Whispers from the dark side: Confronting light new physics with NANOGrav data
MMITP/20-056
Whispers from the dark side: Confronting light newphysics with NANOGrav data
Wolfram Ratzinger and Pedro Schwaller
PRISMA + Cluster of Excellence & Mainz Institute for Theoretical Physics,Johannes Gutenberg-Universit¨at Mainz, 55099 Mainz, GermanyE-mail: [email protected] , [email protected] Abstract:
The NANOGrav collaboration has recently observed first evidence of a grav-itational wave background (GWB) in pulsar timing data. Here we explore the possibilitythat this GWB is due to new physics, and show that the signal can be well fit also withpeaked spectra like the ones expected from phase transitions (PTs) or from the dynamicsof axion like particles (ALPs) in the early universe. We find that a good fit to the datais obtained for a very strong PT at temperatures around 1 MeV to 10 MeV. For the ALPexplanation the best fit is obtained for a decay constant of F ≈ × GeV and anaxion mass of 2 × − eV. We also illustrate the ability of PTAs to constrain the pa-rameter space of these models, and obtain limits which are already comparable to othercosmological bounds. a r X i v : . [ a s t r o - ph . C O ] S e p Introduction
With the first direct observation of gravitational waves (GWs) by LIGO [1], a new era inastrophysics and cosmology has started. Since GWs travel almost undisturbed throughspacetime, they can carry information from before the time of CMB emission, which iswhere our direct observations using electromagnetic radiation end. GWs therefore open anew window to the early Universe.Pulsar Timing Arrays such as EPTA [2], PPTA [3] and NANOGrav [4] are sensitive toGWs with frequencies of 10 − Hz and below. A stochastic background of such low frequencyGWs could be produced in the early universe by a variety of processes, such as inflation,cosmic strings, phase transitions, or scalar field dynamics [5]. The most recent data releaseof the NANOGrav collaboration [6] for the first time shows evidence for such a stochasticGW background, which is well described by a f − / power law spectrum with a GW strainamplitude of 2 × − , or equivalently a GW energy density Ω GW h of order 10 − . This isindeed consistent with the GW density one expects from a variety of cosmological sources,as was discussed for the case of cosmic strings [7–9], phase transitions [10, 11], or primordialblack hole formation [12, 13].So far these studies have focussed on demonstrating that a sufficiently large GW densitycan be achieved in these models in the required frequency range. Here we perform the firstfit to the frequency binned NANOGrav data. Since most cosmological sources of GWs havespecific spectral features, it is important to verify that indeed they agree well with the data.In doing this, we are able to obtain best fit parameter regions for two classes of modelsthat produce primordial GWs, namely phase transitions in the early universe [14–18] andaudible axions [19–21]. We also show that the NANOGrav data already puts constraintson the parameter space of these models, which are comparable to the ones coming fromother astrophysical observations such as big bang nucleosynthesis (BBN) or the constrainton the number of relativistic degrees of freedom, N eff .With more precise data it will become possible to distinguish between different cosmo-logical sources and from the expected background due to supermassive black hole binaries.Our work presents a first step in this direction. It is organised as follows: In the next section,we describe our effort at recasting the NANOGrav data, and re-derive the best fit regionsfor single power law fits. The following two sections introduce the parameterisation of thestochastic GW background produced by audible axions and phase transitions, respectively,and the best fit regions for the model parameters, before we present our conclusions. II Refitting the NANOGrav data
The magnitude of a stochastic GW background is typically described by the dimensionless,frequency dependent characteristic strain amplitude h c ( f ). For a single power law it canbe written as h c ( f ) = A GW (cid:18) ff y (cid:19) α , (II.1)– 2 – .0 1.5 1.0 0.5 0.010 A G W B Figure 1 . Comparison of 1 σ and 2 σ contours for a single power law fit to the 5 lowest frequencybins. Our results are shown with continuous blue lines and the original result with orange dots.The black dotted line sits at α = − /
