## Introduction

In recent years, asteroseismology—the analysis of stellar oscillations—has revolutionised our understanding of galactic and stellar evolution and has become one of the most powerful tools to unravel the interior structures of different types of stars from the core-hydrogen burning stage to the final phases of evolution (e.g. white dwarfs). Since oscillation modes propagate throughout the stellar interior, their frequencies contain information about the inner structure of the stars. Thanks to phenomenal data obtained from space telescopes such as MOST1, CoRoT2, BRITE-Constellation3, Kepler4 and TESS5, the investigation of the oscillation properties of stars in a mass range from about 0.5 to 40 M with effective temperatures between about 3000 and 100,000 K allowed to test and improve our current theories of stellar structure and evolution6. Multiple studies have shown that the uncertainty of theoretical frequencies is nowadays higher than the uncertainty on observationally inferred pulsation frequencies, given that the time-base of observations is long enough.

Asteroseismology of stars before the onset of core-hydrogen burning, i.e. pre-main sequence stars, is a comparably young research field that has advanced due to dedicated observations7 and adaptations of theoretical model calculations8. However, our general theory of early stellar evolution lacks essential physical ingredients that are either not well understood or not accurately included in our theoretical models. As the earliest phases in the lives of stars determine their future fate and the creation and evolution of planetary systems, an improved understanding of these stages is essential.

The often adopted classical picture of early stellar evolution assumes a fully convective star with a large initial radius. In this case, a two solar mass star has an initial radius of 55 solar radii when creating a pre-main sequence model according to our standard input physics (see methods, subsection stellar evolution models) with the standard routine of MESA9, which subsequently contracts along the Hayashi track10. The contraction releases energy from the gravitational potential, heating the stellar interior and fueling the convection. With higher temperatures, the onset of thermonuclear reactions slows the contractions, the stellar interior becomes radiative and the subsequent evolution of the young star in the Hertzsprung-Russell diagram follows the Henyey track11. Further contraction then leads the stellar core to burn the initially present 12C into 14N via the first parts of the CNO cycle (12C(p, γ)13N(β+, ν)13C(p, γ)14N), which is a chain of nuclear reactions catalysed by carbon (C), nitrogen (N) and oxygen (O). The star further continues to contract towards the Zero Age Main Sequence (ZAMS) only after the initial 12C has been lowered to equilibrium abundances, leading to additional heating until the full CNO cycle can operate in equilibrium. While energy production of the CNO cycle dominates for stars discussed in this work, stars with masses below 1.3 solar masses obtain most of their energy from the proton-proton chain. This marks the onset of core-hydrogen burning, the start of the stars’ main sequence lifetime and, with it, the end of the pre-main sequence evolution. The classical concept of stellar evolution assumes further that at the time the stars arrive on the main sequence, their previous stages do not leave a traceable imprint: the stars forget the details of their early evolutionary history.

This simplified picture of early stellar evolution has been overhauled by multiple authors in the last decades12,13,14,15,16,17,18,19,20 but has so far not reached the field of asteroseismology. This includes the wrong but deep-rooted view that stars suddenly become optically visible once passing the stellar birthline16. In contrast to the classical pre-main sequence evolution models that simplified initial conditions, state-of-the-art calculations of these earliest phases in stellar evolution start from protostellar seeds with a few Jupiter masses and about one solar radius. The ZAMS mass is then reached through accretion throughout the first few million years. The evolutionary tracks of accreting protostars in the Hertzsprung-Russell diagram differ significantly dependent on the adopted accretion rates and model assumptions. However, even before the ZAMS is reached, spectroscopic observables are unable to differentiate between these models and classical models8.

Evolutionary calculations of accreting protostars in the literature usually fall in one of two classes. In the constant accretion scenario, the accretion rate is kept at a constant level before dropping exponentially to reach the final mass. Constant accretion rates are predicted in semi-analytic models of spherical cloud collapse, which neglect the formation of circumstellar disks21,22. An exponential decline occurs in the later phases because of a finite mass reservoir of collapsing prestellar clouds23. In the other scenario, more realistic accretion histories from hydrodynamical simulations that take the disk formation phase into account17,20,24,25 are taken to evolve the protostar. In these models, the matter is accreted on the star through a protostellar disk, which alters notably the mass accretion history depending on the physical mechanisms of mass and angular momentum transport that operate in the disk (see methods, subsection accretion histories from hydrodynamical simulations).

