6.1. General Dust Domination
Let us now derive the kernels and some estimates for the induced GWs. First we note that the fact that
in an early Matter Dominated (eMD) era seemingly makes naive calculations of induced GWs easier. Just take a transfer function equal to unity in (
53), which yields
where we took
(
). Now we have to simply integrate (
60) with a constant source. In the limit where
, we have that the averaged kernel squared is just a constant [
51,
63]. In our notation we have
We can then use (
58) to give a naive estimation of the induced GW during the early matter era. However, this is far from being a complete picture. As first realised by Inomata, Kohri, Nakama and Terada, induced GWs from a dust dominated are more subtle and depend on the nature of the transition to radiation domination [
65,
66]. If the transition is gradual, the induced GW spectrum is actually suppressed [
65]. The suppression can be understood from the fact that the initial enhancement in the dust dominated universe found in Ref. [
51] is due to the constant
. If
decays, the enhancement is not as efficient. Instead, if the transition from dust to radiation is instantaneous, the induced GW spectrum gets amplified very much [
66]. We will shortly explain the source of the amplification. Let us first note that the reason why we did not find such behaviour in
Section 4 comes down to the particular behaviour of
in dust domination. In
Section 4, due to the fact that we assumed
, the general solution to
in a
universe is a decaying and oscillating function. After the transition to radiation domination,
is also decaying and oscillating, albeit with different frequency. Thus, the matching between the two periods is well described within the WKB approximation. In this case, only those scales close to
might receive at most a
correction.
What is special about the instantaneous reheating in the dust dominated universe is that we are suddenly going from
to
. Since
did not have time to decay and the frequency of the oscillations is proportional to the ratio of the largest wavenumbers
k with the reheating scale
, the amplitude of
right after the sudden reheating will be huge. A more physical picture was presented in Ref. [
66] and it goes as follows. Density fluctuations grow as
and attain a “large” amplitude by the end of the dust dominated era. These large fluctuations suddenly evaporate and are converted into fluctuations in a radiation fluid, which create strong pressure waves. These waves generate spacetime oscillations and are the source to induced GWs.
Let us explicitly show that
becomes very large right after reheating. To do that, we first consider that
has an arbitrary amplitude during the eMD of
. Then, the eMD ends abruptly and we enter a late Radiation Domination (lRD). Matching the general solution (
73) and its first derivative for
and
we arrive at
where
since we are in a radiation dominated universe,
is defined in (
84) and comes from requiring continuity of
a and
and we defined
If we now compute the variation of
per Hubble time we find that the leading contribution for
is given by
Thus, we see that the amplitude of the time derivative is roughly
This means that if there are fluctuations on scales the amplitude of is very large and so it will produce induced GWs with a large amplitude.
Let us move on to the GWs induced after the sudden transition. For simplicity, we only focus on the largest contribution to the source (
53), which comes from the time derivatives and reads
Inserting Equations (
145) and (
147) into (
60) we arrive at the kernel during the lRD, namely
The integral in (
148) may be done analytically in terms of sine and cosine integrals but it is not so illuminating. The interested reader is referred to Ref. [
66]. Using our experience of
Section 4, we know that the integral often diverges when
. Thus, within our order of magnitude estimates, we shall focus solely on the neighbourhood of the resonance. In this case, the divergent piece is the cosine integral
and the kernel is approximately given by
The calculation of the averaged kernel squared is straightforward and yields
For other approximations regarding the IR tail, such as the large momentum contribution
, we refer the reader to Refs. [
66,
68].
With the analytic approximation for the kernel (
150) we shall turn our attention to the dominant contribution to the tensor spectrum (
58). For clarity, let us explicitly insert (
150) into (
58) to obtain the resonant part of the tensor spectrum, which leads us
Before carrying out the integral, let us analyse some of the assumptions. First, we are assuming that (
151) yields the dominant contribution. This will be the case if there is integration support for large
k. This means that
should peak at large
k. We also argued that there might be a limit to the magnitude of
k due to non-linearities given by (
139). We should use such a limit if there are no stronger motivations to consider
. However, as we shall see, if PBH dominate the universe, there must be fluctuations above
[
69]. For these reasons, let us parameterize the scalar power spectrum as power-law with a UV cut-off
given by [
71]
where
is the amplitude and
n the spectral tilt. Looking at (
151) we see that the integrand peaks at large
k if
. This includes an almost scale invariant spectrum (
) and the PBH density fluctuations (
) [
69,
70].
