Post-inflationary Scalar Tensor Cosmology and Inflationary Parameters

Scalar fields provide attractive modifications of pre-BBN cosmology, which have interesting implications for dark matter abundances. We analyse the effect of these modifications on the number of e-foldings between horizon exit of CMB modes and the end of inflation $(N_k)$, and examine the consequences for inflationary predictions of various models. We find significant effects in the predictions in the $(n_s, r)$ plane. For a large part of the parameter space, the shift in $N_k$ is positive; this ameliorates the tension of $m^{2} \varphi^{2}$ and natural inflation with data.


Introduction
Cosmological observations are the source of some of the biggest challenges in Physics today. Only 5% of the energy density of the universe is baryonic matter, the remaining is dark matter (27%) and dark energy (68%). The Cosmic Microwave Background (CMB) and galaxy distributions exhibit correlations even at super-horizon scales, with distinct patterns in the inhomogeneities. The ΛCDM model together with an inflationary epoch in the early universe has become the standard framework used to address these observations.
In this setting, a popular scenario to address dark matter abundance is the Weakly Interacting Massive Particle (WIMP) paradigm. At early times, dark matter was in thermal equilibrium with the standard model degrees of freedom. As the temperature of the universe decreased, the dark matter interaction rate fell down and its number density became a constant in the comoving frame -the present dark matter abundance is this relic. The required value of the dark matter abundance is obtained for σ th v rel 3 × 10 −26 cm 3 sec −1 , where v rel is the Moller velocity and a thermal average is taken. This gives the typical value of the cross section to be that associated with the weak scale. The emergence of the weak scale from the relic abundance calculation is often cited as the WIMP miracle. Experimentally, there are intense efforts to probe dark matter cross sections [1,2] while future experiments will further probe the annihilation rate for a wide range of dark matter masses [3,4] 1 .
A key input for the standard computation of thermal dark matter abundance is that the energy density of the universe was dominated by radiation at the time that dark matter decoupled. One of the major successes of the hot big bang model is nucleosynthesis (BBN), this requires that the universe was radiation dominated at the time of production of the light nuclei, the corresponding temperature being T BBN 1 MeV (see [24,25] for the related constraints on late time entropy production). But, the history of the universe prior to BBN remains highly unconstrained. A modification of the history of the universe prior to nucleosynthesis, in particular a modified expansion rate of the universe can significantly alter the dark matter relic density computations. Scalar fields with non-trivial evolution provide well motivated modifications of the pre-BBN history with interesting implications for dark matter relic abundances. The simplest scenario with a modified cosmological history is when an extra scalar species is present [26], which does not couple directly to the Standard Model (SM) degrees of freedom or dark matter. The energy density of the decoupled scalar dilutes as ρ φ ∝ a −(4+n) with n > 0 (where a(t) is the scale factor). This form implies that the scalar energy density dominates over radiation at early times. This in turn leads to a modification in the relic abundance calculations. Even this simple model leads to novel features in the phenomenology -dark matter annihilation processes can continue to take place long after the decoupling of dark matter from the thermal bath. This was dubbed as relentless dark matter, as it can be thought of as dark matter trying to get back into thermal equilibrium.
More generally, scalar-tensor (ST) theories of gravity arise in extensions of the standard models of cosmology and particle physics. For example, in higher dimensional models, additional scalar fields arise through the compactification of the extra dimensions and couple to the metric with gravitational strength.
Thus the gravitational interaction is mediated by both the metric and scalar fields so that scalar-tensor gravity models represent a departure from standard General Relativity (GR). In string theory models of particle physics and cosmology, new ingredients such as D-branes can also appear. For such D-branes, the longitudinal fluctuations are identified with the matter fields such as the SM and DM particles, while transverse fluctuations correspond to scalar fields. These scalars couple conformally and disformally to the matter living on the brane and thus can modify the cosmological expansion rate felt by matter and hence the standard predictions for the dark matter relic abundance 2 [39][40][41]. An attractive feature of the ST case, is the possibility of tracker solutions of the scalar field, which ensure that the scalar field becomes inactive at the onset of BBN (and thereafter) as is required for its successes.
