q143/random_k4/random_3.md

Prepared for submission to JHEP
Causal gravitational waves as a probe of free streaming
particles and the expansion of the Universe
Anson Hook,a Gustavo Marques-Tavares,a,b and Davide Raccoc
aMaryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742,
U.S.A.
bDepartment of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, U.S.A.
cPerimeterInstituteforTheoreticalPhysics,31CarolineSt.N.,Waterloo,OntarioN2L2Y5,Canada
E-mail: hook@umd.edu, gusmt@umd.edu, dracco@perimeterinstitute.ca
Abstract: The low frequency part of the gravitational wave spectrum generated by local
physics, such as a phase transition or parametric resonance, is largely fixed by causality,
offering a clean window into the early Universe. In this work, this low frequency end of the
spectrum is analyzed with an emphasis on a physical understanding, such as the suppressed
productionofgravitationalwavesduetotheexcitationofanover-dampedharmonicoscillator
andtheirenhancementduetobeingfrozenoutwhileoutsidethehorizon. Duetothedifference
between sub-horizon and super-horizon physics, it is inevitable that there will be a distinct
spectral feature that could allow for the direct measurement of the conformal Hubble rate
at which the phase transition occurred. As an example, free-streaming particles (such as
the gravity waves themselves) present during the phase transition affect the production of
super-horizon modes. This leads to a steeper decrease in the spectrum at low frequencies
as compared to the well-known causal k3 super-horizon scaling of stochastic gravity waves.
If a sizable fraction of the energy density is in free-streaming particles, they even lead to
the appearance of oscillatory features in the spectrum. If the universe was not radiation
dominated when the waves were generated, a similar feature also occurs at the transition
between sub-horizon to super-horizon causality. These features are used to show surprising
consequences, such as the fact that a period of matter domination following the production
of gravity waves actually increases their power spectrum at low frequencies.
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Contents
| 1 Introduction |              |                   |                   |           |               | 1   |
| -------------- | ------------ | ----------------- | ----------------- | --------- | ------------- | --- |
| 2 The          | spectrum     | of                | causality-limited |           | gravity waves | 4   |
| 2.1            | The          | physical          | intuition         |           |               | 4   |
| 2.2            | Radiation    | domination        |                   |           |               | 7   |
| 2.3            | General      | equation          | of                | state     |               | 7   |
| 3 The          | effect       | of free-streaming |                   | particles |               | 9   |
| 3.1            | Implications |                   | of current        | N limits  |               | 13  |
eff
| 4 Effect     | of              | non-standard |          | expansion  | histories  | 14  |
| ------------ | --------------- | ------------ | -------- | ---------- | ---------- | --- |
| 4.1          | An intermediate |              | period   | of matter  | domination | 14  |
| 4.2          | Using           | the GW       | spectrum | to measure | w(t)       | 18  |
| 5 Conclusion |                 |              |          |            |            | 19  |
A Analytical description of the GW spectrum with an intermediate period of
| matter | domination |     |     |     |     | 20  |
| ------ | ---------- | --- | --- | --- | --- | --- |
1 Introduction
With the first direct detection of Gravitational Waves (GWs) [1], we have gained access to a
new probe of the universe. One especially exciting aspect of GWs is that, because they are so
weakly interacting, they can carry information about the early universe in a very clean and
unpolluted form. The future of GWs is bright as there are many proposed future detectors
such as LISA [2], BBO [3], MAGIS [4] and DECIGO [5]. Using stochastic GW background
1028
searches, these new GW detectors can probe the universe at a time times earlier than
our current best direct probe, the Cosmic Microwave Background (CMB).
There are many exciting possibilities that stochastic GW backgrounds will allows us
to explore. Stochastic GW backgrounds can be produced by a myriad of early universe
phenomena such as inflation [6–9], reheating/preheating [10–15], phase transitions [16] and
topological defects [17, 18] such as cosmic strings or domain walls, or by scalar perturbations
at second order in perturbation theory [19–23]. The detection of such a background would
teach us details about which of these phenomena were active and what were the relevant
| parameters | governing |     | their dynamics. |     |     |     |
| ---------- | --------- | --- | --------------- | --- | --- | --- |
Aside from learning about the actual source of GWs, their discovery also teaches us
about the dynamical evolution of the early universe. There exists a vast literature discussing
– 1 –

these effects and their detectability in various scenarios [16, 24–44] as well as reviews on the
subject [17, 45]. Intuition for this dependence can be most easily developed focusing on a
situation that has been largely overlooked, the low frequency spectrum of a stochastic GW
background produced by a causal source. This is the case for GWs generated by a phase
transition or parametric resonance for example. The low frequency modes that correspond
to length and time scales much longer than the source duration are largely independent of
the details governing the generation of GWs and are instead fixed by causality. The largest
region in causal contact with itself at the time GWs were generated has a size at most of the
order of the Hubble radius, but is generically smaller. These low frequency modes can only
be affected after entering horizon, making the spectrum sensitive to the change of the Hubble
rate as a function of time, or in other words, to the expansion history of the universe.
In this paper we consider the low frequency part of the power spectrum of a stochastic
background of GWs. These modes are typically referred to as the causal part of the spectrum
in order to emphasize the effect of causality, and under standard assumptions have a k3
scaling [46]. Nonetheless, as we will see later, there are effects other than causality that
determine their dynamics. We will refer to these modes as the “causality-limited” part of the
spectrum. The shape of the spectrum for modes which are causality-limited but were sub-
horizon at generation (sub-horizon causality) is completely independent from the evolution of
the universe1. Causality arguments affect modes which were super-horizon at generation in a
two-fold way. The first factor is that the production of these modes is highly suppressed. The
reason for this is that they have frequencies ω (cid:28) H (H being the Hubble rate) at the time
of production, and therefore behave like an over-damped harmonic oscillator, making such
modes very hard to excite. As a result, this effect decreases the power at low frequencies.
The second factor is that once excited, their amplitude stays constant while they are super-
horizon, ratherthandecreasing, leadingtoanoverallincreaseinthepoweratlowfrequencies.
The competition between these two effects determines the shape of the low frequency tail of
GWs.
More mathematically, the physical intuition that we will develop is that the initial con-
ditions for causality-limited modes immediately after generation are h = 0 while h˙ (cid:54)= 0,
ij ij
in much the same way that the initial condition of a hammer hitting a string creates no
displacement but a large velocity. If the GW is sub-horizon, ω (cid:29) H, the mode obtains an
amplitude h ∼ h˙/ω. If the mode is super-horizon, friction is important and the mode obtains
a smaller than expected amplitude h ∼ h˙/H. After this initial displacement, over-damped
modes are frozen in place until Hubble crossing at which point they become the standard
under-damped modes seen in GW detectors.
There are several interesting observations that result from an analysis along the previous
lines. The first is that generically there is a difference between sub-horizon and super-horizon
causality. As a result, there will generically be a feature at horizon crossing. Finding such
a feature would give a model-independent direct measurement of the conformal Hubble rate
1They are however sensitive to the total redshift factor between then and now.
– 2 –

at the time when GWs were created. Upon specifying an expansion history, it is possible to
deduce the Hubble rate, and hence the temperature, at which the GWs were generated. We
willdiscusstwoscenariosinwhichGWmodesthataresuper-horizonorsub-horizon(butata
wavelength larger than the correlation length of the source) at the time of generation behave
differently.
In the first, we study how super-horizon GWs depend on the equation of state parameter
w. We show that while sub-horizon causality forces the spectrum of GWs to fall as k3, that
super-horizon modes fall as k(1+15w)/(1+3w). Thus, as long as w (cid:54)= 1/3, the scaling of GWs
changes between modes that are sub-horizon at the time of generation versus super-horizon.
The change in behavior between sub-horizon modes and super-horizon modes during matter
domination (MD) was previously studied in Ref. [47] and followed up in Refs. [48–50]. We
improve upon these results by developing simple physical intuition for these effects and by
generalizing them to arbitrary equations of state.
Forthesecondscenariowelookattheeffectoffree-streamingparticlesontheproduction
and propagation of GWs. This was originally studied in Ref. [51] for the case where some
species of particles started free-streaming after GWs were produced, and it showed that this
leads to a constant suppression in the amplitude as modes entered the horizon. We focus on
thecontrastingcasewherefree-streamingparticleswerealreadypresentduringthecreationof
theGWs. Wefindthatsub-horizonmodeshavethestandardk3 scaling. Super-horizonmodes
have a scale dependent suppression, and for a sufficiently large fraction of free-streaming
particles there is a a surprising oscillatory feature on top of a suppressed scaling of k4. These
effects come about because in this scenario super-horizon GWs are not frozen out but instead
slowly roll in the potential of the free-streaming particles, suppressing the power in GWs.
This effect is always present, even if at a small level, as the high frequency part of the GW
spectrum itself acts as free-streaming particles.
Another interesting result we find is that an intermediate period of matter domination
after the GW generation actually increases the power at low frequencies. There are well mo-
tivated scenarios where various effects due to reheating generate GWs [10, 11]. In the typical
scenario where reheating occurs at high scales, only the causal part of the GW spectrum is
accessibleasthepeakfrequenciesaretoohigh. Anadditionalfeatureofmanyofthesemodels
is that they generically imply matter domination after the production of GWs. This effect
was previously considered to be undesirable as it dilutes the peak power of the GW. How-
ever, for the low frequencies that are experimentally visible, the effect of a period of matter
domination is to increase the visibility of these scenarios, making it a desirable feature.
In Sec. 2, we discuss the physical intuition behind causality-limited GWs and compare
our physically motivated approximations against exact results. In Sec. 3, we study the effects
of free streaming particles on causal GWs. In Sec. 4, we study how the frequency spectrum
of gravitation waves changes as the universe transitions between matter, radiation, and sub-
horizon modes. Finally, we conclude in Sec. 5.
– 3 –