With these assumptions we shall proceed to perform the integral in (
151). We shall use the fact that
is divergent for
to evaluate the integrand at the resonant point except for
. For simplicity, let us switch integration variables to
which have a Jacobian given by
. Replacing
for
in (
151), evaluating integrand at
except for
and maintaining the
s dependence we arrive at
where we already integrated the cosine integral around the divergence
19 and we have defined [
66]
These bounds on
s come from momentum conservation, i.e.,
, evaluated at the resonant point
. To have an order of magnitude estimate of the amplitude of the peak of the induced GW spectrum, we can evaluate (
154) at
. By doing so, the amplitude of the induced GW spectrum at
due to the sudden reheating is roughly given by
where we used that
and we used that the numerical evaluation of the integral in
s roughly yields an extra factor
[
70]. Let us remind the reader that
since we are in radiation domination. From (
156) we see how if
the amplitude of the induced GWs is tremendously large by a power of 7. Thus, not to backreact, we either need
to be extremely small or we restrict the value of
not to be too large. Now, knowing the amplitude of the peak (
156) we can approximate the induced GW spectrum by
where, although we used the cut-off in the induced GWs, which is at
, the decay after
is so fast that we can safely neglect it.
The resonant peak is not the only contribution after the sudden reheating induced GWs. There is also the IR tail which corresponds to the
region of integration. Nevertheless, this contribution is always suppressed by a factor
. The interested reader can find the corresponding formulas in Refs. [
66,
68]. In addition to that, there is the contribution of the GWs induced during eMD, i.e., using the kernel (
141) into (
58). In the case of the sudden transition, we can take the amplitude of the induced GWs as initial conditions for the subsequent radiation domination era. Then the total spectrum is well approximated by the sum of the two contributions [
66]. This contribution will be small compared to the resonant part (
157) but since it may present a large plateau [
51,
66,
69] it may dominate on the very low frequency band. The case of the gradual transition requires a more careful treatment which can be found in Ref. [
65]. We shall proceed with a very interesting application of (
157).
6.2. PBH Dominated Era
An early matter dominated period could have been due to PBHs. Once formed, PBHs basically behave like a dust fluid, i.e., a fluid with no pressure and no propagation speed of fluctuations, like non-relativistic matter. Furthermore, their energy density practically redshifts as that of non-relativistic matter. You can see this from the following. Since the number density of PBH,
, is conserved (unless they substantially merge or evaporate) we have that
. The total energy density contained in these PBHs is
. Then in periods where PBH evaporation is negligible we have that
. For concreteness, let us assume that PBH formed in a radiation dominated era, with an initial fraction of PBH given by
where in the last step we used the first Friedmann Equation (
A33). If
is large enough, then PBH will eventually dominate the universe since we have that
. Now, for simplicity, let us assume that these PBHs formed by the collapse of large primordial fluctuations with a very peaked primordial spectrum at a scale
. This has two implications. First, the PBH mass function is almost monochromatic. Then, we can use Equation (
30) to estimate the peak mass in terms of the peak scale
by using that
. We refer the reader to
Section 2.3 for more details. Second, the initial fraction of PBH
is determined by the amplitude of the primordial spectrum. In this section, we take for convenience
and
as the free parameters of the model. One can then relate the scale and the amplitude of the primordial spectrum to
and
, if necessary. Interestingly, in this situation the PBH evaporation occurs almost instantaneously [
68] and, therefore, we can use the estimates we derived in
Section 10.2 with some corrections that we describe below. It should be noted that the results we will derive here only apply to a sharply peaked PBH mass function. If we considered an extended PBH mass function, induced GWs would be very much suppressed [
68].
