Exploring Primordial Curvature Perturbation on Small Scales with the Lensing Effect of Fast Radio Bursts

Cosmological observations, e.g., cosmic microwave background, have precisely measured the spectrum of primordial curvature perturbation on larger scales, but smaller scales are still poorly constrained. Since primordial black holes (PBHs) could form in the very early Universe through the gravitational collapse of primordial density perturbations, constraints on the PBH could encode much information on primordial fluctuations. In this work, we first derive a simple formula for the lensing effect to apply PBH constraints with the monochromatic mass distribution to an extended mass distribution. Then, we investigate the latest fast radio burst observations with this relationship to constrain two kinds of primordial curvature perturbation models on small scales. This suggests that, from the null search result of lensed fast radio bursts in currently available observations, the amplitude of primordial curvature perturbation should be less than 8 × 10−2 at the scale region of 105–106 Mpc−1. This corresponds to an interesting mass range relating to binary black holes detected by LIGO–Virgo–KAGRA and future Einstein Telescope or Cosmic Explorer.


INTRODUCTION
The power spectrum of primordial curvature perturbations on large scales has been precisely constrained by a variety of observations.For instance, cosmic microwave background (CMB) and large scale structure (LSS) observations suggest that the amplitude of primordial curvature perturbation should be orders of magnitude larger than O(10 −9 ) at O(10 −4 − 10 0 ) Mpc −1 scales (Planck Collaboration, 2020a).However, most current available cosmological observations are only able to constrain primordial fluctuations at larger scales.Therefore, new probes are in great request to constrain primordial perturbations at smaller scales.Moreover, primordial black hole (PBH) have been a field of great astrophysical interest because they are often considered to make up a part of dark matter.PBH could form in the early Universe through the gravitational collapse of primordial density perturbations (Hawking 1971; Carr zxli918@bnu.edu.cnzhuzh@whu.edu.cn& Hawking 1974;Carr 1975), its formation is closely related to the primordial power spectrum (Sasaki et al. 2018;Green & Kavanagh 2021).Therefore, there are many inflation models, e.g., inflation model with modified gravity (Pi et al. 2018;Fu et al. 2019), multi-field inflation model (Clesse & García-Bellido 2015;Chen & Cai 2019), special single-field inflation model (Cai et al. 2020;Motohashi et al. 2020), to enhance the amplitude of power spectrum of primordial curvature perturbations on small scales which corresponds to PBHs in various mass windows.
Theoretically, the mass of PBHs can range from the Planck mass (10 −5 g) to the level of the supermassive black hole in the center of the galaxy.So far, numerous methods, including both direct observational constraints and indirect ones, have been proposed to constrain the abundance of PBHs in various mass windows (Sasaki et al. 2018;Green & Kavanagh 2021).Gravitational lensing effect is one of direct observational probes to constrain the abundance of PBH over a wide mass range from O(10 −10 M ⊙ ) to O(10 10 M ⊙ ).In general, we can divide the method of lensing effect into four types (Liao et al. 2022): 1. Searching the luminosity variation of persistent sources (MACHO Collaboration, 2001;Griest et al. 2013;Niikura et al. 2019a,b;EROS-2 Collaboration, 2007;Zumalacarregui & Seljak 2018), for example, observing a large number of stars and looking for amplifications in their brightness caused by lensing effect of intervening massive objects could yield constraints on the abundance of deflectors (MACHO Collaboration, 2001;EROS-2 Collaboration, 2007;Griest et al. 2013;Niikura et al. 2019a,b); 2. Searching multiple peaks structures of transient sources (Muñoz et al. 2016;Laha 2020;Liao et al. 2020b;Zhou et al. 2022a,b;Oguri et al. 2022;Krochek & Kovetz 