Imprints of cosmic strings on the cosmological gravitational wave background

The equation which governs the temporal evolution of a gravitational wave (GW) in curved space-time can be treated as the Schrodinger equation for a particle moving in the presence of an effective potential. When GWs propagate in an expanding Universe with constant effective potential, there is a critical value (k_c) of the comoving wave-number which discriminates the metric perturbations into oscillating (k>k_c) and non-oscillating (k<k_c) modes. As a consequence, if the non-oscillatory modes are outside the horizon they do not freeze out. The effective potential is reduced to a non-vanishing constant in a cosmological model which is driven by a two-component fluid, consisting of radiation (dominant) and cosmic strings (sub-dominant). It is known that the cosmological evolution gradually results in the scaling of a cosmic-string network and, therefore, after some time (\Dl \ta) the Universe becomes radiation-dominated. The evolution of the non-oscillatory GW modes during \Dl \ta (while they were outside the horizon), results in the distortion of the GW power spectrum from what it is anticipated in a pure radiation-model, at present-time frequencies in the range 10^{-16} Hz<f<10^5 Hz.


I. INTRODUCTION
The so-called cosmological gravitational waves (CGW) represent small-scale perturbations to the Universe metric tensor [1]. Since gravity is the weakest of the four known forces, these metric corrections decouple from the rest of the Universe at very early times, presumably at the Planck epoch [2]. Their subsequent propagation is governed by the space-time curvature, encapsulating in the field equations the inherent coupling between relic GWs and the Universe matter-content; the latter being responsible for the background gravitational field [3].
In this context, we consider the coupling between CGWs and cosmic strings. They are one-dimensional objects that can be formed as linear defects at a symmetrybreaking phase transition [4], [5]. If they exist, they may help us to explain some of the large-scale structures seen in the Universe today, such as gravitational lenses [6]. They may also serve as seeds for density perturbations [7], [8], as well as potential sources of relic gravitational radiation [9].
In the present article we explore another possibility: A fluid of cosmic strings could be responsible for the constancy of the effective potential in the equation which drives the temporal evolution of a CGW in an expanding Universe. As we find out, a constant effective potential leads to a critical comoving wave-number (k c ), which discriminates the metric fluctuations into oscillating modes (k > k c ) and non-oscillatory (k < k c ) ones. As long as the latter lie outside the horizon, they do not freeze out, resulting in the departure of the inflationary GW powerspectrum from scale-invariance. This would be the case, if there is a short period after inflation where the cosmological fluid is made out of radiation and a sub-dominant component of cosmic strings. As regards the space-time geometry itself, the spatially flat Friedman -Robertson -Walker (FRW) model appears to interpret adequately both the observational data related to the known thermal history of the Universe and the theoretical approach to cosmic string configurations [4]. Consequently, we will assume our cosmological background to be a spatially flat FRW model. This Paper is organized as follows: In Section II we summarize the theory of CGWs in curved space-time. In Section III we demonstrate that, in a radiation model contaminated by a fraction of cosmic strings, the effective potential in the equation which governs the temporal evolution of a CGW in curved space-time is constant. In Section IV we explore the characteristics of a potential contribution of cosmic strings to the evolution of the Universe and in Section V we study the propagation of the non-oscillatory GW modes during this stage. We find that, if the Universe evolution includes a radiationplus-strings stage, then, although it could last only for a short period of time, its presence would lead to a distortion of the stochastic GW background from what it is anticipated in a pure radiation model, at present-time frequencies in the range 10 −16 Hz < f 10 5 Hz.

