Ability of LIGO and LISA to probe the equation of state of the early Universe
The expansion history of the Universe between the end of inflation and the onset of radiation-domination (RD) is currently unknown. If the equation of state during this period is stiffer than that of radiation, , the gravitational wave (GW) background from inflation acquires a blue-tilt at frequencies corresponding to modes re-entering the horizon during the stiff-domination (SD), where is the frequency today of the horizon scale at the SD-to-RD transition. We characterized in detail the transfer function of the GW energy density spectrum, considering both ’instant’ and smooth modelings of the SD-to-RD transition. The shape of the spectrum is controlled by , , and (the Hubble scale of inflation). We determined the parameter space compatible with a detection of this signal by LIGO and LISA, including possible changes in the number of relativistic degrees of freedom, and the presence of a tensor tilt. Consistency with upper bounds on stochastic GW backgrounds, however, rules out a significant fraction of the observable parameter space. We find that this renders the signal unobservable by Advanced LIGO, in all cases. The GW background remains detectable by LISA, though only in a small island of parameter space, corresponding to scenarios with an equation of state in the range and a high inflationary scale GeV, but low reheating temperature MeV (equivalently, ). Implications for early Universe scenarios resting upon an SD epoch are briefly discussed.
Ability of LIGO and LISA to probe the equation of state of the early Universe
Institute of Physics, Laboratory of Particle Physics and Cosmology (LPPC), École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland.
- 1 Introduction
- 2 Gravitational waves
- 3 Inflationary spectrum in the presence of a stiff epoch
- 4 Detection Prospects
- 5 Summary and Outlook
Observations of the cosmic microwave background (CMB) and the large scale structures of the Universe strongly support the idea that the Universe underwent a period of cosmic inflation at its early stages. Apart from solving the horizon and flatness problems, and providing the appropriate initial conditions for primordial density perturbations, inflation is a very natural phenomenon. The most recent measurement of the B-mode polarization anisotropies of the CMB [1, 2] sets a bound on the inflationary Hubble rate which corresponds to an energy scale . At some point below this energy scale, the energy budget of the universe must be converted into a thermal bath of radiation in order to switch to the standard hot Big Bang cosmology, from which point the expansion history is well known. To explain the observed abundance of light elements in the universe, Big Bang Nucleosynthesis (BBN) must take place during the radiation domination (RD) epoch, starting at a temperature of around , when the energy budget is dominated by photons and relativistic neutrinos. For consistency, the RD epoch must begin therefore before the onset of BBN. With the period of inflation ending at an energy scale and the onset of RD occurring at least above , we are left with an unknown intermediary period which may span up to orders of magnitude in energy scale.
During inflation, tensor metric perturbations generated from quantum fluctuations are spatially stretched to scales exponentially larger than the inflationary Hubble radius. This process results in a (quasi-)scale invariant tensor power spectrum at superhorizon scales [3, 4, 5, 6], with a very small red tilt. After inflation, the horizon begins to grow faster than the redshifting of length scales, and the tensor modes re-enter the horizon successively as the Hubble radius catches up with each wavelength. Once a tensor mode crosses inside the horizon, it becomes part of the stochastic background of gravitational waves (GWs). Different tensor modes re-enter the horizon at different times, and hence propagate through different periods of the evolution of the Universe after they have become sub-horizon. For tensor modes crossing during RD, the resulting present GW energy density spectrum is (quasi-)scale invariant, with the same tilt as the original super-horizon inflationary tensor power spectrum. For periods of expansion when the energy budget of the Universe is not dominated by relativistic species, the (quasi-)scale invariance is broken, and the resulting GW spectrum becomes significantly tilted in the frequency range corresponding to the modes crossing the horizon during such period(s). The expansion history of the Universe is imprinted in this way in the power spectrum of the freely-lingering primordial GW background that we can observe today. In other words, we might detect or constrain the post-inflation expansion history by attempting to detect the relic GWs of inflationary origin, see e.g. [7, 8, 9, 10, 11, 12, 13, 14]. In turn, the expansion history will inform us about the matter fields driving the expansion.