| 2 The   | spectrum | of causality-limited |     |     | gravity | waves |     |     |
| ------- | -------- | -------------------- | --- | --- | ------- | ----- | --- | --- |
| 2.1 The | physical | intuition            |     |     |         |       |     |     |
Throughoutthisworkwewillbeexploring“causality-limited”GWs,thatwedefineasfollows.
Consider a source that is active for a short amount of time 1/β. The causality-limited part
of the GW spectrum consists of the waves whose period and wavelength are much longer
than the source’s temporal and spatial correlations respectively. In other words, λ (cid:29) 1/β,
where λ is the wavelength and 1/β is the duration of the process generating the GWs. The
standardexampleofsuchcausality-limitedGWsisthelowfrequencypartoftheGWspectrum
| generated | by a cosmological |     | phase | transition. |     |     |     |     |
| --------- | ----------------- | --- | ----- | ----------- | --- | --- | --- | --- |
Using conformal time τ and conformal Hubble rate H = a(cid:48)/a, the equation of motion for
a comoving mode k of the graviton h is (we follow a notation close to the one of [46])
ij
32πGρ
|     |     | ∂2h |         | +k2h   | a2   |     |             |       |
| --- | --- | --- | ------- | ------ | ---- | --- | ----------- | ----- |
|     |     |     | ij +2H∂ | τ h ij | ij = | Π   | ij ≡ J ij , | (2.1) |
|     |     | τ   |         |        |      | 3   |             |       |
where Π is the dimensionless anisotropic stress of the sector generating the GWs, ρ is the
ij
energydensityofthatsector, andthenormalizationofΠ followsthatof[46]withaprefactor
ij
of 32πG/3. We will project both h and J onto the two independent (+,×) polarisations,
|     |     |     |     | ij  | ij  |     |     |     |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
and assume that the respective amplitudes J and h of the two polarisations are equal. We
will also assume that the GWs are produced on a time scale fast compared to Hubble so
that we can approximate H as a constant over its production and the time over which Π is
non-zero to be small. The time at which the phase transition occurs will be denoted by τ .
(cid:63)
The solution to Eq. (2.1), assuming h(τ) = 0 for τ < τ , can be found using Green’s
(cid:63)
| functions | to be |     |                           |     |          |           |          |     |
| --------- | ----- | --- | ------------------------- | --- | -------- | --------- | -------- | --- |
|           |       |     | (cid:90) e−H(τ−τ(cid:48)) |     |          |           |          |     |
|           |       |     |                           |     | (cid:16) | (cid:112) | (cid:17) |     |
h(k,τ) = dτ(cid:48)√ sin (τ −τ(cid:48)) k2−H2 J(k,τ(cid:48)). (2.2)
k2−H2
We will be interested in the initial conditions right afterwards τ(cid:48) = τ +(cid:15) assuming that J
(cid:63)
occurs fast so that we can take J(k,τ(cid:48)) = J (k)δ(τ(cid:48)−τ ), and we will drop the k dependence
|     |     |     |     |     | (cid:63) | (cid:63) |     |     |
| --- | --- | --- | --- | --- | -------- | -------- | --- | --- |
from here on. The approximation of the source as a Dirac delta simplifies the analysis and
holdsformodeswhoseperiodismuchlongerthanthedurationofthesource, andthuscannot
resolvedetailsofthetimedependence. Therangeofmodesforwhichtheapproximationholds
will vary depending on the exact dynamics of what sources the gravitational wave. Using this
| approximation | and | Eq. (2.2), | we         | find that |         |            |          |       |
| ------------- | --- | ---------- | ---------- | --------- | ------- | ---------- | -------- | ----- |
|               |     | h(k,τ      | +(cid:15)) | = 0,      | ∂ h(k,τ | +(cid:15)) | = J .    | (2.3) |
|               |     |            | (cid:63)   |           | τ       | (cid:63)   | (cid:63) |       |
Afterwards, we have the second order differential equation and initial conditions
|     |     |     |     | ∂2h+2H∂ | h+k2h | = 0, |     |     |
| --- | --- | --- | --- | ------- | ----- | ---- | --- | --- |
|     |     |     |     | τ       | τ     |      |     |     |
(2.4)
|     |     |     | h(k,τ | ) = 0,   | ∂ h(k,τ | ) = J             | .   |     |
| --- | --- | --- | ----- | -------- | ------- | ----------------- | --- | --- |
|     |     |     |       | (cid:63) | τ       | (cid:63) (cid:63) |     |     |
|     |     |     |       |          | – 4 –   |                   |     |     |

TheobservablerelevantforGWdetectorssuchasLISAisdΩgw/dlogk. Thisisobtained
using
1
|     |     |          |     | (cid:88) |     | (cid:10) h(cid:48) | r(x,τ)h(cid:48) | s(x,τ) | (cid:11) |     |       |
| --- | --- | -------- | --- | -------- | --- | ------------------ | --------------- | ------ | -------- | --- | ----- |
|     |     | ρgw(x,τ) |     | =        |     |                    |                 |        | ,        |     | (2.5) |
|     |     |          |     |          | 32π | Ga2                | ij              | ij     |          |     |       |
r,s=+,×
which holds only when the relevant modes have entered the horizon at late times and started
| oscillating. | We will | also | take                 |     |          |                       |     |        |     |     |       |
| ------------ | ------- | ---- | -------------------- | --- | -------- | --------------------- | --- | ------ | --- | --- | ----- |
|              |         |      | (cid:10)             |     | (cid:11) |                       |     |        |     |     |       |
|              |         |      | h(k,τ)h(k(cid:48),τ) |     | ≡        | (2π)3δ3(k−k(cid:48))P |     | (k,τ), |     |     | (2.6) |
h
with P (k,τ) the dimensionful power spectrum of GWs. Fourier transforming gives
h
k5P
|     |     |     | dΩgw |     | 1 dρgw | 1   |     | (k,τ) |     |     |       |
| --- | --- | --- | ---- | --- | ------ | --- | --- | ----- | --- | --- | ----- |
|     |     |     |      | =   |        | =   | h   | ,     |     |     | (2.7) |
c2(2π)3a2G
|     |     |     | dlogk |     | ρ c dlogk | ρ   |     |     |     |     |     |
| --- | --- | --- | ----- | --- | --------- | --- | --- | --- | --- | --- | --- |
where ρ = 3H2/(8πG) is the critical energy density. In the second equality we use the
c
simplification valid at late times that h(cid:48) = kh, and where an additional k3 comes from the
phase space factor. We can then obtain the k scaling of dΩgw/dlogk by taking k5 and
| multiplying | by the | k scaling | of  | h2. |     |     |     |     |     |     |     |
| ----------- | ------ | --------- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
The observable of interest depends on h(k,τ), which comes from solving Eq. (2.4). From
that we can see that there are two contributions to the k scaling of (cid:104)hh(cid:105), the k dependence
of the source J (k) and propagation effects. Here is where causality plays a central role: as
(cid:63)
long as we are interested in wavelengths much longer than the typical spatial correlation of
the sources, the Fourier transformed source J ≈ constant (see e.g. [46]), and hence the k de-
(cid:63)
modes.2
pendence coming from the source is trivial for long wavelength Assuming correlation
k−1
lengths for the source, , that are small compared to the horizon size, we can separate
source
thesolutionsofthecausality-limitedmodesintotworegimes, k (cid:29) k (cid:29) H (sub-horizon)
|     |     |     |     |     |     |     |     | source |     | (cid:63) |     |
| --- | --- | --- | --- | --- | --- | --- | --- | ------ | --- | -------- | --- |
and k (cid:28) H (super-horizon). Note that in the case of the phase transitions, the existence of
(cid:63)
a causal sub-horizon regime depends on whether all the sources of GWs (including in partic-
ular the sound wave contribution) have a short duration compared to the Hubble time (see
| e.g. [52–55] | for recent | studies | on  | this | topic). |     |     |     |     |     |     |
| ------------ | ---------- | ------- | --- | ---- | ------- | --- | --- | --- | --- | --- | --- |
Sub-horizon regime k (cid:29) H : These modes are under-damped by definition. Thus, the
(cid:63)
standard approximation is that of a frictionless solution rescaled by a(τ )/a(τ) to take care
(cid:63)
of Hubble friction as can be seen by using the WKB approximation. This is easily shown to
be
|     |     |     | a(τ  | )J                |        |                 |     |                        |     |     |       |
| --- | --- | --- | ---- | ----------------- | ------ | --------------- | --- | ---------------------- | --- | --- | ----- |
|     |     |     |      | (cid:63) (cid:63) |        |                 |     |                        |     |     |       |
|     |     |     | h ≈  |                   | sink(τ | −τ (cid:63) ) , | (k  | (cid:29) H (cid:63) ). |     |     | (2.8) |
|     |     |     | a(τ) | k                 |        |                 |     |                        |     |     |       |
2Moreexplicitly,ifforseparationsxlargerthanalengthscaleλthetwo-pointfunctionP (x)ofthesource
Π
|     |     |     |     |     | (cid:82) |     | (cid:82)λdx4πxsin(kx)P |     |     | (cid:82)λdx4πx2P |     |
| --- | --- | --- | --- | --- | -------- | --- | ---------------------- | --- | --- | ---------------- | --- |
vanishes, then its Fourier transform P (k) = d3xeik·xP (x) = (x) ≈ (x)
|               |             |     |        | Π             |     | Π   | 0   |     | k Π | 0   | Π   |
| ------------- | ----------- | --- | ------ | ------------- | --- | --- | --- | --- | --- | --- | --- |
| is a constant | independent | of  | k when | k(cid:28)λ−1. |     |     |     |     |     |     |     |
– 5 –

From this, we immediately see that sub-horizon causality automatically gives
dΩgw
|     |     |     |     |     |     | ∝ k3, |     | (k (cid:29) | H ) | (2.9) |
| --- | --- | --- | --- | --- | --- | ----- | --- | ----------- | --- | ----- |
(cid:63)
dlogk
| regardless | of what |     | the equation |     | of state | of the | universe |     | is. |     |
| ---------- | ------- | --- | ------------ | --- | -------- | ------ | -------- | --- | --- | --- |
Super-horizon regime k (cid:28) H : There are two competing factors controlling the dy-
(cid:63)
namics of the k (cid:28) H modes. The first is that one is attempting to excite an over-damped
(cid:63)
harmonic oscillator. This effect suppresses the power in these modes. The second is that
super-horizon modes are frozen in place until they enter the horizon, increasing the rela-
tive power in these modes. The competition between these two effects is what controls the
| behavior | of k | (cid:28) H | modes. |     |     |     |     |     |     |     |
| -------- | ---- | ---------- | ------ | --- | --- | --- | --- | --- | --- | --- |
(cid:63)
| Very | quickly | (roughly |     | after | a single | e-fold) | the     | initial  | conditions |        |
| ---- | ------- | -------- | --- | ----- | -------- | ------- | ------- | -------- | ---------- | ------ |
|      |         |          |     |       | h(k,τ )  | = 0     | ∂ h(k,τ |          | ) = J      | (2.10) |
|      |         |          |     |       | (cid:63) |         | τ       | (cid:63) | (cid:63)   |        |
evolve into
J
(cid:63)
|     |     |     |     | h(k,τ | (cid:63) ) ∼ |     | , ∂ | τ h(k,τ | (cid:63) ) ≈ 0 | (2.11) |
| --- | --- | --- | --- | ----- | ------------ | --- | --- | ------- | -------------- | ------ |
H
(cid:63)
due to Hubble friction, and remain frozen while the mode is super-horizon. The O(1) number
outfrontdependsontheexactequationofstateoftheuniverse. Thisscalingcanbeconfirmed
∂2h
by solving ij +2H (cid:63) ∂ τ h ij = 0 with a non-zero initial velocity. Notice that the amplitude
τ
of the oscillation is suppressed compared to what one would expect based on the amplitudes
for sub-horizon modes: ∼ J (cid:63) /k. This suppression is the effect of attempting to excite an
| over-damped |     | harmonic | oscillator. |     |     |     |     |     |     |     |
| ----------- | --- | -------- | ----------- | --- | --- | --- | --- | --- | --- | --- |
At this point, the mode is frozen out and its amplitude is constant horizon entry, which
| occurs | at a conformal |     | time | τ k given | by  |     |     |     |     |        |
| ------ | -------------- | --- | ---- | --------- | --- | --- | --- | --- | --- | ------ |
|        |                |     |      |           |     | H(τ | ) ≈ | k.  |     | (2.12) |
k
Afterthemodeentersthehorizon,theGWcanbeapproximatedascompletelyunder-damped
| and redshifts |     | away | as a(τ | )/a(τ) | giving |     |     |     |     |     |
| ------------- | --- | ---- | ------ | ------ | ------ | --- | --- | --- | --- | --- |
k
|     |     |     |     |     | a(τ ) | J        |     |     |                        |        |
| --- | --- | --- | --- | --- | ----- | -------- | --- | --- | ---------------------- | ------ |
|     |     |     |     |     | k     | (cid:63) |     |     |                        |        |
|     |     |     |     | h ≈ |       | sinkτ    | ,   | (k  | (cid:28) H (cid:63) ). | (2.13) |
|     |     |     |     |     | a(τ)  | H        |     |     |                        |        |
(cid:63)
We have been cavalier about the phase information just putting in sinkτ, as the phase is
unimportant for computing the power due to averaging over many oscillations.
– 6 –