One of the major successes of the inflationary paradigm is the explanation for the approximately scale invariant spectrum of the CMB. Given a model of inflation, the predictions for the scalar spectral tilt (n s ) and the tensor to scalar ratio (r) are determined by the number of e-foldings, N k , between horizon exit of the CMB modes and the end of inflation. Typically N k is taken to be in the range 50 to 60.
It is important to keep in mind the inputs that go into deriving this. The derivation makes use of two aspects of the physics of inflation: the condition for horizon exit of the CMB modes (k = a k H k ; where k is the wavenumber of the CMB modes, a k and H k are the scale factor and Hubble constant at the time of horizon exit) and the relationship between the strength of the scalar perturbations and the energy density at the time of horizon exit. Taking these inputs, the evolution of the energy density from the time of horizon exit to the present epoch and entropy evolution from the end of reheating to the present epoch are tracked. For the evolution, the post-inflationary history is taken to be the standard history associated with the hot big-bang model. This tracking then yields the equation that determines N k : where r is the tensor to scalar ratio and w re , N re are the (effective) equation of state parameter and number of e-foldings during the reheating epoch. In the canonical reheating scenario, w re 0, but more general cases can give 1/3 < w re < 0 [42,43]. Given the uncertainties associated with reheating and the fact that the tensor mode has still not been detected, equation (1) does not fix the value of N k exactly, but under the assumption that the reheating epoch is not highly extended it serves as the motivation for the usual range of 50 to 60 for N k . The input that the post-inflationary history is the standard history associated with the hot big bang model is central to the derivation of (1), any modification of the post-inflationary history has an effect on the preferred range for N k . As we enter the era of precision cosmology 3 , it is very important to determine N k accurately, so that theoretical predictions to confront the data can be made. The models discussed above in the context of dark matter abundance have a non-standard cosmological history prior to BBN. The goal of this paper is to determine N k for these models and obtain the implications for inflationary predictions.
The rest of the paper is organised as follows. We review theories with scalars in pre-BBN cosmology motivated by dark matter considerations in section 2. In section 3, we obtain the expressions for N k in these theories and in section 4 we discuss the implications of our results for the predictions in the (n s , r) plane of various inflationary models. We conclude in section 5.
2 Scalars in pre-BBN Cosmology

Extra scalar species
The simplest scenario to modify the cosmological history before the onset of BBN, is to consider a new scalar species φ, which does not couple to matter, and has a non-trivial cosmological evolution in the pre-BBN history. The defining property of a such decoupled scalar is that its energy momentum tensor is conserved separately: ∇ µ T φ µν = 0. In addition, various components of matter have individually conserved stress tensors ∇ µ T i µν = 0, where the index i runs over the components of matter whose stress tensors are conserved. The consequence of these conservation equations is that the various components of matter dilute independently and standard relationships between the number densities and the temperature 4 hold.
To study alternative scenarios for dark matter production, the authors of [26] considered cosmologies where the scalar energy density redshifts as The dilution can be expressed in terms of the temperature by making use of conservation of entropy where g s (T ) is the effective number of degrees of freedom associated with the entropy density at temperature T , and T r is a reference temperature. The reference temperature is chosen such that the energy 3 It is expected that the next generation of CMB experiments [44][45][46] will bring ∆ns at the 1 − σ confidence level down to 0.002, an improvement by a factor of three in comparison with Planck 2015 [1]. The sensitivity for r will come down to 10 −3 . 4 We will find that these relations do not hold in the Einstein frame when we discuss scalars which have nontrivial couplings to the matter fields. density of the scalar and radiation are equal at T = T r . With this, the deviation from standard cosmology is completely characterised by the two parameters (n, T r ).