To characterize the post-inflation pre-BBN expansion history, we consider an intermediate epoch taking place between the end of inflation and the onset of RD, with an EoS parameter . In scenarios where the inflaton oscillates around the minimum of its potential after inflation, an effective EoS (averaged over inflaton oscillations) emerges [15, 16], lying within the range for , . There is however no reason, neither theoretical nor observational, to exclude a stiff case . A period with a stiff EoS can actually be achieved naturally in a universe dominated by the kinetic energy of a scalar field after inflation, either through inflaton oscillations  under a steep potential (e.g. with ), or simply by an abrupt drop of the potential for large values of the inflaton. In the latter case it is particularly easy and natural to obtain an EoS close to unity . We note that emerges as a natural upper bound from the requirement that the sound speed of a fluid does not exceed the speed of light. Nevertheless, we expect that in general field theory constructions can approach unity only from below, so we will consider only .
Scenarios with a stiff dominated (SD) epoch between inflation and RD are particularly appealing from observational point of view: the stiff period induces a blue-tilt in the GW energy spectrum at large frequencies [7, 8, 17, 18, 19, 11, 20, 21, 22, 23, 24], opening up the possibility of detection of this GW background by upcoming GW direct-detection experiments. Furthermore, the possibility of a stiff epoch is also well motivated on the theory side, by various early Universe scenarios for which the implementation of a post-inflationary stiff period is crucial. For instance, in Quintessential inflation [25, 26, 27, 28, 29, 30, 31, 32, 33] the inflationary epoch is followed by a period where the universe is dominated by the kinetic energy of the inflaton, with the potential adjusted to describe the observed dark energy as a quintessence field. In the original gravitational reheating formulation [34, 35] the Universe is reheated by the decay products of inflationary spectator fields if the Universe undergoes a sufficiently long stiff epoch after inflation. Even though basic implementations of such gravitational reheating scenarios have been shown to be inconsistent with BBN/CMB constraints , unnatural constructions are still viable. Furthermore, variants that can naturally avoid the inconsistency have been also proposed, like the Higgs-reheating scenario , where the Standard Model (SM) Higgs is a spectator field with a non-minimal coupling to gravity and the Universe is reheated into SM relativistic species (decay products of the Higgs), if inflation is followed by an SD period. The same mechanism can actually be realized with generic self-interacting scalar fields , other than the SM Higgs. See  for a recent re-analysis of the idea.
Models with blue-tilted inflationary GW spectrum due to the presence of an SD epoch have been studied in various contexts [7, 8, 20, 40, 19, 18, 9, 11, 41]. The resulting GW energy spectrum can be characterized by three quantities: the Hubble rate during inflation, the equation state parameter during the stiff epoch, and the Hubble/energy scale at the SD-to-RD transition, parametrized by the redshifted frequency today corresponding to such scale. In general, any blue-tilted GW background can be probed with a variety of experiments, see e.g. . The aim of our present work is to assess the ability of the Advanced Laser Interferometric Gravitational wave Observatory (aLIGO), as well as of the (will-be) first generation space-based GW detector, the Laser Interferometric Space Antenna (LISA), to probe the parameter space spanned by , , and . A study in this spirit was initiated in [9, 11], but with the SD-to-RD energy scale fixed to its smallest possible value, , so that was fixed to its lowest possible value Hz. In our present work we present a systematic exploration of the observability of the full parameter space .
As a consistent cosmological history must preserve the success of BBN, we need to prevent the presence of a stiff epoch from changing significantly the expansion rate during BBN. There are two ways by which the latter can happen. First, if the stiff epoch does not end in time before the start of BBN the expansion rate would certainly deviate significantly from that of the CDM. Second, even if RD starts in time, GWs should not carry too much energy, or otherwise the expansion rate during BBN (or CMB decoupling for this matter) would still be sufficiently altered. Thus, in order to be consistent with BBN/CMB, bounds must be put on the parameters , as these control both the duration of the stiff period and the shape of the GW spectrum (and hence the energy carried by the GWs). As we will see, these constraints will restrict severely the ability of detectors to observe the blue tilted GW background due to an SD epoch. We find that the above constraints render the signal completely inaccessible to the observational window of aLIGO, independently of the parameter space. Whilst the background remains detectable by LISA, it is only observable if the inflationary scale is as large as GeV (corresponding to at least of its current upper bound), lies in the range , or equivalently the Universe becomes RD at sufficiently low temperatures MeV (i.e. the SD spoch spans many decades in energy scale), and the stiff EoS is confined within the narrow range . This corresponds to a small island in the parameter space .