Asteroseismic studies allow constraining the physics inside stellar interiors6. Self-driven gravity modes in γ-Doradus stars, for example, allow the determination of the core boundary mixing profile and temperature gradient from asteroseismic modelling of period spacings26,27. A different type of pulsator is δ-Scuti stars, which are located within the classical instability strip28 with effective temperatures of 6500 K ≤ Teff ≤ 10,000 K, stellar luminosities in the range 0.6 $$\le \,\log \left(L/{L}_{\odot }\right)\,\le$$ 2 and surface gravities in the range 3.2 cm s$${}^{-2}\,\le \,\log \left(g\right)\,\le \,4.4$$ cm s−2 that exist in the main sequence29, pre-main sequence7,8,30, and post-main sequence phase29. Pre-main sequence δ-Scuti stars are expected with masses from 1.5 to 3.5 M7,8 while the upper limit for the mass of main sequence counterparts lies around 2.3 M30. δ-Scuti stars, independent of their respective evolutionary stage, pulsate in low order pressure modes driven by the heat engine mechanism, although turbulent pressure and especially time-dependent convection play a major role in the calculation of instability regions8,31,32.

Here, we investigate whether the self-driven pressure modes in δ-Scuti stars can provide similar constraints as γ-Doradus stars, but for the initial model applied in stellar structure and evolution calculations. Our results show that the process of star formation leaves behind an imprint on the stellar structure that is potentially detectable in pulsation frequencies of δ-Scuti stars. This opens a window to investigate early stellar evolution in young pulsating stars that could allow many insights potentially invaluable for related fields such as exoplanet studies.

## Results

### Stellar structure of early stellar evolution

In this work, we take a 2 M star with δ-Scuti pulsations as an example and calculate three different models of its pre-main sequence evolution using the software instrument MESA9 with the aim to explore whether the pulsation frequencies can distinguish between the models. The first model is a classical pre-main sequence model starting from the initial assumptions described above. In addition, we calculate models in the constant accretion scenario similar to the models calculated by Steindl et al.8. Finally, we calculate disk-mediated models using 34 of the accretion histories from Elbakyan et al.20, scaled to reach 2 M after the accretion has finished.

We use the resulting equilibrium models to perform a linear non-adiabatic pulsation analysis with the stellar oscillation code GYRE33 to obtain pulsation frequencies and growth e-folding times, the latter of which provides information on whether the pulsation mode is expected to be self-driven and, hence, observable.

Asteroseismic studies are inherently dependent on the frequencies that can be measured from observations. The expected uncertainties of frequencies obtained from time series photometry are a result of the duration of the light curve available. When studying theoretical prospects, it is common to use a conservative approach26 to estimate the value of observational uncertainties, Δf: the Rayleigh limit,

$${{\Delta }}f=1/T,$$
(1)

where T is the time span of the observation. To investigate whether asteroseismology can distinguish between the different models, we compare the resulting frequencies to the Rayleigh limit corresponding to state-of-the-art observations, motivated by the Kepler4 observations in the nominal mission (4 years) and TESS5 observations in the continuous viewing zone (357 days). This corresponds to Rayleigh limits of 0.000684 cycles per day (c/d) or about 0.0079 μHz for 4 years and 0.0028 c/d or about 0.032 μHz for 357 days according to Eq. 1.

The theoretical calculations performed in this work result in a theoretical sample of 2 M models that range from the pre-main sequence phases until almost the terminal age main sequence, where stars have reached the end of core-hydrogen burning. Figure 1 shows the immense difference in stellar structure between the classical pre-main sequence and the disk-mediated accretion model through a Kippenhahn diagram. The classical model ignites deuterium in the centre at the age of 10,000 years, shown by the first appearance of a yellow burning zone in Fig. 1. After this, the convection delivers new deuterium to the centre, where nuclear burning of deuterium continues until the central deuterium abundance is depleted (i.e. after 100,000 years). In the disk-mediated accretion model, it takes 50,000 years until deuterium is first ignited. The accretion energy added to the outer parts of the stellar model results in a temperature inversion8,19 that leads to off-centre ignition of deuterium burning. In contrast to the classical model, the position of deuterium burning inside the star is dependent on the accretion rates: the convective area retreats towards the surface during strong accretion bursts that are similar in magnitude to FU Orionis-type eruptions34. As the convective area retreats, deuterium added to the outer layers by accretion fails to reach the inner parts of the star. While the classical model keeps burning deuterium in the core, the disk-mediated accretion model has exhausted the central deuterium reservoir. Deuterium burning keeps producing high amounts of energy at the bottom of the convective envelope, the position of which is strongly dependent on the accretion rate. This continues until the star has reached the phase of pre-main sequence evolution, for which almost the whole star is radiative. Once, the star has reached its final mass of 2 M and has evolved onto the ZAMS, the Kippenhahn diagrams of the classical and the disk-mediated accretion model seem identical.