The total energy density can then be written as where g(T ) is the number of degrees of freedom associated with the energy density of the radiation at temperature T . The successes of BBN require that T r > (15.4) 1/n MeV [26]. Typical values for T r considered in [26] are of the order of GeV. The relentless phenomenon occurs for n ≥ 2 and n ≥ 4 for swave and p-wave annihilation, respectively [26]. We will use (4) in section 3 to calculate the modification to the number of e-folds N k due to the pre-BBN non-standard cosmological evolution described above.

Scalar-tensor theories
In this section, we describe the post-inflationary histories associated with a class of scalar-tensor theories, which are motivated from models with extra dimensions. It was shown in [47] that the most general physically consistent relation between two metrics g µν andg µν related by a single scalar field φ includes a conformal C(φ) as well as a disformal relation D(φ) as The presence of the scalar modifies the expansion rate felt by any matter coupled to the "disformal" metricg µν , with interesting implications. The effect of this modified expansion rate for the thermal production of dark matter was recently studied in [40,41]. We refer the reader to the literature [40,41] for details of the general set-up and derivations.
In these models, the total energy momentum tensor in the Einstein frame, which is defined with respect to the metric g µν , is conserved: On the other hand, the energy momentum tensor in the Jordan, or disformal frame is conserved,∇ µT µν = 0, where∇ µ is the covariant derivative computed with respect tog µν andT µν is the energy momentum tensor computed with respect to the disformal metricg µν . The energy densities and pressures in the two frames can be shown to be given by [40,41] where γ = (1 + D C (∂φ) 2 ) −1/2 and the equations of state in both frames are related byω i = ω i γ 2 . In the pure conformal case, γ = 1 and thusω = ω. Finally, the Jordan and Einstein frame frame scale factors are related byã The functions C and D can in principle take any form so long as they satisfy the causality con- Note also that in our conventions, In phenomenological studies, this is the only requirement on C, D and the scalar field takes the standard action of a canonically normalised field. On the other hand, in D-brane scalar-tensor theories, the scalar action takes the DBI form: , and the functions C, D are related as M 4 CD = 1. Moreover, M is given by the tension of the D-brane, and thus to the string scale (see [41] and below). Finally, in this set-up, the inflaton ϕ may, or not, be disformally coupled to the scalar φ, but it will anyhow have some coupling and will thus decay into both φ and the matter fields coupled to it.

E-foldings between Horizon exit of CMB modes and the end of inflation
As discussed in the introduction, the number of e-foldings of the universe between the horizon exit of the CMB modes and the end of inflation (N k ) is a central input for making predictions for the inhomogeneities in the CMB. The quantity is sensitive to the post-inflationary history of the universe and hence is expected to be altered with any modification of the pre-BBN history of the universe. In this section, we compute N k for the theories described in section 2. In spirit, the computations follow the standard procedure to determine N k (see for e.g. [48] for a recent discussion), but the modified post-inflationary histories in these theories lead to non-standard equations for N k .