| 2.2 Radiation |     | domination |     |     |     |     |     |     |     |     |     |     |
| ------------- | --- | ---------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
It is well known that in radiation domination (RD) there is a simple solution for the GWs.
During RD, a ∼ τ and H = a(cid:48)/a = 1/τ. We can solve Eq. (2.4) exactly in this case and find
|     |     |     |     | J τ               |        |               | a(τ )J   |          |     |              |     |        |
| --- | --- | --- | --- | ----------------- | ------ | ------------- | -------- | -------- | --- | ------------ | --- | ------ |
|     |     |     |     | (cid:63) (cid:63) |        |               | (cid:63) | (cid:63) |     |              |     |        |
|     |     |     | h = |                   | sink(τ | −τ (cid:63) ) | =        | sink(τ   | −τ  | (cid:63) ) . |     | (2.14) |
|     |     |     |     | kτ                |        |               | a(τ)     | k        |     |              |     |        |
Thus, the full solution is given by the solution in a friction-less universe times a red-shifting
| factor of | a(τ )/a(τ). |     |     |     |     |     |     |     |     |     |     |     |
| --------- | ----------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
(cid:63)
We now show that the physical intuition given in Sec. 2.1 replicates this exact solution.
The sub-horizon modes given in Eq. (2.8) have the same scalings as the exact solution. The
super-horizon modes given in Eq. (2.13) simplify using k = H(τ ) = H a(τ )/a(τ ) to give
|     |     |     |     |     |         |            |      |                      | k   | (cid:63) | (cid:63) k |        |
| --- | --- | --- | --- | --- | ------- | ---------- | ---- | -------------------- | --- | -------- | ---------- | ------ |
|     |     |     |     |     | a(τ k ) | J (cid:63) | a(τ  | (cid:63) )J (cid:63) |     |          |            |        |
|     |     |     |     | h ≈ |         | sinkτ      | ∼    | sinkτ                |     |          |            | (2.15) |
|     |     |     |     |     | a(τ)    | H          | a(τ) | k                    |     |          |            |        |
(cid:63)
which is again the same scaling as the exact solution, confirming our physical intuition.
| From | these | results | we  | can calculate |      | how | dΩgw/dlogk | scales | with | k:         |     |        |
| ---- | ----- | ------- | --- | ------------- | ---- | --- | ---------- | ------ | ---- | ---------- | --- | ------ |
|      |       |         | 1   |               | dΩgw |     |            |        |      |            |     |        |
|      |       | h(k)    | ∼   | ⇒             |      | ∼   | k5h(k)2    | ∼ k3,  | (k   | (cid:28) H | ).  | (2.16) |
(cid:63)
|     |     |     | k   |     | dlogk |     |     |     |     |     |     |     |
| --- | --- | --- | --- | --- | ----- | --- | --- | --- | --- | --- | --- | --- |
This reproduces the famous fact that causality in a RD universe leads to the spectrum of
| GWs falling | off | as k3    | in the | small    | k limit. |     |     |     |     |     |     |     |
| ----------- | --- | -------- | ------ | -------- | -------- | --- | --- | --- | --- | --- | --- | --- |
| 2.3 General |     | equation |        | of state |          |     |     |     |     |     |     |     |
We now repeat our exercise for a general equation of state p = wρ. In this case, we have
n
|     |     |     |     |     | a   | ∝ τn, | H = | ,   |     |     |     | (2.17) |
| --- | --- | --- | --- | --- | --- | ----- | --- | --- | --- | --- | --- | ------ |
τ
with n = 2/(1+3w), so that for radiation (matter) domination we have n = 1 (n = 2). In
| this case | the exact |     | solution | of Eq.     | (2.4)                | is    |              |        |       |     |           |        |
| --------- | --------- | --- | -------- | ---------- | -------------------- | ----- | ------------ | ------ | ----- | --- | --------- | ------ |
|           |           |     |          | (cid:16)τ  | (cid:17)n−1(cid:104) |       |              |        |       |     | (cid:105) |        |
|           |           |     | kτ2      | (cid:63)   |                      |       |              |        |       |     |           |        |
|           | h(k,τ)    |     | = J      |            |                      | j (kτ | )y           | (kτ)−j | (kτ)y |     | (kτ ) ,   | (2.18) |
|           |           |     | (cid:63) | (cid:63) τ |                      | n−1   | (cid:63) n−1 |        | n−1   | n−1 | (cid:63)  |        |
where j (x) and y (x) are spherical Bessel functions of first and second kind respectively.
| n   |     | n   |     |     |     |     |     |     |     |     |     |     |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
While this is the exact solution, it is useful to consider two limits. The first is the limit of
| sub-horizon | modes: |     |        |     |                 |          |          |        |            |          |     |        |
| ----------- | ------ | --- | ------ | --- | --------------- | -------- | -------- | ------ | ---------- | -------- | --- | ------ |
|             |        |     |        |     | a(τ (cid:63) )J | (cid:63) |          |        |            |          |     |        |
|             |        |     | h(k,τ) | ≈   |                 | sink(τ   | −τ       | ) , (k | (cid:29) H | ).       |     | (2.19) |
|             |        |     |        |     |                 |          | (cid:63) |        |            | (cid:63) |     |        |
a(τ) k
|     |     |     |     |     |     | –   | 7 – |     |     |     |     |     |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |

The second is the limit of super-horizon modes at production after they enter the horizon,
| H (cid:29) | k (cid:29) H: |     |     |     |     |     |     |     |     |     |
| ---------- | ------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
(cid:63)
(cid:0) 1(cid:1)
|     |     |        | Γ n− | J τ         | (cid:18) 2 | (cid:19)n (cid:16) | nπ(cid:17) |      |                        |        |
| --- | --- | ------ | ---- | ----------- | ---------- | ------------------ | ---------- | ---- | ---------------------- | ------ |
|     |     |        |      | √2 (cid:63) | (cid:63)   |                    |            |      |                        |        |
|     |     | h(k,τ) | ≈    |             |            | cos                | kτ −       | , (k | (cid:28) H (cid:63) ). | (2.20) |
|     |     |        |      | 2 π         | kτ         |                    | 2          |      |                        |        |
Again, we will show that the physical intuition given in Sec. 2.1 replicates this exact
solution in the two limits. The sub-horizon solution in Eq. (2.19) has the same parametric
behavior found in Eq. (2.8). For the super-horizon case, note that Eq. (2.13) for a general
| equation | of state | can | be simplified | using |     |     |     |     |     |     |
| -------- | -------- | --- | ------------- | ----- | --- | --- | --- | --- | --- | --- |
(cid:19)1
|     |     |     |     |     |     | (cid:18) a(τ ) |     |     |     |     |
| --- | --- | --- | --- | --- | --- | -------------- | --- | --- | --- | --- |
(cid:63) n
|     |     |     |     | H(τ | ) = H |                | =   | k.  |     | (2.21) |
| --- | --- | --- | --- | --- | ----- | -------------- | --- | --- | --- | ------ |
|     |     |     |     | k   |       | (cid:63) a(τ ) |     |     |     |        |
k
| After some | algebra, |     | we find | that  |              |     |                       |     |     |        |
| ---------- | -------- | --- | ------- | ----- | ------------ | --- | --------------------- | --- | --- | ------ |
|            |          |     |         | a(τ   | ) J (cid:63) |     | J (cid:63) τ (cid:63) |     |     |        |
|            |          |     |         | h ≈ k | sinkτ        | ∼   | sinkτ                 | ,   |     | (2.22) |
(kτ)n
|     |     |     |     | a(τ) | H (cid:63) |     |     |     |     |     |
| --- | --- | --- | --- | ---- | ---------- | --- | --- | --- | --- | --- |
reproducing the parametric behavior of the exact solution for the super-horizon modes.
The scaling of dΩgw/dlogk can be directly obtained from the scaling of the solutions.
| For sub-horizon |     | modes | k (cid:29) | H   |     |     |     |     |     |     |
| --------------- | --- | ----- | ---------- | --- | --- | --- | --- | --- | --- | --- |
(cid:63)
dΩgw
|     |     |     |     | ∼ k5h(k)2 |     | ∼ k3, | (k  | (cid:29) H ). |     | (2.23) |
| --- | --- | --- | --- | --------- | --- | ----- | --- | ------------- | --- | ------ |
(cid:63)
dlogk
As expected, as long as the wavelength of the mode is sufficiently larger than the distances
over which the sources are spatially correlated, sub-horizon causality forces these modes to
| fall as | k3. For | the super-horizon |     | modes, | we  | have |     |     |     |     |
| ------- | ------- | ----------------- | --- | ------ | --- | ---- | --- | --- | --- | --- |
dΩgw
|     |     |     |     | ∼ k5h(k)2 |     | ∼ k5−2n, | (k  | (cid:28) H ). |     | (2.24) |
| --- | --- | --- | --- | --------- | --- | -------- | --- | ------------- | --- | ------ |
(cid:63)
dlogk
Weseethatthereismorepoweratlowk foranyequationofstatewithw < 1/3. Inparticular,
| for the | case | of MD, | n = 2, we | find that |     |     |     |     |     |     |
| ------- | ---- | ------ | --------- | --------- | --- | --- | --- | --- | --- | --- |
dΩgw
|     |     |     |     | ∼   | k   | Matter | domination. |     |     | (2.25) |
| --- | --- | --- | --- | --- | --- | ------ | ----------- | --- | --- | ------ |
dlogk
From the discussion above we find that there is a distinct kink in the spectrum at horizon
crossing as the modes transition from sub-horizon to super-horizon causality. This kink in
the spectrum occurs for all cosmologies, except for exact RD where the spectrum scales as
k3 for long-wavelength modes. Identification of this generic feature in a signal allows for a
model independent measurement of the value of conformal Hubble at which the GWs were
generated.
We can use the exact analytic results of Eq. (2.18) to obtain dΩgw/dlogk. To do this,
– 8 –

