Inflation Without a Trace of Lambda

We generalise Einstein's formulation of the traceless Einstein equations to $f(R)$ gravity theories. As in the traceless version of general relativity, we find that in traceless $f(R)$ gravity there is no vacuum energy cosmological constant problem. The cosmological constant is a mere integration constant that is unconnected to any vacuum energy density or constant term in the gravitational lagrangian. We obtain the integrability condition for the cosmological constant to appear as an integration constant without a connection to any vacuum energy density. We show that, in $D$% -dimensional spacetime, traceless higher-order gravity is conformally equivalent to general relativity and a scalar field $\phi $ with a potential given by the scale-invariant form: $V(\phi )=\frac{D-2}{4D}Re^{-\phi }$, where $\phi =[2/(D-2)]\ln f^{\prime }(R)$. Unlike in the conformal equivalent of full general relativity, flat potentials are found to be possible in all spacetime dimensions for polynomial lagrangians of all orders. Hence, we can solve the cosmological constant vacuum energy problem and have accelerated inflationary expansion in the very early universe with a very small cosmological constant at late times for a wide range of traceless theories. Fine-tunings required in traceless general relativity or standard non-traceless $f(R)$ theories of gravity are avoided. We show that the predictions of the scale-invariant conformal potential are completely consistent with microwave background observational data.


I. INTRODUCTION
The vacuum Einstein equations R µν − 1/2g µν R = 0 are scale invariant: each of the two terms on the left-hand side are homogeneous of degree two in the derivatives of the metric and there is no length with respect to which one can measure the size of any dimensionful constant. The introduction of a cosmological constant into the energy density of the (Lorentzinvariant) vacuum breaks scale invariance by adding an energy-momentum tensor of the form T µν = −Λg µν to the vacuum field equations [1] and permits the usual inflationary cosmological solutions. However, it also creates a scale given by Λ = (8πG) −1/2 , and hence a discrepancy between the observed energy density of empty space, which is measured to be around 10 −47 GeV 4 , and the expected vacuum field energy Λ 4 /16π 2 ≈ 2 × 10 71 GeV 4 . This is sometimes called the 'cosmological constant problem' [2][3][4][5]. By introducing Λ into the vacuum Einstein equations we introduce the possibility of inflationary (accelerating) de Sitter solutions at the expense of an anomalously large vacuum energy density. In this paper, we suggest an approach that can retain the presence of inflationary solutions in the theory without automatically admitting the possibility of a vacuum energy density. We describe formulation of the traceless version of the Einstein equations, first proposed by Einstein, before calculating the traceless version of the field equations for vacuum f (R) gravity theories. We will show that, in D-dimensional spacetime, traceless higher-order gravity is conformally equivalent to general relativity and a scalar field φ with a potential given by a scale-invariant form: V (φ) = D−2 4D Re −φ , where φ = [2/(D − 2)] ln f ′ (R). Unlike in the conformal equivalent of full general relativity, flat potentials are found to be possible in all spacetime dimensions for polynomial lagrangians of all orders. Traceless higher-order gravity, like traceless general relativity, avoids a cosmological constant problem associated with an energy density of the vacuum. We will calculate the slow-roll parameters and show that traceless higher-order gravity provides many possible inflationary models in agreement with microwave background data, and avoids fine tuning problems.

II. TRACELESS GENERAL RELATIVITY
We seek a situation where the cosmological constant problem is absent but the possibility of inflation remains. We note that there is a way, first suggested by Einstein [6], to cancel the effects of a cosmological constant by using only the traceless parts of the Einstein equations to define the theory of gravitation [3,[6][7][8]. Before extending this approach to higher-order lagrangians, we summarize the structure of traceless general relativity introduced by Einstein and clearly expounded by Ellis in ref. [8].
For any tensor L µν , its traceless part in D spacetime dimensions is defined to be the new tensor, so that L = Tr L µν = 0. For the Einstein tensor, defined by G µν = R µν − (1/2)g µν R, and setting the Einstein constant 8πG/c 4 = 1, we find that in D dimensions its traceless part is [6], (since 1/D = 1/2 + (2 − D)/2D). The traceless Einstein equations are then postulated to be where T µν is the usual energy-momentum tensor of matter, and T its trace. All the vacuum solutions of general relativity remain unchanged and the nine traceless field equations are a subset of the full (ten) Einstein equations. Crucially, however, although the stress tensor T µν can have a trace, only the tracefree part gravitates 1 .
These traceless Einstein equations are scale invariant, like the standard vacuum Einstein equations. Hence, it follows that for T µν = −Λg µν , the terms on the right-hand sides cancel each other and leave a vacuum theory, G µν = 0. This means that in the traceless theory, unlike in general relativity, the cosmological constant has nothing to do with an energy density of the vacuum or a constant term in the gravitational action. In fact, it can easily be shown to be simply an arbitrary integration constant, λ, that appears in the Einstein equations, given by [3,7,8], In these equations, λ plays the role of an arbitrary initial condition, its value picks out different solutions with specific properties, and should not be considered as values of a parameter appearing in the gravitational action, like Λ. In this case the field equations (4) are not generally covariant (like (3) or the standard Einstein equations), as they are only invariant under the restricted group of volume-preserving (unimodular) coordinate transformations [9], which is equivalent to the introduction of a new scalar field (the metric determinant) into the theory [3].