II. GRAVITATIONAL WAVES IN CURVED SPACE-TIME
The far-field propagation of a weak CGW (|h µν | ≪ 1) in a curved, non-vacuum space-time is determined by the differential equations [10] under the gauge choice which brings the linearized Einstein equations into the form (1). In Eqs (1) and (2), Greek indices refer to the four-dimensional space-time, R αµνβ is the Riemann curvature tensor of the background metric, h µ µ is the trace of h µν and the semicolon denotes covariant derivative.
In the system of units where c = 1, a linearly-polarized, plane GW propagating in a spatially flat FRW cosmological model, is defined as [9] where τ is the conformal-time coordinate, Latin indices refer to the three-dimensional spatial section and δ ij is the Kronecker symbol. The dimensionless scale factor R(τ ) is a solution to the Friedmann equation (where, the prime denotes differentiation with respect to τ and G is Newton's constant), with matter-content in the form of a perfect fluid, T µν = diag(ρ, −p, −p, −p), which obeys the conservation law and the equation of state where ρ(τ ) and p(τ ) represent the mass-density and the pressure, respectively. The linear equation of state (6) covers most of the matter-components considered to drive the evolution of the Universe [11] - [13], such as quantum vacuum (m = 0), a network of domain walls (m = 1), a gas of cosmic strings (m = 2), dust (m = 3), radiation (m = 4) and Zel'dovich ultra-stiff matter (m = 6). For each component, the continuity equation (5) yields where M m is an integration constant, associated to the initial mass-density of the m−th component. Provided that the various components do not interact with each other, a mixture of them obeys [11] where, now, Eq (5) holds for each matter-constituent separately.
In the case of an one component fluid, the Friedman equation (4) reads and, for every type of matter-content other than cosmic strings (m = 2), it results in where, the time-parameter η is linearly related to the corresponding conformal one, by η = m−2 2 τ and we have set η m = ( 8πG 3 M m ) −1/2 . Notice that, for m = 0 (De Sitter inflation) and 0 < τ < ∞, we obtain −∞ < η < 0.
The general solution to Eq (1) in the curved space-time (3) is a linear superposition of plane-wave modes where h k (τ ) is the time-dependent part of the mode denoted by k and ε ij is the polarization tensor, depending only on the direction of the comoving wave-vector k j . Accordingly, for a fixed wave-number k 2 = k 2 j , the timedependent part of the corresponding GW mode satisfies the second-order differential equation [14], [15] Eq (12) can be treated as the Schrödinger equation for a particle moving in the presence of the effective potential and, in a cosmological model of the form (10), is written in the form yielding where, now, a prime denotes derivative with respect to η, c 1 and c 2 are arbitrary constants to be determined by the initial conditions and k m = 2 m−2 k, so that k m η = kτ . Finally, H (1) |ν| and H (2) |ν| are the Hankel functions of the first and the second kind, of order [16] Therefore, different types of matter-content (reflecting different periods in the evolution of the Universe) admit different Hankel functions (see also [17]).

III. CONSTANCY OF THE EFFECTIVE POTENTIAL
A. Implications on CGWs' propagation A case of particular interest, involved in the timeevolution of a primordial GW, is when the effective potential (13) is constant for every τ , namely where M is a non-negative constant of dimensions L −4 . In this case, Eq (12) is written in the form where is the (constant) frequency of the wave. According to Eq (19), a critical value of the comoving wave-number arises, through the condition This critical value discriminates the primordial GWs in modes with k > k c , which oscillate for every τ , and modes with k < k c , which grow exponentially for every τ , (the exponentially decaying solutions are neglected).

B. Cosmological models of constant effective potential
Now, the question arises on whether there exists a spatially flat FRW cosmological model in which the effective potential is constant. To answer this question, we set Upon consideration of Eqs (4) and (23), Eq (17) results in the ordinary differential equation which admits the solution where C is an arbitrary integration constant of dimensions L −4 . In comparison to Eqs (7) and (8), we distinguish the following cases: (i) M = 0 and C = 0: This case corresponds to vacuum and flat space-time and it will not be considered further.
(ii) M = 0 and C = 0: This choice results in the radiation-dominated Universe where the critical wave-number vanishes (k c = 0).
(iii) M = 0 and C = 0: Hence, which corresponds to a string-dominated Universe [18]. It is worth noting that the constant M appearing in the effective potential (17) is, in fact, the initial mass-density of the strings, M 2 [clf Eq (7)]. A string-dominated Universe does not seem likely [19], [20] and, therefore, this case is of no particular interest.
(iv) Finally, if both C and M differ from zero, then, the function ρ(τ ) consists of two parts: One evolving as R −4 and the other as R −2 . By analogy to Eq (8), this type of matter-content can be met in a cosmological model filled with relativistic particles (radiation) and a fluid of cosmic strings, without interacting with each other, as it should be the case shortly after the dynamic friction between them [4] became unimportant [5]. Therefore, in this case, Once again, the constant M , appearing in the effective potential, is associated to the initial amount of strings in the mixture. It appears that, whenever the effective potential acquires a non-zero constant value, this value always involves the initial density of a cosmic-string gas. We conclude that, in the presence of cosmic strings the effective potential is reduced to a non-vanishing constant and, therefore, oscillation of the metric perturbations is possible only if their comoving wave-number is larger than a critical value, depending on the mass-density of the linear defects In other words, a cosmic-string network discriminates the primordial GWs predicted by inflation into oscillating and non-oscillating modes, something that should be reflected in the power-spectrum of the stochastic GW background. We shall attempt to illustrate how, in a realistic setting.

