1088/0264-9381/32/1/015024 [arXiv:1406.4998[hep-th]].[38] Schwartz MD. Quantum Field Theory and the StandardModel. 2013[39] Claus Kiefer. Quantum gravity, Volume 155 of Interna-tional series of mono- graphs on physics. Oxford Univ.Pr., Oxford, UK, 2012[40] C. Gao, “Generalized modified gravity with the sec-ond order acceleration equation,” Phys. Rev. D (2012) 103512 doi:10.1103/PhysRevD.86.103512[arXiv:1208.2790 [gr-qc]].[41] S. Deser and R. P. Woodard, “Nonlocal Cos-mology,” Phys. Rev. Lett.99