2022;Connor & Ravi 2023;Blaes & Webster 1992;Nemiroff et al. 2001;Ji et al. 2018;Lin et al. 2022), such as searching echoes due to the milli-lensing effect of fast radio bursts (FRBs) were proposed to put constraints on the PBH abundance (Muñoz et al. 2016;Laha 2020;Liao et al. 2020b;Zhou et al. 2022a,b;Oguri et al. 2022;Krochek & Kovetz 2022;Connor & Ravi 2023;Kalita et al. 2023); 3. Searching multiple images produced by milli-lensing of possible persistent sources like the compact radio sources (CRSs) can be used to constrain the supermassive PBH (Press & Gunn 1973;Kassiola et al. 1991;Wilkinson et al. 2001;Zhou et al. 2022c;Casadio et al. 2021); 4. Searching the waveform distortion caused by the lensing effect of distant sources (Jung & Shin 2019;Liao et al. 2020a;Urrutia & Vaskonen 2021;Wang et al. 2021;Basak et al. 2022;Zhou et al. 2022d; LIGO Scientific and VIRGO and KAGRA Collaborations, 2023;Urrutia et al. 2023;CHIME/FRB Collaboration, 2022;Leung et al. 2022;Barnacka et al. 2012), for example, distorting the GW waveform as the fringes were proposed to constrain PBH with stellar mass (Jung & Shin 2019;Liao et al. 2020a;Urrutia & Vaskonen 2021;Wang et al. 2021;Basak et al. 2022;Zhou et al. 2022d; LIGO Scientific and VIRGO and KAGRA Collaborations, 2023;Urrutia et al. 2023).In this work, based on a relationship for applying constraints with the monochromatic mass distribution (MMD) to a specific extended mass distribution (EMD), we proposed to use the lensing effect of fast radio bursts to study the primordial curvature perturbations on small scales which have not been achieved by other observations.This paper is organized as follows: Firstly, we introduce formation of PBHs from the primordial curvature perturbation model in Section 2,.In Section 3, we carefully analyzed constraints on PBHs from the lensing effect.In Section 4, we present the results of constraints on power spectrum.Finally, we present discussion in Section 5. Throughout, we use the concordance ΛCDM cosmology with the best-fit parameters from the recent Planck observations (Planck Collaboration, 2020b).

FORMATION OF PRIMORDIAL BLACK HOLES
The power spectrum of primordial curvature perturbations determines the probability of PBH production, the mass function of PBHs, and the PBH abundance (Sasaki et al. 2018;Green & Kavanagh 2021).The phenomena of critical collapse could describe the formation of PBHs with mass m PBH in the early Universe, depending on the horizon mass m H and the amplitude of density fluctuations δ (Carr et al. 2016): where K = 3.3, γ = 0.36, and δ th = 0.41 (Harada et al. 2013), and the horizon mass m H is related to the horizon scale k (Nakama et al. 2017) where g * is the number of relativistic degrees of freedom.
The coarse-grained density perturbation is given by where W (q/k) is the Gaussian window function, P ζ (q, p mf ) is the power spectrum of primordial curvature perturbation, and T (q, k) is the transfer function (Young et al. 2014;Ando et al. 2018) where x = q √ 3k .To convert σ 2 (k) to the mass function of PBHs, we calculate the probability of PBH production by considering the Press-Schechter formalism (Press & Schechter 1974) where P mH (m H ) denotes a Gaussian probability distribution of primordial density perturbations at the given horizon scale, The PBH energy fraction is calculated from Eq. ( 5) as where M eq = 2.8 × 10 17 M ⊙ is the horizon mass at the time of matter-radiation equality (Nakama et al. 2017).
In addition, the mass function of PBHs ψ(m PBH , p mf ) can be obtained by differentiating Ω PBH with the PBH mass where p mf represents the parameters from the power spectrum of primordial curvature perturbation P ζ (q, p mf ).The corresponding total PBH abundance is defined as where Ω DM is dark matter density parameter at present universe (Planck Collaboration, 2020a).In order to distinguish f PBH from the observational constraints, we label the f PBH obtained from the power spectrum of primordial curvature perturbation is written as f PBH,th in Eq. ( 9).