| we take | the late | time      | limit              | of Eq. | (2.18) | and find            |     |     |          |                     |     |
| ------- | -------- | --------- | ------------------ | ------ | ------ | ------------------- | --- | --- | -------- | ------------------- | --- |
|         |          | (cid:16)τ | (cid:17)n(cid:104) |        |        | (cid:16) nπ(cid:17) |     |     | (cid:16) | nπ(cid:17)(cid:105) |     |
(cid:63)
| h(k,τ)   | ≈ J      | τ                 |                   | j (kτ                       | )sin     | kτ −         | −y  | (kτ )cos     | kτ      | − .  | (2.26) |
| -------- | -------- | ----------------- | ----------------- | --------------------------- | -------- | ------------ | --- | ------------ | ------- | ---- | ------ |
|          |          | (cid:63) (cid:63) | τ                 | n−1                         | (cid:63) | 2            |     | n−1 (cid:63) |         | 2    |        |
| From the | equation | above             | we                | quickly                     | see      | that         |     |              |         |      |        |
|          |          |                   |                   | (cid:16)τ (cid:17)2n(cid:0) |          |              |     |              |         |      |        |
|          |          | |h|2              | J2τ2              | (cid:63)                    |          | )|2+|y       |     | )|2(cid:1)   | sin2(kτ |      |        |
|          |          | =                 |                   |                             | |j n−1   | (kτ (cid:63) | n−1 | (kτ (cid:63) |         | +φ), | (2.27) |
|          |          |                   | (cid:63) (cid:63) | τ                           |          |              |     |              |         |      |        |
where φ is a constant phase factor. Finally, we arrive at an exact expression for the energy
| in GWs | (after averaging |     | over | the                      | oscillations) |           |            |     |            |     |        |
| ------ | ---------------- | --- | ---- | ------------------------ | ------------- | --------- | ---------- | --- | ---------- | --- | ------ |
|        |                  |     | dΩgw |                          |               | k5(cid:0) |            |     | )|2(cid:1) |     |        |
|        |                  |     |      | ∝ k5(cid:104)h2(cid:105) | ∝             | |j        | (kτ )|2+|y |     | (kτ        |     | (2.28) |
|        |                  |     |      |                          |               | n−1       | (cid:63)   | n−1 | (cid:63)   |     |        |
dlogk
where we have neglected unimportant proportionality constants. In Fig. 1 we use these exact
analytic results to show how sub-horizon scaling becomes super-horizon scaling for various
| equations | of state. |     |     |     |     |     |     |     |     |     |     |
| --------- | --------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
103
|     |     |     |     |     | Hubblecrossingatτ |     | →   |     |     |     |     |
| --- | --- | --- | --- | --- | ----------------- | --- | --- | --- | --- | --- | --- |
|     |     |     | 102 |     |                   | ★   |     |     |     |     |     |
101
100
w=0
10-1
1/3
|     |     |     | 10-2 | w   | =   |     |     |     |     |     |     |
| --- | --- | --- | ---- | --- | --- | --- | --- | --- | --- | --- | --- |
1
w =
10-3
|     |     |     | 10-1 |     |     |     | 100 |     | 101 |     |     |
| --- | --- | --- | ---- | --- | --- | --- | --- | --- | --- | --- | --- |
Figure 1. Plot of the scaling of dΩgw/dlogk versus k for different equations of state w. The
different cosmologies are all normalized so that their sub-horizon modes are of the same size in order
| to emphasize | their  | super-horizon |                | differences. |           |     |     |     |     |     |     |
| ------------ | ------ | ------------- | -------------- | ------------ | --------- | --- | --- | --- | --- | --- | --- |
| 3 The        | effect | of            | free-streaming |              | particles |     |     |     |     |     |     |
In this section, we show how free-streaming relativistic particles induce not only a dampening
of GWs at horizon crossing but also change the shape of the causality-limited part of the
GW spectrum. We specialize to the case of a radiation-dominated universe with a fraction
ffs = ρfs/ρ in free-streaming relativistic particles since for other equations of state the
total
fraction ffs changes significantly with the expansion of the universe. As was derived in
|     |     |     |     |     |     | – 9 – |     |     |     |     |     |
| --- | --- | --- | --- | --- | --- | ----- | --- | --- | --- | --- | --- |

Ref. [51], GWs are affected by the presence of particles whose free-streaming length is larger
than the Hubble radius. In this case, the propagation of the free-streaming particles along
| geodesics | is affected |     | by and | affects | the | propagation | of  | GWs. |     |     |     |
| --------- | ----------- | --- | ------ | ------- | --- | ----------- | --- | ---- | --- | --- | --- |
It is well known that during BBN and CMB a large fraction of the energy density in
radiation was free-streaming, since neutrino interactions freeze out at temperatures below
approximately 2MeV. Little is known about our Universe prior to this era, and therefore the
existence of other relativistic free-streaming species at earlier times remains an interesting
possibility in the early Universe. In fact, since GWs themselves are free-streaming, the high
frequency part of the GW spectrum itself is an irreducible contribution to the population of
free streaming particles. We will see that if such particles were present during the generation
of GWs, they would lead to striking features in the causality-limited part of the spectrum.
| The          | equation | of          | motion | for        | GWs               | in this | scenario | is [51] |          |          |       |
| ------------ | -------- | ----------- | ------ | ---------- | ----------------- | ------- | -------- | ------- | -------- | -------- | ----- |
| u2∂2h(u)+2u∂ |          | h(u)+u2h(u) |        |            | =                 |         |          |         |          |          |       |
| u            |          | u           |        |            |                   |         |          |         |          |          |       |
|              |          |             |        | (cid:90) u | (cid:18) sin(u−x) |         | cos(u−x) |         | sin(u−x) | (cid:19) |       |
|              |          | −24ffs      |        | dx         | −                 |         | −3       |         | +3       | ∂ h(x),  | (3.1) |
|              |          |             |        |            | (u−x)3            |         | (u−x)4   |         | (u−x)5   | x        |       |
u(cid:63)
where u = kτ. As indicated before, we are interested in causality-limited GWs. Thus, we are
| solving | these equations |     | subject |       | to the       | boundary | condition |              |              |     |       |
| ------- | --------------- | --- | ------- | ----- | ------------ | -------- | --------- | ------------ | ------------ | --- | ----- |
|         |                 |     |         | h(k,τ | (cid:63) ) = | 0,       | ∂ τ h(k,τ | (cid:63) ) = | J (cid:63) . |     | (3.2) |
There are two important effects that free-streaming particles have on GWs: post-pro-
duction dynamics and horizon entry. The effects of free-streaming particles on horizon entry
is the standard effect emphasized in the case of neutrinos. Note that the LHS of Eq. (3.1) is
larger than the RHS by a factor of u2 and hence the largest effect is expected to be at early
(cid:46)
times when u = kτ 1. From this we see that sub-horizon modes are completely unaffected
by free-streaming particles. Conversely, super-horizon modes are hit by a uniform frequency
independent suppression when they enter the horizon, e.g. GW amplitudes are suppressed
by a factor of 0.8 when f = 0.4, as originally shown in Ref. [51]. Amusingly, we will see that
for GWs generated by phase transitions, this frequency-independent suppression is in fact a
| subdominant | effect. |     |     |     |     |     |     |     |     |     |     |
| ----------- | ------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
The novel effect here is that free-streaming particles also cause a suppression in the
production of GWs. What we mean by this is that the transition of super-horizon GWs from
a large velocity with a small amplitude to a large amplitude with no velocity discussed in
Eq. (2.11) is significantly affected by the presence of free-streaming particles. The suppressed
production can be computed analytically using the techniques of the previous sections.
We first consider the limit u = kτ (cid:28) 1, where the modes are very super-horizon. In this
| limit, Eq. | (3.1) | can       | be simplified |             | to  |        |          |     |          |         |       |
| ---------- | ----- | --------- | ------------- | ----------- | --- | ------ | -------- | --- | -------- | ------- | ----- |
|            |       |           |               |             |     |        | (cid:16) |     | (cid:17) | 8f      |       |
|            | u2∂   | 2h(u)+2u∂ |               | h(u)+u2h(u) |     |        |          |     |          | fs      |       |
|            |       |           |               |             |     | = −ffs | h(u)−h(u |     | ) ,      | ffs = . | (3.3) |
|            |       | u         |               | u           |     |        |          |     | (cid:63) | 5       |       |
|            |       |           |               |             |     | –      | 10 –     |     |          |         |       |

The only difference between this equation and Eq. (3.1) is that we have taken the kτ (cid:28) 1
limitof theRHS, eliminating theintegral. This approximationallows usto studythe effectof
free-streaming radiation on the evolution of super-horizon GWs. The approximation breaks
down near horizon entry, however the effect of horizon entry has been studied in the past and
approximately amounts to a k-independent reduction of the amplitude, proportional to ffs
for ffs (cid:28) 1, and so it can be treated separately.
The suppression of GW production by free-streaming particles can be directly seen from
theRHSofEq.(3.3),wherethefree-streamingparticlesactlikeaHubblescalemass. Thus,in
contrast to the normal scenario where wave amplitudes freeze out and remain constant while
outside of the horizon, the effective Hubble scale mass is constantly reducing the amplitude
of the GW due to a slow roll type effect. Note that this effect is absent if the wave starts out
with negligible velocity like in previously studied scenarios, in which cases the RHS vanishes.
Eq. (3.3) can be solved exactly with our initial conditions to give
(cid:112)
h(τ) =
J
(cid:63)
u2
(cid:63) [j (u )y (u)−j (u)y (u )] α =
−1+ 1−4ffs
α (cid:63) α α α (cid:63)
k 2
(3.4)
J u2 (cid:104) (cid:16) απ(cid:17) (cid:16) απ(cid:17)(cid:105)
≈ − (cid:63) (cid:63) j (u )cos u− −y (u )sin u− ,
α (cid:63) α (cid:63)
uk 2 2
where j and y are spherical Bessel functions and we have taken the large time limit in the
α α
second line. From this, we find that the suppression factor is simply
h amplitude (ffs) ≈ (cid:112) u2j (u )2+u2y (u )2, (3.5)
(cid:63) α (cid:63) (cid:63) α (cid:63)
h
amplitude
(ffs = 0)
which follows directly from Eq. (3.4) (and holds even when α is imaginary).
While Eq. (3.5) is rather un-illuminating in and of itself, it can be simplified in two
interesting limits 4ffs (cid:28) 1 and 4ffs > 1. In the first limit, we find that
h
amplitude
(ffs)
≈ (kτ
(cid:63)
)ffs (cid:0) ffs (cid:28) 1 (cid:1) . (3.6)
h
amplitude
(ffs = 0)
In the other limit, 4ffs > 1, we find
(cid:115)
h amplitude (ffs) (cid:112)
(cid:18)(cid:113) (cid:19)
≈ kτ
(cid:63)
C
1
+C
2
sin 4ffs−1log(kτ
(cid:63)
)+C
3
, (3.7)
h
amplitude
(ffs = 0)
where C
1
, C
2
, C
3
are unenlightening functions of ffs. Interestingly, we find that in this limit
there is an overall suppression of the amplitude that is independent ffs and that there is an
additional oscillatory feature on this suppressed amplitude. The appearance of an oscillation
as ffs increases comes from when the super-horizon modes go from being over-damped to
under-damped even while super-horizon. In this limit, the mass coming from free-streaming
particles overcomes Hubble friction and induces oscillations.
– 11 –

