Spectrum of gravitational waves from long-lasting primordial sources

We discuss long-lasting gravitational wave sources arising and operating during radiation-dominated stage. Under a set of assumptions, we establish the correspondence between cosmological evolution of a source and the resulting gravitational wave spectrum. Namely, for the source energy density ρs falling as a power law characterized by the exponent β, i.e., ρs ∝ 1/aβ , where a is the Universe scale factor, the spectrum takes the form Ωgw ∝ f 2β-8 in certain ranges of values of constant β and frequencies f. In particular, matching to the best fit power law shape of stochastic gravitational wave background discovered recently by Pulsar Timing Array collaborations, one identifies β ≈ 5. We demonstrate the correspondence with concrete examples of long-lasting sources: domain walls and cosmic strings.


Introduction
Recently various Pulsar Timing Array (PTA) collaborations including NANOGrav, PPTA, EPTA with InPTA, and CPTA, reported a signal consistent with stochastic gravitational wave (GW) background in the nHz frequency range [1,2,3,4,5].The spectral index p of the best-fitting power law to the measured frequency f dependence of the relic GW abundance, is given by p = 1.8 ± 0.6 at 68% CL [6] 1 .The latter is in tension with the value characteristic for GW-driven supermassive black hole binaries, p = 2/3 [7], or the spectral index describing infrared tails of some typical primordial sources, p = 3 (see below).One should note, however, that current observational and modelling uncertainties are large enough, so that "canonical" GW sources like supermassive black hole binaries [8,9,10,11] or first order phase transitions [6,12,13] still have a great potential of explaining the signal.(See [14] discussing various sources of primordial GWs in light of PTA data and the reference list there.)Yet it appears timely to speculate on the possibility that the value p ≈ 2 persists in the future with the increased sensitivity of PTA data, and discuss generic properties of GW sources yielding the desired spectral index.We focus on cosmological GW sources operating at radiation domination and fully relying on post-inflationary physics 2 .In this situation, p = 3 is perhaps the most common outcome supported by causality arguments, with an important loophole that it assumes a very short (much shorter than a Hubble time) duration of a source [17,18,19].
This suggests a way to obtain the spectral index p ≈ 2 by switching to enduring GW sources lasting for many Hubble times.One example of such a source is represented by the network of melting domain walls characterized by a time-dependent tension [20,21,22,23].While this GW source relies on a concrete particle physics scenario, only a few of its properties are relevant for obtaining the spectral index p = 2, with the most non-trivial one being cosmological evolution of its energy density ρ s ∝ 1/a 5 (t).In this paper, we establish this on general grounds without resorting to melting domain walls and details of microscopic physics.More broadly, we establish the correspondence between the spectral index p and the time-dependence ρ s = ρ s (t) chosen to be of the form We show that in a certain range of constant β and a range of frequencies f (infinitely broad for an everlasting GW source), one has under a set of reasonable assumptions.The correspondence (2) typically gets violated starting from3 β = 11/2, when the standard spectral index p = 3 is recovered: for β > 11/2 GW source effectively acts as instantaneous, and the value p = 3 persists.