The paper is divided as follows. In Section 2, we briefly review the form of the spectrum of GWs from inflation. In Section 3, we derive the energy density spectrum of the inflationary GW background in the presence of an SD epoch, considering two possible modelings for the SD-to-RD transition: an ’instantaneous’ transition with a sharp jump in the EoS, and a ’smooth’ transition modeled by the evolution of two fluid components, one made of radiation and another by a scalar field dominated by its own kinetic energy. In Section 4, we first quantify the full parameter space that can be probed by LISA and aLIGO. We then restrict such parameter space to be consistent with upper bounds on stochastic GW backgrounds from BBN and CMB considerations. We also extend our analysis to include possible changes in the number of relativistic degrees of freedom (), and the presence of a small red tensor tilt, as motivated in slow-roll inflation. In Section 5, we summarize our findings and discuss briefly the implications for some early Universe scenarios resting upon the presence of an SD epoch.
From now on, GeV is the reduced Planck mass, is the scale factor, is the conformal time, and we use the Friedman-Lemaître-Roberson-Walker (FLRW) metric as the background metric.
2 Gravitational waves
A tensor-perturbed FLRW metric can be written as
where we assume the perturbations to be transverse and traceless , so that they can be identified with gravitational waves (GWs). Expanding the Einstein equations to linear order gives
where primes denote derivatives with respect to the conformal time . To bring (2.2) to a more useful form, we perform a spatial Fourier- and polarization-mode decomposition
where stands for the polarization states and are a basis of polarization tensors satisfying , , , , and . These bring us to the GW equation of motion
where we have suppressed the polarization indices , as we assume that the GW spectrum is unpolarized . Assuming that the background metric is isotropic, we also write , with .
A useful quantity to characterize a GW background is the tensor power spectrum , defined through the following relation
where denotes a statistical ensemble average.
2.1 Inflationary spectrum
In this paper we focus on the tensor modes generated during inflation as they were spatially stretched past the inflationary Hubble radius. At the end of inflation, these modes represent superhorizon tensor perturbations with an almost scale invariant power spectrum 
with a spectral tilt, a pivot scale, and the Hubble rate when the mode exited the horizon during inflation. The presence of the tilt stems from the fact that the inflationary phase cannot be perfectly de Sitter. Nevertheless, the spectrum is expected to be only slightly red-tilted in slow-roll inflation, with the spectral index ’slow-roll suppressed’ as
where is the tensor-to-scalar ratio evaluated at the scale , constrained by the most recent analysis of the B-mode polarization anisotropies of the CMB at a scale , as [1, 2]. This bound actually implies an upper bound on the energy scale of inflation, which must be constrained as
Furthermore, the upper bound on also implies that the red-tilted spectral index can only be very small . The tensor spectrum is therefore very close to be exactly scale-invariant, at least at around the CMB scales. Actually, in the absence of running of the spectral index, the amplitude of the tensor spectrum would fall only by a factor during the 26 orders of magnitude separating the CMB scales and the Hubble radius at the end of inflation. Therefore, for simplicity, we will consider from now on an exact scale-invariant inflationary spectrum, as this gives an excellent approximation. We will comment on deviations from this assumption in Sect. 4.3.1.
From a theoretical perspective, it is convenient to work with the power spectrum , as it is precisely this quantity that is predicted by inflation to be approximately scale invariant. During the evolution of the Universe after inflation, when the tensor modes cross inside the Hubble radius, they become a stochastic background of gravitational waves (GWs). In order to quantify the ability of GW direct detection experiments to measure the inflationary GW background, it is customary to express the amount of GWs in terms of their energy density spectrum (at sub-horizon scales) , defined as the GW energy density per unit logarithmic comoving wavenumber interval, normalized to the critical density ,
which characterizes the expansion history between the moment of horizon re-entry of a given mode , defined as with , , and a later moment . For the power spectrum today we will use the notation . Note that the factor in Eq. (2.10) is simply due to averaging over harmonic oscillations of the modes deep inside the horizon.
If we assume that immediately after inflation, the Universe became radiation dominated (RD) with equation of state , the resulting present-day GW energy density spectrum is (quasi-)scale invariant, for the frequency range corresponding to the modes crossing the Hubble radius during RD. Setting and averaging over oscillations, the amplitude of the characterizing the energy density spectrum today is
where we have used and introduced the RD transfer function 
In the of Eq. (2.11) we have used , , , , and (so that ). For simplicity we have considered equal to the Standard Model (SM) degrees of freedom () before the electroweak symmetry breaking, and hence independent of . In reality the number of SM relativistic change with the scale, but we postpone the discussion of this spectral distortion to Section 4.3.2. For the time being we simply consider an identical suppression for all the modes as .