CONSTRAINTS ON F PBH FROM THE LENSING EFFECT
For a lensing system, Einstein radius is one of the characteristic parameters and, taking the intervening lens with mass m as a point mass, it is given by where D S , D L and D LS represent the angular diameter distance to the source, to the lens, and between the source and the lens, respectively.The lensing cross section due to a PBH lens is given by an annulus between the maximum and minimum impact parameters (y ≡ β/θ E , β stands for the source angular position), It is worth emphasizing that the maximum impact parameter y max and minimum impact parameter y min generally depends on the observing instruments or the nature of the lensing source.For example, the maximum impact parameter y max and minimum impact parameter y min of FRBs micro-lensing system are determined by the maximum flux ratio of two lensed peaks and the width of signals, respectively.For a single source, the optical depth for lensing due to a single PBH is where H(z L ) is the Hubble expansion rate at z L , H 0 is the Hubble constant, and n L (f PBH,obs , m) is the comoving number density of the PBHs with the monochromatic mass distribution (MMD) where ρ c is critical density of universe.Correspondingly, the f PBH obtained from the lensing effect is written as f PBH,obs in Eq. ( 13).According to the Poisson law, the probability for the null detection of lensed event is If we have detected a large number of astrophysical events N tot , and none of them has been lensed, the total probability of unlensed event would be given by If none lensed detection is consistent with the hypothesis that the universe is filled with the PBHs to a fraction f PBH,obs at 100Π% confidence level, the following condition must be valid For a null search of lensed signals, then the constraint on the upper limit of f PBH,obs can be estimated from Eq. ( 16).For the optical depth τ i ≪ 1, we can obtain the expected number of lensed events (17) It should be pointed out that the above formalism is only valid for the simple but widely used MMD, where δ(m PBH −m) represents the δ-function at the mass m.In fact, there is a specific EMD which corresponds to different the power spectrum of primordial curvature perturbation from different inflation models.Therefore, it is important and necessary to derive constraints on PBH with some theoretically motivated EMDs, which are closely related to realistic formation mechanisms of PBHs.For the above-mentioned EMDs, the lensing optical depth for a given event can be written as, where is the comoving number density of the PBHs at EMD ψ(m PBH , p mf ) Then we can derive a universal formula for connecting the constraints on the f PBH,obs for applying constraints with the MMD to EMD.Firstly, we must respectively note the f PBH,obs under MMD and EMD as f MMD PBH,obs and f EMD PBH,obs .In addition, we can obtain the optical depth of single lensing source in MMD and EMD framework and note them as (21) From the relationship of the optical depth of MMD and EMD (see Eq. ( 19) and Eq. ( 12)), we can obtain that In addition, we can obtain the upper limits of f max PBH,obs from Eq. ( 16) and Eq. ( 21) in the MMD and EMD framework as .
Finally, we can integrate Eq. ( 3) with the same mass distribution ψ(m, p mf ) over m to obtain that This relationship indicates that constraints on the f PBH,obs from lensing effect can be perfectly consistent with the formula for applying constraints with the MMD to specific EMD (Carr et al. 2017).Furthermore, the same relationship in Eq. ( 24) can be derived from Eq. ( 17).

RESULTS
In this section, we consider two kinds of power spectrum of primordial curvature perturbation P ζ (k, p mf ).The first case is a δ function of ln k, i.e.