Generalities
Focusing on spin-2 linear perturbations of the FLRW metric, one writes where h ij is the transverse traceless tensor.Hereafter, we switch to conformal time τ .The equation governing production of GWs in the presence of a source is given by where G is the Newton's constant.As in Ref. [17], we performed separation of the source into the homogeneous background part ρ s (τ ) and dimensionless inhomogeneous part Π ij (x, τ ), which is a transverse traceless tensor.Let us switch to Fourier transform fields: and perform splitting over polarizations: Here e A ij is a pair of polarization tensors normalized as e A ij e A ′ ij = 2δ AA ′ , where A, A ′ = 1, 2. We assume that no GW production takes place before τ i , when the source is switched on.By continuity of the field h(k, τ ) and its time derivative ∂h(k, τ )/∂τ at τ i , one should impose initial conditions h(k, τ i ) = 0 and ∂h(k, τ )/∂τ | τ =τ i = 0.Then, solution of Eq. ( 5) in the background dominated by radiation reads: The energy density of GWs today is given by where the subscript ′ 0 ′ refers to the present Universe, and by ⟨...⟩ we mean the ensemble average over many realizations of the stochastic source.Assuming that the source is terminated at some time τ f ≪ τ 0 , using Eqs.( 6) and (8), and summing over polarizations of GWs, we obtain for the GW energy density: Note that we limit both τ i and τ f to be within radiation era, with no much loss of generality.Indeed, PTAs as well as other current or planned GW observatories operate in a range of frequencies characteristic for GW emission taking place at radiation domination for not very small reheating temperatures.Assuming statistically homogeneous and isotropic sources, so that and dropping the fast oscillating cosine function in Eq. ( 10), we obtain the spectral energy density: (12) Here we introduced the power spectrum of the source P (k, τ ′ , τ ′′ ), which will play a key role in what follows.

Spectral index of gravitational waves
The form of the power spectrum P (k, τ ′ , τ ′′ ) can be fixed from its mass dimension and scaling properties.Namely, let us perform a transformation upon the scale factor a → ζa supplemented by coordinate transformations k → ζk and τ → τ /ζ with ζ > 0 being some constant, so that the physical quantities k/a and aτ remain invariant.It is straightforward to check that the function P (k, τ, τ ′ ) transforms under the rescaling as The latter can be traced back to the fact that the dimensionless stress Π ij is invariant under rescaling, which in turn can be inferred from Eq. ( 5).Keeping in mind mass dimension of the power spectrum P (k, τ, τ ′ ), which is minus three, one can write: where P is the reduced dimensionless power spectrum depending only on combinations of coordinates kτ ′ and kτ ′′ .However, expression ( 14) is not the most generic one, because the function P may depend on the initial time of GW emission τ i as well as mass parameters M n (τ ) (which can vary with time, e.g., if they have thermal origin) characteristic for a concrete model: To obtain Eq. ( 14), we first eliminate dependence on τ i by simply stating that This is a non-trivial assumption, and in case of topological defects it is a manifestation of scaling regime.While the scaling regime is indeed observed for cosmic strings [24,25,26] and domain walls [27,28], one should be careful when applying Eq. ( 16) to generic sources.Regarding dependence on the masses M n (τ ), it is enough to consider the regime k/(aM n ) ≪ 1; then, dependence on masses M n (τ ) can be ignored, provided that P remains finite in the limit k/(aM n ) → 0, while kτ ′ and kτ ′′ are kept fixed, i.e., Note that the set M n (τ ) does not include the Hubble rate H(τ ), which would be the smallest mass scale in the problem.The reason is that k/(aH) ∼ kτ , and consequently Eq. ( 14) is not affected.The same is true for any mass scale with the same time dependence as H(τ ).