Eq. (2.11) describes the amplitude of the of the inflationary GW (quasi-)scale invariant energy density spectrum today, corresponding to the modes that crossed the horizon during RD. However, if after inflation there is a transient period of evolution with EoS , before RD is established, the resulting GW energy density spectrum today will no longer remain scale-invariant. As we will see next, the spectrum today will actually consist of two parts: a high-frequency branch, corresponding to the modes that crossed the horizon during the transient epoch, and a (quasi-)scale invariant branch corresponding to the modes that crossed the horizon during RD111There is yet another part of the spectrum, corresponding to modes that crossed the Hubble radius after matter-radiation equality, which behaves as . This corresponds to very small frequencies today Hz, and hence we will not be concerned with such low frequency end of the spectrum, as it only affects the CMB and it cannot be probed by direct-detection GW experiments..
3 Inflationary spectrum in the presence of a stiff epoch
Let us consider now that there is a period in the early Universe, spanning from the end of inflation till the onset of RD, with EoS (possibly depending on time). In standard single field slow-roll scenarios the inflaton exhibits a minimum in the potential around which it oscillates in the period following inflation. For an inflaton potential of the form , an effective (oscillation averaged) EoS emerges as . For () we obtain a RD period with , while for () we obtain instead a matter dominated (MD) era with . Interestingly, for we obtain a stiff dominated (SD) period with EoS .
In general the effective EoS in the epoch following immediately after inflation must fall in the range . Even though it is common to assume that , there is no reason (theoretical or observational) to exclude the stiff case . In this paper we are particularly interested in exploring this latter possibility. In fact, a post-inflationary period with a stiff EoS can be realized easily in a generic model of inflation. For example, in scalar singlet driven inflation, the slow-roll condition is achieved by simply demanding , where and are the inflaton potential and kinetic energy densities. Inflation cannot be sustained however if the potential drops to . Furthermore, if a feature in the inflaton potential allows its value to drop much below the kinetic energy , the EoS can become stiff after inflation, .
A simple realization of an SD regime is obtained by assuming a rapid transition of the potential from during inflation to some small value after inflation. The transition itself would actually trigger the end of inflation, leading to a post-inflationary EoS . In general we expect that the EoS can approach unity from below, but never achieve exactly, as this would require an exactly flat direction with . A natural scenario where inflation is followed by a KD phase is that of Quintessential-Inflation , where the inflaton potential is engineered so that the necessary transition occurs at the end of inflation, and the potential is also adjusted to describe the observed dark energy as a quintessence field, see e.g. [26, 27, 28, 29, 30, 31, 32, 33] for different proposals. As mentioned before, a stiff period can be also engineered through the oscillations of the inflaton with potential , . In this case we note however that the stiff period cannot be sustained for very long, as self-resonant effects lead eventually to a fragmentation of the coherent oscillating inflaton condensate .
Considering the presence of an SD period before the onset of RD is actually not only theoretically well motivated but also phenomenologically interesting. As we have mentioned in the Introduction, and as we will show in detail, an SD period with equation of state induces a large blue-tilt in the high frequency branch of the inflationary GW energy density spectrum , making this signal possibly observable with direct detection GW experiments. The aim of this paper is to quantify precisely our observational ability to measure such GW background, i.e. to determine the observable parameter space (we will also present a brief discussion on the implications for particle physics models). For our purposes, the details of the SD model implementation are unimportant. Hence, from now on we will focus on the phenomenology of SD assuming that for some unknown reason there is such a phase following the end of inflation. The background energy density of the inflationary sector evolves after inflation as , with the initial energy density at the end of inflation. In general the EoS is determined by the inflaton potential and is a function of time. However we expect it to change only adiabatically during SD, and in any case we can always describe the scaling of the energy density in terms of an effective (logarithmic-averaged) value of the EoS during the stiff period, , so that .
Let us consider therefore an SD period between the end of inflation and the onset of RD, with effective EoS deep inside the SD (i.e. way before reaching RD). Once the Universe enters into RD, we match the expansion history with that of a Universe with energy budget dominated by the SM radiation , according to the standard hot Big Bang picture. We sketch the different epochs of the expansion history we consider in Figure 1. From now on, the subscripts , , and , stand for “evaluation at” or “corresponding to” the end of inflation/beginning of SD epoch, end of SD epoch/beginning of RD period, and onset of BBN, respectively. In order to not sabotage the success of BBN, a minimal requisite that we need to impose over the assumed expansion history is that the stiff epoch must end before the beginning of BBN, i.e. .