where p mf,1 ≡ [A δ , k 0 ], A δ and k 0 are dimensionless amplitude and constant wave number, respectively.In Fig. 1, we show several examples for the mass function ψ(m, p mf,1 ) which correspond to the δ-function power spectrum of primordial curvature perturbation as Eq. ( 25).Specifically, we choose the constant wave number to be k 0 = [2, 3, 4, 5, 6] × 10 5 Mpc −1 with fixed dimensionless amplitude A δ = 5 × 10 −2 presented in the left panel of Fig. 1.Similarly, we show the mass function with different dimensionless amplitude A δ = [1, 3, 5, 7, 9] × 10 −2 and fixed constant wave number k 0 = 5 × 10 5 Mpc −1 presented in the right panel of Fig. 1.The second case is a nearly scale invariant shape of the form where p mf,2 ≡ [A ns , n s ], A ns and n s are dimensionless amplitude and spectral tilt, respectively.In addition, We take k min and k max as 10 5 Mpc −1 and 10 6 Mpc −1 , which approximately correspond to PBH mass in the range of 10 M ⊙ to 10 3 M ⊙ .In Fig. 2, we show several examples for the mass function ψ(m, p mf,2 ) which correspond to the nearly scale invariant power spectrum of primordial curvature perturbation as Eq. ( 26).The same as the first case, we choose the spectral tilt n s = 0.8 In order to discuss the constraints on the power spectrum of primordial curvature perturbation from the lensing effect, we take FRBs as an example.At present, we use 593 publicly available FRBs compiled by zhou et al. 2022 works (Zhou et al. 2022b).These sources consist of more than five hundred FRB events from 2018 July 25 to 2019 July 11 (CHIME/FRB Collaboration, 2021).The distance and redshift of a detected FRB can be approximately estimated from its observed dispersion measure (DM), which is proportional to the number density of free electron along the line of sight and is usually decomposed into the following four ingredients, where DM host and DM src represent DM from host galaxy and local environment, respectively.We adopt the minimum inference of redshift for all host galaxies, which corresponds to the maximum value of DM host + DM src to be 200 pc/cm 3 .DM MW is the contribution from the Milky Way.In addition, DM IGM represents DM contribution from intergalactic medium (IGM).The DM IGM −z relation is given by (Deng & Zhang 2014) and it is approximately expressed as DM IGM ∼ 855z pc/cm 3 by considering the fraction f IGM of baryon in the IGM to f IGM = 0.83 and the He ionization history (Zhang 2018).DM and redshift measurements for several localized FRBs suggested that this relation is statistically favored by observations (Li et al. 2020).We present the inferred redshifts of 593 available FRBs in the left panel of Fig. 3.For milli-lensing of FRBs, the critical value R f,max and the width (w) of the observed signal determine the maximum and minimum value of impact parameter in the cross section.To ensure that both signals are detectable, the maximum value of impact parameter y max can be obtained by requiring that the flux ratio of two lensed images is smaller than a critical value R f,max , (2020b), we take R f,max = 5 for cases when we study lensing of the whole sample of all currently public FRBs.
In addition, the minimum value of impact parameter y min can be obtained from the time delay between lensed signals and pulse widths of all FRBs are presented in the left panel of Fig. 3.After determining the the maximum and minimum value of impact parameter in the optical depth from Eqs. (28-29), we can combined 593 FRBs and Eq. ( 15) to obtain the upper limit of f PBH,obs at 100Π% confidence level.In the right panel of Fig. 3, we demonstrate the constraints on f PBH,obs when the MMD is considered.In the ≳ 10 3 M ⊙ large-mass end, the constraint on f PBH saturates to 9.8 × 10 −2 at 68% confidence level.Then, From the relationship in Eq. ( 24), we derive the upper limit on f PBH,obs corresponding to above two primordial curvature perturbation P ζ (k, p mf ) models, and the results are shown in the left panel of Figs.(4-5).For the first primordial curvature perturbation, we vary the constant wave number k 0 from 10 5 Mpc −1 to 10 6 Mpc −1 , which roughly correspond to PBHs with ≲ 10 3 M ⊙ .And, the dimensionless amplitude A δ is greater than 0.01.The regions where 20% and 50% of f PBH,obs have been marked the red solid lines in the left panel of Fig. 4. In addition, it should be that the white regions in the left panel of Fig. 4 represent that f PBH,obs is more than 1.In this model, less k 0 and larger A δ corresponds to the smaller peak of the mass distribution function, which leads to improve the constraints on the f PBH,obs .In the right panel of Fig. 4, the parameter space p mf,1 ≡ [A δ , k 0 ] can be allowed to exist in the brown area within the red solid line.There are two conditions under which a parameter space can be allowed to exist: The first condition means that the density parameter of PBH from theory can not be larger than the one of dark matter.The second condition means that the theoretical prediction of PBH abundance should be lower than the upper limit of observational constraints.We find that the amplitude of primordial curvature perturbation is less than 6 × 10 −2 at the scale region of k 0 ≥ 2 × 10 5 Mpc −1 .For the second case, we assume that the value of n s varies from 0.5 to 1.5 and A ns is greater than 0.01.The regions where 15% and 20% of dark matter can consist of PBHs are denoted by red solid line in the left panel of Fig. 5.In this model, less n s corresponds to the lagrer peak of the mass distribution function.In addition, larger A ns corresponds to the broader mass distribution.The quantitative competition between the two above-mentioned effects of broadening the mass distribution, hence the constraints on the f PBH,th is more complicated.Based on the above conditions of Eq. ( 30), we present the allowed parameter space p mf,2 ≡ [A ns , n s ] in the brown area within the red solid line at the right panel of Fig. 5.We find that the amplitude of primordial curvature perturbation is less than 5 × 10 −2 at the scale invariant range (n s ∼ 1).