N k for decoupled scalars
We start from the condition for horizon exit of modes during the inflationary epoch: k = a k H k . Where k is the comoving wavenumber of the mode, a k and H k are the scale factor and Hubble constant at the time of horizon exit of the mode. We write the above as: where a end , a re are the scale factors at the end of inflation and end of the reheating epoch. a 0 is the scale factor today. We follow the convention of tracking the evolution in terms of the number of e-foldings, (8) then reads Note that if the entire post-inflationary history of the universe is known then N re and ln a re a 0 , can be where A s is the primordial scalar amplitude and r the tensor to scalar ratio. Thus, the last term in (9) is known modulo the value of r; the dependence on r is logarithmic and can be estimated without much uncertainly given a model of inflation. Thus, given a post-inflationary history, (9) essentially determines N k . While it is a perfectly valid approach to determine N k from (9) where ρ end is the energy density of the universe at the end of inflation and ρ re is the energy density at the end of reheating. We simplify the above by writing The first term involves the ratio of the energy densities at horizon exit and the end of inflation, it can be computed explicitly given the inflationary potential. It typically is small as the energy density remains a constant during the inflationary epoch. The second term is related to the primordial scalar amplitude and r (10). The right hand side of (11) also involves the energy density at the end of the reheating epoch (ρ re ). We write this as where g re are the effective number of degrees of freedom at the end of reheating and T re the reheat temperature of the Standard Model degrees of freedom. Usually, the Einstein frame energy density at the end of the reheating epoch is equated to the energy density of the radiation. As discussed in section 2, in the theories with a decoupled scalar proposed in [26] the decoupled scalar also carries a fraction of the energy density -we incorporate this in (13) by including the factor η (which is equal to the ratio of the total energy density and the energy density of the radiation at the end of reheating). Furthermore, we assume that there is no production of entropy 5 after the reheating epoch -entropy conservation can then be used to relate the reheat temperature to the temperature of the CMB Combining equations (11)-(14), we obtain: where Finally, we subtract (15) from (9) to eliminate the dependence on a re /a 0 . This yields an equation for N k : Let us discuss each term in the right hand side. The first term depends on the parametrisation of reheating (N re , w re ). Given the usual uncertainties associated with it, this term is the major source of the uncertainty in N k . The second and third terms are determined in terms in terms A s and r (10). In the fourth term, k/a 0 is the pivot scale for CMB observations and T 0 = 2.73 K. The fifth term depends on the number of effective degrees of freedom at the time of reheating; β ∝ ln g 1/4 re g 1/3 s,re , hence it gives a small contribution. The sixth term involves the ratio of the energy densities at the time of horizon exit and the end of inflation. It is typically small as the energy density does not change much during inflation.
The last term involves the parameter η introduced in (13). In conventional cosmological scenarios, this term vanishes as the entire energy density of the universe is in the form of radiation after reheating.
Plugging in the numerical factors we find 6 As discussed above, in conventional cosmological scenarios η = 1. The term involving the reheating parameters is typically negative, simulations of various scenarios for reheating give w re < 1/3 (see for e.g. [42,43]). The term involving r is manifestly negative. Given this, equation (18) gives motivation for the usual range of 50 to 60 for N k in conventional cosmologies, (i.e. η = 1). We can read off the expression for η (in terms of n and T r ) from the expression for the total energy in the post-inflationary Recall that the η contribution to N k is: δN k ≡ 1/4 ln η (17).
As was shown in [26], the largest effect in the dark matter decoupling is obtained for smaller T r 's, so long as the BBN predictions are not affected, which gave T r 10 MeV. From (19) we see that the largest effect in δN k for a given reheating temperature, will precisely be given for the smallest T r 's as shown in Fig. 1.
Taking for example a reheating temperature of T re = 10 14 GeV, for n = 2 and T r = 1 GeV one obtains δN k ≈ 16. For n = 2, T r = 10 6 GeV, δN k ≈ 9. Since n appears in the exponent in (19), higher values of 6 We take gre ≈ gs,re ≈ 100 and ρ end ρ k ≈ 1. ∼ O(1). Note that in general, this factor is greater than one, hence when relevant, it will lead to a further enhancement of the effect n can lead to much larger shifts in N k . For n = 4 (recall that this is the critical value of n for relentless dark matter if dark matter annihilation is a p-wave process [26]) with T r = 1 GeV yields δN k ≈ 32, while T r = 10 6 GeV yields δN k ≈ 18. Note that since T re T r , the change in δN k is larger for smaller values of T r (see inset in Fig. 1). For example, for a factor m ∼ O(1) change in T r , the change in δN k ∼ n 4 log m, which gives an O(1) change (see inset in Fig. 1).
We will discuss the implications of our results for confronting inflationary models with data in section 4, here would like to emphasise that shifts in the value of N k obtained above can have a significant effect on the (n s , r) predictions of any inflationary model. Also, note that since η is the ratio of the total energy density and the energy density in radiation at the end of reheating, η > 1; hence the effect of the last term in (18) is to increase N k . The positivity of the shift is natural as in the epoch in which the scalar dominates the energy density in the post-inflationary history the energy density dilutes as a −(4+n) , i.e faster than radiation. This is similar to a kination phase [56][57][58][59], which is known to make a positive contribution to N k (see for e.g. [60] for a recent discussion). Of course, the key difference is that this phase occurs after reheating.