A remarkable fact is that, as first realised by Papanikolaou, Vennin and Langlois [
69], due to the inhomogeneous distribution, PBH themselves create density fluctuations which might later source induced GWs. To see this, note that PBH formation by the collapse of large fluctuations is a rare event [
174] and so each PBH form uncorrelated of the others. Additionally, clustering is often negligible. This means that, as a good approximation PBHs will be randomly distributed uniformly in space. In other words, PBHs are distributed according to Poisson statistics. Such Poisson statistics are dictated by the mean inter-PBH co-moving separation. If PBH follow a monochromatic mass function, the mean inter-PBH co-moving separation at formation is given by
Note that in Equation (
159) we identify the co-moving inter-PBH separation as the UV cut-off in Equation (
152). This is because the fluid description of the PBH gas is valid only for
, i.e., at distances much larger than
. In this coarse grained regime the initial dimensionless power spectrum of PBH density fluctuations reads [
69]
We have shown with Equation (
160) that PBHs give rise to density fluctuations. However, in the current model, PBHs form in an early Radiation Dominated stage (eRD) and, therefore, such density fluctuations correspond to isocurvature fluctuations. Thus, they do not source induced GWs yet. Induced GWs are mainly generated by curvature fluctuations. If one is interested in the latter aspect, the general source term in a two fluid system can be found in Ref. [
70]. The isocurvature nature of the PBH density fluctuations can be understood from the following [
70]. PBH formation in the fluid picture may be regarded as a transition of a fraction of the homogeneous radiation fluid into dust. Yet, the total energy density of radiation remains homogeneous, while PBH are distributed randomly. Thus, the inhomogeneity due to PBHs must be isocurvature as the total energy density is homogeneous. Interestingly, such isocurvature is converted into curvature fluctuations when PBH dominate the universe. This type of transition from radiation to dust was studied analytically in detail by Kodama and Sasaki [
285,
286]. Thus, we can directly borrow their results which provide the transfer function for a vanishing initial curvature perturbation
and non-zero isocurvature deep inside the eMD era as
The subscript “eq” refers to the early radiation-PBH equality. For an intuitive re-derivation, see Ref. [
69]. Equation (
161) tells us that the initial isocurvature has been transferred to a constant curvature perturbation. Modes which entered the horizon before the early radiation-PBH equality have decayed substantially and, hence, the suppression factor
. This means that the curvature power spectrum due to early PBH fluctuations in the eMD on the smallest scales, i.e., (
) is given by
It is the curvature perturbation spectrum given by (
162) which sources induced GWs during the eMD. However, we are most interested in the induced GWs generated right at the start of the lRD. As we shall see, since PBH evaporation has a finite duration even for the monochromatic case, we must take into account an important suppression. Then we shall use our estimate Equation (
156). Before that, we need to understand and quantify the PBH domination a bit further.
Let us derive the conditions to have a PBH dominated era in the early universe allowed by current observations. We start with our PBHs given an initial mass
(
30) and an initial fraction
. After formation, these PBHs will evaporate emitting Hawking radiation. This means that
cannot be too small, otherwise PBH evaporate before they dominate. To quantify this requirement it is enough to look at the evaporation time which for a non-spinning black hole reads [
68,
189,
287]
where
are the spin-weighted degrees of freedom and
If we assume only the standard model of particle physics we have that for PBHs which evaporated before BBN then
. If we include a possible PBH spin the evaporation time decreases, reaching a factor
for the extremal BH case [
288,
289,
290,
291,
292]. Thus, we can take into account the PBH spin by replacing
. This effect is considered in the induced GWs by Ref. [
71]. We will not consider PBH spin in what follows. From Equation (
163) we see that the “reheating” temperature, i.e., the temperature of the radiation fluid which dominates the universe after PBH evaporate, is given solely in terms of
. Using the current cosmological parameters, which are given in
Appendix A, we find that the evaporation temperature can be written as
where
are the effective degrees of freedom in the energy density of radiation [
293,
294]. Here we find our first constraint on the model: PBH cannot evaporate too late if they are to reheat the universe. For a successful BBN the reheating temperature must be
[
295,
296,
297,
298]. This puts an upper bound on the PBH mass as
Using Equation (
165), the co-moving scale corresponding to the horizon size at the time of evaporation is given by
where
are the effective degrees of freedom in the entropy. With the above formulas we have that the end of the PBH domination era is fixed by the PBH mass. The second constraint on the model comes from requiring that PBHs dominate before they evaporate. This is a constrain on the initial fraction of PBHs in terms of the PBH mass, which is found to be [
68,
70]
Before we compute the final amplitude of the induced GW spectrum, it is very important to estimate the effects of the finite duration of the evaporation to the perturbations. Although the transition to a radiation dominated universe takes place within less than
of e-fold, perturbations in very small scales are very much affected by the finite width of the transition, as was discovered by Inomata et al. [
65,
66,
68]. The main reason is clear if we neglect the expansion of the universe, which is justified as we are interested in very subhorizon scales. As PBHs evaporate their energy density decays as [
68]
Now, we are interested in any effect that this decay may have on
. The Newtonian potential is related to fluctuations by the Poisson equation which in Fourier space reads
where
is the density contrast. We will do a big step here that can be easily confirmed by looking at the formulas in the appendix of Ref. [
70] or in Ref. [
68]. If we completely neglect the expansion of the universe we find that
is a solution to the equations of motion of PBH density fluctuations. With this solution, we can estimate the size of the fluctuations in the radiation fluid sourced by the evaporation. With the Green’s method to solve the equations of motion of radiation fluctuations, we find that
where
is the evaporation rate of PBHs and it is given by
What (
171) tell us is that fluctuations in the radiation fluid
with
are very suppressed with respect to
due to radiation pressure. This means that for such modes the dominant contribution to
is given by
, even when PBH do not dominate the universe. Then by Equation (
170) we see that
until
, at which point the fluctuations in radiation dominate and perturbations behave as in a radiation dominated universe. The temporal decay of
due to the decay of
yields a relative suppression given by [
68]
where
refers to the value of
if the transition is instantaneous.
