According to the Tidal Torque Theory TTT the angular momentum AM of dark matter haloes arises from the tidal torque suffered by ellipsoidal protohaloes due to their surrounding mass distribution. Unfortunately the practical implementation of the TTT in the peak model of structure formation where protohaloes are indeed ellipsoidal patches around triaxial peaks maxima in the Gaussiansmoothed density field faces some fundamental issues. First the mass and collapse time of patches marked by Gaussian peaks are poorly determined. Second the growth of the resulting AM is only known in linear regime. This Paper is the first of two devoted to revisiting the TTT and accurately calculating the rotational halo properties it predicts. Here we use the CUSP formalism fixing the above mentioned indeterminacy in the protohalo AM. The peak viewpoint is fully accomplished in the sense that not only is the protohalo identified to a peak but also the mass fluctuations causing the tidal torque are identified to peaks or holes local minima. This way we obtain a very simple analytic expression for the mean and median Lagrangian protohalo AM which can be readily implemented in galaxy formation models and be compared to the results of simulations.
['Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - Astrophysics of Galaxies']
The Tidal Torque Theory Revisited. I. Protohalo Angular Momentum
{'The Tidal Torque Theory Revisited. I. Protohalo Angular Momentum': "Eduard Salvador-Sol'e 1 ⋆  \nand Alberto Manrique  \nInstitut de Ci'encies del Cosmos i Departament de F'ısica Qu'antica i Astrof'ısica. Universitat de Barcelona, Mart'ı i Franqu'es 1, E-08028 Barcelona, Spain  \n15 October 2024", 'ABSTRACT': 'According to the Tidal Torque Theory (TTT), the angular momentum (AM) of dark matter haloes arises from the tidal torque suffered by ellipsoidal protohaloes due to their surrounding mass distribution. Unfortunately, the practical implementation of the TTT in the peak model of structure formation, where protohaloes are, indeed, ellipsoidal patches around triaxial peaks (maxima) in the Gaussian-smoothed density field, faces some fundamental issues. First, the mass and collapse time of patches marked by Gaussian peaks are poorly determined. Second, the growth of the resulting AM is only known in linear regime. This Paper is the first of two devoted to revisiting the TTT and accurately calculating the rotational halo properties it predicts. Here we use the CUSP formalism fixing the above mentioned indeterminacy in the protohalo AM. The peak viewpoint is fully accomplished in the sense that not only is the protohalo identified to a peak, but also the mass fluctuations causing the tidal torque are identified to peaks or holes (local minima). This way we obtain a very simple analytic expression for the mean (and median) Lagrangian protohalo AM, which can be readily implemented in galaxy formation models and be compared to the results of simulations.  \nKey words: methods: analytic - galaxies: haloes - cosmology: theory, dark matter - dark matter: haloes', '1 INTRODUCTION': "It is commonly accepted that dark matter (DM) haloes acquire their angular momentum (AM) as a consequence of the tidal torque caused by neighbouring mass fluctuations, as suggested by Hoyle et al. (1949).  \nPeebles (1969) applied this Tidal Torque Theory (TTT) to estimate the growth of the AM in spherical protohaloes (see also Thuan & Gott 1977). Doroshkevich (1970) refined the theory taking into account that protohaloes are ellipsooidal and White (1984) completed this picture by taking into account that the tidal torque is due to the also non-spherical gravitational potential of the neighbouring mass distribution, being the principal axes of the inertia and tidal tensors not aligned in general.  \nTaylor expanding to second order the peculiar gravitational potential Φ( x ) at the centre of mass (c.o.m.) of the protohalo and integrating over its volume, White (1984) found that, in linear regime, the AM grows, keeping the spin orientation fixed, according to  \nJ i ( t ) ≈ -a 2 ( t ) ˙ D ( t ) ϵ ijk T jl I lk , (1)  \nwhere J i ( t ) is the i th Cartesian component of the angular momentum J of the protohalo at the cosmic time t , ϵ ijk is the fully antisymmetric Levi-Civita rank-three tensor, T ij ≡ ∂ 2 Φ /∂x i ∂x j is the tidal (or deformation) tensor, i.e.  \nthe Hessian of the potential Φ, at the protohalo c.o.m. and I is the inertia tensor of the protohalo with respect to that point, with both quantities appropriately scaled so as to be independent of the arbitrary time when they are calculated. The growth of J i is thus encoded in the factor a 2 ( t ) ˙ D ( t ), where a ( t ) and D ( t ) are the cosmic scale and linear growth factor, respectively. This way, J i grows as t in the Einstein-de Sitter (EdS) cosmology.  \nFactor a 2 ( t ) ˙ D ( t ) arises from the time dependence of the integral of the cross product of r × ˙ r over the protohalo mass particles, taking into account that in linear regime the space stretches proportionally to a ( t ) and the peculiar particle velocity ˙ r is proportional to their peculiar acceleration, which factorises as ˙ D ∇ Φ. The remaining time-independent quantity is the Lagrangian AM of the linear protohalo.  \nStrictly speaking, equation (1) holds for protohaloes in linear regime only. However, it also approximately holds in mildly non-linear regime (Catelan & Theuns 1996b). In addition, simulations show that the AM of relaxed haloes is roughly consistent with such a growth until the final accelerated collapse of the protohalo (e.g. Sugerman, Summers & Kamionkowski 2000; Porciani, Dekel, & Hoffman 2002a,b). But one cannot be more precise because the translation from the protohalo AM to the final AM of relaxed haloes involves highly non-linear effects, such as the accelerated collapse and virialisation (through shell-crossing) of the objects and the presence of major mergers, that are very hard to properly address.  \nMoreover, even focusing on protohaloes in linear regime, the TTT is not a closed subject. In the PS (Press & Schechter 1974) and 'excursion set' (ES; Bond et al. 1991) models of structure formation where protohaloes are overdense patches undergoing top-hat spherical collapse, the protohalo AM does not grow. It only does provided protohaloes are ellipsoidal, as in the more realistic peak model introduced by Doroshkevich and White. This motivated Heavens & Peacock (1988), Hoffman (1988) and Catelan & Theuns (1996a,b) to implement the TTT in such a model where protohaloes correspond to triaxial density maxima in the Gaussian-smoothed random Gaussian density field. Unfortunately, this implementation faced the fundamental issue that the exact extension (or mass) and collapse time of the collapsing patches encompassed by Gaussian peaks are unknown, which forced those authors to adopt some uncertain assumptions. In addition, calculations were quite complex and the final results could only be in numerical form. In particular, Hoffman (1988) used Monte Carlo simulations, while the derivations by Heavens & Peacock (1988) and Catelan & Theuns (1996a) are analytical, though involving numerical integrations. Consequently, their predictions are difficult to compare with the results of simulations and to implement in analytic models.  \nBut all these shortcomings can in principle be overcome thanks to the ConflUent System of Peak trajectories (CUSP) formalism introduced in Manrique & Salvador-Sol'e (1995, 1996) and Manrique et al. (1998). This formalism allows one to determine the mass and collapse time of patches marked by Gaussian peaks and to deal with the effects of collapse and virialisation and of major mergers as well (Salvador-Sol'e & Manrique 2021). Thus, it should hopefully also enable one to properly address the halo AM in the TTT. This is precisely the aim of the series of two Papers started here.  \nIn the present Paper, we concentrate in revisiting the TTT and accurately calculate the AM of linear protohaloes in the peak model. Making use of the CUSP formalism, we solve the above mentioned fundamental issues of this model. In addition, the problem is dealt with from a full peak viewpoint in the sense that, not only is the protohalo identified to a peak but also the main source of the tidal torque acting on it is treated as a peak or a hole (local density minima). As a bonus, we find a very practical fully analytic expression of the Lagrangian protohalo AM. These results are used in Paper II to accurately derive all rotational halo properties.  \nThe layout of the Paper is as follows. In Section 2 we remind the protohalo properties in the CUSP formalism. The properties of the neighbouring mass excess or default responsible for the tidal torque acting on a protohalo are derived in Section 3. In Section 4 we use the previous results to obtain the (time-invariant) Lagrangian protohalo AM which determines the growth of the Eulerian counterpart in linear and mildly non-linear regimes. Our results are discussed and summarised in Section 5.", '2 PROTOHALO': "In the peak model, protohaloes are collapsing patches traced by peaks in the linear random Gaussian density field. These collapsing patches are triaxial (like their associated peaks) rather than spherical, so they undergo ellipsoidal collapse. Their collapse time (along all three axes) t then depends  \nnot only on the mass (or scale R ) and density contrast δ of the peak, like in spherical collapse, but also on its curvature (or sharpness) x , defined as -∇ 2 δ ( x ) scaled to the rms value of that quantity, as well as on its shape, i.e. ellipticity e and prolateness p , i.e. t = t ( δ, x, e, p, R ). Nonetheless, as we will see shortly, the PDF of e , p and, particularly, x of peaks with density contrast δ at scale R in the Gaussiansmoothed density field at t i are sharply peaked, so we can assume them with fixed values at the respective maxima, e max , p max , x max . Therefore, given any halo mass definition (see below) fixing the mass M of haloes at t , the corresponding ellipsoidal patches at t i traced by triaxial peaks with a positive δ at the suited scales R ( M,t ) will collapse essentially at the same time, with t a monotonic decreasing function of δ like in top-hat spherical collapse. In other words, in Gaussian ellipsoidal collapse like in top-hat spherical one, there is a correspondence between haloes with M at t and peaks with δ and R in the Gaussian-smoothed density field at t i . 1  \nThe relations between δ and R of peaks and M and t of the corresponding haloes can be obtained by enforcing the consistency conditions that all the DM in the Universe at any time t is locked in haloes of different masses and that the mass M of a halo is equal to the volume-integral of its density profile times the squared radius (Juan et al. 2014). Specifically, if we write the density contrast δ and scale R of peaks at t i in the general form  \nδ ( t, t i ) = r δ ( t ) δ th c ( t ) D ( t i ) D ( t ) (2)  \nσ 0 ( R,t, t i ) = r σ ( M,t ) σ th 0 ( M,t ) D ( t i ) D ( t ) , (3)  \nwhere δ th c ( t ) is the critical linearly extrapolated density contrast for top-hat spherical collapse at t , equal to 3(12 π ) 2 / 3 / 20 ≈ 1 . 686 in the EdS cosmology, D ( t ) is the linear growth factor, equal to the cosmic scale factor a ( t ) in that universe, and σ i ( R,t, t i ) is the Gaussian i th-order spectral moment on scale R , then the functions r δ and r σ are wellfitted, in all cases analysed, by the analytic expressions (see Salvador-Sol'e & Manrique 2024)  \nr δ ( t ) ≈ a d D ( t ) ( t ) D ( t ) D ( t ) = 1 -d 0 a 0 . 435 /a ( t ) ( t )  \n(4)  \nr σ ( M,t ) ≈ 1 + r δ ( t ) S ( t ) ν th ( M,t ) S ( t ) = s 0 + s 1 a ( t )+log [ a s 2 ( t ) 1+ a ( t ) /A ] , (5)  \nwhere ν th ( M,t ) ≡ δ th ( t, t i ) /σ th 0 ( M,t i ) = δ th c ( t ) /σ th 0 ( M,t ) is the peak height in top-hat smoothing. The values of the coefficients d , s 0 , s 1 , s 2 and A for some relevant cosmologies (Table 2) and halo mass definitions are given in Table 1. The previous relations imply in turn the following t i -independent relation between the Gaussian and top-hat peak heights  \nν ( M,t ) = r δ ( t ) r σ ( M,t ) ν th ( M,t ) . (6)  \n1 This correspondence is not one-to-one because some peaks lie within others with identical density contrast at a higher scale, so they are swallowed by their host (and become subhaloes) before accomplishing full collapse. However, that is irrelevant for our purposes here.  \nTable 1. Coefficients in the halo-peak relations.  \n∗ M vir and M 200 are the masses inside the region with a mean inner density equal to ∆ vir ( t ) (Bryan & Norman 1998) times the mean cosmic density and 200 times the critical cosmic density, respectively.  \nEquation (3) defines the Gaussian scale R ( M,t,t i ) that corresponds to the mass M of haloes at t in terms of the tophat 0th-order spectral moment at t , σ th 0 ( M,t ) and it shows that the Gaussian scale R is not simply proportional to the top-hat one R th satisfying M = 4 π/ 3( R th ) 3 ¯ ρ ( t i ). However, in the case of power-law power spectra, P ( k, t ) = P 0 ( t ) k n , there is a more direct relation between the Gaussian scale R and of the top-hat one R th delimiting physically the protohalo of mass M . Indeed, the definition of the j th-order spectral moment for Gaussian and top-hat filters, denoted with index f equal to 'blank' and to 'th', respectively,  \n( σ f j ) 2 ( R f ) = P 0 D ( t 0 ) /D ( t ) 2 π 2 ( R f ) n +3+2j ∫ ∞ 0 d xx n +2(1+j) ( W f ) 2 ( x ) , (7)  \nleads to  \nR ( M,t,t i ) = r R ( M,t ) R th ( M,t i ) , (8)  \nwith  \nr R ( M,t ) = [ Q 0 r σ ( M,t )] -2 n +3 , (9)  \nwhere Q 2 0 ≡ ∫ ∞ 0 d xx n +2 W 2 ( x ) / ∫ ∞ 0 d xx n +2 ( W th ) 2 ( x ), and W and W th are the Fourier transforms of the Gaussian and top-hat filters, respectively. Thus, taking into account that the CDM spectrum in the galaxy mass range is locally of the power-law form (with n ≈ -1 . 