IV. A UNIVERSE WITH COSMIC STRINGS
The presence of cosmic strings in a unified gauge theory is purely a question of topology. The simplest SO(10) model, for example, predicts strings [21]. Many superstring-inspired models also result in the formation of linear topological defects [22], [23]. Cosmic strings are formed at a symmetry-breaking phase transition, within the radiation-dominated epoch where τ cr is the time at which the Universe acquires the critical temperature below which the strings are formed and we have normalized R(τ cr ) to unity. In particular, after inflation (and reheating) the Universe enters in an early-radiation epoch [24], during which the background temperature drops monotonically (T ∼ R −1 ). For τ ≥ τ cr , this cooling process results in the breaking of a fundamental U(1) local gauge symmetry, leading to the formation of linear defects (for a detailed analysis see [4] and/or [5]).
By the time the cosmic strings are formed, they are moving in a very dense environment and, hence, their motion is heavily damped due to string-particle scattering [25] - [28]. This friction becomes subdominant to expansion damping at [25] where, µ is the mass per unit length of the linear defect. For τ ≥ τ * , the motion of long cosmic strings can be considered essentially independent of anything else in the Universe and soon they acquire relativistic velocities. Therefore, we may consider that, after τ * the evolution of the Universe is driven by a two-component fluid, consisting of relativistic particles (dominant) and cosmic strings (sub-dominant). Consequently, Eq (28) holds and τ * marks the beginning of a radiation-plus-strings stage. During this stage, the Friedman equation (4) yields Nevertheless, the scale factor (32) can drive the Universe expansion only for a short period of time after τ * , since cosmic strings should (at any time) be a small proportion of the Universe energy-content. This means that the equation of state considered in (28) should have validity only for a limited time-period, otherwise cosmic strings would eventually dominate the overall energydensity [18].
In fact, a radiation-plus-strings stage (if ever existed) does not last very long. Numerical simulations [29] - [32] suggest that, after the friction becomes unimportant, the production of loops smaller than the Hubble radius gradually results in the scaling of the long-string network. Accordingly, the linear defects form a self-similar configuration, the density of which, eventually, behaves as R −4 [4]. In this way, apart from small statistical fluctuations, at some time τ sc > τ * the Universe re-enters in the (late) radiation era before it can become string-dominated. The duration (∆τ = τ sc −τ * ) of the radiation-plus-strings stage is quite uncertain, mostly due to the fact that numerical simulations can be run for relatively limited times. For example, the longest run of [32] suggests that τ sc ≃ 4.24 τ * (corresponding to a factor of 18 in terms of the physical time), while [31] raise this value to τ sc ≃ 6.48 τ * (t sc ≃ 42 t * ).
In what follows, we explore the evolution of GW modes with k < k c through the radiation-plus-strings stage.