Other relevant mass scales M n are assumed to be considerably larger than H, i.e., M n ≫ H, while the momenta k/a of interest are comparable to the Hubble rate.Assuming Eqs. ( 16) and ( 17), we proceed with Eq. ( 14).Substituting Eqs. ( 2) and ( 14) into Eq.( 12) and defining ξ ≡ kτ , we obtain Note that the expression in front of the integral is independent of τ i ; this is clear from Eq. ( 2) and the fact that a(τ ) ∝ τ during radiation domination.Nevertheless, dependence on τ i and τ f may enter through integration limits ξ i and ξ f .One can ignore this dependence, provided that the integral is saturated for ξ ′ and ξ ′′ inside the region (ξ i , ξ f ), so that This implies that the integral on the r.h.s. of Eq. ( 19) is finite The constant C here is independent of k, -this property can be traced back to our assumptions ( 16) and (17).Hence, in the the range of momenta (26), we can write in terms of GW relic abundance, which explains Eq. ( 3).Here Ω is defined as where ρ tot,0 is the energy density of the Universe today, and the frequency f is related to the conformal momentum k by Fixing β = 5, we end up with which is the behaviour favoured by the recent PTA data.Were the condition (20) violated, the spectral energy density would depend on ξ i and ξ f , which would yield the additional frequency f dependence eventually spoiling (22).Validity of Eq. ( 20) implies restriction on the range of momenta k, or equivalently frequencies f , which cannot be arbitrary for fixed τ i and τ f .Generically, one writes where the coefficient α ≳ 1 relates the characteristic frequency of GW emission at some time τ to the Hubble rate H(τ ) (we give a more precise definition shortly).We keep α constant explicitly assuming that the characteristic frequency is pinned to H(τ ), which is a rather common property of cosmological GWs.Note that Eq. ( 26) defines desired duration of a GW source: the resulting range of frequencies ( 26) should be broad enough to cover frequency bands of observational facilities, e.g., PTAs.
The conditions ( 20) and ( 21) can be reinterpreted as constraints on the constant β.For this purpose, let us focus on GWs emitted during the short time interval τ 1 ≤ τ ≤ τ 2 , so that τ 2 − τ 1 ≪ 1 for all relevant momenta.We approximate this transient GW emission by the broken power law described by the peak frequency f p ≃ α/(2πa 0 τ ) (this expression defines the coefficient α entering Eq. ( 26)) separating IR and UV tails: where q is a constant characterizing the UV tail.The latter is not robust against details of a particular GW source, and this is the reason, why we choose to do not specify q.In particular, (constant tension) domain walls [28] or bubble collisions during first order phase transitions [13] lead to q = −1; at the same time acoustic waves triggered by phase transitions lead to q = −3 [29,30].On the other hand, the IR tail ∝ f 3 fixed by causality considerations is a rather generic property of transient sources.This universality is, however, violated in cases, cf.Refs.[18,19,31], and we will encounter with one situation of that kind in what follows.Note that if the actual IR tail is steeper than in Eq. ( 27), our discussion below is still applicable in a conservative sense.Having this said, we continue with Eq. (27).
In a similar manner, one could infer the lower bound on the constant β from the behaviour of the UV tail.Again restricting to the instantaneous source obeying ξ 2 − ξ 1 ≪ 1, but now assuming ξ 2 ≫ α, ξ 1 ≫ α, we get Hence, convergence of the integral (19) in the limit ξ f → ∞ gives β > 3 + q/2.We conclude with the following range of constants β: Thus, for a not very blue UV tail of the transient source, q < 2, correspondence between the value β = 5 and the best fit PTA spectral index p = 2 is safe.Note that the cases correspond to the situations, when GW emission is strongly saturated around the initial moment τ i (unless there are departures from the IR tail ∝ f 3 , see the comment after Eq. ( 27)) or the endpoint τ f , respectively.In both cases, the source can be approximated as transient, and one results with GW spectrum (27).

Examples
Let us apply the p = 2β−8 to some concrete examples of long-lasting sources.Perhaps the most well-known source of this type is represented by constant tension cosmic strings [32] resulting from spontaneous symmetry breaking of global or gauge continuous symmetries.In the scaling regime, the energy density of the network of cosmic strings during radiation domination mimics that of relativistic species, i.e., ρ s ∝ 1/a 4 (τ ).Hence, β = 4 and we result with the flat spectrum Ω gw (f ) ∝ f 0 [33,34,35].This conclusion can be extrapolated to a wide range of different topological defects with the same redshift behaviour as cosmic strings [36,37,38].