A. Conformal equivalence
Since for a scalar field φ with potential V (φ), the traceless part of its energy-momentum tensor, T µν = ∂ µ φ∂ ν φ − (1/D)g µν (∂φ) 2 , does not contain the potential V (φ) and we have T = 0, it follows that in the traceless theory given by (3) the scalar field φ gravitates but without a potential. Therefore, any property of the solutions of the traceless field equations (3) which depends implicitly on the potential V (φ), such as the possibility of slowroll inflation, should probably be considered as an 'off-shell' effect requiring some sort of fine-tuning.
To expound this, we can restrict the metric variations δg µν used in the action principle by moving to a metric conformally related to g µν (that enters in the field equations (3)), by introducing an analytic function f (R) by setting [15] Then, following the steps in the Appendix (Part A), we find that the vacuum traceless Einstein equations transform into, where,T 2 In the present case of traceless general relativity, we could of course have used the standard parametrization of the conformal factor, namely, e φ = Ω 2 . However, for the purpose of easily comparing the two cases, we prefer to continue to use the conformal factor appearing in the case of f (R) theory. with and the trace of the matter tensor T satisfies the integrability condition (5). Here, λ plays the role of an arbitrary initial condition, but any constraint to be satisfied (such as the flatness conditions) will lead to the necessity of fine-tuning λ. The potential is also scale invariant; III. TRACELESS f (R) GRAVITY Adding a cosmological constant term is not the only way to break the scale invariance of the vacuum Einstein equations. The addition of higher-order curvature terms to the lagrangian also introduces a scale (see for example, [10], p. 153), and can lead to inflation in a natural way [11][12][13][14]. We consider the theory derived from a gravitational lagrangian that is an arbitrary analytic function of the scalar curvature, f (R), [13]. The field equations in vacuum, with f ′ = df /dR, derived by varying the action formed from the lagrangian f (R), are These equations are fourth order in the spacetime derivatives of the metric, g µν . The most interesting property they possess for our present purposes is that they miss being scale invariant simply because of the presence of the term proportional to g µν f . If this term were absent, Eq. (12) would be uniform in scale in the same way that the vacuum Einstein equations are for the spacetime metric g µν -but now containing only fourth-order instead of second-order terms in the spacetime derivatives of the metric.
There is also the more suggestive form for these vacuum equations (assuming f ′ = 0 for all R), first introduced in Ref. [15], for M µν which picks out the Einstein tensor explicitly: and also introduces the term proportional to Rf ′ − f which can provide an inflationary potential. This is important in the conformal picture of general relativity with a self-interacting scalar field as source [16], as it allows an inflationary phase in cosmological solutions of these theories [15,17]. However, with regard to the cosmological constant problem and the contributions to the energy of the vacuum, Eq. (13) -although not scale invariant -behaves exactly like the vacuum Einstein equations when adding a term proportional to Λ, and so the same comments apply as were made above for the full general relativity theory.
The question then arises as to whether the traceless version of Eq. (12) can solve the cosmological constant problem of the standard f (R) gravity while maintaining the inflationary character of solutions of the theory.
In contrast to the original higher-order field equations for f (R) gravity (12), we see that there is now no term proportional to g µν f , and so in terms of the notional scalar 'field' It is a homogeneous polynomial of degree two in the spacetime derivatives of the field Φ.
Following the traceless general relativity development, we postulate the full traceless f (R) field equations to be where the left-hand side is given by (14). The stress tensor of matter and radiation T µν on the right-hand side of this equation will in general have a nonzero trace but only its traceless part will gravitate. We note that by setting T µν = −Λg µν , Eq. (15) becomes the vacuum equation Both equations (15) and (16) are scale invariant. We will show that Eq. (15) is related to the standard f (R)-matter equations through a nontrivial integrability condition and, as in traceless general relativity, eq. (5), the cosmological constant becomes an arbitrary integration constant. In this respect, the traceless f (R) situation is completely analogous to that of traceless general relativity [3,8]. From (15), we find by covariant differentiation that since we assume the usual conservation law, ∇ µ T µν = 0. However, since (12) gives it follows from the definition of the traceless tensor M µν that so that, which means that M − T is a constant: When we set f (R) = R and D = 4, we find precisely the standard integrability result of ref. [3], (note that from (14) it follows that M = −R in that case while, in general, ). Using (21), the traceless f (R) equations (15), become or, Since our whole formulation is generally covariant, it is unnecessary to restrict the general covariance of our scheme, hence we are not going to use Eqns. (23) from now on. We note that a violation of general covariance by using Eq. (23) leads to a fine-tuning of the integration constant λ, something that was to be avoided in the first place: this equation is exactly like the basic f (R) equation only having the extra λ term on the left-hand side, and so a conformal transformation leads to the same potential as was found in Ref. [15], but now with an extra term proportional to λ. For example, when D = 4 and f (R) is quadratic, the potential is V (φ) = e −φ (e 2φ − e φ ) 2 + λ . Hence, when calculating the slow-roll parameters the most dominant terms will contain a λ term needs to be finely tuned. For instance, for the slow-roll parameter ǫ = 1 and so we need to impose a fine-tuning condition on the integration constant λ, at the end of inflation, so as to ensure that the denominator in Eq. (24) tends to 1. Our previous results, in Ref. [15], about the possibility of inflation in all spacetimes with quadratic lagrangians when D = 4, and their subsequent agreement with the microwave data correspond to the special choice of initial conditions λ = 0, in the present case.
This reveals that in all those models the cosmological constant problem remains. We believe that the great advantage of traceless f (R) theory in the conformal frame, over the traceless version of general relativity or the standard f (R) theory considered in the conformal frame, is that it allows us to express the scalar curvature R by inverting f (R) to by-pass the explicit use of the integrability condition (which introduces lambda and leads to the fine-tuning problem discussed above), as well as avoiding a second fine-tuning problem for initial conditions which is present in the standard f (R) theory.

