Imprint of early dark energy in stochastic gravitational wave background

Gravitational waves (GWs) search technologies [1–5] have strong ongoing research interest. Implications from these developments focus not only on discovering new astrophysical objects [6, 7], but also probing gravitation physics [8–13] and non-standard cosmologies [14–21]; and GW observations has been used to measure the current Hubble rate H0 [22–25]. Although not yet sufficiently precise to resolve the Hubble tension, estimates will be continuously improved. Increasingly attention is focusing on possible solutions of Hubble tension, a Hubble rate discrepancy between local observations, such as supernovae signals [26–31], lensing time delays [32–38] and non-local searches, e.g. the cosmic microwave background (CMB) [39–42]. Local measurements confirm H0 is approximately five sigma statistical significance above observation in the CMB with assuming standard cosmological model (ΛCDM). This strong disagreement has been closely examined using many statistical [43–46] and measurement methods [31, 36, 47–50], and re-examining potential technical issues [51–56] in both local and CMB measurements (see reviews [48, 57–59]). Various evidence suggests the discrepancy arises due to a currently unknown physical phenomenon outside conventional ΛCDM predictions. V. Poulin et al. [60–62] showed that early dark energy (EDE) behaves as a cosmological constant when comoving scaling factor a(t) is smaller than critical ac ∼ 10−4, and is then diluted faster or equal to radiationlike component to relieve the Hubble tension. EDE contributes up to 20% energy density of the universe (modeldependent) at ac, consequently accelerating the universe expansion and hence slightly delaying the universe entering the matter-domination era. This framework brings H0 in estimated CMB to be consistent with local measurements and also with measured high and low redshifts [63–66]. We adopted the simplest EDE model, i.e., slow-rolling potential V (φ) ∝ φ with scalar field φ [67–69]. Effective mass for φ is lighter than the Hubble rate on the early universe, and hence Hubble friction overdamps scalar field motion and freezes it. Consequently, the scalar field behaves as a subdominant cosmological constant until the Hubble rate decreases to approximately scalar effective mass, namely, the driving force overcomes the Hubble friction. Subsequently, the field starts oscillating as a fluid with equation of state wφ = (n− 1)/(n+ 1). Cosmic strings, one dimension long-lived topological defects, are stable and predictable sources for stochastic GW background (SGWB) and hence ideal creator sources for GW as a messenger that carries new signal from the early universe [70–72]. Such stable objects arise from beyond standard model theories, such as spontaneously broken U(1) [73–76] or superstring theories [77– 80]. The GW frequency spectrum formed by the cosmic string network is an approximate plateau over a broad frequency range with parameter Gμ dependence, where G is the Newtonian gravitational constant and μ is string tension. GW experiments EPTA [81] and PPTA [82] provide strong bounds on Gμ . 2× 10−11 [17], but this remains in tension with the NANOGrav 12.5yr result [83] Gμ ∈ (2, 30) × 10−11 in 95% C.L. [84, 85]. The tension may be due to the different noise analysis methods [86]. This paper shows that EDE imprints a GW signal in the SGWB formed from the cosmic string network that is distinguishable from other astrophysical and cosmological signals in the GW frequency spectrum. Such a unique spectrum could be detected in future GW experiments LISA [87–89] and SKA [90]. We also contour the SKA and LISA sensitivities on EDE parameter space.