3, the expected slope for the signal of SMBHs. where A GW is the amplitude, α is the slope and f y = 1 / year is a reference frequency atwhich the amplitude is fixed. An important related quantity is the energy density in GWsas a fraction of the critical energy density, Ω GW , which is given by [4]Ω GW ( f ) h = 2 π H f h c ( f ) , (II.2)where H = 100 km/s/Mpc and H = h H is the Hubble rate today with h ≈ . α = 3 /
2, which is corroboratedby the broken power law fit. Instead the 5 lowest frequency bins contribute 99.98% of thesignificance of the potential GW signal.In the following, we will therefore fit our signal models to the 5 lowest frequency bins,assuming that the remaining data points are explained by white noise. The results ofthe free spectrum fit are given in terms of the timing residual, which is related to thecharacteristic strain as residual( f ) = 14 π f y (cid:18) ff y (cid:19) − / h c ( f ) , (II.3)in units of seconds. Note that we have chosen the prefactor in this formula such thatby fitting a single power law to the data, we can reproduce the best fit contours of [6],see Fig. 1. In the following sections, we will fit this data with signal templates motivatedby concrete new physics scenarios. – 3 – II Audible axions and NANOGrav
The audible axion is a simplified model where an axion-like particle a couples to a darkphoton X through a term of the form L ⊃ − q F aX µν ˜ X µν , (III.1)where F is the axion decay constant, i.e. the scale where the global symmetry in the UVis broken and gives rise to the light pseudoscalar a , q is a dimensionless charge, and X µν and ˜ X µν are the dark photon field strength tensor and its dual. The axion has a potential V ( a ) = m a F (1 − cos( a/F )), such that its mass is given by m a .As usual in the axion misalignment mechanism, we assume that after the end of infla-tion, the axion is displaced from the minimum of V ( a ) by θF , with θ an order one angle.The axion remains displaced until the Hubble rate becomes of order m a , at which pointit starts to oscillate around the origin. It was shown in [22] that the presence of a darkphoton leads to a suppression of the axion dark matter abundance, making larger values of F consistent with observations. An efficient energy transfer to the dark photons is possibledue to a tachyonic instability that develops while the axion rolls. The same process alsoamplifies quantum fluctuations in the dark photon field, which grow to macroscopic scalesand source a detectable GW background [19].The GW spectrum produced by audible axions is peaked at the frequency correspond-ing to the dark photon momentum mode that grows the fastest, and is closely related tothe axions mass m a . In terms of the model parameters, the peak frequency, redshifted totoday, can be estimated as f peak0 ≈ . × − Hz (cid:18) qθ (cid:19) (cid:16) m a − meV (cid:17) . (III.2)The amplitude of the GW signal is determined by the strength of the source, i.e. the energythat is initially carried by the axion. This is mostly influenced by the size of the decayconstant F . The peak amplitude of the signal can be estimated asΩ h ≈ . × − (cid:18) Fm pl (cid:19) (cid:18) θ q (cid:19) / . (III.3)To perform our fits we use the signal shape provided in [20]Ω ( f ) h = Ω h . (cid:16) f / (2 f peak0 ) (cid:17) / (cid:16) f / (2 f peak0 ) (cid:17) / exp (cid:104) . (cid:0) f / (2 f peak0 ) − (cid:1)(cid:105) . (III.4)In Fig. 2 we show on the left the best fit of an audible axion compared to the five firstfrequency bins from NANOGrav. On the right we show the one and two sigma contoursin the F - m a plane with θ = 1 and q = 50 fixed. To get such a strong signal the energyin the axion that is transmitted to the dark photon has to be quite significant. The darkphoton is a form of dark radiation and therefore contributes to the number of relativistic– 4 – Frequency / Hz10 G W h Runaway PTNon-Runaway PTAudible Axion 10 m a / eV10 F / G e V F > m pl N eff Figure 2 . Left: Signal of the best fits of a runaway and a non-runaway phase transition as well asan audible axion compared to the first frequency bins of NANOGrav in the frequency-Ω GW h plane.Right: 1 σ and 2 σ regions in the F - m a plane parameterizing the audible axion. The horizontal linesindicate the bounds originating from the decay constant F having to be smaller than the Planckmass m pl and from the dark photon relic density not violating the bounds on N eff . degrees of freedom N eff . From Fig. 2 it becomes clear that this excludes approximately halfof the parameter space in the best fit region. That is, if there are no further mechanismsto reduce the energy in the dark photon.Values of F and m a which lie above the green contours predict a GW signal which istoo large, i.e. this region is excluded by the NANOGrav data. While the N eff is slightlystronger, it is worth noting that PTAs are already able to put competitive bounds on thisscenario. IV Phase transitions and NANOGrav