In this section we will solve (2.4) for the aforementioned cosmological scenario, propagate the solution to the present era when it can be detected, and express the result in terms of the GW energy density spectrum today (2.9). Our focus is mainly on the modes that re-enter the horizon during the stiff phase (SD-reentering modes), as they constitute the part of the GW spectrum that can be probed observationally. The expression for the present-day GW energy spectrum depends on the assumption on how the transition from the SD to RD era takes place. There are two natural cases to consider:
Instant transition. Here the transition from SD to RD is modeled as ’instantaneous’, i.e. it occurs in a very short time interval compared to the Hubble timescale at the moment of the transition. The scale factor and Hubble rate are continuous during the transition, but we consider a sudden jump in the effective EoS from to . For example, this could happen if the stiff fluid decays into radiation at some point (which marks the end of the SD epoch) by some process characterized by a timescale much shorter than the instantaneous Hubble time at that moment.
Smooth transition. Here the SD spoch is driven by a fluid component (typically formed by the inflaton field itself) which dominates the energy budget of the Universe and has a stiff equation of state , but there is also a relatively small amount of radiation present at the end of inflation / onset of SD. The energy densities of the two components scale freely as the Universe expands, i.e. we assume no interaction between the two sectors. Due to the different scalings, as for radiation and as for the stiff fluid, the radiation component eventually dominates the energy budget of the Universe. The transition from SD to RD then occurs smoothly, over a few Hubble times, as the energy density of radiation gradually overtakes that of the stiff fluid.
Since the time-dependence of the scale factor around the SD-to-RD transition is different in the two cases, the GWs that have entered the horizon during SD will evolve differently. In turn, this means that the GW energy spectrum for the modes that re-entered the horizon before and around the SD-to-RD transition will differ in the two cases. The GW energy spectrum can be computed fully analytically in the instant transition case. However, in the smooth transition case we can only compute analytically the asymptotic high and low frequency branches of the GW spectrum, corresponding to the modes that entered the horizon deep inside SD and RD, respectively. From an observational point of view, this is not a problem, as only the high frequency branch of the spectrum can be potentially probed by GW detectors. For completeness, in any case, we will provide a numerical computation of the full GW spectrum in the smooth transition case.
3.1 Instant transition
We consider in this subsection the expansion history shown in Figure 1, assuming the transition from the SD to RD epoch occurs instantaneously, that is, in a time much shorter than the instantaneous Hubble time at the moment of transition. We consider the scale factor and the Hubble rate around the transition as continuous smooth functions, but we model the EoS as a discrete function
where is the step-function. The energy density of the background is therefore continuous, and scales as
where . From here the scale factor during each period can then be solved exactly as
and, correspondingly, the Hubble rate as
where denotes the time at the radiation-mater equality, and we have defined
For convenience, we have introduced the time variables and via
From now on, and without loss of generality, we fix and at the end of inflation (onset of SD).
where denotes derivatives with respect to (since , we do not distinguish between the derivatives with respect to or ). Requiring that the tensor mode function must match the inflationary spectrum and in the superhorizon limit , the solution to (3.9) during the stiff period when is
where is the Bessel function of the first kind. Using the small argument limit of the Bessel function, for , we obtain when , as it should. The superscript indicates that the solution applies in the stiff epoch. We will use analogous notations in what follows.
where this time denotes derivatives with respect to (again simply because ). The solution during the RD era to (3.11) , is
where are Bessel functions of the first and second kind and the superscript indicates that the solution applies during RD. At this solution must match with solution (3.10) [and hence simultaneously with if the mode is superhorizon]. Continuity of the tensor modes and their derivatives requires
from which we get
with , and
The sub-horizon and limit of (3.12) is
where we have used in the second line, and defined
It can be shown that subhorizon modes always scale as regardless of how the scale factor evolves with time. Thus, even though the second expression of c.f. Eq. (3.20), was derived in the RD epoch, once its time evolution (i.e. the damping of the tensors due to the expansion of the Universe) is written as the expression remains valid for the subsequent epochs as well. This is, however, only true for modes that became subhorizon before the moment of matter-radiation equality at , i.e. for modes .