N k for coupled scalars
In this section, we derive the expression for the number of e-foldings between horizon exit of CMB modes and the end of inflation (N k ) for the coupled case where the scalar arises from a scalar-tensor theory and thus couples both conformal and disformally to matter. The crucial difference from the discussion in the previous section is that the standard relations for energy and entropy conservation now hold in the Jordan frame 8 . The analogous expression for (9) in the present case is given by whereÑ and since the scalar is not active at the end of inflation, we can takeã end = a end (that is, C end = 1). Moreover, since standard GR evolution must be reached at the onset of BBN to avoid spoiling its predictions, we also haveã 0 = a 0 . Now, using energy conservation in the Jordan frame, we can writẽ We gave an expression for ρ end in (12) in terms of ρ k . On the other hand, in the Jordan frame we have [40,41] where g re is the effective number of degrees of freedom (which in principle depend onT ) at the end of reheating andT re is the the Jordan frame reheat temperature. Furthermore, assuming as before that there is no production of entropy after the reheating epoch -entropy conservation in the Jordan frame can be used to relate the reheat temperature to the temperature of the CMB: Using these equations, we arrive at an expression for N k in terms ofÑ re : Using (21), and considering thatω re ∼ ω end , we arrive at In general C(φ) depends on time (or temperature). However, as studied in [41], a pure disformal term with gives a non-trivial enhancement of the expansion rate, and thus an earlier dark matter freeze-out than in the standard case. This causes a relentlessness effect, similar to the decoupled case discussed in the previous section. Therefore a constant C 0 is the most interesting example. Furthermore, we can also assume a Jordan frame equation of stateω re to be roughly constant.
In the case of a D-brane scalar tensor model [41], the disformal and conformal functions arise from the DBI action as discussed above. The scale M introduced there is related to the tension of a D3-brane, which is given by the string scale as M = M s (2πg −1 s ) 1/4 . Defining the scaleM = M C 1/4 0 , the interesting cases for the modified dark matter thermal scenarios occurred forM ∈ (10−300) GeV in [41]. Considering a typical string scale, which arises in string theory models of inflation, M s ∼ (10 14 − 10 16 ) GeV (with g s ∼ 0.1) allows us to find the value of C 0 and thus the modification to N k . The net effect is a positive modification to N k , hence increasing the required number of efolds by about δN k ∼ 15 (forω re ∼ 0), and thus similar to the decoupled example with e.g. n = 2, T r = 1 GeV and T re = 10 14 GeV. In terms of a D3-brane picture, this can be interpreted as the D-brane moving at the tip of a warped throat.
The case C 0 = 1 on the other hand, would correspond to an unwarped geometry. In this case,M = M and therefore an effect in the thermal dark matter production is tied to the string scale. In this case the relevant values of M for a relentless dark matter effect, require extremely low string scales [41]. In the phenomenological case on the other hand, the values of C 0 and D 0 are in principle unconstrained and therefore can again give large positive corrections to δN k .

Implications for Inflationary Predictions
In this section, we discuss the implications of the shifts in N k derived in section 3 for the predictions of inflationary models in the (n s , r) plane. Inflationary models fall into universality classes characterised by the relationship between n s , r and N k [61]. It is natural to use this classification to study how the shift in N k can affect various models. Class I models obey the relations where λ is a constant which depends on the inflationary model. Note that a increase N k implies that models with higher values of η are preferred (similarly higher values of N k imply higher values of η). The prototypical models for Class I are hilltop [62] and Starobinsky [63]. Amongst the models obtained from string theory, it includes the fibre inflation model [64][65][66]. The Starobinsky model has which corresponds to λ = −1. For these models lowering of N k (as found in [52]) can be interesting as this pushes up the value of r, the increase in N k found for the models makes the size of r even smaller.