A. Conformal equivalence of traceless f (R) gravity
We can now uncover the conformal relative of the traceless theory given by (16). If we introduce a new scalar field φ as in Eq. (6) and following the steps detailed in the Appendix, Part B, we find that the conformally transformed field equation (16) becomes, with the effective scalar field potential given by This is the self-interacting scalar field potential for the traceless vacuum f (R) field equations (16) in the conformal frame. Its sign is determined by the inversion of f (R) and by (6). We note that in the variables (g, f ′ ) the potential given by (28) is scale-invariant (something which is not true of the potential in the conformal transformation of the 'standard' (nontraceless) higher-order gravity theory derived in [15]).
It is important to notice that in all traceless higher-order polynomial gravity theories, flat potentials occur quite generally in the conformal traceless sub-theory. For the generic leading-order lagrangian, we find the potential to be: where the constant c ≡ ((D − 2)/4D)(1/nA) 1/(n−1) . This leads to a huge variety of flat potentials in all dimensions and any leading degree of the gravitational lagrangian. We recall that in the standard conformal correspondence between f (R) gravity theory and general relativity plus a scalar field first studied in [15] there is a different structure: flat plateaux in V (φ) arise in D spacetime dimensions only when the highest polynomial term in R in the f (R) lagrangian is 1 2 D, and so for D = 4 this allows at most a quadratic term. By contrast, here we see that a flat potential conducive to exponential inflation is easy to achieve for the theory in (29) in all dimensions. Therefore, we conclude that slow-roll inflation is possible in traceless nonlinear f (R) theories. Quite a large family of underlying models lead to observationally compatible outcomes for inflation without requiring any fine-tuning.
We note that the exponential factor present in the potential (28) means it is unnecessary to assume that the minimum of the potential is very close to zero. Even for values of the scalar field comparable to the Planck mass, the potential gets arbitrarily close to zero because the exponential factor is O(1), so ensuring a small present value of the cosmological constant.
This factor is also responsible for the ease with which our theory keeps the vacuum energy below 10 −48 GeV 4 : The flatness conditions, as well as the requirement to keep V (φ) below the Planck energy density, are easily satisfied by a scalar field with mass below 10 −12 GeV, as required from general theory (cf., eg, [3], Section VI).