ToobtaintheshapeoftheGWspectrumasafunctionofk,wenumericallysolveEq.(3.1).
The results for the suppression can be found in Fig. 2, which shows the numerical results as
|     |      | 100   |         |     |         |         |               |       | 100   |     |
| --- | ---- | ----- | ------- | --- | ------- | ------- | ------------- | ----- | ----- | --- |
|     |      |       |         | ● ● | ● ● ● ● | ● ● ● ● | ● ● ● ● ● ● ● | ●● ●● | ●●●   |     |
|     |      | ● ● ● | ● ● ● ● |     |         |         | ● ●           | ●     |       |     |
|     |      |       |         |     |         |         | ● ● ● ●       | ●     |       |     |
|     |      |       |         |     |         | ● ●     |               |       |       |     |
|     |      |       |         |     |         | ● ●     | ●             |       |       |     |
|     |      |       |         |     | ● ●     |         | ●             |       | 1     |     |
|     |      |       |         |     | ●       |         | ●             |       | 1 0 - |     |
|     |      |       |         | ●   | ●       |         |               |       |       |     |
|     |      |       | ●       | ●   |         |         | ●             |       |       |     |
|     |      |       | ● ●     |     |         |         |               |       |       |     |
|     |      |       | ●       |     |         |         | ●             |       |       |     |
|     |      | ● ●   |         |     |         |         |               |       |       |     |
|     | 10-1 | ●     |         |     |         | ●       |               |       | - 2   |     |
|     |      |       |         |     |         | ●       |               |       | 1 0   |     |
●
●
●
●
●
|     |     |     |     |     | ●   |     |     |     | 10-3 |     |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | ---- | --- |
●
●
●
|     | 10-2 |     |     |     |     |     |     |     | 10-4 |     |
| --- | ---- | --- | --- | --- | --- | --- | --- | --- | ---- | --- |
●
|     |     | 10-5 | ● 10-4 |     | 10-3 | 10-2 | 10-1 | 100 |     |     |
| --- | --- | ---- | ------ | --- | ---- | ---- | ---- | --- | --- | --- |
Figure 2. The suppression effect of free streaming particles on the production of GWs, for free
streaming fractions 0.01, 0.1 and 0.4 respectively. The dots are numerical data points while the solid
lines are the analytical approximations. The dotted purple line is a k4 scaling shown to highlight the
| oscillations | taking | place. |     |     |     |     |     |     |     |     |
| ------------ | ------ | ------ | --- | --- | --- | --- | --- | --- | --- | --- |
well as the analytical approximation, Eq. (3.5). As can be clearly seen, the analytic results
and the numerical agree very well for ffs = 0.01,0.1 and 0.4. This agreement holds despite
the fact that the transition region, where u = kτ ∼ 1, is not accurately captured by our
approximation, and hence, the traditional effect of horizon entry is not entirely taken into
account.
We can re-express our results for the amplitude of the GWs in terms of the observable
| dΩgw/dlogk. | For | sub-horizon | modes, |     | we have | the previous | result |     |     |     |
| ----------- | --- | ----------- | ------ | --- | ------- | ------------ | ------ | --- | --- | --- |
dΩgw
k3
|                   |     |        |         | ∼   |     | (k (cid:29) | H ).     |     |     | (3.8) |
| ----------------- | --- | ------ | ------- | --- | --- | ----------- | -------- | --- | --- | ----- |
|                   |     |        | dlogk   |     |     |             | (cid:63) |     |     |       |
| For super-horizon |     | modes, | we have |     |     |             |          |     |     |       |
dΩgw
k3+1 6ffs
|                          |     |           | ∼   | 5   |     | (k (cid:28) | H , ffs (cid:28) | 1)  |     | (3.9) |
| ------------------------ | --- | --------- | --- | --- | --- | ----------- | ---------------- | --- | --- | ----- |
|                          |     | dlogk     |     |     |     |             | (cid:63)         |     |     |       |
| for small free-streaming |     | fractions |     | and |     |             |                  |     |     |       |
(cid:18) (cid:18)(cid:113) (cid:19)(cid:19) (cid:18) (cid:19)
| dΩgw |     |     |     |     |     |     |     |     | 5   |     |
| ---- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
∼ k4 C +C sin 4ffs−1log(k/H )+C k (cid:28) H , ffs (cid:38) (3.10)
|       |     | 1 2 |     |     |     | (cid:63) 3 |     |     | (cid:63) |     |
| ----- | --- | --- | --- | --- | --- | ---------- | --- | --- | -------- | --- |
| dlogk |     |     |     |     |     |            |     |     | 32       |     |
for large free-streaming fractions, where C are constants. This shows that the presence
1,2,3
|     |     |     |     |     | – 12 | –   |     |     |     |     |
| --- | --- | --- | --- | --- | ---- | --- | --- | --- | --- | --- |

of free-streaming radiation during the generation of GWs leads to a spectrum that decreases
fasteratlowfrequency,withthespectralindexgrowinglinearlyinffs from3untilitsaturates
| at 4 for | ffs ≥ | 5 . |     |     |     |     |     |     |     |
| -------- | ----- | --- | --- | --- | --- | --- | --- | --- | --- |
32
As mentioned before, the high k (sub-horizon) modes of the GWs themselves necessarily
play the role of free-streaming particles. The derivation of Eq. (3.1) assumes that the free-
streamingparticlesfollowgeodesics, whichGWsdoaswell. Thus, theonlypossibledifference
betweenthehighfrequencyGWmodesandfree-streamingparticlesisthatthehighfrequency
GW modes were produced by the phase transition itself. Imagine that the GWs are produced
| at a time    | (cid:15) after | the phase   | transition, |                   | then Eq. | (3.1) becomes |          |          |        |
| ------------ | -------------- | ----------- | ----------- | ----------------- | -------- | ------------- | -------- | -------- | ------ |
| u2∂2h(u)+2u∂ |                | h(u)+u2h(u) |             | =                 |          |               |          |          |        |
| u            |                | u           |             |                   |          |               |          |          |        |
|              |                | (cid:90)    | u           | (cid:18) sin(u−x) |          | cos(u−x)      | sin(u−x) | (cid:19) |        |
|              | −24ffs         |             | dx          | −                 |          | −3            | +3       | ∂ h(x).  | (3.11) |
|              |                |             |             | (u−x)3            |          | (u−x)4        | (u−x)5   | x        |        |
u(cid:63)+k(cid:15)
As long as H (cid:15) (cid:28) 1, we recover Eq. (3.3) and we see that the high frequency GW modes
(cid:63)
| behave           | exactly | like any | other   | free-streaming | particles. |     |     |     |     |
| ---------------- | ------- | -------- | ------- | -------------- | ---------- | --- | --- | --- | --- |
| 3.1 Implications |         | of       | current | N eff          | limits     |     |     |     |     |
ResultsfromCMBandBBNmeasurementslimittheamountofnonStandardModelradiation
present in the Universe at those times. These results are presented as limits on the effective
| number | of neutrinos, |     |     |     |                           |       |     |     |     |
| ------ | ------------- | --- | --- | --- | ------------------------- | ----- | --- | --- | --- |
|        |               |     |     |     | 8 (cid:18) 11 (cid:19)4/3 | (ρ +ρ | )   |     |     |
ν X
|     |     |     |     | N = |     |     | ,   |     | (3.12) |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | ------ |
|     |     |     |     | eff | 7 4 | ρ   |     |     |        |
γ
where ρ and ρ are respectively the energy density in photons and neutrinos and ρ repre-
|           | γ   | ν            |     |            |         |               |     |     | X   |
| --------- | --- | ------------ | --- | ---------- | ------- | ------------- | --- | --- | --- |
| sents any | new | contribution | to  | the energy | density | in radiation. |     |     |     |
A combined analysis of BBN and CMB measurements leads to a bound ∆N (cid:46) 0.3
eff
[56]. This implies that at neutrino decoupling a new type of radiation could contribute to
at most 4.4% of the radiation energy. However, due to the decoupling of massive degrees
of freedom, the relative temperature between the standard model plasma and any decoupled
species changes as the Universe evolves. In particular, a ∆N (cid:46) 0.3 at BBN translates to
eff
| ffs (cid:46) 0.09 | for temperatures |     | above | the | weak scale. |     |     |     |     |
| ----------------- | ---------------- | --- | ----- | --- | ----------- | --- | --- | --- | --- |
This shows that if below the weak scale there are no modifications to ΛCDM, the maxi-
mum allowed ffs is 0.09, and if a non-zero N is measured in future CMB experiments, the
eff
nature of this radiation at high energies could be tested using GWs.3 Nevertheless larger
ffs are possible with simple modifications of early cosmology, such as late equilibration of
the radiation with the SM with subsequent decay or through an entropy injection into the
standard model plasma, which dilutes the relative energy density in the new species. The
3WhileBBNmeasurementsalonecannotdistinguishfree-streamingradiationfromaninteractingrelativistic
fluid, CMB measurements are now precise enough to differentiate between them (see e.g. [57–59]). Note that
the CMB measurements are only sensitive to the nature of the relativistic species at low energies, O(eV).
– 13 –

latter case would also lead to a temporary change in the equation of state that changes the
| GW  | signal as discussed | in           | the next | section.  |           |     |     |
| --- | ------------------- | ------------ | -------- | --------- | --------- | --- | --- |
| 4   | Effect of           | non-standard |          | expansion | histories |     |     |
In this section we show how the causality-limited spectrum of GWs can be used to probe
the expansion rate of the Universe after the GWs were generated. There are two main
effects of a non-standard expansion history in the spectrum of GWs: an overall re-scaling of
the spectrum due to the change in expansion history; and a change in the power-spectrum
of causality-limited modes that enter the horizon during an era of non-radiation domination.
Thefirstfeaturehasbeenwidelydiscussedintheliterature,whilethesecondhasbeenstudied
in the case of tensor modes for inflation [31, 33–36] and cosmic strings [32, 38, 40, 41, 60]
but has been mostly overlooked with regard to phase transitions (see [47–50] for some recent
| studies | in this direction). |     |        |           |            |     |     |
| ------- | ------------------- | --- | ------ | --------- | ---------- | --- | --- |
| 4.1     | An intermediate     |     | period | of matter | domination |     |     |
Inthissubsectionweconsidertheeffectofanintermediateperiodofmatterdomination(MD).
We will compare two scenarios. In the first scenario, GWs are produced at a temperature T
(cid:63)
and the universe is radiation dominated (RD) until CMB. In the second scenario, GWs are
again producedat a temperatureT but there is an intermediate period of matter domination
(cid:63)
between the temperatures Tr→m, where the universe goes from RD to MD, and Tm→r, where
| it goes | from MD | to RD. |     |     |     |     |     |
| ------- | ------- | ------ | --- | --- | --- | --- | --- |
A summary of the results in this section can be seen in Fig. 3 where we compare the
numerical solutions for the spectrum in both cases. We find that if one is interested in modes
that enter the horizon before Tr→m, then the extra red-shifting induced by the period of MD
simply decreases the amplitude of these modes. If one considers instead the modes that enter
thehorizonafterTm→r thentheiramplitudeincreasesduetothered-shiftingofthefrequencies
movingpowerfromhighfrequenciestolowerfrequencies. Inwhatfollows, wegiveanintuitive
explanation of the results while in App. A we give a more analytical derivation.
Matterdominationgivesanentropydumpthatincreasesthered-shiftbetweenthesource
and present day observers yielding two distinct effects on the spectrum. The first is that the
extra expansion dilutes the energy density, decreasing the overall power. The second is that
the frequencies themselves are red-shifted, an effect that increases the power at low frequency
as it shifts the entire spectrum to lower frequencies. The competition of these two effects
| determines | the | behavior | of the spectrum. |     |     |     |     |
| ---------- | --- | -------- | ---------------- | --- | --- | --- | --- |
The net effect of MD can be understood intuitively and is shown visually in Fig. 4. We
first discuss the effect of the dilution of the energy density from the additional red-shifting.
FormodesthatenterthehorizonbeforeTr→m,theoverallpowerisredshiftedbyanadditional
(∆a)−4
|     | due to | the additional | expansion. |       | Here we take         |     |     |
| --- | ------ | -------------- | ---------- | ----- | -------------------- | --- | --- |
|     |        |                |            |       | (cid:18) (cid:19)1/3 |     |     |
|     |        |                |            | (a /a | ) Tr→m               |     |     |
(cid:63) 0 1
|     |     |     | ∆a  | =     | =      | > 1 | (4.1) |
| --- | --- | --- | --- | ----- | ------ | --- | ----- |
|     |     |     |     | (a /a | ) Tm→r |     |       |
(cid:63) 0 2
|     |     |     |     |     | – 14 – |     |     |
| --- | --- | --- | --- | --- | ------ | --- | --- |