DISCUSSION
Although CMB and LSS observations have yielded strict bounds on the primordial curvature perturbation at Mpc scale and higher, effective constraints on primordial fluctuations on the small scales are still rare.Fortunately, the gravitational lensing effects, such as echoes of transient sources, can be used as powerful probes to constrain PBHs.Therefore, we proposed using lensing effect to constrain the primordial curvature perturbation on small scales.In this paper, we first derive the relationship that connects constraints of f PBH,obs from the MMD to EMD for all lensing effect.Then, taking FRB as an example, we propose that its lensing effect can be used to exploring the primordial curvature perturbation.By combining 593 FRB samples (Zhou et al. 2022b) and two kinds of primordial curvature perturbation models, we present the constraints on f PBH,obs and the allowed regions of parameter space of primordial curvature perturbations in Figs.(4-5).In general, null search result of lensed FRB in the latest 593 events would constrain the amplitude of primordial curvature perturbation to be less than 8 × 10 −2 at the scale region of 10 5 − 10 6 Mpc −1 .Moreover, there are two significant aspects in our analysis: • When comparing the abundance of PBHs calculated by any theoretical model with the observational constraints, we should transform the results from MMD to the EMD under the corresponding theoretical framework.For observational constraints from the lensing effect, we can use Eqs.( 24) to translate MMD and EMD results.
• Since the primordial power spectrum determines the mass distribution (ψ(m, p mf )) and theoretical abundance of PBHs (f PBH,th ), it suggests that the primordial curvature perturbation parameters p mf are degenerate with the abundance of PBHs f PBH theoretically.Therefore, if there is a tension between the predicted range of f PBH,obs and f PBH,th from the future lensing signals, we should consider the following possible reasons: 1.Whether the primordial perturbation model is correct; 2. Whether there are other compact dark matter, such as axion mini-clusters (Hardy 2017) and compact mini halos (Ricotti 2009), participating in the observation process; 3. Whether PBHs exist evolutionary processes, such as accretion (Ricotti 2007) and halo structure (Delos & Franciolini 2023), to change the theoretical f PBH,th or observed physical processes.
There are several factors contributing to the uncertainties in our analysis.For example, the values for δ th depends on the profile of perturbations, the threshold value of the comoving density could vary from 0.2 to 0.6 (Musco & Miller 2013;Harada et al. 2013;Yoo et al. 2018;Musco et al. 2021).Moreover, non-Gaussian due to the nonlinear relationship between curvature and density perturbations would lead to the amplitude of the power spectrum of primordial curvature perturbation P ζ (q, p mf ) might be a factor of O(2) larger than if we assumed a linear relationship between ζ and δ (Gow et al. 2021;De Luca et al. 2019;Young et al. 2019).Finally, our analysis are based on the Press-Schechter theory.It should be noted that the statistical methods, e.g., Press-Schechter or peaks theory, would slightly affect the results (Gow et al. 2021).It is foreseen that these constraints will be of great importance for exploring PBHs with their formation mechanisms relating to the physics of the early universe.

Figure 3 .Figure 4 .Figure 5 .
Figure 3. Left:Two-dimensional distribution of inferred redshifts and widths for the latest 593 FRBs.Right: Constraints on the upper limits of fraction of dark matter in the form of PBHs with the MMD from the fact that no lensed signal has been found in 593 FRBs data.