75 and Q 0 ≈ 0 . 6 in the galaxy mass range), we can also adopt the relations (8)-(9) as a useful approximation in CDM. It is worth mentioning that, in the case of virial masses ( M vir defined in the caption of Table 1), r σ is a function of M alone, r σ ( M ) = 1 + c/σ th 0 ( M ), where σ th 0 ( M ) is the top-hat 0th order spectral moment at the present time and c = 0 . 14 and 0.10 in the WMAP7 and Planck14 cosmologies, respectively (Salvador-Sol'e & Manrique 2024). Thus, R is, in this case, a function of M alone of the order of R ≈ 2 R th .  \nIn what fallows, we will be concerned with to different cosmic times apart from the arbitrary initial one t i : one to describe the evolution of the protohalo AM and the other one referring to the protohalo collapse time. To avoid any confusion in the notation, t will be reserved to the former, while for the latter, which appears much less frequently, we will use t c .  \nThe average number density of peaks at t i with ν , x , e and p and oriented according to the Euler angles α , β and κ at the scale R per infinitesimal range of all these arguments is  \nTable 2. Cosmological Parameters.  \n- a Komatsu et al. (2011).  \n(BBKS)  \nN pk ( ν, x, e, p, α, β, κ, R )= e -ν 2 2 (2 π ) 2 R 3 ⋆ F ( x )e -( x -γν ) 2 2(1 -γ 2 ) [2 π (1 -γ 2 )] 1 2 × P ep ( e, p ) P E ( α, β, κ ) , (10)  \nwhere γ ≡ σ 2 1 / ( σ 0 σ 2 ) R ⋆ ≡ √ 3 σ 1 /σ 2 ( γ = [( n +3) / ( n +5)] 1 / 2 ( R ⋆ = [( n +5) / 6] -1 / 2 R for power-law spectra),  \nF ( x )= ( x 3 -x ) { erf [ ( 5 2 ) 1 2 x ] +erf [ ( 5 2 ) 1 2 x 2 ]} 2+ ( 2 5 π ) 1 2 [( 31 x 2 4 + 8 5 ) e -5 x 2 8 + ( x 2 2 -8 5 ) e -5 x 2 2 ] , (11)  \nP E ( α, β, κ ) is the isotropic PDF of Euler angles and  \nP ep ( e, p ) ≈ 1 2 πσ e σ p e -( e -¯ e ) 2 2 σ 2 e -( p -¯ p ) 2 2 σ 2 p (12)  \nis the joint probability function of ellipticities, e = ( λ 1 -2 λ 2 + λ 3 ) / (2 x ) ( e ⩾ 0), and prolatenesses p = ( λ 1 -λ 3 ) / (2 x ) ( -e ⩽ p ⩽ e ), where λ i ⩾ λ 2 ⩾ λ 3 are the eigenvalues, with positive sign, of the second order derivative tensor ( λ 1 + λ 2 + λ 3 = x , so there are 2 degrees of freedom only), which, for large x as corresponding to galactic haloes, are nearly Gaussian with corresponding means ¯ e = 1 / { √ 5 x [1 + 6 / (5 x 2 )] 1 / 2 } and ¯ p = 6 / { 5 x 4 [1 + 6 / (5 x 2 )] 2 } and dispersions σ e = ¯ e/ √ 6 and σ p = ¯ e/ √ 3 (BBKS).  \nAs mentioned, the x -PDF given by the second factor on the right of equation (10) is very sharply peaked and all peaks with ν at R have nearly the same curvature, 2 equal to its maximum or mean value  \n⟨ x ⟩ ( R,ν ) = G 1 ( γ, γν ) G 0 ( γ, γν ) , (13)  \nwhere G i is the i th x moment,  \nG i ( γ, γν )= ∫ ∞ 0 d xx i F ( x ) e -( x -γν ) 2 2(1 -γ 2 ) [2 π (1 -γ 2 )] 1 / 2 . (14)  \nIn addition, the average of any function f ( x ) of the peak curvature is very nearly equal to f ( ⟨ x ⟩ ), which can be used in the above mentioned mean peak ellipticity and prolateness and the corresponding dispersions.  \nThus, we can marginalise the curvature and work with the average number density of peaks at t i with C = { e, p, α, β, κ } at R ,  \nN pk ( ν, C, R )= G 0 ( γ, γν ) (2 π ) 2 R 3 ⋆ e -ν 2 2 P ep ( e, p ) P E ( α, β, κ ) , (15)  \n2 Even though the e - and p -PDFs are also quite sharply peaked, the protohalo AM crucially depends on the triaxial shape and orientation of the corresponding peaks. Thus, we cannot make such an approximation for these quantities.  \nand with the joint conditional PDF of such peak properties,  \nP pk ( C | δ, R ) = P ep ( e, p ) P E ( α, β, κ ) . (16)  \nAs mentioned, collapsing patches are ellipsoidal, but the previous quantities e and p are the ellipticity and prolateness of peaks, not of protohaloes. The physical (top-hat) semiaxes, a 1 , a 2 and a 3 of the protohalo satisfy the condition a 1 a 2 a 3 = ( R th ) 3 (there are also two degrees of freedom in the a i values) and are related to the eigenvalues of the correspondng peak through  \na i = R th ( Λ λ i ) 1 / 2 , (17)  \nwhere Λ is the geometrical mean of the eigenvalues, Λ = ( λ 1 λ 2 λ 3 ) 1 / 3 . Thus, the semi-axes of a protohalo are related to the ellipticity and prolateness of the associated peak through  \ne = 1 2 (1 -ˆ a -2 2 ) and p = 1 2 (ˆ a -2 1 -ˆ a -2 3 ) , (18)  \nwhere we have used the scaled semi-axes, ˆ a i , defined as  \nˆ a i ≡ a i R th ( x Λ ) 1 / 2 = 3 a i R th { (1 -2 e )[(1 + e ) 2 -9 p 2 ] } 1 / 3 . (19)  \nConsequently, the Lagrangian inertia tensor I at t of the ellipsoidal protohalo with uniform density ¯ ρ ( t i ) relative to its c.o.m. in Cartesian coordinates oriented along the principal axes is  \nI = M 5 a 2 ( t i )   ι 1 0 0 0 ι 2 0 0 0 ι 3   , (20)  \nwhere ι 1 ≡ a 2 2 + a 2 3 , ι 2 ≡ a 2 1 + a 2 3 and ι 3 ≡ a 2 2 + a 2 1 . We remark that, for simplicity, we have assumed that protohaloes are linear at all relevant scales, which implies that their density is everywhere essentially equal to ¯ ρ ( t i ). Strictly speaking, protohaloes are not homogeneous, their typical density profile increases downwards until becoming non-linear at a small enough scale (Salvador-Sol'e et al. 2012a; Salvador-Sol'e & Manrique 2021), so their real inertia tensor may substantially deviate from that given in equation (20). This point will be properly addressed in Paper II.", '3 DOMINANT TIDAL TORQUE': "The deformation tensor at the c.o.m. of the protohalo may be due to a mass excess identified to a peak or a mass default identified to a hole, i.e. a local minimum in density contrast, with identical statistial properties than peaks but for the opposite signs of the density contrast, curvature and eingenvalues of the secoond derivative tensor. For simplicity we will concentrate for the moment in the case of peaks, but we will come back to the most general case of peaks and holes at he end.  \nThere are, of course, many (actually infinite) nested peaks of different scales in the neighbourhood of the peak with δ at R tracing the protohalo and we cannot sum up all their individual tidal torques because the contribution of many of those peaks is included within that of a few others, that are dubbed here as the /tieaccentlowercase dominant peak. From now on, the properties of those dominant peaks are denoted with subindex 'd' to  \ndistinguish them from the properties of the peak associated with the protohalo itself.  \nMore specifically, as shown by Manrique & Salvador-Sol'e (1995), peaks are often nested within each other in continuous sets in the R -δ plane of the smoothed linear density field at t i , the so-called peak trajectories, tracing the mass growth of accreting haloes, 3 Peak trajectories satisfy the differential equation  \nd δ ' d R ' = -⟨ x ⟩ [ δ ( R ' ) , R ' ] σ 2 ( R ' ) R ' , (21)  \nwhere ⟨ x ⟩ ( δ ' , R ' ) is the mean curvature of peaks with δ ' at R ' . For large R ' as corresponding to the mass excesses able to cause a substantial tidal torque to galaxy mass protohaloes, ⟨ x ⟩ is very approximately equal to γν (BBKS), so equation (22) becomes  \ndln δ ' d lnR ' = m, (22)  \nwhere m = -( n + 3) / 2. Since all peaks along a trajectory remain essentially at the same position (Manrique & Salvador-Sol'e 1995) and the mass excess (peculiar mass) of the ellipsoidal patch with density contrast δ ' and scale R ' is 4 π/ 3 [( R th ) ' ] 3 δ ' ¯ ρ ( t i ), to choose the peak along any given trajectory causing the strongest tidal torque we must find the scale R ' maximising δ ' ( R ' )[( R th ) ' ] 3 or δ ' ( R ' )( R ' ) 3 , with δ ' ( R ' ) equal to the solution of equation (22)  \nδ ' ( R ' ) = δ R ( R ' R ) m (23)  \nfor the boundary condition δ ' ( R ) = δ R with any given δ R , where R is the scale of the peak marking the protohalo. From equation (23) we see that, for the CDM spectrum in the relevant mass range where m ≈ -0 . 63, δ ' ( R ' ) diminishes with increasing R ' less rapidly than ( R ' ) -3 , so δ ( R ' )[( R ' )] 3 increases monotonically with R ' . Therefore, there is no maximum δ ' ( R ' )( R ' ) 3 value over a peak trajectory; what limits tidal torque is the scale ( R th ) ' ≈ R ' / 2 which cannot exceed the separation r to the c.o.m. of the protohalo. Indeed, for larger scales, the ellipsoidal shells lying beyond do not contribute to the tidal force at the c.o.m. of the protohalo (see below), while δ ' ( R ' ) is increasingly smaller, so the maximum tidal torque setting the dominant peak is found for the scale R ' = r . That is thus the scale R d of the dominant peak in each trajectory determined by the value of δ R and its density contrast δ d is the corresponding value of δ ' ( R d ). Note that, since r > R th and, hence, R d > R , δ d ≡ δ ' ( R d ) is substantially smaller than the density contrast δ of the protohalo. Thus, when the protohalo will collapse, the dominant mass excess causing its tidal torque will still be in linear regime. This result will be used in Paper II.  \nThe conditional PDF of peaks with ν d and C d = { e d , p d , α d , β d , κ d } properties in infinitesimal ranges at the dominant scale R d lying at a distance r from the peak with  \nδ and C = { e, p, α, β, κ } at R is  \nP d ( ν d , C d , R d | δ, C, R, r )=4 πr 2 ∫ ∞ -∞ d δ ' N pk ( ν d , C d , R d | δ ' , R ) × [1+ ξ pk ( r ; ν, R, ν d , R d )] P ( δ ' , R | δ, C, R, r ) , (24)  \nwhere N pk ( ν d , C d , R d | δ ' , R ) is the conditional number density of peaks with ν d = δ d /σ 0 ( R d ) and C d at R d subject to lying at a point with δ ' at R or the point with δ ' = δ R at R setting the boundary condition of the corresponding trajectory and P ( δ ' , R | δ, C, R, r ) is the probability of finding such a point at r from a peak with δ and C at R . The factor 1 + ξ pk ( r, ν d , R d , ν, R )], where ξ pk ( r ; δ d , R d , δ, R ) is the cross-correlation between peaks with δ at R and with δ d at R d , enhances the local average conditional number density N pk ( ν d , C d , R d | δ ' , R ) of peaks with δ d at R d at a distance r from a peak with δ at R with respect to the global average conditional number density (i.e. with no constraint relative to the separation from any other peak).  \nThe conditional peak number density appearing in the relation (24) takes the form (BBKS)  \nN pk ( ν d , C d , R d | δ ' , R ) = G 0 (˜ γ, ˜ γ ˜ ν ) (2 π ) 2 R 3 ⋆ ( R d ) e d e -( ν d -ϵ d ν ' ) 2 2 e 2 d × P ep ( e d , p d ) P E ( α d , β d , κ d ) , (25)  \nwhere P E ( α d , β d , κ d ) is the usual PDF of Euler angles and P ep ( e d , p d ) is the ellipticity- and prolateness-PDF of the form (12) for the dominant peak causing the tidal torque, but with ⟨ x d ⟩ replacing x d for the above mentioned reason and G 0 (˜ γ, ˜ γ ˜ ν ) given by equation (14) for i = 0 but with γ and ν replaced by  \n˜ γ = γ d [ 1 + ϵ 2 d (1 -r 1 ) 2 (1 -ϵ 2 d ) ] 1 / 2 (26)  \n˜ ν = γ d ˜ γ d (1 -r 1 ) (1 -ϵ 2 d ) [ ν d (1 -ϵ 2 d r 1 ) (1 -r 1 ) -ϵ d ν ' ] , (27)  \nrespectively, with ν ' ≡ δ ' /σ 0 , ϵ d ≡ σ 2 0h / [ σ 0 ( R d ) σ 0 ( R )], being σ j h defined as σ j but with R replaced by R h = [( R 2 d + R 2 ) / 2] 1 / 2 , e d ≡ √ 1 -ϵ 2 d and r 1 ≡ σ 2 1h σ 2 0 /σ 2 0h σ 2 1 .  \nOn the other hand, the conditional probability P ( δ ' , R | δ, C, R, r ) is Gaussian with mean (BBKS)  \n⟨ δ ( r ) | C ⟩ = γδ 1 -γ 2 ( ψ γ + ∇ 2 ψ 3 ) -⟨ x ⟩ σ 0 1 -γ 2 ( γψ + ∇ 2 ψ 3 ) + 5 2 ⟨ x ⟩ σ 0 ( ψ ' r -∇ 2 ψ 3 ) A ( e pk , p pk ) , (28)  \nfor some known function A that is not relevant here, and variance 4  \n⟨ [∆ δ ( r )] 2 | C ⟩ = σ 2 0 { 1 -1 1 -γ 2 [ ψ 2 + ( 2 γψ + ∇ 2 ψ 3 ) ∇ 2 ψ 3 ] -5 ( ψ ' r -∇ 2 ψ 3 ) 2 -3( ψ ' ) 2 γ 2 + 1 (1 -γ 2 ) 2 ( γψ + ∇ 2 ψ 3 ) 2 } , (29)  \nwhere ψ is the matter correlation function ξ ( r ) on scale R normalised to ξ (0) and ψ ' is its r -derivative, being r in units  \n4 The units of r are different from those used in Manrique et al. (1998). Note also that, according to the normalisation used in the definition of x , the curvature variance ⟨ [∆ x ] ⟩ is unity.  \nof R ⋆ . For high peaks, as corresponding to galactic haloes, ⟨ x ⟩ approaches γν (BBKS) and, for power-law spectra (with n < -1), the gradients of ψ can be neglected in front of unity and of ψ , which in turn is of order unity ( r is small, of the order of R ). Consequently, ⟨ [∆ δ ( r )] 2 | C ] ⟩ approaches ˆ σ 0 ≡ σ 2 0 [1 -ψ 2 ] ≈ 0 and ⟨ δ ( r ) | C ⟩ ≡ δψ ≈ δ . Thus, the conditional probability P ( δ ' , R | δ, C, R, r ) approaches  \nP ( δ ' , R | δ, C, R, r ) ≈    e -( δ ' -δ ) 2 2ˆ σ 2 0 √ 2 π ˆ σ 0    ˆ σ 0 ≈ 0 ≈ δ D ( δ ' -δ ) , (30)  \nwhere δ D ( δ ' -δ ) is the Dirac delta function and it does not depend neither on the properties C nor on r because they do not correlate with δ ' . Thus, we can write P ( δ ' , R | δ, C, R, r ) ≡ P ( δ ' , R | δ, R ).  \nWith this approximate expression for the conditional probability P , the integral on the right of equation (24) leads to  \nP d ( ν d , C d , R d | δ, R, r ) ≈ 4 πr 2 [1+ ξ pk ( r ; ν d , R d , ν, R )] ×N pk ( ν d , C d , R d | δ, R ) = 4 πr 2 [1+ ξ pk ( r ; ν, R )] N pk ( ν, C d , R ) , (31)  \nwhere δ d and R d of the dominant peak are functions of δ and R of the peak at r from that marking the protohalo through the relations δ d = ( R d /R ) m δ = ( r/R th ) m δ (eqs. [8][9]). The second equality on the right of equation (31) arises from writing the number density of peaks with ν d at R d whose trajectory has a peak with δ at R directly as the unconditional number density of peaks with ν at R . This in turn implies that the cross-correlation function of peaks with ν d at R d and with ν at R reduces to the autocorrelation function, ξ pk ( r ) = B 2 ( R,ν ) ξ m ( r ), of peaks with ν at R . Given the form of the right hand member of equation (31), P d ( ν d , C d , R d , r | δ, R ) is simply P d ( C d , r | δ, R ). Note the properties C do not appear in the conditional PDF of the dominant peak, P d , though they appear, of course, in the joint probability of the 'protohalo-dominant tidal torque source' properties used in Section 4.  \nTaking ξ pk ( r ; ν, R ) given by the leading order of the perturbative bias expansion, ξ pk ( r ; ν, R ) ≈ B 2 ( R,ν ) ξ ( r ), where B ( R,ν ) ≈ 1 . 6 ( ν -γ ⟨ x ⟩ ) / [ σ 0 (1 -γ 2 )] is protohalo linear bias in the galactic halo mass range (Salvador-Sol'e & Manrique 2024), and ξ ( r ) = ( r/s 0 ) -˜ γ , with ˜ γ ≈ 2 and s 0 ∼ 15 h -1 Mpc (Abdullah et al. 2024), is the matter autocorrelation, the probability P d takes the form  \nP d ( C d | δ, R, r ) = 4 πr 2 [1 + B 2 ( R,ν ) ξ ( r )] N pk ( ν, C d , R ) . (32)  \nAs mentioned at the beginning of this Section, the previous derivation could also be followed for tidal torques caused by mass defaults instead of mass excesses. Following the same derivation for peaks leading to equation (31) but for holes (minima of the linear density field, i.e. like peaks but with negative ν and x ), we would be led to a hole density contrast δ h = ˜ Bδ m with the linear bias of holes with -δ at R , ˜ B , equal to minus the linear bias of peaks with δ at R , B . This implies that, while the peak-peak correlation is given by ξ pk ( r ) = B 2 ( R,ν ) ξ m ( r ), where ξ m ( r ) is the matter correlation, the peak-hole cross-correlation is given by ξ pkh ( r ) = -B 2 ( R,ν ) , ξ m ( r ). Consequently, in the probability of finding the main source (peak or hole) of the tidal torque of the same modulus at a distance r of the protohalo,  \nP d ( C d | δ, R, r ) (eq. [32]), the term with the linear bias parameter B 2 arising from peaks cancels with that arising from holes, and we are led simply to  \nP d ( C d | δ, R, r ) = 8 πr 2 N pk ( ν, C d , R ) . (33)  \nThe peculiar (excess of default) mass of the dominant peaks or holes along trajectories with boundary condition | δ ( R ) | in the neighbourhood of the protohalo is M d = 4 π/ 3 δ d ( R th d ) 3 δ d ¯ ρ ( t i ), with R th d = r . (For simplicity in the notation we skip the arguments of δ d ( t c , t i ), where t c is the collapse time of the main tidal torque source.) Consequently, the corresponding tidal torque is just proportional to δ d = δ ' ( R d ). And, since δ d decreases with increasing R d ∝ r according to equation (23) with δ R = δ , the main dominant peak or hole causing the strongest tidal torque on the protohalo is simply that lying the closest to it, i.e. with the smallest R d and, hence, the highest δ d . Thus, we will concentrate from now in the nearest dominant peak or hole, taking into account that the probability that the nearest dominant peak or hole with ν d and C d on scale R d is at a distance r from the c.o.m. of the protohalo is the probability P d of finding one of such dominant peaks or holes at r times the probability that there is no other dominant peak or hole closer to the c.o.m. of the protohalo,  \nP st ( C d | δ, R, r ) ≈ 4 πr 2 N pk ( ν, C d , R ) [ 1 -4 π 3 r 3 N pk ( ν, C d , R ) ] . (34)  \nOn the other hand, the peculiar gravitational potential Φ d of the corresponding patch assumed with uniform peculiar density δ d ¯ ρ ( t i ) at x = ( x 1 , x 2 , x 3 ) in physical Cartesian coordinates with origin at the c.o.m. of the protohalo aligned with the principal axes of its ellipsoid is (e.g. Chandrasekhar 1987)  \nΦ d ( x ) = πa d1 a d2 a d3 Gδ d ¯ ρ ( t i ) × [ ∫ ∞ S ( x ) d s ∆( s ) -3 ∑ i =1 x 2 i ∫ ∞ S ( x ) d s ( a 2 d i + s )∆( s ) ] , (35)  \nwhere we have written the peculiar (positive or negative) mass in terms of the semi-axes, M d = 4 π/ 3 a d1 a d2 a d3 δ d ¯ ρ ( t i ). In equation (35), G is the gravitational constant, a d1 , a d2 and a d3 are the top-hat semi-axes of the ellipsoidal mass excess,  \n∆( s ) ≡ [ ( a 2 d1 + s )( a 2 d2 + s )( a 2 d3 + s ) ] 1 / 2 (36)  \nand the function S ( x ) is defined as the positive root of equation  \n3 ∑ i =1 x 2 i a 2 d i + S = 1 . (37)  \nEquation (35) presumes that the point x is external to the ellipsoidal mass excess or defect, which is indeed the case at the c.o.m. of the protohalo since, by construction, R th d = r . From that potential, we can derive the Lagrangian tidal tensor acting at t on the protohalo at r = R th d with all timevarying quantities scaled to the values at t , which is very approximately given by (see App. A)  \nT d ij = a 3 ( t i ) D ( t i ) 4 π 3 Gδ d ¯ ρ ( t i ) [ 3 x i x j a 2 d i a 2 d j a 2 d -δ ij ] , (38)  \nwhere a 2 d ≡ 1 / 3 ∑ 3 k =1 a 2 d k . Without saying, the preceding  \nMNRAS 000 , 000-000 (0000)  \nderivation holds for positive (mass excesses) or negative (mass defaults) δ d values.", '4 PROTOHALO ANGULAR MOMENTUM': "Once we have determined the Lagrangian protohalo inertia tensor I (Sec. 2) and tidal tensor T d due to a dominant peak or hole (Sec. 3), we can proceed to calculate the Lagrangian AM, J L . That is simply the tensor product of those two tensors, equal to ϵ ijk T d jl I lk in a given Cartesian coordinate system. Of course, the principal axes of those tensors are not necessarily aligned with each other (if they are the tidal torque is null), but they show different spatial rotations with respect to the coordinate system of reference. Thus, we have  \nJ L d i = -ϵ ijk [ R d T d R T d ] jl [ RIR T ] lk , (39)  \nwhere I and T d are the tensors oriented along their own principal axes, in the form (20) and (38), respectively, and R and R d are the respective rotation (or direction cosine) matrices, functions of the Euler angles { α, β, κ } and { α d , β d , κ d } , index T denoting transpose.  \nEquation (39) is for one particular spatial configuration of the system. To find the typical Lagrangian protohalo AM we must average the absolute magnitude of components J L d i over all possible configurations of the 'protohalo-main tidal torque source' system for the joint PDF of all its properties,  \nP ( C d , C | δ, R, r ) = P st ( C d | δ, R, r ) P pk ( C | δ, R ) , (40)  \nwith C d = { e d , p d , α d , β d , κ d } and C = { e, p, α, β, κ } , being the PDFs P pk and P st given by equations (16) and (34), respectively.  \nActually, what we are interested in is the average of the J d modulus rather than the average of the absolute magnitude of the components J d i . However, by isotropy of the universe, the averaged absolute magnitude of any component | J d i | must be the same, implying ⟨ J d ⟩ 2 = 3 ⟨| J d i |⟩ 2 . Consequently, the average modulus can be calculated from the average absolute magnitude of any given component. Moreover, since for any spatial configuration of the two objects, there is another one yielding an AM with the same modulus and opposite sign, the average of | J d i | over the whole composite angular surface, (8 π 2 ) 2 strad 2 , is the same as the average of J d i over half that angular surface. We can thus average expression (39), say, over α , β and κ in 8 π 2 strad 2 and over α d , β d and γ d in 8 π 2 / 2 strad 2 .  \nBy doing this (see App. B), we arrive at the following approximate analytic expression for the average Lagrangian protohalo AM  \nJ L d ≈ 2 G ¯ ρ 1 / 3 0 δM 5 / 3 5 × 3 1 / 2 D ( t i ) ( r R th ) m H ( e d , p d ) ( 1 -2 e 1+ e ) 2 / 3 p, (41)  \nwhere ¯ ρ 0 is the current cosmic mean density and H ( e d , p d ) is a function of the ellipticity and prolateness of the peak or hole marking the tidal torque source, defined in equation (B9).  \nNext step is to average over the ellipticity and prolateness of the two objects for the corresponding PDFs P ep (eq. [12]). The result for virial masses to leading order is (see App. C)  \nJ L d ≈ 0 . 347 G ¯ ρ 1 / 3 0 δ g ( γ, γν ) M 5 / 3 D ( t i ) ( r R th ) 5 m/ 3 , (42)  \nwhere  \ng ( γ, γν ) ≡ [ ( γν ) 1 / 3 ⟨ x ⟩ 2 +6 / 5 ] 2 . (43)  \nSince m = -( n +3) / 2 is negative, the peak or hole marking the mass excess or default causing the strongest tidal torque on the protohalo is that lying the closest to the protohalo c.o.m. for which r is the smallest. Therefore, the mean Lagrangian AM, J L , of protohaloes collapsing at t c in haloes with M is the average of J L d given by equation (42) over the separation r inside the sphere of radius r st centred at the c.o.m. of the protohalo and statistically harbouring only one such dominant peaks or holes,  \nJ L = ∫ r st ( δ,R ) 0 d rJ L d ( r, δ, M ) P ( δ, R, r ) , (44)  \nwhere, after the previous averages, P ( δ, R, r ) is given by equation (40), with P pk = 1 and P st with no arguments C d and C , and, as mentioned, r st ( δ, R ) ( > R th ) is the solution of the implicit equation  \n1 = 8 πR 3 N pk ( ν, R ) ∫ r st R 0 d s s 2 , (45)  \nwhere N pk ( ν, R ) results from marginalising N pk ( ν, C, R ) (eq. 15) with respect to C , which leads to  \nr st R = [ 2 3 π ( n +5 6 ) 3 / 2 G 0 ( γ, γν ) e -ν 2 2 ] -1 3 . (46)  \nThe typical r st /R th value ( ≈ 2 r st /R for virial masses) as a function of M is shown in Figure 1. Note that r st /R th is not only constant in linear regime where r st and R th are proportional to a ( t ), but it is also independent of the halo collapse time because ν is time-invariant and weakly dependent on M , being r st /R th ∼ 4 . 5 for all halo masses except at the very high-mass end where the halo number density rapidly falls off and s goes to infinity ( s 5 m/ 3 vanishes).  \nIt is worth noting that, in the case of peaks conditioned to lie in a background δ m at a scale R m > 3 R , the same procedure followed in Salvador-Sol'e & Manrique (2024) would lead to a conditional peak number density N pk that differs from that appearing in equation (45) by an additional term (to first order in δ m ) proportional to νδ m /σ 0 , i.e. of the same amount but opposite sign (through ν ) for peaks and holes. Consequently, the change in the number density of both kinds of objects would balance and r st /R th would remain the same. This collateral result is used in Salvador-Sol'e, Manrique & Agull'o (2024).  \nAfter performing the integral (44), we arrive at the following mean Lagrangian protohalo AM  \nJ L ≈ 0 . 521 5 m/ 3 + 3 G ¯ ρ 1 / 3 0 ( r st R th ) 5 m 3 g ( γ, γν ) M 5 / 3 δ D ( t i ) , (47)  \nplotted in Figure 2 as a function of the virial mass M of haloes at t c equal to the present time. We see that the dependence on M of factors ( r st /R th ) 5 m/ 3 and g ( γ, γν ) is very week so that J L is essentially proportional to M 5 / 3 .  \nSimilarly, by integrating J L (eq. [42]) from R th to the median separation r med between the protohalo and the nearest dominant tidal torque source, solution of the equation  \n0 . 5 = ∫ r med 0 d r r 5 m 3 +2 ∫ r st 0 d r r 5 m 3 +2 = ( r med r st ) 5 m 3 +3 (48)  \nFigure 2. Mean Lagrangian protohalo AM as a function of the virial mass of present haloes in the Planck14 cosmology (solid red line). The straight dashed blue line shows the best fit to a simple M 5 / 3 law.  \n<!-- image -->  \n/circledot  \nFigure 1. Typical separation (independent of the redshift) between the protohalo and main mass fluctuation responsible for the tidal torque, scaled to the protohalo radius, as a function of the virial mass of current haloes in the Planck14 cosmology.  \n<!-- image -->  \n/circledot  \nimplying r med = r st / 2 1 / (5 m/ 3+3) , we are led to the median Lagrangian protohalo AM, J L med = J L / 2.", '5 SUMMARY AND DISCUSSION': "Using the TTT, we have derived the Lagrangian protohalo AM in CDM cosmologies within the framework of the peak model of structure formation where haloes are believed to form from the collapse of patches around triaxial peaks in the Gaussian-smoothed Gaussian random density field at an (arbitrary) initial time t i . This has been done by accounting for the first time for the accurate extension (mass) and ellipsoidal collapse time of such patches. These improvements have been possible thanks to the use of the CUSP formalism making the link between haloes with M at the cosmic time t c and Gaussian peaks with density contrast δ on scale R .  \nThe approach we have followed is innovative in the sense that not only are protohaloes identified to peaks, but the mass excesses or defaults responsible for the torques are also identified to peaks or holes. Specifically, we have found that, among all it nested (positive or negative) density fluctuations of different scales around a protohalo, traced by continuous δ ( R ) peak or hole trajectories, the dominant ones are the largest ones fitting in the neighbourhood of the protohalo (i.e. with top-hat scale R th d equal to the separation r ) whose peak at scale R is twin (possibly except for the sign) to the peak marking the protohalo itself. Those 'dominant peaks' are indeed the most probable ones due to the correlation between peaks (or between peaks and holes) of the same scale (e.g. Salvador-Sol'e & Manrique 2024). And, among all those dominant tidal torque sources, the strongest tidal torque is due to that lying the closest to the protohalo. Of course, there are many more sets of nested (positive or negative) density fluctuations in the neighbourhood of the protohalo that cause small additional tidal torques. But these additional density fluctuations are very numerous and typically half of them slightly boost the tidal torque of the main source and the other half mitigates it (their tidal torque point to the opposite direction), so their global effect basically cancels.  \nAnother important accomplishment has been that the predicted mean Lagrangian AM J L ( M,t c ) of protohaloes with virial mass M collapsing at t c has the simple analytic expression  \nJ L ≈ 0 . 521 5 m/ 3 + 3 G ¯ ρ 1 / 3 0 s 5 m 3 g ( γ, γν ) M 5 / 3 δ D ( t i ) , (49)  \nso it can be readily checked against simulations and implemented in analytic models of structure formation. In equation (47) m is -( n + 3) / 2 in terms of the effective power index n ≈ -1 . 