103
100 f3
10-3
f
10-6
f3
RD MD RD
10-9
10-1 100 101 102 103 104
Figure 3. The low-frequency spectrum of causality-limited GWs in the case with an intermediate
period of matter domination (light blue) and the case with just radiation domination (orange). The
intermediate period of matter domination is present between the frequencies fm→r to fr→m. The
dashedblacklinesshowthescalingasf (f3)duringMD(RD).Highfrequencymodeshavelesspower
due to the presence of MD while low frequency modes have more power.
Figure 4. A qualitative depiction of the two physical effects determining the shape of the causality-
limited range of the GW spectrum in pure RD (scenario 1) and with an intermediate epoch of MD
(scenario2). Left: impactofthedilutionofρgw withrespecttotheambientenergydensityduringMD
for sub-horizon modes. Right: effect of the different redshift of the scale factor in the two scenarios.
See the text for more details.
to be the difference between scenarios 1 (pure RD) and 2 (intermediate MD) in the amount
of red-shifting between the phase transition and us. For modes that re-enter the horizon after
Tm→r, the modes were frozen out during matter domination and so their overall power is
not modified. Intermediate frequencies interpolate between the two results with a f scaling.
We now discuss the effect of the red-shifting of momenta. The observable GW frequency f
– 15 –

corresponds to the physical momentum k/a of the GW, which in scenario 2 redshifts more by
a factor ∆a. The f3 fall off of the power spectrum during RD means that when comparing
thetwospectraatthesamephysicalmomenta, thepoweris enhancedbyafactor of(∆a)3 for
all modes entering during RD. In totality, for modes that re-enter the horizon before Tr→m,
these two effects combine to give a total suppression of 1/∆a while for modes that re-enter
the horizon after Tm→r, they combine to give a total enhancement of (∆a)3. Thus we find
that an intermediate period of MD increases the power at low frequencies but suppresses it
at high frequencies.
These estimates can be made more concrete as follows. In the first scenario where the
| universe | was always | radiation |     | dominated, |     | the   | spectrum | of GWs | is         |     |     |       |
| -------- | ---------- | --------- | --- | ---------- | --- | ----- | -------- | ------ | ---------- | --- | --- | ----- |
|          |            |           |     | dΩgw,1     |     |       | f3       |        | T T        |     |     |       |
|          |            |           |     |            | =   | Agw,1 |          | f ∼    | (cid:63) 0 |     |     | (4.2) |
(cid:63),1
|     |     |     |     | dlogf |     |     | f3         |     | M   |     |     |     |
| --- | --- | --- | --- | ----- | --- | --- | ---------- | --- | --- | --- | --- | --- |
|     |     |     |     |       |     |     | (cid:63),1 |     | P   |     |     |     |
where Agw,1 is the amplitude of the GW spectrum at its peak and this scaling holds up to
the peak frequency f . Frequencies higher than this are sensitive to the properties of how
(cid:63),1
| the signal | was generated. |     |     |     |     |     |     |     |     |     |     |     |
| ---------- | -------------- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
As discussed before, there are two main effects of a period of intermediate matter domi-
nation. Firstly, the amplitude of the peak signal is decreased by (∆a)−4,
|     |     |     |     |       |     |       | (cid:18) Tr→m | (cid:19)−4/3 |     |     |     |       |
| --- | --- | --- | --- | ----- | --- | ----- | ------------- | ------------ | --- | --- | --- | ----- |
|     |     |     |     | Agw,2 | =   | Agw,1 |               |              | .   |     |     | (4.3) |
Tm→r
Secondly, the frequencies are shifted by 1/∆a so that the new peak frequency is
|     |     |     |            |            | (cid:18) | (cid:19)−1/3 |     | (cid:18)          | (cid:19)−1/3 |     |     |       |
| --- | --- | --- | ---------- | ---------- | -------- | ------------ | --- | ----------------- | ------------ | --- | --- | ----- |
|     |     |     |            |            | Tr→m     |              | T   | (cid:63) T 0 Tr→m |              |     |     |       |
|     |     |     | f =        | f          |          |              | ∼   |                   |              | .   |     | (4.4) |
|     |     |     | (cid:63),2 | (cid:63),1 |          |              |     |                   |              |     |     |       |
|     |     |     |            |            | Tm→r     |              | M   | P Tm→r            |              |     |     |       |
Finally, the universe transitions to and from MD at the critical frequencies
|     |     |      |     |       | (cid:18) |      | (cid:19)−1/3 |      |       |     |     |       |
| --- | --- | ---- | --- | ----- | -------- | ---- | ------------ | ---- | ----- | --- | --- | ----- |
|     |     |      |     | Tr→mT | 0        | Tr→m |              |      | Tm→rT | 0   |     |       |
|     |     | fr→m | ∼   |       |          |      |              | fm→r | ∼     | .   |     | (4.5) |
|     |     |      |     | M     |          | Tm→r |              |      |       | M   |     |       |
|     |     |      |     | P     |          |      |              |      |       | P   |     |       |
f3
From this starting point, the entire spectrum can be found by scaling as as long as the
| Universe | is in RD       | and | as f1   | as long | as  | it is | in MD. |     |     |     |     |     |
| -------- | -------------- | --- | ------- | ------- | --- | ----- | ------ | --- | --- | --- | --- | --- |
| The      | final spectrum |     | is then |         |     |       |        |     |     |     |     |     |

|     |     |         |     |     |        | (cid:18) | (cid:19)−1/3 |     |     |     |        |     |
| --- | --- | ------- | --- | --- | ------ | -------- | ------------ | --- | --- | --- | ------ | --- |
|     |     |        | f 3 | d Ω | g w ,1 | T r      | m            |     |     |     |        |     |
|     |     |  Agw,2 |     | =   |        | →        |              |     |     | f   | > fr→m |     |
 
|     |     |    | f 3         | d   | lo g f | T m | r   |     |     |     |     |     |
| --- | --- | --- | ----------- | --- | ------ | --- | --- | --- | --- | --- | --- | --- |
|     |     |   | (cid:63) ,2 |     |        |     | →   |     |     |     |     |     |

| d Ω |        |         | f 3         | f     |       |     |          |          |          |     |        |       |
| --- | ------ | --------- | ----------- | ----- | ----- | --- | -------- | -------- | -------- | --- | ------ | ----- |
|     | g w ,2 |           | r →         | m     |       |     |          |          |          |     |        |       |
|     | =      | Agw,2     |             |       |       |     |          |          | fr→m     | > f | > fm→r | (4.6) |
| d   | lo g f |           | f 3         | fr    |       |     |          |          |          |     |        |       |
|     |        |          |             | →m    |       |     |          |          |          |     |        |       |
|     |        |         | (cid:63) ,2 |       |       |     |          |          |          |     |        |       |
|     |        |          |             |       |       |     |          | (cid:18) | (cid:19) |     |        |       |
|     |        |         | f 3         | f     | f     | 3   | d Ω      | T        |          |     |        |       |
|     |        |          | r →         | m m → | r     |     | g w ,1   | r→ m     |          |     |        |       |
|     |        |   Agw,2 |             |       |       | =   |          |          |          | f   | < fm→r |       |
|     |        |          | f 3         | f r→  | m f 3 |     | d lo g f | T m r    |          |     |        |       |
|     |        |           | (cid:63) ,2 |       | m     | → r |          | →        |          |     |        |       |
|     |        |           |             |       |       | –   | 16 –     |          |          |     |        |       |

We see all of the effects that we had mentioned before. At high frequencies, there is a
suppression of 1/∆a = (Tr→m/Tm→r)−1/3 while at low frequencies there is an enhancement
of (∆a)3 ∼ Tr→m/Tm→r.
10-2 10-1 100 101 102 103
103
k3
100 k
10-3
k3
10-6
RD MD
10-9
10-5 10-4 10-3 10-2 10-1 100 101
Figure5. ThespectrumofaGWproducedduringmatterdomination. Sub-Hubblecausalityenforces
a k3 scaling, while super-Hubble scaling gives k during matter domination and k3 during radiation
domination.
A limit of the previous scenario is when the period of matter domination extends to the
point in time when the GWs were produced. As shown in Fig. 5, in this case the power
in modes that are sub-horizon when the spectrum is generated scales as k3 while for those
outside it goes as k. Thus even though this also has a transition from a k3 scaling to a k
scaling,theoriginforthek3 scalingisdifferentbetweenthetwoscenarios,sub-horizonphysics
instead of radiation domination. It is then an interesting question to see if it is possible to
differentiate between the two.
As the transition from sub-horizon to MD and the one from RD to MD both lead to
identical scaling away from the transition region, we cannot differentiate between the two
based on their spectrum alone. However, the two transitions are not identical. Thus, in
principle, if one sees the transition between a k3 scaling to a k1 scaling, one could use its
shape to determine whether the k3 scaling was due to radiation domination or modes being
sub-horizon.
WeillustratethedifferencebetweenthesetwoscenariosinFig.6. Fromthis,oneseesthat
thetwodifferenttransitionsareinfactdistinguishable,butthattheirdeviationisrathersmall.
Without extra information from shorter wavelengths that are sensitive to the generation
mechanism for the waves, one would need an extremely precise measurement of the transition
region in order to determine which kind of transition it is.
– 17 –