Figure 2 shows the asteroseismic properties of disk-mediated model #27 in terms of the Brunt-Väisälä frequency (also referred to as the buoyancy frequency) and the Lamb frequency for l = 1 as an example. These frequencies are important indicators of the oscillatory structure of a star. While it is impossible to perform a direct comparison between the disk-mediated and the classical pre-main sequence models (among other things, because the disk-mediated models have different masses), we can compare snapshots of the classical model at early stages with the same stellar radius as the disk-mediated models. During the time of highest accretion (panel c of Fig. 2), the internal structure of the disk-mediated model is significantly different to the corresponding snapshot of the classical model with the same radius. While the latter is almost entirely convective, the disk-mediated model shows radiative areas in which the Brunt-Väisälä frequency is non-zero. In addition, the Lamb frequency is shifted to lower frequencies and obtains extra features in the radiative regions. But even during a quiescent phase (panel d of Fig. 2), the differences are evident: While at that moment, the classical model has obtained a radiative core that provides a non-zero Brunt-Väisälä frequency, the internal structure of the disk-mediated model still shows some characteristic patterns. Towards the end of the accretion phase (panel e of Fig. 2), the characteristic frequencies look qualitatively similar. However, changes in the internal structure still lead to changes in both the Brunt-Väisälä frequency and the Lamb frequency, especially towards the centre of the star. In the last snapshot, directly after mass accretion has finished, the characteristic frequencies of both models become similar, but small deviations in the internal structure of the models still lead to visible differences in the Brunt-Väisälä frequency.

Including the effects of accretion also significantly alters the structure and the position of the evolutionary tracks and hence the spectroscopic observables of pre-main sequence stars, shown in a Hertzsprung-Russell diagram in Fig. 3 (a Kiel diagram is available as Supplementary Fig. 1). Neither the constant accretion model nor the disk-mediated accretion model reaches the high values of stellar luminosity that is a consequence of the large radius assumed for the classical initial model.

### The imprint of star formation on stellar pulsations

We calculated pulsation frequencies at the pre-main sequence and the zero-age main sequence with the aim to compare the pulsation spectra based on the different modelling approaches (see methods, subsection stellar evolution calculations and subsection stellar pulsation frequencies). Figure 4 presents the frequency differences in these two snapshots of the evolution. During the pre-main sequence phase, the frequency differences are well above the Rayleigh limits for both the Kepler and TESS datasets. Furthermore, there is also a clear difference between the constant accretion model and the disk-mediated model. As such, the pulsation frequencies of pre-main sequence δ-Scuti stars are not only sensitive to the difference between classical models and accreting models, but also to the applied accretion rate. Although the difference between the disk-mediated model and the classical model is less prominent at the zero-age main sequence, it still allows the distinction between the two models for all pulsation modes with the precision reached by data from the Kepler space mission and for a few modes with the observational precision achieved for TESS data. The spread between the constant accretion model and the disk-mediated model is larger compared to the model during the pre-main sequence phase, again pointing to the sensitivity of the pressure modes to the details of the accretion process during early stellar evolution.

The different mass accretion rates for the 34 disk-mediated accretion models (see Supplementary Fig. 2) lead to different evolutionary tracks and internal structures during the pre-main sequence phase, which manifests themselves as frequency differences (see Supplementary Figs. 336). Figure 5 and Supplementary Fig. 37 give an overview of the resulting changes in pulsation frequencies as a direct consequence of the accretion histories. The region in the Hertzsprung-Russell diagram occupied by the different evolutionary tracks is enormous compared to the classical pre-main sequence model or the constant accretion model.

These frequency differences essentially result from differences in the stellar radius. The radii of the disk-mediated accretion models are up to a few parts per thousand (ppt) smaller than that of the classical model at our pre-defined pre-main sequence stage. This stems from a slightly more massive inner region, as is presented in Fig. 6 and Supplementary Fig. 38. Pressure mode pulsations are sensitive enough to feel this size difference and translate them into frequency differences. Once the models are at our pre-defined stage of the ZAMS, the sizes are on average equal, but there is still a spread in both stellar radii and frequency difference comparable to the Rayleigh limit for TESS data. This spread of pulsation frequencies within the 34 different accretion models provides an additional uncertainty in theoretically calculated pressure mode pulsation frequencies for 2 M δ-Scuti stars only from the (unknown) accretion past.

Considering every excited mode for all 34 of the 2 M δ-Scuti star models with our standard input physics, we find that currently available space photometry provides sufficient precision to distinguish between classical and more realistic pre-main sequence models. During the pre-main sequence phase, the differences in pulsation frequencies are larger than Kepler precision for all excited modes in our models, while they are larger than TESS precision for about 73% of all excited modes (see Table 1).