Now, we can use our estimate for an instantaneous transition (
156) but taking into account the suppression of
until the transition to radiation domination is completely achieved. This yields a spectrum of curvature fluctuations at the start of the lRD given by
Comparing (
174) with (
152) we see that
and
The final ingredient to derive the induced GW spectrum is the hierarchy of the scales involved. First, note that the ratio between the cut-off
and
is independent of
and it is given by [
70]
The
independence of the ratio is due to the fact that
only depends on the PBH mass and
because it is a dust dominated universe. The hierarchy is closed with the second and third relations respectively given by [
70]
Now, inserting Equations (
174), (
176) and (
177) into (
157) yields an amplitude of induced GWs at the peak of approximately
Note that the estimate Equation (
178) is the amplitude of induced GWs right after evaporation. It should also be noted that (
178) is valid for very sharp PBH mass functions. As the mass function widens, the induced GW counterpart gets suppressed [
68]. On top of that, we have that scales
(
139), specially for
, PBH density fluctuations
become
or larger. However, our estimate (
178) has been derived in the linear regime and might receive corrections due to non-linear effects. Nevertheless, we expect it to be a crude order of magnitude estimate since the main source of induced GWs, the curvature perturbation, remains always smaller than unity,
20 i.e.,
. Numerical relativity simulations are needed to derive a preciser estimate of the induced GWs generated in the non-linear regime. We show the resulting induced GW spectrum in
Figure 8. Using (
167) and (
176) we find that the peak of induced GWs today lies at a frequency given by
Quite surprisingly, the frequency of the peak enters the observational range of LISA, DECIGO and LIGO for . Thus, this scenario is testable in the future.
To compute the amount of induced GWs measured today we must consider the redshift of GWs until today. We did this in
Section 5 with Equation (
115). However, if we want to evaluate the amplitude of induced GWs at BBN, we only need to know the fraction of GWs that compose the total radiation after reheating. This can be done by considering the degrees of freedom of the standard model, which leads us to
Since these induced GWs act as additional radiation, they contribute to the effective number of relativistic species
at BBN. The contribution of GWs to
is by convention
21 written as [
29]
where in the last step we used that BBN [
299,
300] sets an upper bound
.
22 In this way, we can constrain the amplitude of GWs from existing BBN bounds. Note that to be precise the BBN bound is a constrain on the total energy density of relativistic particles. Thus, we should have actually integrated our induced GW spectrum over
and then compare it with the bound (
181). However, since the GW spectrum is very peaked and we are already working with order of magnitude estimates, it is justified to just look at the peak of the GW spectrum. If we use the current constraints from BBN, then we obtain that
This is a new constraint on early epochs of PBH dominations which cannot be obtained by any other means.
Interestingly, PBH evaporation also emits “gravitons”, i.e., very high frequency GWs. While these gravitons are not observable by GW detectors, they might be seen as effective relativistic particles at BBN [
288,
289,
290,
291,
292]. Future CMB probes such as CMB-S4 [
303] will be able to improve current bounds down to
. Unfortunately, the graviton contribution of non-spinning PBHs to
is out of reach for future CMB probes. Nevertheless, if PBH have a large spin (and dominate the universe at some point), the graviton contribution to
will be accessible to future experiments [
291,
292]. This signature from graviton emission together with induced GWs will be an interesting probe of PBH dominated eras. While the graviton contribution to
tells us about PBH spin, the induced GW counterpart has information on the formation mechanism [
71]. As a clarification, note that the graviton emission and induced GWs are two very different effects. While the former is emitted by Hawking evaporation, the latter is generated by the time dependence of density fluctuations. The composition of such density fluctuations is not important, e.g., if a fraction are gravitons, as long as it behaves as fluctuations of a radiation fluid.
Before ending this section, we should note that we also have the induced GWs generated in the PBH dominated stage by using the Kernel (
141). This is studied in detail in Ref. [
69]. However, as we discussed at the beginning of this section this contribution is sensitive to the detailed transition to radiation domination. In the present case, where the transition is almost instantaneous, we can continue the tensor mode amplitude build up during eMD to the lRD. Nevertheless, this contribution is always subdominant compared to the one after reheating [
66]. Thus, our estimate (
178) provides the dominant peak of the induced GW spectrum.