It has been known for many years that a cosmological phase transition (PT), such as fromthe spontaneous breaking of a global or gauge symmetry through a scalar field that ac-quires a vacuum expectation value, produces a stochastic GW background if the transitionis strongly first order [14–16]. While a large variety of models exists that predict such atransition at different scales, the GW signal of a strong first order PT is universally de-scribed by only four parameters, the ratio between the vacuum and total energy density α = ρ vac /ρ tot , the time scale of the transition β/H , where H is the Hubble scale at the timeof the transition, the temperature T ∗ at which the transition takes place and the bubblewall velocity v w [17, 23].We use the signal templates in terms of these parameters as given in [24]. The peakfrequencies and amplitudes of the two most important contributions to the signal scale as f p ≈ × − Hz (cid:18) βH (cid:19) (cid:18) T ∗ GeV (cid:19) , (IV.1)Ω GW h ≈ − v w (cid:18) βH (cid:19) − n (cid:18) α α (cid:19) , (IV.2)– 5 – T * / GeV10 / H BB N T * / GeV0.20.40.60.81.0 BB N / H = 100/ H = 10 Figure 3 . Left: Regions favoured by the NANOGrav signal for a vacuum PT, with v w = 1,shown as a function of the transition temperature T ∗ and the PT timescale β/H . Right: Same fora strong first order PT in a plasma, with v w = 1 and fixed values of β/H , as function of T ∗ andthe energy budget α . The vertical line at one MeV indicates the onset of BBN, below which strongconstraints apply to any models that alter the expansion rate of the Universe. where n = 1 for the sound wave contribution and n = 2 for the scalar field contribution, andwe neglect order one numbers which are not relevant for the qualitative discussion. Verystrong transitions are characterised by α > . v w →
1. The NANOGrav signal corresponds to an energy density Ω GW h > − at a frequency around 10 − Hz, so that only a strong transition will be able to explain thedata. Furthermore we immediately see that T ∗ should be of order 10 − − − GeV, i.e.the PT should happen at a very low scale. The implications of this for concrete modelswill be discussed in more detail below.We consider two scenarios. If the PT takes place at a temperature significantly belowthe critical temperature, the Universe will be dominated by vacuum energy, i.e. the α dependence drops out of Eq. (IV.2). In such a supercooled PT, no friction acts on thebubble wall, so that v w = 1. Furthermore in the absence of a plasma, the only source ofGWs is the scalar field itself, i.e. n = 2 in Eq. (IV.2). In that case, a good fit to thedata requires relatively small values of β/H (cid:46)
50, and transition temperatures aroundor below the MeV scale, as shown in Fig. 3. Above the peak frequency, the GW strainamplitude of the PT signal falls as f − / . Therefore if the peak frequency lies below thelowest frequency probed by NANOGrav, the signal will look like a single power law to thedetector. This explains the flat direction in the fit towards lower temperatures and lowervalues of β/H . However lower values of β/H are increasingly difficult to obtain in realisticmodels, therefore this region should be considered less favoured.If the PT is very strong but not supercooled, the bubble walls will still reach a rela-tivistic terminal velocity, so for simplicity we again set v w = 1. In this case sound wavesin the plasma induced by the PT are the dominant source of GWs, and the amplitude isonly suppressed by one power of β/H . As expected, in Fig. 3 we see that a good fit to thedata in the T ∗ − α plane is found both for β/H = 10 and β/H = 100, where in the second– 6 –ase the suppression of the signal is compensated by a larger energy budget α . Again wealso find a flat direction, where the peak of the PT signal is shifted below the NANOGravfrequency range, and data is fit by the high frequency tail.In both