where in the last step we have introduced inflationary tensor power spectrum (2.6) (with ), and used , , and . Since here we consider an instant SD-to-RD transition, the radiation energy density is equal to the critical density at the start of RD222As we will see later on, an analogous relation in the smooth transition case differs by a factor of 2., . This and the scaling law of radiation energy density implies
where is the frequency corresponding to the horizon scale at the onset of RD, , is the window function defined in Eq. (3.21), and we have introduced the constant
which ranges as for . The window function varies smoothly around the frequencies , and its asymptotic limits at large frequencies (corresponding to modes crossing the horizon during SD) and small frequencies (corresponding to modes crossing the horizon during RD), determine the asymptotic behaviour of the energy density spectrum. In particular we obtain
What matters from the point of view of detection prospects of this signal is the fact that the high-frequency branch of the spectrum rises with frequency, exhibiting a significant blue tilt for a stiff EoS ,
which approaches unity as we take . It is precisely this large tilt that lead us to consider the ability of GW detectors to measure this signal: as we will discuss later, a significant fraction of the parameter space characterizing the shape of the spectrum, leads to the high-frequency part of the spectrum being above the sensitivity of LISA and LIGO at their corresponding key frequencies.
The window function characterizes, in a sense, the ’interpolation’ around the central frequencies , of the two asymptotic regimes at large and small frequencies. In the next section, we will actually compute numerically the exact frequency dependence of the window function when the SD-to-RD transition is not modeled as instantaneous, but rather a smooth transition resulting from the gradual domination of the energy budget in the Universe of an initially small radiation component. From an observational point of view, given the current upper bound on the amplitude of the small frequency [c.f. Eq. (2.11)], the actual frequency dependence of around is irrelevant, as it cannot be observed by direct detection experiments. Nevertheless, as the expansion history of the Universe changes depending on the expansion rate assumed around the SD-to-RD transition, the high frequency branch of the GW spectrum will experience a slightly different expansion history once the corresponding modes cross inside the horizon. As we will show next, this translates into a correction of the normalization constant characterizing the rising high-frequency branch of the spectrum.
3.2 Smooth transition
Let us consider now a situation where at the end of inflation , the total energy density is split between a dominant stiff fluid with energy density , , and a subdominant amount of radiation with energy density . We assume that the stiff fluid has an equation of state parameter , which we take as constant for simplicity. The expansion of the Universe is then driven by the energy densities of the two fluids, each of which scale freely, as and as . The outline of the expansion history will roughly follow the sketch shown in Figure 1, but this time, the transition from SD to RD is slow and smooth, instead of ’instantaneous’. While the stiff fluid dominates, , the equation of state of the Universe remains approximately constant and equal to . Since scales down faster than , there is always a moment, which we denote as , at which . This point marks the end of the SD epoch and the beginning of the RD epoch, although the expansion history around this moment is neither purely SD nor RD, but rather dictated by a mixture of the two fluids. In a few Hubble times after , the radiation component becomes the energy-dominating fluid, and from then on the expansion history follows that of standard RD embedded in the usual CDM scenario.
In such smooth SD-to-RD transition, the scale factor cannot be solved analytically, let alone the GW equation of motion. It is, however, possible to work out analytically the blue-tilted high-frequency branch of the GW spectrum corresponding to modes entering the horizon deep inside the SD epoch at , when the equation of state of the Universe is approximately constant. Fortunately, these are the modes that can actually be probed by GW detectors. Far before the SD-to-RD transition takes place, the difference between the previously considered instant transition and the presently considered smooth transition considered is not yet apparent and the tensor mode function is given by the same expression as in the instant transition case (3.10). In order to avoid having to deal with the part of the expansion history close to the SD to RD transition where the evolution of the scale factor is not analytically solvable, we employ the trick we used earlier, namely rewriting the tensor power spectrum in terms of the scale factor . Once we do that, the resulting expression will be valid in all the subsequent epochs.
In order to proceed, we need first to obtain the value of in the two fluid approach. The condition at implies and the scaling of the energy density of the stiff fluid gives . Together, they yield
Taking the sub-horizon limit of expression (3.10), squaring it, and averaging over oscillations, we arrive at
Now that the solution is expressed in terms of the scale factor, it remains valid in all the subsequent epochs, and we can omit the superscript . Building the present-day tensor power spectrum with (3.20), and plugging this into (2.9), leads to the present-day energy spectrum for the modes re-entering the horizon during the SD,
where in the second step we have introduced the inflationary tensor power spectrum (2.6) (with ) and used , and , whereas in the third step we have used that , the definition of [c.f. Eq. (3.25)] and of the inflationary [c.f. (2.11)], and the fact that in a smooth transition which implies