|     |     | 0.03 0.1 | 0.3 1 | 3 10 |     |
| --- | --- | -------- | ----- | ---- | --- |
RD
|     |     | 3   |     | 3   |     |
| --- | --- | --- | --- | --- | --- |
|     |     | 2   |     | 2   |     |
|     |     | 1   |     | 1   |     |
MD
|     |     | 0        |       | 0    |     |
| --- | --- | -------- | ----- | ---- | --- |
|     |     | 0.03 0.1 | 0.3 1 | 3 10 |     |
Figure 6. Comparison between the evolution of the tilt of GW spectra in two different scenarios:
transition from sub-horizon modes to MD super-horizon modes (green) and from RD super-horizon
modes to MD super-horizon modes (purple). Sub-horizon and RD super-horizon modes both scale as
dΩgw/dlogk ∼ k3 while MD super-horizon modes scale as dΩgw/dlogk ∼ k so that both scenarios
havetheexponenttransitionfrom3to1. Asisshowninthepicture, thetransitionregionisdifferent,
| showing   | that the two scenarios | can in principle | be differentiated. |     |     |
| --------- | ---------------------- | ---------------- | ------------------ | --- | --- |
| 4.2 Using | the GW spectrum        | to               | measure w(t)       |     |     |
In the early universe, it is highly likely that w was not constant and was a changing function
of time. In this section, we consider how to measure w(t). We will demonstrate that w(t)
can be to a very good approximation read off directly from the slope of dΩgw/dlogk.
In an expanding universe with a constant w, the slope of dΩgw/dlogk for super-horizon
| modes is | given by Eq. (2.24) | and repeated | below for convenience |     |     |
| -------- | ------------------- | ------------ | --------------------- | --- | --- |
dΩgw 1+15w
|     |     |     | ∼ k . |     | (4.7) |
| --- | --- | --- | ----- | --- | ----- |
1+3w
dlogk
When one considers a generic w(τ), there is no longer an exact solution. However, the
intuition developed before still applies. As mentioned before, we can approximate the GW as
having a suppressed production coupled with being frozen out until k = H. These combined
| to give | Eq. (2.13), repeated | below for | the sake of convenience |     |     |
| ------- | -------------------- | --------- | ----------------------- | --- | --- |
a(τ ) J
k (cid:63)
|     |     | h = | sinkτ | .   | (4.8) |
| --- | --- | --- | ----- | --- | ----- |
a(τ) H
(cid:63)
– 18 –

| As  | long as | w(cid:48)(τ) (cid:28) | H, then | we can proceed | as  | before   | and obtain |     |     |     |     |       |
| --- | ------- | --------------------- | ------- | -------------- | --- | -------- | ---------- | --- | --- | --- | --- | ----- |
|     |         |                       |         | dΩgw           |     | 1+15w(τ) |            |     |     |     |     |       |
|     |         |                       |         |                | ∼   | k        | ,          |     |     |     |     | (4.9) |
1+3w(τ)
dlogk
where, for each momentum k, w(τ) should be taken at the time τ when k re-enters the
horizon, k = H(τ). We see that in this approximation, one can simply read off w(τ) straight
| from | the slope | of dΩgw/dlogk. |     | More explicitly  |     |     |          |      |     |     |     |        |
| ---- | --------- | -------------- | --- | ---------------- | --- | --- | -------- | ---- | --- | --- | --- | ------ |
|      |           |                |     | dlog(dΩgw/dlogk) |     |     | 1+15w(τ) |      |     |     |     |        |
|      |           |                |     |                  |     | =   |          | .    |     |     |     | (4.10) |
|      |           |                |     | dlogk            |     |     | 1+3w(τ)  |      |     |     |     |        |
|      | 10-2      | 10-1 100       | 101 | 102 103          |     |     | 10-2     | 10-1 | 100 | 101 | 102 | 103    |
|      | 3         | RD             |     |                  |     | 3   |          |      |     | RD  |     |        |
|      | 2         |                |     |                  |     | 2   |          |      |     |     |     |        |
MD
|     | 1   |     |     |     |     | 1   |     |     |     |     |     |     |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
MD
|     | 0   |     |     |     |     | 0   |     |     |     |     |     |     |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
10-5 10-4 10-3 10-2 10-1 100 101 10-5 10-4 10-3 10-2 10-1 100 101
Figure 7. A comparison between the exact equation of state of the universe versus what is measured
from the slope of the GW spectrum. In the solid blue line, we have the exact numerical result while
in the dashed blue line, we have the approximation as measured from the slope of the GW spectrum.
Following this expression, one could in principle extract w(τ) by just measuring the tilt
of the low-frequency tail of the GW spectrum. The approximations made to obtain this
simple expression are not valid during the transition region between matter and radiation
domination. To see if they work in practice, we compared this procedure to several w(τ)
numerically. The results are shown in Fig. 7. We see that this procedure is surprisingly
accurate.
5 Conclusion
In this paper, we have presented a physical picture of how to understand causality-limited
gravitational waves. This picture allows one to easily estimate and predict the shape and be-
havior of low frequency gravity waves. As an example, we showed that generally sub-horizon
and super-horizon modes behave very differently, allowing one to make model independent
– 19 –

measurements of the conformal Hubble rate at which the gravity waves were created. Addi-
tionally, we showed how an intermediate period of MD has the effect of suppressing power at
high frequencies and enhancing power at low frequencies.
Perhaps the most surprising result is that we showed that free-streaming particles change
the scaling of the GW power spectrum. Unlike the usual case where particles start free-
streaming after GWs are produced, if there are free-streaming particles present when GWs
are produced, they lead to a suppression in the production of super-horizon gravitational
waves, bringing the scaling from k3 down to a maximum of k4. In the limit of a large number
of free-streaming particles, oscillatory features are present demonstrating that interesting
effects unrelated to causality can sculpt the spectrum of gravity waves.
There is much still left to be done with causality-limited gravitational waves. This range
of the spectrum is special because its shape can be calculated from first principles. However,
because causality sends the power to zero at low frequencies, a dedicated analysis would
be needed to see to what extent gravity wave detectors such as LISA would be able to
detect changes on this falling spectrum. On a separate note, since preheating and reheating
generated GWsare typically followed by aperiod ofmatter domination, they are morevisible
than previously expected. It would be interesting to see if these models naturally generate
signals detectable by existing or future gravitational wave detectors.
Acknowledgments
We thank Dani Figueroa and Raman Sundrum for stimulating discussions. We also thank
G´eraldine Servant and Peera Simakachorn for comments on the draft. The authors also
thank the KITP institute (From Inflation to the Hot Big Bang) where part of this work was
conducted, and the support of the NSF under the Grant No. PHY-1748958. This research
was supported in part by the NSF under Grants No. PHY-1914480, PHY-1914731 and by
the Maryland Center for Fundamental Physics (MCFP). Research at Perimeter Institute is
supported in part by the Government of Canada through the Department of Innovation,
Science and Economic Development Canada and by the Province of Ontario through the
Ministry of Colleges and Universities.
A Analytical description of the GW spectrum with an intermediate period
of matter domination
WestartbyusingthephysicalintuitiondescribedinSec.2.1tofindanapproximateanalytical
form of the GW spectrum in cosmologies with a non-standard expansion history between
generation and observation of the GWs. We can approximate Eq. (2.4) for super-horizon
modes by
2n
h(cid:48)(cid:48)+ (cid:63) h(cid:48) = 0
τ (A.1)
h(τ ) = 0, h(cid:48)(τ ) = J ,
(cid:63) (cid:63) (cid:63)
– 20 –

where n = 2/(1+3w ) is fixed by the equation of state at τ . As long as n > 1/2, these
|         | (cid:63)   |         | (cid:63) |        |     |     |     |     | (cid:63) |     | (cid:63) |     |
| ------- | ---------- | ------- | -------- | ------ | --- | --- | --- | --- | -------- | --- | -------- | --- |
| initial | conditions | quickly |          | evolve | to  |     |     |     |          |     |          |     |
J
|     |     |     |     |     | h(τ) | =      | (cid:63) | .   |     |     |     | (A.2) |
| --- | --- | --- | --- | --- | ---- | ------ | -------- | --- | --- | --- | --- | ----- |
|     |     |     |     |     |      | (2−1/n | )H       |     |     |     |     |       |
(cid:63) (cid:63)
Afterwards, h remains constant while k (cid:28) H, in analogy with Eq. (2.11), even if the equation
| of state | (EOS) | of  | the Universe |     | changes. |     |     |     |     |     |     |     |
| -------- | ----- | --- | ------------ | --- | -------- | --- | --- | --- | --- | --- | --- | --- |
Assuming a constant EOS during horizon entry, kτ ∼ 1, we must solve
2n
|     |     |     |     | h(cid:48)(cid:48)+ | k h(cid:48)+k2h |     | = 0, |     |     |     |     |     |
| --- | --- | --- | --- | ------------------ | --------------- | --- | ---- | --- | --- | --- | --- | --- |
τ
(A.3)
J
|     |     |     |     |       |        | (cid:63) |     | h(cid:48)(τ |      |     |     |     |
| --- | --- | --- | --- | ----- | ------ | -------- | --- | ----------- | ---- | --- | --- | --- |
|     |     |     |     | h(τ i | ) =    |          | ,   | i )         | = 0, |     |     |     |
|     |     |     |     |       | (2−1/n |          | )H  |             |      |     |     |     |
(cid:63) (cid:63)
where the initial time, τ (cid:28) 1/k, is chosen such that the equation of state is constant between
i
τ and horizon entry, and n is fixed by the equation of state of the Universe when the mode
| i   |     |     |     | k   |     |     |     |     |     |     |     |     |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
enters the horizon. The solution to this equation in the τ (cid:28) 1/k limit is given by
i
|        |     |               |          |        |          |          | (cid:18)  | (cid:19)n |      |          |     |       |
| ------ | --- | ------------- | -------- | ------ | -------- | -------- | --------- | --------- | ---- | -------- | --- | ----- |
|        |     |               |          |        | 2J       | Γ(n      | +1/2)     | 2         | k −1 |          |     |       |
|        |     |               |          |        | (cid:63) | k√       |           |           |      |          |     |       |
|        |     |               | h(τ) =   |        |          |          |           |           | j n  | −1 (kτ). |     | (A.4) |
|        |     |               |          | (2−1/n | )H       |          | π         | kτ        | k    |          |     |       |
|        |     |               |          |        | (cid:63) | (cid:63) |           |           |      |          |     |       |
| Taking | the | kτ (cid:29) 1 | limit of | the    | equation | above    | we arrive | at        |      |          |     |       |
(cid:18) (cid:19)n
|     |     |      |     | J (cid:63) |     | Γ(n k√ +1/2) | 2   | k      |     |          |     |       |
| --- | --- | ---- | --- | ---------- | --- | ------------ | --- | ------ | --- | -------- | --- | ----- |
|     |     | h(τ) | ≈   |            |     |              |     | cos(kτ |     | −n π/2). |     | (A.5) |
k
|     |     |     | (2−1/n |     | (cid:63) )H (cid:63) |     | π kτ |     |     |     |     |     |
| --- | --- | --- | ------ | --- | -------------------- | --- | ---- | --- | --- | --- | --- | --- |
a−1,
Once k (cid:29) H the amplitude falls as h ∝ and hence, focusing only on the amplitude of h
we have
|     |     |     |     |        |            |          |          | (cid:18) | (cid:19)n |     |     |       |
| --- | --- | --- | --- | ------ | ---------- | -------- | -------- | -------- | --------- | --- | --- | ----- |
|     |     |     |     |        | J (cid:63) | Γ(n      | k√ +1/2) | 2        | k a(τ     | r ) |     |       |
|     |     |     | |h| | ≈      |            |          |          |          |           | ,   |     | (A.6) |
|     |     |     |     | (2−1/n |            | )H       |          | kτ       |           | a   |     |       |
|     |     |     |     |        | (cid:63)   | (cid:63) | π        | r        |           |     |     |       |
where τ is some fixed reference time at which the EOS was still determined by n . Fixing
|     | r   |     |     |     |     |     |     |     |     |     |     | k   |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
the reference time τ allows us to compare different modes that enter horizon during an era
r
of fixed n.
We will now use the result in Eq. (A.6) to compare two scenarios to show the effects
of non-standard expansion histories. As before in the first scenario, GWs are produced at
a temperature T at a scale factor a and the universe is radiation dominated until the
|     |     | (cid:63) |     |     |     | (cid:63) |     |     |     |     |     |     |
| --- | --- | -------- | --- | --- | --- | -------- | --- | --- | --- | --- | --- | --- |
CMB. In the second scenario, GWs are produced at the same temperature as before, T , but
(cid:63)
there is an intermediate period of MD between the temperatures Tr→m (where the universe
becomes matter dominated) and Tm→r (where the universe becomes radiation dominated).
For simplicity we will take the scale factor at production to be the same a , which implies
(cid:63)
that the scale factor today will differ due to the difference in expansion history.
As discussed in Section 4.1, one of the major effects of an intermediate period of MD is
that the frequency spectrum of GWs is shifted by the additional expansion due to the matter
|     |     |     |     |     |     | –   | 21 – |     |     |     |     |     |
| --- | --- | --- | --- | --- | --- | --- | ---- | --- | --- | --- | --- | --- |