IV. SLOW-ROLL INFLATION PREDICTIONS FOR THE CMB
We can calculate the simple first-order predictions that our theory makes for the CMB in the (n s − r)-plane ('tilt' and 'tensor-to-scalar' ratio). We define the potential slow-roll parameters in the usual way, [19,20], by and the e-folding function that gauges the length of inflation by, Then, for example for the potential with D = 4, n = 2, namely, and the spectral index and tensor to scalar ratio are, hence, we find, and so, to first order in slow roll, so that (b) D = 2n : In this case for large φ, we have an asymptotically exponential potential of the form, Hence, in that limit, so that Introducing a new parameter, p, defined by we have, We conclude that when D − 2n → 0, we have p → ∞, corresponding to de Sitter spacetime asymptotically. On the other hand, when D − 2n ∼ 0, then p ≫ 1, and the slow-roll conditions for ǫ, η are satisfied. Finally, when D and n take values such that the difference D − 2n is neither exactly 0, nor near zero, nor asymptotic to 0, then we can always rescale φ and V (φ) to obtain a potential which has a flat plateau in some finite neighborhood of zero.
We note the special situation that arises globally when D = 2n and there is no need to rescale the (scale-invariant) potential to obtain the characteristic flat plateau in a finite neighborhood of the minimum. This mirrors the flat plateau condition found in conformal relations of ordinary f (R) gravity earlier in ref. [15].

V. CONCLUSIONS
In this paper we have generalised an idea, originally due to Einstein, to consider the traceless version of the gravitational field equations. Einstein applied this idea to general relativity but we have extended it to f (R) gravity theories. We have shown that if we add a cosmological constant term T µν = −Λg µν , no matter how large the value of Λ, to the higher-order gravity equations (12), then passing to the traceless higher-order theory, these equations become the vacuum equations (16), which can be written in the form (23) with zero right-hand side. After deriving the traceless f (R) field equations for the first time in this paper, we have shown that in vacuum these higher-order equations are conformally equivalent to general relativity plus a self-interacting scalar field, Eq. (26) with a scale-invariant potential given by Eq. (28). This potential gives rise to flat plateaux which offer the possibility of exponentially rapid, slow-roll inflation with values of n s and r that agree well with observational bounds from Planck [21,22]. We have delineated the outcomes of slow-roll inflation in the two distinctive situations defined by the relation between the spacetime dimension and the highest polynomial order of f (R). Specialising to four-dimensional spacetimes, we have derived the slow roll parameters for these types of inflation and shown how the results are compatible with microwave limits on inflationary parameters without requiring any finetuning or choosing special values of initial conditions. This considerable enlarges the range of observationally acceptable inflationary models significantly beyond those mapped in ref. [22]. At the end of this inflationary phase, the separation of the cosmological constant term from any quantum vacuum energy density means that it can be arbitrarily small, and remain small for billions of years, consistent with current observations. For the conformal transformation (6), we use the standard relations for the Ricci tensor, and the scalar curvature, (Tracing both sides we recover '0 = 0' as expected.) Since the original theory G µν = 0 is scale invariant, we can use the harmonicity condition (cf. [18], Eq. D. 13) for f ′ , to rewrite the last term in (A.3), so that using the field equation G µν = 0, Eq. (A.3) becomes, In terms of the scalar field φ, we find that which is the field equation (8), (9). Here, we have used the following results which follow from the definition of the conformally related metrics: from (6), f ′ = e D−2 2 φ , and so the Calculating again the covariant derivative of the log derivative of f ′ using the quotient rule, we find Substituting in Eq. (A.5) and expressing the first two terms on the right-hand side in terms of φ, we arrive at Eq. (A.6).
Part B: Traceless f (R) theory First, write the vacuum equation (16) in a form containing the Einstein tensor. Since g µν R, (A. 10) and assuming that f ′ has constant sign, substituting in equation (14) we find We then take this equation to the conformal frame. We do this in two steps, first expressing everything in terms of f ′ . The conformal relation of the tilded Einstein tensor to the untilded one is found using the transformations (A.1), (A.2), Solving this for the untilded Einstein tensor G µν and substituting back to the field equation (A.11), we obtainG g µν (f ′ ) −2 (∇f ′ ) 2 We can also simplify the last two terms, which gives (26)-(28), after we express the second term on the right-hand side using the identity g µν (∇φ) 2 =g µν (∇φ) 2 (which holds when multiplying and dividing by e φ and noting that e −φ (∇φ) 2 = e −φ g ρσ ∇ ρ φ∇ σ φ =g ρσ ∇ ρ φ∇ σ φ), and likewise for the last term, g µν R = g µν e −φ R.