The mixing of chemical elements in the radiative envelope plays an important role in stellar evolution, feeding the convective core with additional fuel and thus changing the evolutionary path27. To investigate the dependence of our findings on the choices of this input physics, we repeated our calculations with different values for the amount of envelope mixing and overshooting. The additional mixing in the stellar envelope results in a larger radius and frequency differences during the pre-main sequence stage of evolution (see Table 1, Fig. 7, Supplementary Figs. 39 and 40 and methods, subsection stellar evolution calculations) as compared to our standard input physics. As all excited modes show frequency differences larger than the precision yielded by Kepler and TESS observations, it is evident that the improved modelling of the early stellar evolution results in changes which are detectable by means of pulsations in pre-main sequence δ-Scuti stars. The calculations with more overshooting lead to smaller differences during the pre-main sequence and at the ZAMS (see Supplementary Figs. 41 and 42), but the frequency differences remain large enough to be observable with Kepler precision.

Pre-main sequence δ-Scuti stars become unstable to pulsation even before the evolutionary stage at which we have presented our results above8. To compare the different models at earlier evolutionary stages, we additionally calculated the frequency spectra of models with a fixed radius of 2.8 R, which the stars obtain at an effective temperature of around 7300 K. While this does not present an equal evolutionary stage, it serves as a guide to investigating differences in stellar structure. As expected, the frequency differences between the classical and the disk-mediated pre-main sequence models for some radial orders are even larger at this earlier evolutionary stage (see Table 1 and Fig. A39).

## Discussion

Our results are in strong contrast with the common belief that the physical details of early stellar evolution (i.e. star formation, mass accretion process, etc) have no influence on the stellar structure and suggest that the accretion history should be included in our asteroseismic interpretation of pre-main sequence δ-Scuti stars. Although the frequency differences are larger than the expected precision of state-of-the-art observations, we find that the effects described in this work are of lesser extent compared to other input parameters in stellar structure (stellar mass, metallicity, convective overshooting, etc.) and thus can only be applied once a sufficiently constrained model has been found already. As such, this manuscript provides a similar analysis to Michielsen et al.26, which constitutes the basis for many impressive recent results regarding the mixing profile of gravity mode pulsators27. In a similar fashion, this work demonstrates that forward asteroseismic modelling of pre-main sequence δ-Scuti stars should provide an opportunity to test the imprint of star formation on their pulsation frequencies (see methods, subsection stellar evolution calculations).

Zwintz et al.7 showed that multiple well-known pre-main sequence stars would be in the perfect evolutionary stage to launch an in-depth study and perform asteroseismic modelling with disk-mediated pre-main sequence models. Unfortunately, young open clusters have been avoided by the main Kepler mission, and the TESS continuous viewing zones point far off the galactic plane. The latter is the region of the sky where most young open clusters are located. Hence, we are currently unable to perform such a study due to the lack of suitable observations. Predictions from this work can only be verified observationally once long time series photometry from space of pre-main sequence δ-Scuti stars become available. We remain hopeful that the next prime mission for such observations, ESA’s PLATO mission, will provide observations that close this gap. Besides that, the ideal solution for this problem could be a dedicated small-scale space mission, providing long baseline observations of young open clusters. Once applicable observations are available, we propose analysis according to the statistical procedure of forwarding asteroseismic modelling described by Aerts et al.35, which predominantly relies on the observed frequency values. Classical constraints (such as Teff, $$\log (g)$$, $$\left[{{{{{{{\rm{M}}}}}}}}/{{{{{{{\rm{H}}}}}}}}\right]$$, and $$\log \left(L/{L}_{\odot }\right)$$) are of significant importance to limit the parameter space of initial calculations as well as to eliminate models a posteriori (see methods, subsection forward asteroseismic modelling of pre-main sequence δ-Scuti stars). Observational proof of the potentially detectable signatures of the imprint on star formation of stellar pulsations hence also relies on successful mode identification and accurate spectroscopic parameters (for example, ±150 K in Teff and ±0.1 dex in $$\log (g)$$.

The issue of unrealistic initial models is not unique to δ-Scuti stars but applies to all types of stars across the observable mass range. Hence, a similar investigation of, for example, higher mass stars that are expected to reach the main sequence long before ending their respective evolution as accreting objects will be important for progress, in particular for asteroseismology of β-Cephei or Slowly Pulsating B stars.

## Methods

### Accretion Histories from hydrodynamical simulations

The mass accretion rates were taken from Elbakyan et al.20 based on the numerical hydrodynamics model of Vorobyov & Basu25. This model computes the formation and long-term evolution of circumstellar disks starting from the gravitational collapse of rotating prestellar clouds, taking disk self-gravity, stellar irradiation heating, disk radiative cooling and turbulence into account. The effects of turbulence were considered using the α-model of Shakura & Sunyaev36 with a constant α-value set equal to 0.005. The use of the thin-disk limit allows computing the disk dynamical evolution for up to 1–2 Myr, which is not achievable with full 3D models. The mass accretion rates were computed as the mass of gas passing per unit time through the inner sink cell. The radius of the sink cell was set equal to 5 au to reduce the stringent requirements on the hydrodynamic timestep imposed by the Courant-Friedrichs-Levy condition. The accretion rates in the late disk evolution phase (after 1–2 Myr) were manually tapered off to zero to take the disk photoevaporation into account37. The model reproduces the observed slope of the mass accretion rate vs. stellar mass dependence38,39 in young star-forming regions and also can explain the luminosity problem for young low-mass stars40. We note that the accretion rates derived with the model of Vorobyov & Basu38,39 are qualitatively similar to those of Bae et al.41 and Vorobyov et al.42, who used a smaller sink cell and the layered model of magnetised disks of Armitage et al.43. The main feature of all these models is that the matter is transported from the prestellar cloud to the forming star through the circumstellar disk, unlike the spherical collapse models22 in which the disk phase is absent. As a result, the mass accretion rates are disk-mediated and can exhibit complex behaviour such as accretion bursts similar to those observed in FU Orionis and EX Lupi-type objects34.