scenarios, we find that the PT should happen at a temperature around 1 MeV,with only a small viable region slightly above 10 MeV. Since extensions of the SM at suchlow scales are almost impossible to hide from laboratory experiments, it is clear that the PTshould take place in a dark sector, with only very weak interactions with the SM [24–31].Nevertheless it was shown in [24] that also PTs in a dark sector are subject to strongconstraints, in particular if they happen close to the scale of BBN. The reason is that BBNis a sensitive probe of the Hubble scale at temperatures below the MeV scale, which inturn depends on the total energy density in the Universe, since gravity is universal. Eitherthe energy density in the hidden sector should be transferred to the SM before the onset ofBBN at T ∼ N eff .Viable models should therefore have few degrees of freedom, and still feature a verystrong first order PT. The simplest scenario is probably a single scalar field with a non-renormalizable potential, such as a very light radion or dilaton. Indeed for these modelsit is known that a strongly supercooled first order PT can occur and produce a largeGW background [32–36]. For renormalizable scenarios, the most minimal models thatwere found in [24] consist of either two real singlet scalars or a U (1) gauge boson with acomplex scalar charged under the gauge symmetry. While the majority of the parameterspace of these models features a weaker PT, there are benchmark points with α > . β/H (cid:46) N eff .Finally also here it should be noted that PTs with T ∗ ∼ V Discussion and Outlook
The first hint of a GWB observed by NANOGrav is very intriguing. While the data canbe well explained with a single power law, consistent with the expected background fromsupermassive black hole binaries (SMBHBs), we show here that also broken power lawspectra, which are predicted in various extensions of the SM, can well describe the signal.In both new physics scenarios we considered, the peak of the GW signal is stronglycorrelated with the relevant mass scale of the new physics, either the axion mass or themass scale of the new sector that undergoes a phase transition. The PTA data thereforealready allows us to narrowly constrain the potential mass range.Since the data suggests very light new physics, it is already clear that these newparticles have to be part of a dark sector that is only very weakly coupled to the SM,– 7 –therwise laboratory experiments would have uncovered them already. Yet astrophysicaldata on BBN and N eff constrain the parameter space of such dark sectors.For the audible axion scenario, we find parameter regions consistent with N eff formasses around 10 − eV and a decay constant of 5 × GeV. This region may be probedin the future by the CASPEr-wind experiment [41], and also by future black hole binarymerger data through the superradiance effect [42].A first order PT can explain the data if the transition is very strong and happens attemperatures between 1-10 MeV, or slightly below, if BBN and N eff constraints can beevaded. We have briefly illustrated some dark sector models that are known to satisfyall requirements. Here it will of course be interesting to ask whether concrete realisationscan also explain the observed dark matter abundance, and whether they leave observableimprints elsewhere.Already this first hint of a stochastic GW background in the PTA range providesus with a deep insight into possible new physics explanations of the signal. With moreprecise frequency binned data it will be possible to distinguish between different modelsand astrophysical backgrounds such as the one from SMBHBs. It would also be interestingto directly fit a broader range of GW templates to the pulsar timing data, possibly includingpolarised signals such as the one expected from audible axions. Exciting times lie ahead! Acknowledgments
We would like to thank Moritz Breitbach, Michael Gellar, Joachim Kopp, Eric Madge, LisaMichaels, Toby Opferkuch, Daniel Schmitt and Ben Stefanek for useful discussions. Ourwork is supported by the Deutsche Forschungsgemeinschaft (DFG), Project ID 438947057.We also acknowledge support by the Cluster of Excellence “Precision Physics, Funda-mental Interactions, and Structure of Matter” (PRISMA+ EXC 2118/1) funded by theGerman Research Foundation (DFG) within the German Excellence Strategy (Project ID39083149).
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