75 of the CDM spectrum at galactic halo masses, ¯ ρ 0 is the present mean cosmic density, s ≡ r st /R th is the time-invariant and approximately mass-independent ( r st /R th ≈ 4 . 5) typical separation between the c.o.m. of the protohalo and the main tidal torque source scaled to the protohalo size, factor g is also a very weak function of M given by equation (43) and δ ( t c , t i ) ∝ D ( t i ) (eq. [2]) is the density contrast of Gaussian peaks at t i that undergo ellipsoidal collapse at t c . On the other hand, the median Lagrangian protohalo AM is J L ( M,t c ) / 2.  \nBoth the mean and median Lagrangian AMs are independent of the (arbitrary) initial time t i and of the cosmic time  \nt , as expected, while they depend on the protohalo collapse time t c through δ ( t c , t i ) /D ( t i ) and on the halo virial mass M essentially through factor M 5 / 3 , in agreement with the results of simulations (e.g. Sugerman, Summers & Kamionkowski 2000). In turn, their Eulerian counterparts depend t through the additional factor a 2 ( t ) ˙ D ( t ).  \nNo need to say that the Lagrangian protohalo AM derived here only holds for objects in linear regime. Moreover, it presumes the protohalo is linear at all relevant smaller scales, implying that they are roughly homogeneous with inner local density equal, to leading order, to the mean cosmic density. In Paper II we will extend those results to protohaloes endowed with their real typical density profile (Salvador-Sol'e et al. 2012a) and infer the typical spin and specific AM profile of virialised haloes.", 'ACKNOWLEDGEMENTS': "This work was funded by the Spanish MCIN/AEI/ 10.13039/501100011033 through grants CEX2019-000918-M (Unidad de Excelencia 'Mar'ıa de Maeztu', ICCUB) and PID2022-140871NB-C22 (co-funded by FEDER funds) and by the Catalan DEC through the grant 2021SGR00679.", 'DATA AVAILABILITY': 'The data underlying this article will be shared on reasonable request to the corresponding author.', 'REFERENCES': "Abdullah M. H., Klypin A., Prada F., Wilson G., Ishiyama T., Ereza J., 2024, MNRAS, 529, L545 Bardeen J. M., Bond J. R., Kaiser N., Szalay A. S., 1986, ApJ, 304, 15 (BBKS) Bond, J.R., Cole, S., Efstathiou, G., & Kaiser, N. 1991, ApJ, 379, 440 Bryan G.L. & Norman M. L., 1998, ApJ, 495, 80 Catelan P., Theuns T., 1996a, MNRAS, 282, 436 Catelan P., Theuns T., 1996b, MNRAS, 282, 455 Chandrasekhar S., 1987, efe..book Doroshkevich, A. G. 1970, Astrophysics, 6, 320 Henry, J. P., 2000, ApJ, 534, 565 Hoffman Y., 1988, ApJ, 329, 8 Hoyle F., Burgers J. M., van de Hulst H. C., 1949, eds., in Problems of Cosmical Aerodynamics, Central Air Documents Office, Dayton, p. 195 Juan E., Salvador-Sol'e E., Dom'enec G., Manrique A., 2014, MNRAS, 439, 719 Komatsu E., Smith K. M., Dunkley J., Bennett C. L., Gold B., Hinshaw G., Jarosik N., et al., 2011, ApJS, 192, 18L137 Manrique A. & Salvador-Sol'e E., 1995, ApJ, 453, 6 Manrique A. & Salvador-Sol'e E., 1996, ApJ, 467, 504 Manrique A., Raig A., Solanes J. M., Gonz'alez-Casado G., Stein, P., Salvador-Sol'e E., 1998, ApJ, 499, 548 Heavens A., Peacock J., 1988, MNRAS, 232, 339 Peebles P. J. E., 1969, ApJ, 155, 393 Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2014, A&A, 571, AA16 Porciani C., Dekel A., Hoffman Y., 2002a, MNRAS, 332, 325  \nPorciani C., Dekel A., Hoffman Y., 2002, MNRAS, 332, 339  \nPress, W. H., & Schechter, P. 1974, ApJ, 187, 425  \nSalvador-Sol'e, E., Vi˜nas, J., Manrique, A., & Serra, S. 2012a, MNRAS, 423, 2190 Salvador-Sol'e E., Manrique A., 2021, ApJ, 914,141 Salvador-Sol'e E., Manrique A., 2024, accepted for publication in ApJ Salvador-Sol'e E., Manrique A., Agull'o E., 2024, accepted in ApJ Sugerman B., Summers F. J., Kamionkowski M., 2000, MNRAS, 311, 762 Thuan T. X., Gott J. R., 1977, ApJ, 216, 194 White, S. D. M. 1984, ApJ, 286, 38", 'APPENDIX A: TIDAL TENSOR': 'The Hessian of the Lagrangian gravitational potential (35) is  \nT d ij ≡ a 3 ( t i ) D ( t i ) ∂ 2 Φ d ∂x j ∂x i = a 3 ( t i ) D ( t i ) 4 π ∆[ S ( 0 )] ∆[ S ( x )] Gδ d ¯ ρ ( t i )   x i x j [ a 2 d i + S ( x )][ a 2 d j + S ( x )] { 3 ∑ k =1 x 2 k [ a 2 d k + S ( x )] 2 } -1 -δ ij ∫ ∞ S ( x ) ∆[ S ( x )] d s ∆( s )( a 2 d i + s )   , (A1)  \nwhere δ ij is the Kronecker delta, ∆[ S ( x )] is given by equation (36) but with s replaced by S ( x ) defined in equation (37).  \nSince R d is limited by the condition R th d = r , the c.o.m. of the protohalo is at the edge of the ellipsoidal mass excess or default, i.e. ∑ 3 k =1 x 2 k /a 2 d k = 1 and S ( x ) = 0. Consequently, the tidal tensor is simply  \nT d ij = a 3 ( t i ) D ( t i ) 4 πGδ d ¯ ρ ( t i )   x i x j a 2 d i a 2 d j ( 3 ∑ k =1 x 2 k a 4 d k ) -1 -δ ij U ( a 2 d k )   , (A2)  \nwhere  \nU ( a 2 d k ) ≡ 1 2 ∫ ∞ 0 d s/a 2 d j 1 + s/a 2 d j [ 3 ∏ k =1 ( 1 + s a 2 d k ) ] -1 / 2 = 1 2 ∫ ∞ 1 d y y 3 / 2 [ 3 ∑ k =1 a d j + s ( y ) 2 a 2 d k + s ( y ) ] -1 . (A3)  \nTaking into account that the ellipticity and prolateness of peaks is moderate (BBKS), we have  \nU ( a 2 d k ) ≈ 1 6 ∫ ∞ 1 d y y 3 / 2 = 1 3 . (A4)  \nIn addition, we can neglect the terms in 1 -a 2 d k /a 2 d , where a 2 d ≡ 1 / 3 ∑ 3 k =1 a 2 d k , in the Taylor expansion of ∑ 3 k =1 x 2 k /a 4 d k around ∑ 3 k =1 x 2 k / ( a 2 d k a 2 d ),  \n3 ∑ k =1 x 2 k a 4 d k = 3 ∑ k =1 x 2 k a 2 d k a 2 d [ 1 + ( 1 -a 2 d k a 2 d ) + ... ] ≈ 1 a 2 d 3 ∑ k =1 x 2 k a 2 d k = 1 a 2 d . (A5)  \nThus, we arrive at the almost fully accurate expression  \nT d ij ≈ a 3 ( t i ) D ( t i ) 4 π 3 Gδ d ¯ ρ ( t i ) [ 3 x i x j a 2 d i a 2 d j a 2 d -δ ij ] . (A6)  \nNote that in the limit a d k = R th d , where the two approximations above hold exactly, expression (A6) recovers the Hessian of the gravitational potential at the edge ( r = R th d ) of a spherically symmetric system, as expected.', 'APPENDIX B: ORIENTATION AVERAGES': 'As argued in Section 4, to find the modulus of the Lagrangian AM of a protohalo we must simply perform the average of any arbitrary component, e.g. for i = 1, of such an AM, equation (39), over the two sets of Euler angles giving the orientation of the ellpsoidal protohalo and of the ellipsoidal mass excess or default. To do that average we adopt the simplest initial configuration of the system, i.e. that in which the principal axes of both objects are aligned with each other though with different indexes in general (see below for the interest of that freedom) and with the position vector r of the c.o.m. of the protohalo from that of the mass excess or default.  \nSince the averages over the two sets of Euler angles may be carried out independently, we can start by averaging over α , β and κ in the whole angular space 8 π 2 strad 2 . The scaled inertia tensor (eq. [20]) then becomes  \n⟨ I ⟩ α,β,κ = M 15 a 2 ( t i )   ι 1 0 0 0 ι 2 0 0 0 ι 3   , (B1)  \nand the component i of the AM reads  \n⟨ J L d i ⟩ α,β,κ = -M 15 a 2 ( t i ) ϵ ijk [ R d T d R T d ] jk ι k . (B2)  \nor, equivalently,  \n⟨ J L d i ⟩ α,β,κ = -M 15 a 2 ( t i ) [ R d T d R T d ] jk ( ι j -ι k ) , (B3)  \nwith i , j and k following the natural (ciclic) order.  \nAs mentioned, the principal axes of I may not coincide with those of T d . Thus, regardless of the orientation of T d , we  \nmay assume ι j -ι k ≡ a 2 k -a 2 j in equation (B3) equal to a 2 3 -a 2 1 . This is very convenient because then we simply have (see eqs. [18]-[19])  \na 2 3 -a 2 1 = 1 a -2 3 -1 a -2 1 = a -2 1 -a -2 3 a -2 3 a -2 1 = ˆ a -2 1 -ˆ a -2 3 ˆ a -2 3 ˆ a -2 1 = 2 p [ (1 -2 e ) 2 (1 + e ) 2 -9 p 2 ] 1 / 3 ≈ 2 ( 1 -2 e 1 + e ) 2 / 3 p, (B4)  \nwhere we have neglected | (3 p ) / (1 + e ) | in front of unity (not only is | p | ⩽ e , but, for peaks of galactic mass, | p | ≪ 1).  \nConsequently, we have  \n⟨ J L d i ⟩ α,β,κ ≈ 2 M ( R th ) 2 15 a 2 ( t i ) [ R d T d R T d ] jk ( 1 -2 e 1 + e ) 2 / 3 p. (B5)  \nin terms of the ellipticity and prolateness of the peak associated with the protohalo.  \nWe must now average over the Euler angles α d , β d and κ d in 8 π 2 / 2 strad 2 from any arbitrary initial orientation of the position vector r d of the c.o.m. of the protohalo (e.g. in the a d i direction). We remark that, even though the top-hat scale R th d of the ellipsoidal mass excess or default was chosen to be bounded to r so that we must not worry about possible outer shells not contributing to the tidal force, when R th d is close to r , the c.o.m. of the protohalo may be found inside or outside the ellipsoidal mass excess or default depending on the orientation of r d . In these circumstances the average over the Euler angles α d , β d and γ d will slightly overestimate the real AM of the protohalo when the peak lies inside the ellipsoidal mass excess and slightly underestimate it otherwise (or the opposite for a mass deffault) because the scale of the real ellipsoidal mass excess (default) responsible of the tidal torque could have been taken somewhat larger. Both effects should essentially balance each other, however, so we can forget about that complication and keep the tidal tensor with the form (B3) regardless of the orientation. On the other hand, for the reasons mentioned in Section 4, we can concentrate in the orientation average over 4 π 2 strad 2 only of any particular component, say component 1, of the AM vector.  \nAfter a lengthy calculation, we arrive at the following average over α d , β d and κ d in 4 π 2 strad 2 of [ R d T d R T d ] 23 with T d given by equation (A6)  \n⟨ [ R d T d R T d ] 23 ⟩ α d ,β d ,κ d = a 3 ( t i ) G ¯ ρ ( t i ) δ 3 D ( t i ) ( r R th ) m H ( a d i ) (B6)  \nH ( a d i ) ≡ 4 π -9 36 + a 2 d ⟨ x d ⟩ Λ d { 5 π -8 30 ˆ a -4 d1 + ( 8 9 π + 26 + 75 π 120 ) ˆ a -2 d1 ˆ a -2 d2 -1 10 ˆ a -4 d2 -1 5 ˆ a -2 d1 ˆ a -3 d3 -( 1 5 π + 3 π 16 ) ˆ a -2 d2 ˆ a -2 d3 + 4 5 ˆ a -4 d3 } , (B7)  \nwhere we have taken into account the equality δ d = ( R d /R ) m δ with m = -[( n + 3) / 2] 1 / 2 and R d /R = r/R th . Taking into account the equality ˆ a -2 d i = λ d i / ⟨ x d ⟩ , H ( a d i ) adopts the following approximate form as a function of e d (or ˜ e d = -(1 -2 e d ) / 3) and p d  \nH ( e d , p d ) ≈ 1 10 ( 1+ p -2 / 3 d 3 ( ˜ e 2 / 3 d +˜ e 8 / 3 d -˜ e 14 / 3 d ) { 1+ 0 . 6 p d ˜ e -1 d 1+˜ e d [ 90 -( 1 ˜ e d -2 ) 3 ]}) ≈ 1 10 [ 1+ p -2 / 3 d 3 ( ˜ e 2 / 3 d +˜ e 8 / 3 d -˜ e 14 / 3 d ) ( 1 + 54 p d ˜ e -1 d 1 + ˜ e d ) ] . (B8)  \nTaking into account that | ˜ e d | ≲ 0 . 3, H ( e d , p d ) can be further approximated to  \nH ( e d , p d ) ≈ 1 10 { 1 + (˜ e d p d ) -1 / 3 [ 16 + 1 3 p 4 / 3 d (˜ e d p d ) ]} = 1 10 { 1 -3 1 / 3 [(1 -2 e d ) p d ] -1 / 3 [ 16 -3 -2 (1 -2 e d ) p -1 / 3 d ]} . (B9)  \nThus, having performed those two averages, the typical value of e.g. the first Cartesian component of the Lagrangian proptohalo AM takes the form  \nJ L d1 ≡ ⟨⟨ J L d1 ⟩ α,β,κ ⟩ α d ,β d ,κ d ≈ 2 3 G ¯ ρ 1 / 3 0 δ M 5 / 3 5 D ( t i ) ( r R th ) m H ( e d , p d ) ( 1 -2 e 1 + e ) 2 / 3 p, (B10)  \nwhere ¯ ρ 0 is the current mean cosmic density, implying an average modulus of the Lagrangian AM equal to  \nJ L d = √ 3 J L d i ≈ 2 5 × 3 1 / 2 D ( t i ) G ¯ ρ 1 / 3 0 δ M 5 / 3 ( r R th ) m H ( e d , p d ) ( 1 -2 e 1 + e ) 2 / 3 p. (B11)', 'APPENDIX C: ELLIPTICITY AND PROLATENESS AVERAGES': 'We must average J L d given by equation (B11) over the ellipticity and prolateness of the peaks or holes associated with the protohalo and the dominant mass excess or default causing the torque.  \nTaking into account the following approximate averages of powers of p d from -e d to e d holding for massive haloes, with e d ≲ 0 . 3 and p d ≪ σ dp ,  \np -1 / 3 d ( e d ) ≈ 6 e 2 / 3 d √ 2 πσ dp e -¯ p 2 d 2 σ 2 dp [ e d ¯ p d 5 σ 2 dp + O ( e 3 d ) ] ≈ 6 e 5 / 3 d ¯ p d 5 √ 2 πσ 3 dp (C1)  \np -2 / 3 d ( e d ) ≈ 6 e 1 / 3 d √ 2 πσ dp e -¯ p 2 d 2 σ 2 dp [ 1 -e 2 d (¯ p 2 d -σ 2 dp ) 14 σ 2 dp + O ( e 3 d ) ] ≈ 6 e 1 / 3 d √ 2 πσ dp [ 1 + e 2 d 14 ] , (C2)  \nthe averages of J L d over p and p d lead to  \nJ L d ≈ 2 5 √ 3 G ¯ ρ 1 / 3 0 δ M 5 / 3 D ( t i ) ( r R th ) m H ( e d ) ( 1 -2 e 1 + e ) 2 / 3 ¯ p, (C3)  \nwhere  \nH ( e d ) ≈ 1 10 [ 1 -24 × 3 4 / 3 5 √ 2 πσ 3 dp (1 -2 e d ) -1 / 3 e 5 / 3 d ¯ p d + 2 × 3 -2 / 3 √ 2 πσ dp (1 -2 e d ) -2 / 3 e 1 / 3 d ( 1 + e 2 d 14 ) ] , (C4)  \nwhere ¯ p and ¯ p d are much smaller than unity, e and e d are smaller than unity and so are also O ( σ dp ) = O ( σ ep ) = O (¯ e d ). Thus, to leading order in such small quantities we have  \nJ L d ≈ 1 25 √ 3 G ¯ ρ 1 / 3 0 δ M 5 / 3 D ( t i ) ( 1 + 2 × 3 -2 / 3 √ 2 πσ d p e 1 / 3 d ) ( r R th ) m ¯ p, (C5)  \nwhere we have kept the term e 1 / 3 d /σ dp because it is of O (¯ e -2 / 3 d ).  \nAnd, taking into account the following approximate average from -(¯ e d + 3 σ de ) to ¯ e d + 3 σ de = 1 . 40¯ e d (even though e d is positive, it is convenient to extend the integral to negative e d values to find the right normalisation constant of the approximately normal PDF)  \ne 1 / 3 d = 6 √ 2 πσ de (1 . 4¯ e d ) 4 / 3 e -¯ e 2 d 2 σ 2 de [ 1 . 4¯ e d 7 ¯ e d σ 2 de + O (¯ e 3 d ) ] ≈ 6 × 1 . 4 7 / 3 7 √ 2 π ¯ e 10 / 3 d σ 3 de (C6)  \nand the equalities σ de = ¯ e d / √ 6 and σ dp = ¯ e d / √ 3, we find, to leading order, the following average over e d  \nJ L d ≈ 0 . 169 G ¯ ρ 1 / 3 0 δ M 5 / 3 D ( t i ) ( r R th ) m ¯ e -2 / 3 d ¯ p = 5 . 07 G ¯ ρ 1 / 3 0 δ M 5 / 3 D ( t i ) ( r R th ) m ¯ e -2 / 3 d ¯ e 4 . (C7)  \nLastly, expressing ¯ e and ¯ e d as functions of ⟨ x ⟩ and ⟨ x d ⟩ (see the comment on eq. [12]) and, taking into account that for large masses, as it is the case for the dominant mass excess or default, the average curvature is ⟨ x d ⟩ ≈ γν d = ( r/R th ) m γν , where r/R th substantially larger than unity, and adopting M vir masses so that R d /R ≈ r/R th , we obtain  \nJ L d ≈ 0 . 347 G ¯ ρ 1 / 3 0 δ M 5 / 3 D ( t i ) ( r R th ) 5 m/ 3 [ ( γν ) 1 / 3 ⟨ x ⟩ 2 + 6 5 ] 2 . (C8)'}