V. CGWS IN THE PRESENCE OF COSMIC STRINGS
A. Evolution of modes outside the horizon CGWs are produced by quantum fluctuations during inflation (e.g. see [33]). Some of them escape from the visible Universe, once their reduced physical wavelength [λ ph = λ 2π R(τ )] becomes larger than the (constant) inflationary horizon [ℓ H = H −1 dS , H dS being the Hubble parameter of the de Sitter space]. Eventually, every CGW with k ≤ k max = H dS R dS is exiled from the Hubble sphere and freezes out, acquiring the constant amplitude [34], [35] where m P l = G −1/2 is the Planck mass. After inflation, i.e. within the subsequent radiation epoch, analytic solutions for α k (τ ) can be expressed in terms of the Bessel function J 1 2 (kτ ) [clf Eq (15) for c 1 = c 2 ]. Accordingly, when kτ ≪ 1, the perturbation's amplitude evolves slowly and is approximately constant. Once kτ ≈ 1, the amplitude decays away rapidly before entering in an oscillatory phase with slowly decreasing amplitude, when kτ ≫ 1. Physically, this corresponds to a mode that is (almost) frozen beyond the horizon, until its physical wavelength becomes comparable to the Hubble radius, at which point it enters in the visible Universe (e.g. see [36]). In other words, as the Universe expands, a fraction of the modes that lie beyond the horizon re-enters inside the Hubble sphere. At the time of re-entry their amplitude is given by Eq (34), while, afterwards, they begin oscillating. The k-dependence of their amplitude implies a scale-invariant power-spectrum [37].
However, if the cosmological evolution includes a radiation-plus-strings stage, then, during this stage, the effective potential is a non-vanishing constant. In other words, k c = 0 and the equation which governs the temporal evolution of the GW modes with k < k c does not admit the solution (35), but Eq (22). As a consequence, even if they lie outside the horizon, these modes do not freeze out.
At the beginning of the radiation-plus-strings stage, the GW modes that fit inside the visible Universe obey the condition while, modes of k < k * = τ −1 * lie outside the horizon. In order to examine whether k c ≷ k * we need to determine the initial mass-density of the linear defects, since, by definition, Let us consider a network of cosmic strings characterized by a correlation length ξ(τ ). This may be defined as the length such that the mass within a typical volume ξ 3 , is µξ [4]. In this case, at τ = τ * , the cosmic strings contribute to the Universe matter-content a mean density where γ * is a numerical constant of the order of unity, representing the number of correlation lengths inside the horizon at τ * . Accordingly, and hence For GUT-scale strings we have (Gµ) ∼ 10 −6 and γ * ≃ 7 (e.g. see [4]), so that k c ≃ 2 × 10 −2 k * ≪ k * . In other words, at the beginning of the radiation-plus-strings stage, the GW modes of k < k c do not fit inside the horizon.
On the other hand, for τ * < τ ≤ τ sc , the condition of fitting inside the Hubble sphere is written in the form Since the hyperbolic cotangent on the rhs is larger than unity for every τ , Eq (41) suggests that the GW modes of comoving wave-numbers k < k c remain outside the horizon during the whole radiation-plus-strings stage.
Nevertheless, by virtue of Eq (22), for τ * < τ ≤ τ sc their amplitude continues to evolve as (22) and (34)]. This behavior ends at τ sc , when the scaling of the long-string network is completed and the Universe re-enters in the (late) radiation era. For τ > τ sc the GW modes of k < k c are no longer influenced by the radiation-plus-strings stage and therefore, just like the rest of the metric perturbations outside the horizon, (re)freeze out. As a consequence, their amplitude acquires the constant value