For the initial conditions, Elbakyan et al.20 have chosen a span of initial cloud masses from 0.061 to 1.69 M and initial ratios of rotational to gravitational energy from 0.25 to 11.85%. These numbers lie within the observationally inferred values for prestellar clouds44. The resulting accretion histories lead to stellar masses between 0.0468 and 1.319 M. All mass accretion rates include an early part with high mass accretion rates (corresponding to the pre-disk stage), an abrupt drop to no accretion (manifesting the formation of a centrifugally balanced disk), and subsequent continuation of mass accretion rates that gradually decline with time while experiencing bursts up to 10−3M/yr in several models. The calculation of stellar structure models with these mass accretion rates sometimes experiences convergence problems in the first few thousand years. To withstand this issue, we use the beginning of #28 for every mass accretion history, the consequence of which is that the mass accretion rate until the first drop to zero is the same for every model calculated in this work. During the remaining accretion time, the accretion rate is scaled up to ensure that the final model obtains 2 M. Given the different final masses in the original mass accretion rates, this scaling factor is different for every calculation. Original mass accretion rates that lead to very low-mass stars are subject to larger scaling factors as a consequence. For the lowest final mass, model #17, the scaling results in mass accretion being too high at the early stages for the MESA model to converge and is hence omitted from this study. All adjusted mass accretion rates are shown in Supplementary Fig. 2. The scaling procedure assumes that the mass accretion rate histories for stars of 2 M do not differ qualitatively from their lower-mass counterparts, which we expect to be the case. An increase in the stellar mass results in a more stable disk against gravitational instability, but this effect can be counterbalanced by a higher mass infall rate on the disk24 in more massive initial clouds. Furthermore, similar but scaled-up mass accretion rates, including accretion bursts, were recently obtained in 3D numerical gravito-radiation hydrodynamics simulations of massive protostars45, implying a qualitative similarity in the mass accretion histories across the stellar mass spectrum.

### Stellar evolution models

Stellar evolution calculations in this work have been performed with the software instrument Modules for Experiments in Stellar Astrophysics, MESA9,46,47,48,49 version v-12778. MESA is a Henyey style code with a coupled solution for the structure and composition equations9. This work hence relies on the calculation of one-dimensional spherically symmetric stellar models for which we use the standard MESA equation of state9 and the OPAL opacity tables50. Our calculations of the equilibrium stellar model ignore rotation and magnetic fields. While stellar rotation is an important ingredient in stellar evolution and stellar pulsations, computational asteroseismology is currently not able to include these effects in the calculation of the equilibrium model since the physical descriptions are missing. Hence, we apply the standard approach in asteroseismic studies26,27 and apply the effect of stellar rotation on the pulsation frequencies during the pulsation analysis of the equilibrium model with GYRE.

The mixing of chemical elements by convection is treated in MESA as a diffusive process governed by a diffusion coefficient D. In convective areas, the diffusion coefficient Dconv,0 is calculated via the mixing length description of Cox & Giuli51 with a mixing length of αMLT = 2.2 and the Ledoux criterion as well as the description of convective premixing to find convective boundaries48. Beyond these boundaries, we employ exponential overshooting according to

$${D}_{{{{{{{{\rm{OV}}}}}}}}}=\, {D}_{{{{{{{{\rm{conv}}}}}}}},0}\exp \bigg(\frac{-2z}{f{H}_{p}}\bigg),$$
(2)