Class II models are characterised by where again λ is a constant depending on the inflationary model. These are large field models. The prototypes for these are the monomial potentials: V (ϕ) = 1 2 m 4−α ϕ α (α = 2 corresponds to the famous m 2 ϕ 2 model [67][68][69], α = 2/3, 1 arise in the axion monodromy examples [70,71]). The predictions in the (n s , r) plane are For the m 2 ϕ 2 model, N k in the range of 50 to 60, gives n s in the range 0.960 to 0.966 and r in the range 0.13 to 0.16. Recall that Planck 2015 (TT+TE+EE) + low P + lensing for the ΛCDM + r model gives n s = 0.9688 ± 0.0061 and r < 0.114 at 1 − σ [1]). The major tension with data arises from the high value of the prediction for r. This can be ameliorated with an increase in the value in N k . For e.g. N k = 72 gives r = 0.11 and n s = 0.972, which is within the Planck 2015 1 − σ range for n s and r. In scenarios for which the reheating epoch is not extended, this corresponds to δN k ≈ +16. Such a shift is possible for a large range of the model parameters both the decoupled and coupled scalar cases; particularly in the case of relentless dark matter and the D-brane scalar-tensor model [41]. We note that although the shift brings both n s and r within their 1 − σ values, the model only enters the the 2 − σ region in the marginalised joint distribution in the (n s , r) plane 9 . But overall the shift in the predictions is in the direction favoured by the present data. For the linear potential (α = 1), N k in the range of 50 to 60 gives n s in the range 0.980 to 0.983 and r in the range 0.080 to 0.066. In this case the prediction for n s is on the higher side, and increase in N k further increases the tension in the n s prediction. The situation for α = 2/3 is similar.
Another interesting class of models is that of natural inflation [72]. In this case the dependence on N of the cosmological parameters (n s , r) is non-perturbative with:

Conclusions
Scalar fields are ubiquitous in theories beyond standard models of cosmology and particle physics. During the early pre-BNN cosmological history, these can appear as an extra species, decoupled from matter, or in scalar-tensor theories, where the gravitational interaction is mediated by both the metric and scalar field, coupling to matter conformal and disformally.
These scalars evolve during the pre-BNN epoch, giving rise to interesting implications for dark matter relic abundances [26,[39][40][41]. The modified post-inflationary history implies a shift in the number of efoldings between the horizon exit of CMB modes and the end of inflation, N k . At the same time, this implies a change in the inflationary parameters (n s , r). We have computed this shift for theories with decoupled scalars and scalar-tensor theories and examined the effect that this can have on the predictions in the (n s , r) plane for inflationary models. In particular we considered the models discussed recently in [26] when an extra scalar species modifies the DM relic abundance and the disformal models arising from D-brane scalar-tensor theories discusses in [40,41], where the scalar couples conformal and disformally to matter, modifying the expansion rate and thus the DM relic abundances.
For decoupled scalars, we found that the shift in N k is positive definite. The case of relentless DM gives the largest effect with δN k ≈ 15 or larger, with a reheating temperature of around T re ∼ 10 14 GeV (see section 3.1). For the scalar-tensor case, we derived δN k for a disformal enhancement of the expansion rate as studied in [41]. This corresponds to constant conformal and disformal functions C = C 0 , D 0 = M −4 = (C 0 M 4 ) −1 . For the values ofM and C 0 relevant for the modifications of the expansion rate and thus relic DM abundance [41], we found a large positive effect with δN k ≈ 15 (see section 3.2).
We used this to study the implications for the predictions of (n s , r) for some interesting inflationary models. For example, the increase in N k implies the possibility of a decrease in the prediction for r for the m 2 ϕ 2 model, which reduces the tension with data. The change in N k has also an important effect for natural inflation, by decreasing r and moving n s towards the allowed values of Planck. Analysis in the spirit of this work will become more and more important as we probe the CMB even more minutely.