dominated phase. Since we normalized the scale factor at generation a to be the same in
(cid:63)
both scenarios, the relation between the scale factor and plasma temperature is identical in
both cases up until the temperature reaches Tr→m, when matter domination starts in the
second scenario. The expansion history of both scenarios differ from that point on until
the temperature gets to Tm→r, after that point the scale factors corresponding to the same
temperature (and therefore the scale factors at observation) are related by
|     |     |     |     |     |     | (cid:18) | (cid:19)1/3 |     |     |     |     |     |
| --- | --- | --- | --- | --- | --- | -------- | ----------- | --- | --- | --- | --- | --- |
Tr→m
|     |     |     |     |     | a = | a   |     | .   |     |     |     | (A.7) |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ----- |
2 1
Tm→r
Following Eq. (A.6), the amplitude for modes that have entered the horizon in the first
| scenario | is  |     |     |     |                 |      |     |                 |     |     |     |     |
| -------- | --- | --- | --- | --- | --------------- | ---- | --- | --------------- | --- | --- | --- | --- |
|          |     |     |     | J   | (cid:18) am→r,1 | Hm→r |     | (cid:19) am→r,1 |     |     |     |     |
(cid:63)
|     |     |     |     | h ≈ |     |     |     |     | .   |     |     | (A.8) |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ----- |
|     |     |     |     | H   |     | k   |     | a   |     |     |     |       |
(cid:63)
| Therefore, | the | differential | energy | density |         | is given | by                      |     |           |     |     |       |
| ---------- | --- | ------------ | ------ | ------- | ------- | -------- | ----------------------- | --- | --------- | --- | --- | ----- |
|            |     |              |        |         |         | (cid:18) | (cid:19)2(cid:16)am→r,1 |     |           |     |     |       |
|            |     |              | dρ     |         | H 2     | J        |                         |     | (cid:17)4 |     |     |       |
|            |     |              |        | 1       | m →r    |          | (cid:63)                |     | k3.       |     |     |       |
|            |     |              |        | =       |         |          |                         |     |           |     |     | (A.9) |
|            |     |              | dlogk  |         | 2(2π)3G | H        |                         | a   |           |     |     |       |
(cid:63)
For the second scenario we need to separate between the modes that entered the horizon
before, during and after the period of matter domination. The amplitude for modes that
enteredthehorizonaftertheendofmatterdomination(k < Hm→ram→r,2 )canbeimmediately
| found | from Eq. | (A.6), |     |     |          |      |     |          |     |     |     |     |
| ----- | -------- | ------ | --- | --- | -------- | ---- | --- | -------- | --- | --- | --- | --- |
|       |          |        |     |     | (cid:18) |      |     | (cid:19) |     |     |     |     |
|       |          |        |     | J   | am→r,2   | Hm→r |     | am→r,2   |     |     |     |     |
(cid:63)
|     |     |     |     | h ≈ |     |     |     |     | .   |     |     | (A.10) |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ------ |
|     |     |     |     | H   |     | k   |     | a   |     |     |     |        |
(cid:63)
Modes that entered the horizon during matter domination have a different form due to the
| difference | in EOS | and | their amplitude |     | is       | given | by  |           |     |     |     |        |
| ---------- | ------ | --- | --------------- | --- | -------- | ----- | --- | --------- | --- | --- | --- | ------ |
|            |        |     |                 |     | (cid:18) |       |     | (cid:19)2 |     |     |     |        |
|            |        |     |                 | J   | am→r,2   | Hm→r  |     | am→r,2    |     |     |     |        |
|            |        |     | h               | ≈   | (cid:63) |       |     |           | .   |     |     | (A.11) |
|            |        |     |                 | H   |          | k     |     |           | a   |     |     |        |
(cid:63)
Finally, modes that entered the horizon before matter domination are given by
|     | (cid:18)   |     | (cid:19) |     | (cid:32) |      | (cid:33) | (cid:18) |             | (cid:19) |     |        |
| --- | ---------- | --- | -------- | --- | -------- | ---- | -------- | -------- | ----------- | -------- | --- | ------ |
|     | J ar→mHr→m |     | ar→m     |     | a2       | T 2  |          | J        | am→r,2 Hm→r | am→r,2   |     |        |
| h   | ≈ (cid:63) |     |          | ≈   | r→m      | r    | →m       | (cid:63) |             |          | .   | (A.12) |
|     | H          | k   |          | a   | a2       | T2   |          | H        | k           | a        |     |        |
|     | (cid:63)   |     |          |     | m→r,2    | m→r  |          | (cid:63) |             |          |     |        |
|     |            |     |          |     |          | – 22 | –        |          |             |          |     |        |

| From | which | we find | the differential |                         | energy    | density |     |     |     |     |     |     |
| ---- | ----- | ------- | ---------------- | ----------------------- | --------- | ------- | --- | --- | --- | --- | --- | --- |
|      |       |         | (cid:18)         | (cid:19)2(cid:16)am→r,2 |           |         |     |     |     |     |     |     |
|      | dρ    | H 2     | J                |                         | (cid:17)4 |         |     |     |     |     |     |     |
|      | 2     | = m →r  |                  | (cid:63)                |           | k3×     |     |     |     |     |     |     |
|      | dlogk | 2(2π)3G | H                |                         | a         |         |     |     |     |     |     |     |
(cid:63)

|     |     |     |     |     |     | 1 , | k   | < am→r,2 | Hm→r |     |     |     |
| --- | --- | --- | --- | --- | --- | --- | --- | -------- | ---- | --- | --- | --- |

 
|     |     |     |     |    | (cid:16) |      | (cid:17)2 |      |     |                |     |          |
| --- | --- | --- | --- | --- | -------- | ---- | --------- | ---- | --- | -------------- | --- | -------- |
|     |     |     |     |    | am→r,2   | Hm→r | , am→r,2  | Hm→r |     | < k < ar→mHr→m |     |          |
|     |     |     |     | ×   |          |      |           |      |     |                |     | . (A.13) |
k
|     |     |     |     |     | (cid:18) |            | (cid:19)2 |            |     |     |     |     |
| --- | --- | --- | --- | --- | -------- | ---------- | --------- | ---------- | --- | --- | --- | --- |
|     |     |     |     |   | 2        | 2          |           |            |     |     |     |     |
|     |     |     |     |    | a r →    | m T r → m  | , k       | > ar→mHr→m |     |     |     |     |
|     |     |     |     |   | 2        | 2          |           |            |     |     |     |     |
|     |     |     |     |     | a m → r  | ,2 T m → r |           |            |     |     |     |     |
Comparing this to the differential energy density for the RD only scenario, we can immedi-
(∆a)4
ately see the dilution from the extra expansion discussed in the main text. Now, this
distribution is in terms of conformal momentum k, which, due to the difference in scale factor
normalization at late times correspond to distinct frequencies in the two cases. In order to
compare the spectra of the two scenarios and their relation to experimental searches, it is
more useful to compute the differential spectrum in terms of frequency, f = k/(2πa):
|     |     |     |     |      |     | 2    | (cid:18) (cid:19)2(cid:16)am→r,1 |     |           |     |     |        |
| --- | --- | --- | --- | ---- | --- | ---- | -------------------------------- | --- | --------- | --- | --- | ------ |
|     |     |     |     | dρ 1 | H   | m →r | J (cid:63)                       |     | (cid:17)4 |     |     |        |
|     |     |     |     | (f)  | =   |      |                                  |     | a3f3.     |     |     | (A.14) |
1
|     |     |     | dlogf |     |     | 2G  | H (cid:63) | a   |     |     |     |     |
| --- | --- | --- | ----- | --- | --- | --- | ---------- | --- | --- | --- | --- | --- |
Using the fact that scale factors after the end of MD satisfy am→r,1 /a = am→r,2 /a , we find
|     |     |     |     |     |     |     |     |     |     | 1   |     | 2   |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |

|     |     |     |     |     |     |     | 1   | ,   | f < | fm→r |     |     |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ---- | --- | --- |
 
|     |     |         | (cid:18) | (cid:19)3 |     |    | (cid:16) | (cid:17)2 |      |            |     |        |
| --- | --- | ------- | -------- | --------- | --- | --- | -------- | --------- | ---- | ---------- | --- | ------ |
|     |     | dρ      | am→r,2   |           | dρ  |    | fm →r    |           |      |            |     |        |
|     |     | 2 (f) = |          |           | 1   | (f) |          | ,         | fm→r | < f < fr→m | ,   | (A.15) |
f
|     | dlogf |     | am→r,1 |     | dlogf |    | (cid:16)  | (cid:17)−4 |     |      |     |     |
| --- | ----- | --- | ------ | --- | ----- | --- | --------- | ---------- | --- | ---- | --- | --- |
|     |       |     |        |     |       |    | a         |            |     |      |     |     |
|     |       |     |        |     |       |   | m → r , 2 | ,          | f > | fr→m |     |     |
a
|     |     |     |     |     |     |     | m → r , 1 |     |     |     |     |     |
| --- | --- | --- | --- | --- | --- | --- | --------- | --- | --- | --- | --- | --- |
where fr→m = ar→mHr→m/(2πa ) and fm→r = am→r,2 Hm→r/(2πa ), which reproduces the
|        |           |     |          |       | 2   |     |     |     |     | 2   |     |     |
| ------ | --------- | --- | -------- | ----- | --- | --- | --- | --- | --- | --- | --- | --- |
| result | discussed | in  | the main | text. |     |     |     |     |     |     |     |     |
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