where z is the distance to the radiative layer and Hp is the pressure scale height. An additional parameter f0 describes the distance f0Hp from the convective boundary into the convective zone at which the switch to overshooting occurs. The parameters for overshooting are set to f = 0.01 and f0 = 0.005 at the top of any convective zone and to f = 0.005 and f0 = 0.0025 at the bottom. The minimum mixing coefficient throughout the star is set to $${D}_{\min }=1\,{{{{{{{{\rm{cm}}}}}}}}}^{2}{{{{{{{{\rm{s}}}}}}}}}^{-1}$$ for our standard input physics. We use a uniform initial composition of X = 0.734, Y = 0.252 and an initial metallicity of Z = 0.014. Relative abundances for the chemical elements beyond Helium are set according to the solar composition presented in Asplund et al.52, with additional updates to some key elements based on Nieva & Przybilla53 and Przybilla et al.54. This set-up is one of the pre-defined solar compositions in MESA. The updated abundances in the form $${{{{{{{\rm{eps}}}}}}}}\_{{{{{{{\rm{EL}}}}}}}}={\log }_{10}({{{{{{{\rm{EL/H}}}}}}}})+12$$, where EL is a placeholder for elements, are: eps_C = 8.33, eps_N = 7.79, eps_O = 8.76, eps_Ne = 8.09, eps_Mg = 7.56, eps_Al = 6.30, eps_Si = 7.50, eps_S = 7.14, eps_Ar = 6.50 and eps_Fe = 7.52. The composition of the models calculated in this work includes 20 parts per million (ppm) of 2H and 85 ppm of 3He. In terms of atmospheric boundary conditions, we found that the temperature-opacity (T-τ) relation by Eddington55 successfully recreates the instability region for pre-main sequence δ-Scuti stars8. Hence, we use the Eddington atmosphere for our calculations in this work.

We build our initial model by creating a pre-main sequence model with a mass of 0.04 M according to the standard MESA routing. After that, 0.03 M are removed by the standard mass-relax scheme. We then follow the contraction of the remaining 0.01 M protostellar seed until its radius reaches 1.5 R. This procedure is identical to the one used in previous studies18 and leads to initial models that represent an evolved second Larson core with high entropy value18,56.

To model the accretion onto a one-dimensional stellar model, we follow a description applied in the literature16 in the view of a non-spherical accretion process14,15 that allows the star to radiate its energy over most of the photosphere. The energy budget of accretion is governed by gravitational energy per unit of accreted mass − GM/R and the internal energy per unit mass of the accreted material + ϵGM/R, where G is the gravitational constant, M and R are the mass and radius of the accreting star and the value 0 ≤ ϵ ≤ 1 describes the geometry of accretion15,57. In this study, we use ϵ = 0.5, corresponding to accretion from a thin disk around the equator also applied in the previous studies15,19,57. The total energy budget is therefore

$$\frac{{{{{{{{\rm{d}}}}}}}}{E}_{{{{{{{{\rm{acc}}}}}}}}}}{{{{{{{{\rm{d}}}}}}}}t}=\,(\epsilon -1)\frac{GM\dot{M}}{R},$$
(3)

where $$\dot{M}$$ is the mass accretion rate. A second parameter, β, then controls how much of the net energy is absorbed by the stellar envelope $$\big({L}_{{{{{{{{\rm{add}}}}}}}}}\,=\beta \epsilon \frac{GM\dot{M}}{R}\big)$$ and how much is radiated away as accretion luminosity $$\big({L}_{{{{{{{{\rm{acc}}}}}}}}}=(1-\beta )\epsilon \frac{GM\dot{M}}{R}\big)$$. For β = 0, no energy is absorbed by the star and the process is referred to as cold accretion, while the case of 0 < β ≤ 1 is referred to as hot accretion. β is a free parameter that is expected to vary with the time-dependent-accretion rate17,19,20,56. We choose a constant value of β = 0.1 for the constant accretion scenario and functional dependence on the mass accretion rate previously applied in the literature19 for the disk-mediated accretion models. The latter is given by a step function with a smooth transition:

$$\beta \big(\dot{M}\big)=\frac{{\beta }_{{{{{{{{\rm{L}}}}}}}}}\exp \left(\frac{{\dot{M}}_{{{{{{{{\rm{m}}}}}}}}}}{{{\Delta }}}\right)+{\beta }_{{{{{{{{\rm{U}}}}}}}}}\exp \left(\frac{\dot{M}}{{{\Delta }}}\right)}{\exp \left(\frac{{\dot{M}}_{{{{{{{{\rm{m}}}}}}}}}}{{{\Delta }}}\right)+\exp \left(\frac{\dot{M}}{{{\Delta }}}\right)}.$$
(4)

Here, βL = 0.005 and βU = 0.2 are the lower and upper bound of β and $${\dot{M}}_{{{{{{{{\rm{m}}}}}}}}}=6.2\times {10}^{-6}{M}_{\odot }/{{{{{{{\rm{yr}}}}}}}}$$ and Δ = 5.95 × 10−6M/yr are the midpoint and the width of the crossover between the lower and upper limit.