On the other hand, the network of constant tension domain walls [39] resulting from spontaneous breaking of discrete symmetries has a different cosmological evolution ρ s ∼ σ wall H ∝ 1/a 2 (τ ) in the scaling regime, where σ wall = const is a wall tension (wall mass per unit area) related to the expectation value v of the scalar field experiencing symmetry breaking, σ wall ∝ v 3 .Consequently, one gets β = 2, which does not obey (31), unless |q| is chosen to be unnaturally large.As it follows from the discussion after Eq. ( 32), this source effectively acts as instantaneous one and the GW spectrum is given by Eq. ( 27) with the value q = −1 fixed by numerical simulations [28].It is worth stressing that despite the low frequency tail ∝ f 3 is disfavoured by the PTA data, the broken power law (27) with a sufficiently smooth behaviour around the peak frequency fits well observations [6] given present error bars and modelling uncertainties.This motivates building particle physics scenarios, which involve domain walls, cf.Refs.[40].
Topological defects can also be realized in scale-invariant models [21,22,41], in which case their tension decreases with cosmic expansion rather than set by constant mass parameters, as in the examples above.In particular, melting domain walls mentioned in the introduction are characterized by the tension σ wall ∝ T 3 (τ ) [20,21,22].In this case, the energy density of domain wall network behaves as ρ s ∼ σ wall H ∝ 1/a 5 (τ ) in the scaling regime, so that β = 5 and Eq. ( 3) gives p = 2, which is in agreement with PTA data [23].Notably also, the energy density ρ s (τ ) decreases fast enough relative to radiation, so that the infamous domain wall problem is avoided in this scenario.
Finally, we would like to mention the case of melting cosmic strings [41] characterized by the time-dependent tension (string mass per unit length) µ ∝ T 2 (τ ).The energy density of the melting string network evolves with time according to ρ s ∼ µH 2 ∝ 1/a 6 (τ ) corresponding to β = 6.The latter does not fulfil the condition (31); the caveat, however, is that transient GW emission does not lead to the IR tail ∝ f 3 in the case of cosmic strings, at least for the choice of loop production function assumed in Ref. [41].That is, Eq. ( 3) is still applicable and one has Ω gw ∝ f 4 .This reiterates the statement that the constraint (31) should be regarded as a conservative one, and its violation does not immediately invalidate the correspondence (3).

Conclusions
We have shown that the GW spectrum created by the enduring with the energy density evolving as ρ s ∝ 1/a β has a power law behaviour described by the spectral index p = 2β − 8 in a certain range of constants β indicated in Eq. ( 31) (with typical values −3 ≲ q ≲ −1) and frequencies (26), provided that the condition ( 14) is fulfilled.Soft breaking of the condition ( 14) is expected to induce small corrections into Eq.(3).Furthermore, departures from Eq. ( 3) are inevitable due to dependence on the initial and latest emission times, τ i and τ f , of the integration limits in Eq. (19).While such departures are small within assumptions made, they can be important in view of future precision measurements of stochastic GW background.Furthermore, throughout the note we neglected the change of relativistic degrees of freedom g * (T ) during GW emission.This change of g * (T ) is also expected to impact the spectral shape (22), which can be particularly prominent at the times of QCD phase transition [42].
Note that the assumption ( 16) underlying the functional form ( 14) is perhaps the most restrictive one.In the future, it would be interesting to find situations, where it holds, besides topological defects in the scaling regime.In particular, long lasting effects triggered by first order phase transitions deserve a special attention.These effects are known to yield the GW spectrum Ω gw ∝ f in a particular range of frequencies [29,30,43,44,45].This may suggest time evolution of the corresponding source ρ s ∝ 1/a 9/2 (τ ), if the assumptions ( 16) and (17) are approximately valid in this case.
Finally, we wish to note that the present discussion motivated mainly by recent findings of PTA collaborations, may be also useful in view of forthcoming GW searches with LISA [46] and Einstein Telescope [47].In this regard, it would be interesting to generalize results of this work to an arbitrary equation of state of the Universe w.While we focused on radiation domination with w = 1/3, LISA and Einstein Telescope can be sensitive to earlier evolution stages for relatively low reheating temperatures.