B. The distorted power-spectrum
Within the late-radiation era these modes remain frozen until the time τ c . At that time the mode k c enters inside the visible Universe, since its physical wavelength (λ c ph ∼ τ c ) becomes smaller than the corresponding Hubble radius (ℓ H ∼ τ 2 c ). In accordance, for τ > τ c , GW modes of k < k c also enter inside the Hubble sphere. After entering inside the horizon, the GW modes under consideration begin oscillating, thus producing a part of the power-spectrum we observe today (or at some time in the future). However, since they have experienced the influence of the radiation-plus-strings (rps) stage, their amplitude is no longer given by Eq (34), but by Eq (43), thus resulting in the distortion of the GW powerspectrum (P 2 k ∼ k 3 α 2 k ), from what it is anticipated by pure-radiation (rad), at comoving wave-numbers k < k c . Namely, which, upon consideration of Eq (32), is written in the form where, we have set 0 < k kc = x = f fc < 1 and f is the frequency attributed to the GW mode denoted by k. According to Eq (40), coth(k c τ * ) ≃ 5 and Eq (45) results in P rps k<kc P rad Clearly, for ∆τ = 0 (i.e. in the absence of the radiationplus-strings stage) P rps k<kc = P rad k<kc , while, for ∆τ = 0 the inflationary-GW power-spectrum is no longer scaleinvariant.
The spectral function Ω gw , appropriate to describe the intensity of a stochastic GW background [38], is related to the power-spectrum as Ω gw ∼ P 2 k (e.g. see [39]). Therefore, Eq (46) yields Notice that, for every 0 ≤ x ≤ 1, we have Ω rps gw ≤ Ω rad gw , with the equality being valid only for ∆τ = 0. In other words, the involvement of a radiation-plus-strings stage in the evolution of the Universe reduces the stochastic GW intensity to lower levels than those expected by pure radiation. To give some numbers, we take into account the numerical results of [31], as well as those of [32]. Accordingly, a reasonable estimate on the duration of radiation-plus-strings stage would be τ sc = 5.5 τ * and therefore, k c ∆τ ≃ 9 × 10 −2 . In this case, Eq (47) is written in the form from which it becomes evident that, for f < f c , the value of Ω gw is no longer 8 × 10 −14 , as it is predicted by pure radiation [2], [9], but rather Such a distortion reflects a change in the distribution of the GW energy-density among the various frequency intervals, probably due to the coupling between metric perturbations and cosmic strings. The question that arises now is, whether these results are observable by the detectors currently available. To answer this question, we should determine explicitly both f c (the critical frequency) and t c (the physical time at which the GW modes of f < f c begin entering inside the horizon). In what follows, c = 1.
During the early-radiation epoch, the physical time is defined as GUT-scale strings, t cr ∼ 10 −31 sec [5] and therefore f pr c ≃ 1.5 × 10 5 Hz. Clearly, this value is far outside of the range where both the ground-based and the spacebased laser interferometers may operate. A GW of this frequency could be detected only by a system of coupled super-conducting microwave cavities [40], [41].
However, one should have in mind that, this is only the upper bound of the distorted GW power-spectrum. In fact, if cosmic strings contribute to the evolution of the Universe, the GW power-spectrum will decline from what it is anticipated by pure radiation at every presenttime frequency in the range 10 −16 Hz < f f pr c (see Fig.  1). The lower bound of this range arises from the GWs that began entering inside the horizon after the Universe has become matter-dominated [39].
On the other hand, for electro-weak-scale strings, t cr ∼ 10 −11 sec [5] and hence, f pr c ≃ 1.5 × 10 −5 Hz, while, for cosmic strings created at some time in between the GUT-and the electro-weak-symmetry breaking (e.g. t cr ∼ 10 −21 sec), we obtain f pr c ≃ 1.5 Hz. Therefore, a potential detection of CGWs, among other things, would give us valuable information on the epoch (and therefore on the physical mechanism, as well) at which the cosmic strings were formed.

VI. CONCLUSIONS
The equation which governs the temporal evolution of a CGW in a Friedman Universe can be treated as the Schrödinger equation for a particle moving in the presence of the effective potential V ef f = R ′′ /R. In the present article we show that, if there is a period where the effective potential is constant, this would lead to a critical value (k c ) in the comoving wave-number of the metric fluctuations, discriminating them into oscillating (k > k c ) and non-oscillating (k < k c ) modes. As a consequence, when the non-oscillatory modes lie outside the horizon do not freeze out, something that should be reflected in the inflationary-GW power-spectrum.
This property is met in a radiation model contaminated by a fraction of cosmic strings. Therefore, if the cosmological evolution includes a radiation-plus-strings stage, some of the long-wavelength GW modes (although being outside the Hubble sphere) continue to evolve.
However, this stage (if ever existed) does not last very long, since, the cosmological evolution gradually results in the scaling of the cosmic-string network and, after some time (∆τ ), the Universe enters in the late-radiation era.
In a radiation-dominated Universe the metric perturbations of k < k c can enter the horizon, which now expands faster than their physical wavelength. However, the evolution of the non-oscillatory GW modes during ∆τ (while they were outside the horizon) has modified their amplitude and, therefore, oscillation of these modes within the Hubble sphere, results in the distortion of the scale-invariant GW power-spectrum at present-time frequencies in the range 10 −16 Hz < f 10 5 Hz.

Note added in proof
The day after this article was accepted for publication, it came to our attention a recent paper [42] dealing with the scaling of a cosmic-string network in an updated numerical fashion. According to it, the duration of a potential radiation-plus-strings stage in terms of the conformal time (the dynamical range -as it is referred to) is τ sc = 17 τ * (corresponding to a factor of 300 in the physical time). Adaption of this result would lead to a more evident distortion of the inflationary GW spectrum, modifying Eqs (48) and (49) to Ω rps gw (f < f c ) Ω rad gw (f < f c ) ≃ 0.14 × e 0.64 and Ω rps gw ≃ 0.2 Ω rad gw ≃ 1.6 × 10 −14 respectively. The authors would like to thank Professor Mairi Sakellariadou for pointing that out.