The energy absorbed by the star is added as extra heat to the MESA models. The distribution of extra heat within the star is an additional free input choice for the model. Applied methodologies in the literature include uniform distribution57 which is most likely nonphysical, a step function in the outer layers of the star19, and a linear increase as a function of the mass coordinate18. We follow the last approach as we believe this to be the most physical of the three, such that we add the extra heat according to

$$l=\frac{{L}_{{{{{{{{\rm{add}}}}}}}}}}{M}\max \left\{0,\frac{2}{{M}_{{{{{{{{\rm{outer}}}}}}}}}^{2}}\left(\frac{{m}_{r}}{M}-1+{M}_{{{{{{{{\rm{outer}}}}}}}}}\right)\right\}.$$
(5)

Here, Mouter defines the fractional mass of an outer region in which the extra heat l is deposited and mr corresponds to the mass coordinate. We choose a value of Mouter = 0.01 for the disk-mediated accretion models, meaning that the accretion energy absorbed by the star is entirely deposited within the outer 1% of the stellar model. For the constant accretion model, we follow the approach of Steindl et al.8 and choose Mouter = 0.1. The composition of the accreted material is assumed to be invariable with time and corresponds to the initial composition of the star.

We save the stellar model at specific points of the evolution for the subsequent calculation of pulsation mode frequencies and analysis. These points are pre-defined by the central composition of carbon (pre-main sequence) and hydrogen (main sequence). The temperature inversion resulting from the accretion process makes a comparison between the accreting models and the classical model unreasonable during the early phases of evolution. Only after this temperature inversion has vanished, we can safely compare the two evolutionary calculations. We define one point on the pre-main sequence, where the central carbon mass fraction drops to a value of 10−4. This corresponds to the phase of the first CNO ignition: the stellar core is hot enough to burn carbon to nitrogen but is not yet able to follow through with the remainder of the CNO cycle. This is an evolutionary stage in which pre-main sequence δ-Scuti stars can be observed typically7,8,58. After the initial abundance of carbon is depleted, the stellar core again heats up until the full CNO cycle can operate in equilibrium - the main sequence phase of evolution. In the lack of a better definition of the ZAMS, we define the ZAMS where the central hydrogen abundance, Xc, has dropped by 0.01 compared to the initial value. At this point, however, the star is, in fact, already on the main sequence for a short period of time. In order to verify that the origin of the difference in stellar pulsation frequencies does not lie in slightly different evolutionary stages, we ensured that the difference in the hydrogen mass fraction is smaller than 2 × 10−6 and the difference in the carbon mass fraction is smaller than 2 × 10−9. Furthermore, we also checked that the change in pulsation frequency for one timestep is smaller than Kepler precision.

The individual models are presented at two specific evolutionary stages in the main text: (1) The pre-main sequence, where the central carbon mass fraction drops to a value of 10−4 due to the first onset of the CNO cycle. (2) The zero-age main sequence, for this purpose defined as the point when the central hydrogen mass fraction has dropped by 0.01 compared to the initial value: Xc = Xc,init − 0.01.

When calculating the stellar evolution model, we tried to minimise the influence of numerical errors as much as possible. This includes using higher than standard spatial and temporal resolution throughout the stellar model, where we also additionally increased the spatial resolution around specific regions such as the overshooting region. In addition, we performed various simple tests, to investigate the set-up with regard to numerical stability when applying small changes. As such, we can confirm that the results are stable against small changes in the applied accretion factor or the temporal and spatial resolution. Additionally, the set-up produces the same results when saving a MESA model when the accretion has finished and reloading it within an identical copy of the inlists we used to calculate the classical evolution model.

The main result of this work is that the pulsation frequencies of evolutionary calculations, including the mass accretion phase, are significantly different from the pulsation frequencies of the classical evolution model. We verified that this is the case independent of the choice of input physics for chemical mixing by changing one parameter at a time. We perform calculations with five times the envelope mixing ($${D}_{\min }=5$$ cm2s−1) as well as five times the convective overshooting (f = 0.05 and f0 = 0.025 at the top and with f = 0.025 and f0 = 0.0125 at the bottom). The corresponding results are presented in Table 1, Fig. 7 and Supplementary Figs. 39, 40, 41 and 42. The differences in pulsation frequencies between the accreting models and the classical models change, dependent on the input physics chosen, but the discrepancy remains. All different sets of models include pulsation modes with differences larger than the expected uncertainty for observed frequencies, which verifies the main result as stated above. We furthermore note that the frequency differences are larger for models calculated with a higher amount of envelope mixing. The latter is a promising result since the value of envelope mixing chosen in the standard input physics for this work represents very low values, compared to i.e. the SPB star KIC 8324482 for which asteroseismic modelling leads to an envelope mixing coefficient of $$\log {D}_{\min }=3.12{5}_{-0.250}^{+0.125}\,\log ({{{{{{{{\rm{cm}}}}}}}}}^{2}{{{{{{{{\rm{s}}}}}}}}}^{-1})$$59.

We additionally present results at an earlier evolutionary stage in the calculations where the stellar radius reaches 2.8 R after entering the instability region for pre-main sequence δ-Scuti stars8. Because the different evolutionary pathways lead to different stellar radii throughout the evolution, this does not present an independent measure of the evolutionary stage. Nevertheless, it allows us to present the differences between the classical and disk-mediated models further away from the ZAMS. For the calculations, similar to the case of central carbon or hydrogen abundance, we ensured that the difference in radius is smaller than 1 × 10−8R. Supplementary Figs. 43 and 44 present the resulting frequency differences for this case. The discrepancy between the classical and disk-mediated models is even larger compared to the stages described by the central carbon and hydrogen abundance. The larger imprint from the accretion process onto the stellar structure is expected, as less time has passed since the former occurred. The spread between the frequencies of the disk-mediated models themselves is also much larger, reaching more than 0.3 c/d for the fundamental dipole mode.

### Stellar pulsation frequencies

The equilibrium models resulting from the MESA calculations are used as input for the subsequent calculation of non-adiabatic theoretical pulsation frequencies with version 6.0.1 of the stellar oscillation code GYRE33,60,61. Pulsations of δ-Scuti stars are most commonly observed in radial and dipole modes62 as higher order modes suffer from cancellation effects29. In this work, we present results for dipole modes (harmonic degree l = 1, azimuthal order m = 1), but the results for radial modes are virtually identical. We calculate pulsation modes with radial orders 0 < n ≤ 50 in the frequency range 5 ≤ f ≤ 150 c/d. GYRE calculates the oscillation frequencies as solutions to the equations of linear stellar oscillations and assumes a time dependence proportional to $$\exp \left(-i\sigma t\right)$$, where t is the time, i is the imaginary unit and σ = σR + iσI the complex eigenfrequencies. Hence, the period of oscillation is given by $${{\Pi }}=\frac{2\pi }{{\sigma }_{{{{{{{{\rm{R}}}}}}}}}}$$ and the imaginary part of the eigenfrequency gives rise to the growth e-folding time of the pulsation amplitude $$\tau=\frac{1}{{\sigma }_{{{{{{{{\rm{I}}}}}}}}}}$$. For positive growth e-folding times, we expect the pulsation mode to be excited, while negative growth e-folding times correspond to stable modes. The effect of rotation is included by means of a first-order perturbative approach known as Ledoux splitting26,29. For the calculation of the perturbation, we assume uniform rotation at 20% of the critical rotation velocity. At this rotational velocity, the centrifugal acceleration and, hence, the flattening of the star can be ignored26. Furthermore, the rotation frequency is much smaller than the pulsation frequencies, rendering such a first-order perturbative approach adequate. This computational set-up is certainly not adequate for fast rotating stars, but many pre-main sequence δ-Scuti stars are found to be moderately or slowly rotating stars7 for which the prescription of rotation used in this work is adequate.

### Forward asteroseismic modelling of pre-main sequence δ-Scuti stars

Once observations of a suitable pre-main sequence δ-Scuti star are available, forward asteroseismic modelling, according to Aerts et al.35 can be undertaken to investigate which types of pre-main sequence models (disk-mediated, constant accretion scenario, classical) provide the best accordance with the observed pulsation frequencies. The methodology has been developed for gravity mode pulsations and has recently been applied to constrain the internal mixing of such stars27, but the statistical techniques presented in Aerts et al.35 are applicable to different types of pulsators as well. The input needed to perform asteroseismic modelling includes, next to classical constraints such as Teff, $$\log (g)$$, $$\left[{{{{{{{\rm{M}}}}}}}}/{{{{{{{\rm{H}}}}}}}}\right]$$, and $$\log \left(L/{L}_{\odot }\right)$$27, additional mode identification. For pre-main sequence δ-Scuti stars, mode identification can be obtained from regularities in echelle diagrams, as has been shown in the past63,64,65.

In the methodology described by Aerts et al.35, the calculation of the merit function, preferably the Mahalanobis distance, does not include the classical constraints. In contrast, these are used a posteriori to evaluate the goodness of the model35. In applications to gravity mode pulsators, they are typically used to determine the range of initial parameters for the calculation of a model grid or to a posteriori eliminate stellar models outside of the observed regime27,66,67.

Nevertheless, such modelling work would be an immense computational effort as thousands of evolutionary calculations would have to be calculated. Disk-mediated mass accretion rates pose a challenge to the stellar evolution code MESA. As a consequence, calculations of such evolutionary models typically take at least 100 hours of execution time (on 10 cores of Intel Xeon X5650 processors). In this regard, accurate knowledge of the classical constraints Teff, $$\log (g)$$, $$\left[{{{{{{{\rm{M}}}}}}}}/{{{{{{{\rm{H}}}}}}}}\right]$$, and $$\log \left(L/{L}_{\odot }\right)$$ would mean an immense reduction of the computational burden. The prospect of constraining the early phases of star formation, however, would make such an endeavour worthwhile.